Transcription

1 Title A Mathematical Approach to Behavioral nd Structure of Biological Communities Author(s) 池川, 雄亮 Editor(s) Citation Issue Date 2015 URL Rights

2 A Mathematical Approach to Behavioral Variation in Prey and Predator and Structure of Biological Communities ( 被食者と捕食者の行動の変化が生物群集の構造に与える影響に関する数理的研究 ) Yusuke Ikegawa 池川雄亮 Osaka Prefecture University 大阪府立大学 2015

3 Table of Contents General Introduction 1 Chapter 1 Effects of two types of adaptive defense by prey on population density of species in an intra-guild predation system 5 Figures (Chapter 1) 32 Chapter 2 Effects of adaptive defense by pests and switching predation by omnivorous natural enemies on efficiency of biological control 42 Figures (Chapter 2) 71 Conclusion 76 Acknowledgement 78 References 79

4 General Introduction Extinction of species and loss of biodiversity caused by some external factors, such as drastic changes in environments and invasions of alien species, and their detrimental effects on ecosystem functioning has been recent concern (Hooper et al. 2005), and revealing how the biodiversity is maintained is one of the central themes for ecologists. The external factors dramatically alter species composition and interspecific interactions in biological communities (Snyder and Evans 2006; Kenis et al. 2009). Community ecologists have recognized that interspecific interactions are a key factor to determine persistence of species and community structure and, and tried to examine how the changes in the interspecific interactions affect persistence and abundance of species and stability of the communities. To identify the underlying mechanisms of the effect of the interspecific interactions on community structure is needed for resolving the biodiversity problems, and might provide useful implications in biological control of pests in agricultural systems (Bianchi et al. 2006; Letourneau et al. 2009; Griffin et al. 2013) and conservation of rare species in natural systems (Soulé et al. 2003). Many community ecologists have studied effects of the interspecific interactions on community structure using mathematical models. They have considered some community modules consisted of three species and examined indirect effects from the first species to the second, not directly interacting with the first, through the third species (see Holt and Lawton (1994) and Wootton (1994) for reviews). For example, in a system of two predator species sharing one prey species, consumption of prey by one predator reduces gain of the other, and leads to the so-called exploitative competition (Stewart and Levin 1973), mutually negative indirect effects between the 1

5 predators. If one predator feeds on two prey species, increases in the predator by the presence of one prey are harmful to the other prey. This is the so-called apparent competition between the two prey (Holt 1977; Holt and Kotler 1987). In a three species food-chain, control of the intermediate predator by the top predator releases the basal prey from predation pressure and increases the prey, which in turn supports the intermediate predator and subsequently the top predator. This is the so-called trophic cascade (Paine 1980; Oksanen et al. 1981; Carpenter et al. 1985). All the above indirect effects appear when one species indirectly impacts on focal species through variation in density of the third species that directly interacts with both species (Strauss 1991; Holt and Lawton 1994; Wootton 1994), which is the so-called density-mediated indirect effects (Abrams et al. 1996). Recently, some community ecologists have hypothesized that interactions between two species can be modified by the presence of the third species, and demonstrated it in nature (Peacor and Werner 1997; Werner and Peacor 2003; Schmitz et al. 2004). This is called the trait-mediated indirect effects in contrast to the density-mediated indirect effects (Abrams et al. 1996). For example, in a two predator-one prey system, if the prey employs predator-specific defenses (effective only against one type of predators), the presence of one predator can be beneficial for the other because the prey cannot prevent predation from both predators simultaneously. This is the so-called exploitative mutualism (Matsuda et al. 1993). In the absence of defense by the prey, indirect effects between two predators are mutually negative (exploitative competition) as explained in the previous paragraph. Thus, indirect effects between two species can be altered by the behavioral variation of the third species and may eventually influence the community structure. 2

6 As many empirical studies have demonstrated prevalence and importance of behavioral variation in species in natural communities, theoreticians have begun to examine their effects on coexistence of species and stability of the system (Sih et al. 1998; Bolker et al. 2003; Werner and Peacor 2003). For example, the optimal foraging theory (Krebs et al. 1977) predicts that the predator should always prey on the more profitable prey while prey on the less profitable one only when the more profitable one becomes scarce. This type of foraging cannot only maximize the gain of the forager but also enhance three species coexistence (Fryxell and Lundberg 1994; Křivan 2000; Křivan and Diehl 2005) and stabilize the system (Mougi and Nishimura 2009). However, few studies have investigated how the behavioral variation influence population density of species even in relatively simple communities although examining the effect of behavioral variation on the population density might be useful for ecological applications (e.g., biological pest control by natural enemies, conservation of rare species in natural communities). Here, I aimed to examine the effects of the behavioral variation on the population density of each species in a relatively simple community. I focused on the intra-guild predation (IGP) in which two predators compete for one shared prey species and one of them also preys on the other. IGP is widely observed in nature (Polis et al. 1989; Arim and Marquet 2004). Because of the presence of both the omnivorous and monophagous predator, the IGP system includes three density-mediated indirect effects (exploitative competition, apparent competition, and trophic cascade). Their effects on persistence and density of species and stability of the system have been examined by many theoreticians (Holt and Polis 1997; McCann and Hastings 1997; Mylius et al. 2001; Tanabe and Namba 2005; Namba et al. 2008). Holt and Polis (1997) showed that 3

7 a necessary condition for three species coexistence in the IGP system was satisfied only in a limited situation; the omnivorous or monophagous predator went extinct at low and high productivity of basal prey, respectively, and three species could coexist only at intermediate one. Recently, some theoretical studies considered adaptive defense by the shared prey (Kimbrell et al. 2007; Nakazawa et al. 2010) or diet choice by the omnivorous predator (Křivan 2000; Křivan and Diehl 2005; Pal et al. 2014) and examined their effects on coexistence of species and stability of the system. By including such behaviors into the classical model, realization of three species coexistence in the IGP system at high productivity was partially explained. However, their effects on population density of each species are not well-understood despite urgent needs for applications in biological pest control and conservation. To reveal the various effects of behavioral variation on population density of component species in biological communities, in the first chapter, I examined joint effects of predator-specific and -nonspecific adaptive defenses by the shared prey on population density of each species as well as on coexistence of species and stability of the system. Next, in the second chapter, including both the adaptive defense by the shared prey and switching predation (a kind of diet choice allocating disproportionately more effort to more abundant prey) by the omnivorous predator, I examined sole and joint effects of them on population density of the shared prey and discuss whether two predators in combination can suppress the shared prey more than either one of the predators, from the perspective of biological control in agricultural systems. 4

8 Chapter 1 Effects of two types of adaptive defense by prey on population density of species in an intra-guild predation system 5

9 Abstract Intra-guild predation (IGP), predation on intermediate predators which share common prey with the predators, is an important community module to understand a mechanism for persistence of complex food webs. Recently, adaptive defense by shared prey has been recognized to enhance coexistence of species and stability of the system. Some organisms having multiple predators in IGP systems employ two types of defenses; generalized defense that is effective against multiple predators and specialized one that is effective against only a specific predator species. We consider an IGP model including shared prey that can use the two types of defenses in combination against the intermediate predator or omnivore. Assuming that the shared prey can change the allocation of defensive effort to increase its fitness, we show that the joint use of two types of adaptive defenses promotes three species coexistence and enhances stability of the IGP system when the specialized defense is more effective than the generalized one. When the system is unstable, a variety of oscillations appear and both the population densities and defensive efforts or only the population densities oscillate. Joint use of defenses against the intermediate predator tends to increase the population density of the shared prey with the defense efficiencies. In contrast, efficient generalized and specialized defenses against the omnivore often decrease the prey population. Consequently, adaptive defense by the shared prey may not necessarily heighten the population size of the defender but sometimes increases densities of both the attackers and defender in IGP systems Introduction 6

10 Classical theoretical studies on IGP predicted that coexistence of intermediate predators and omnivores was possible only at intermediate productivity and that omnivores or intermediate predators were excluded at low or high productivity, respectively (Holt and Polis 1997; Borer et al. 2003), while empirical studies showed that intermediate predators could coexist with omnivores at high productivity (Amarasekare 2008; Abrams and Fung 2010). Accordingly, theoreticians have modified the classic model to resolve the gap between the empirical observations and theoretical predictions (Heithaus 2001; Mylius et al. 2001; HilleRisLambers and Dieckmann 2003; Holt and Huxel 2007). Some of them have focused on adaptive behavior of organisms, such as diet choice by predators (Křivan 2000; Křivan and Diehl 2005) or induced defense by prey (Kimbrell et al. 2007; Nakazawa et al. 2010; Urbani and Ramos-Jiliberto 2010). Induced defense is morphological or behavioral shifts of prey to avoid predation, which is widely observed in nature (Lima and Dill 1990). If there are multiple predators, prey may develop a single defensive trait effective against many predators (generalized defense) or different traits against different predators (specialized defense) (Sih et al. 1998; Relyea 2003). Examples of the former are reducing activity for mating or foraging to avoid encounters with predators (Huang and Sih 1991; Krupa and Sih 1998; Relyea and Werner 1999; van Buskirk 2001), or changing morphology to prevent predation (van Buskirk and McCollum 2000; Relyea 2001; van Buskirk 2001). Examples of the latter are taking different behavior or morphology at different locations or against different modes of predation (Rahel and 7

11 Stein 1988; Soluk and Collins 1988; Gonzalez and Tessier 1997; Krupa and Sih 1998; Hoverman and Relyea 2007). Theoretical studies focused on generalized defense showed that predators less competitive in exploitative competition for the common prey were often excluded by generalized defense (Lima 1992; Matsuda et al. 1993, 1996; Kimbrell et al. 2007). Kimbrell et al. (2007) studied an IGP system and suggested that coexistence of intermediate predators and omnivores by virtue of adaptive generalized defense was realized only when the predation rate on the intermediate predators by the omnivores was intermediate. In contrast, theoretical studies focused on specialized defense suggested that, since prey tended to allocate more effort toward more consumptive predators, competitive exclusion could be prevented (Lima 1992; Matsuda et al. 1996; Kondoh 2007; Nakazawa et al. 2010). Nakazawa et al. (2010) studied an IGP system with two kinds of specialized defenses by shared prey effective against each of intermediate predators and omnivores and suggested that coexistence of intermediate predators and omnivores was realized by adaptive specialized defense when the predation rate on intermediate predators was low as well as intermediate. Consequently, joint use of the specialized defenses against intermediate predators and omnivores may enhance the three species coexistence more than the generalized one does. However, specialized defense often makes the IGP system unstable, especially in highly productive environments (Nakazawa et al. 2010). Some organisms use these two types of defenses in combination. For example, anuran tadpoles (Rana pirica) become a bloated bulgy morph to avoid predation by swallowing type predators, salamander larvae (Hynobius retardatus), or increase tail fin depth and become a high-tail morph to avoid predation from biting type predators, 8

12 dragonfly larvae (Aeshna nigroflava) by improving their swimming performance (Kishida and Nishimura 2005; Kishida et al. 2009). The former defense is effective only against salamander larvae, while the latter is effective against both predators, and thus functions as a generalized defense (Kishida and Nishimura 2005). Although such joint use of two types of defenses is observed in nature, most of previous theoretical works have focused on either one of the specialized or generalized defense, and rarely studied both of them. In this study, we extend the model of Nakazawa et al. (2010), considering both generalized and specialized adaptive defenses by shared prey in an IGP system. We describe population dynamics by using a Lotka-Volterra model with dynamics of defense efforts of the shared prey. We assume that shared prey allocate defensive efforts toward specialized and generalized defenses so as to increase its fitness. Questions to answer in this article are 1) whether joint use of two types of adaptive defenses promotes three species coexistence at high environmental productivity, 2) how the joint use affects stability and dynamics of the system, and 3) how the two types of defenses shape the population density of each species Model In this article, we extend the model of Nakazawa et al. (2010) which considers predator-specific adaptive defenses by shared prey in an IGP system. We assume that shared prey can employ both generalized defense against intermediate predators and omnivores and either one of two kinds of specialized defenses effective against 9

13 intermediate predators or omnivores. Population dynamics are described by a Lotka-Volterra model as follows: dr dt = (rc R k D Na NR N D P a PR P) R dn dt = (b NRD N a NR R a PN P m N )N dp dt = (b PRD P a PR R + b PN a PN N m P )P where R, N and P represent the population densities of shared prey, intermediate predators and omnivores, respectively. The parameter k is the inverse of density (1-A) (1-B) (1-C) dependence of the shared prey and closely related to the carrying capacity (we consider this as a measure of productivity). r is the intrinsic growth rate of shared prey. a ij is the attack rate of species i on species j (i {N, P}, j {R, N}). b ij is the conversion efficiency of species i consuming species j (i {N, P}, j {R, N}). m i is the density-independent mortality of species i (i {N, P}). D i represents effects of defense by shared prey against predator species i, or fractional decrements in the attack rate. We assume that the reduction in attack rates of predators due to each defense is described as follows: D i = 1 f gi e g f si e si (i {N, P}) (2) where e g and e si represent the efforts toward the generalized defense and the specialized one against species i, respectively (i {N, P}) and f gi and f si represent the efficiencies of the generalized defense and the specialized one against species i, respectively (0 f si, f gi 1, i {N, P}). We assume that shared prey can invest effort to each of the defenses within a limit, 0 e si + e g 1 (i {N, P}). Although attack rates from predators decrease with increasing effort to defense, shared prey incurs costs in the intrinsic growth rate. We assume that the cost of defense is described as follows: 10

14 C = 1 c g e g c si e si (3) where c g and c si represent the coefficients of costs of the generalized defense and the specialized one against species i (0 c g, c si 1, i {N, P}). Here, we assume that shared prey can adaptively change the effort toward each defense to increase its own fitness described by W. We define the fitness as the per-capita growth rate of the shared prey (W = (dr dt) R). The dynamics of each effort to reduce the attack rates is expressed by the replicator equation, which is used in evolutionary game theory (Hofbauer and Sigmund 1998), and described as follows (Matsuda et al. 1996; Kondoh 2007; Nakazawa et al. 2010); de si dt de g dt = v sie si { W e si (e si W e si + e g W e g )} = v ge g { W e g (e si W e si + e g W e g )} (4-A) (4-B) where v g and v si are adaptive rates of the generalized and specialized defenses against species i (i {N, P}), respectively. The greater the adaptive rate is, the faster the shared prey changes its effort to increase its own fitness (W). When the system approaches an equilibrium state, the defensive efforts e g and e si approach either a positive value or zero. If e g (resp. e si ) becomes zero, the generalized defense (resp. specialized defense against species i) vanishes. Even if both predators survive, shared prey may abandon either or both of the generalized and specialized defenses. If the adaptive rates of the generalized and specialized defenses are equal, v si = v g = v, the total defensive effort, e si (t) + e g (t) satisfies the following equation: d (e W W dt si + e g ) = v (e si + e e g ) (1 e si e si e g ). (5) g If e si (0) > 0, e g (0) > 0 and e si (0) + e g (0) < 1, the total effort e si (t) + e g (t) can never exceed 1, since the rate of change is zero at e si + e g = 1. Asymptotic states of defensive efforts are characterized as follows; (i) both efforts asymptotically approach 0, (ii) the 11

15 total effort approaches the maximum (e si + e g = 1), or (iii) the total effort varies between 0 and 1 (0 < e si + e g <1). In the latter two cases, either or both of the efforts toward the generalized and specialized defenses may fluctuate temporally. It depends on parameter values and population densities which of the three cases results. Hereafter, we assume v si = v g. To examine the effects of joint use of generalized and specialized defenses on persistence of intermediate predators and omnivores and stability of the IGP system, we analyze the following two cases; (i) the case in which the shared prey employs both generalized and specialized defenses against the intermediate predator (species N) and (ii) the case in which the shared prey employs both generalized defense and specialized one against the omnivore (species P). We assume that the intermediate predator is superior at exploiting the shared resource in the absence of defense, which is a necessary condition for persistence of the intermediate predator (Polis and Holt 1992; Holt and Polis 1997). Then, the omnivore becomes extinct if the rate of IGP a PN or the productivity k is low, while the intermediate predator does so if a PN or k is high (Holt and Polis 1997). In the intermediate range of the IGP rate or productivity, it is known that the relative efficiency of the direct energy path from the shared prey to the omnivore to the indirect one through the intermediate predator is important to determine the dynamics of IGP in the absence of defense by the shared prey (Takimoto et al. 2007; Namba et al. 2008). When the direct path is more efficient (b PR > b PN b NR ), bistability appears and either the intermediate predator or the omnivore becomes extinct depending on the initial populations. If the indirect path is more efficient (b PR < b PN b NR ), stable coexistence or complex oscillatory coexistence becomes possible (Tanabe and Namba 2005). We 12

16 distinguish three (intermediate predator-dominant, bistable, and omnivore-dominant) cases when the direct path is more efficient and four (intermediate predator-dominant, stable coexistence, oscillatory coexistence, and omnivore-dominant) cases when the indirect path is more efficient, and investigate effects of the adaptive defense by the shared prey. We derive equilibrium population densities and defensive efforts by setting the right-hand sides of eq. (1) and (4) to zero. Then, we evaluate their local asymptotic stability by the Routh-Hurwitz criterion. We numerically calculate mean population densities and defensive efforts when the equilibrium is unstable and the asymptotic solution is periodic. To verify that the periodic state is unique, we attempt numerical calculations for some different initial values. We consider that a predator species is persistent if an equilibrium state in which the predator is existent is stable, or the positive equilibrium state at which the intermediate predator and omnivore coexist is unstable and a stable limit cycle exists. In the following, first, we investigate whether the two types of adaptive defenses promote three species coexistence, in comparison with the case of defenses with fixed efforts, at high environmental productivity. Second we examine how persistence of each species, allocation of defensive efforts to two types of defenses, and stability of the system depend on the IGP rate on intermediate predators and the efficiencies of two types of defenses. Finally, we investigate effects of adaptive defense on population density of shared prey to know whether the instantaneous fitness gain leads to increases in the population density and prosperity of the defensive species. Population densities of intermediate predator and omnivore will also be examined. 13

17 1.3. Result Generalized defense and specialized defense against intermediate predator We examine effects of adaptive defense on persistence of the intermediate predator and omnivore and stability of the system when the shared prey can use the generalized defense and the specialized defense against the intermediate predator Comparison between adaptive and fixed defenses First, we examine how adaptive defense affects persistence of intermediate predators and omnivores (Fig. 1-1a) in comparison with the case of fixed defense (Fig. 1-1b, and c). The intermediate predator and omnivore can coexist in a broader and higher range of environmental productivity (region s.-rnp in Fig. 1-1a) than in the case of fixed effort toward each defense (region s.-rnp in Fig. 1-1b and c). This is mainly because the shared prey allocates the more effort toward generalized defense to defend against the more abundant predator (omnivore) and less effort toward the specialized defense against the less abundant predator (intermediate predator) due to a trade-off between the defenses. Since the generalized defense is more effective against the omnivore than against the intermediate predator (f gn = 0.2, f gp = 0.5) and the efficiency of generalized defense against the intermediate predator is lower than that of the specialized one (f gn = 0.2, f sn = 0.5), generalized defense more effective against the omnivore could indirectly enhance persistence of the intermediate predator. 14

18 When the shared prey adaptively varies defensive efforts, the prey employs only the specialized defense against the intermediate predator at low productivity (s.-rn in Fig. 1-1a) because the intermediate predator is dominant and the specialized defense is more effective than the generalized one. The omnivore can invade as the productivity increases (region s.-rnp in Fig. 1-1a) because the shared prey and intermediate predator increase. Then, the shared prey can save the effort toward the specialized defense against the intermediate predator because the omnivore feeds on and reduces the intermediate predator. As the omnivore abundance increases with the increasing productivity, defense against the omnivore becomes more important and the specialized defense against the intermediate predator is switched to the generalized one which is also effective against the omnivore. Since the efficiency of the generalized defense against the intermediate predator is relatively low, joint use of the specialized defense becomes necessary for the shared prey to suppress the intermediate predator as productivity increases. However, further increases in productivity forces the shared prey to allocate more efforts toward the generalized defense that is effective against the more threatening predator (omnivore), and the effort toward the specialized defense decreases due to the constraint on the total effort (0 e sn + e g 1). After all, increases in productivity make the system unstable (u.-rnp in Fig. 1-1a) and finally the intermediate predator goes extinct (s.-rp in Fig. 1-1a) How do the efficiencies of defenses affect the coexistence of predators? The intermediate predator is driven to extinction when the efficiency of the generalized defense against it becomes high (region s.-rp-e g in Fig. 1-2), because it is 15

19 suppressed by efficient generalized defense by the shared prey and exploitation by the omnivore. The intermediate predator can persist and three species coexistence is realized when the generalized defense is moderately effective against the intermediate predator (region s.-rnp-e g in Fig. 1-2). When the efficiency of the generalized defense against the intermediate predator is low, dynamics of population densities and defensive efforts depend on the relative efficiency of the two energy paths (b PR and b PN b NR ) and the strength of IGP (a PN ). When the direct path is more efficient than the indirect one (b PR > b PN b NR ) and generalized defense is not so effective against the intermediate predator, equilibrium states are determined by the efficiency of specialized defense. When the specialized defense is not effective, the omnivore is excluded (region s.-rn-e g, and s.-rn in Fig. 1-2a, and b). If the efficiency of the specialized defense increases, it is employed by the shared prey although the omnivore is still extinct (region s.-rn-e sn ). Further increases in the efficiency of the specialized defense (Fig. 1-2a, and b) or high rate of IGP (Fig. 1-2c) enables the omnivore to invade. However, the three species coexistence is unstable under the sole use of the generalized defense (Fig. 1-2c, Fig. 1-3a, and b) or the joint use of the specialized and generalized defense (Fig. 1-2a, b, Fig. 1-3c, and d). Because the IGP system tends to be bistable when the direct path is more efficient, if one of the intermediate predator and omnivore decreases, then the other predator increases and defense against it becomes necessary. Thus, both defensive efforts and population densities tend to oscillate (Fig. 1-3a, b, c, and d). Finally, when the specialized defense becomes effective enough, joint use of two types of specialized defenses (on the vertical axes) or specialized and generalized defense make the three species coexistence stable (region s.-rnp-e sn e g in Fig. 1-2a, b, and c) by suppressing 16

20 rapid rebounds of intermediate predators against which the generalized defense is relatively inefficient Thus, joint use of the specialized and generalized defenses has two kinds of stabilizing effects; when a system including two kinds of specialized defenses (on vertical axes in Fig. 1-2a, b, and c) is unstable, increasing efficiency of the generalized defense can save the omnivore which is extinct under the sole use of generalized defense (on horizontal axes in Fig. 1-2a, b, and c). However, the stabilizing effects of joint use of two types of defenses become less marked as IGP becomes weaker. When the indirect path is more efficient than the direct one (b PR < b PN b NR ) and IGP is intermediate or strong, ineffective generalized defense against the intermediate predator destabilizes three species coexistence (Fig. 1-2e, f, and g). Because the generalized defense is more effective against the omnivore, it enhances the oscillatory tendency inherent in systems where the indirect path is more efficient (Tanabe and Namba 2005; Namba et al. 2008). Thus, even if the maximum effort is allocated toward the generalized defense, the population densities oscillate (Fig. 1-3e, and f). However, stable three species coexistence is realized by the joint use of two types of defenses if IGP is weak and the specialized defense is more efficient than the generalized one (Fig. 1-2d). We have also found that bistability appeared in systems without any defenses (Fig. 1-2b) when the direct path is more efficient disappears by transient allocation of defensive efforts. The omnivore is definitely excluded by initial allocation of effort toward the generalized defense by the shared prey, and finally the shared prey abandons the effort after extinction of the omnivore when both defenses against the intermediate predator are ineffective (region s.-rn in Fig. 1-2b and Fig. 1-3g, and h). 17

21 Long-term behavior of shared prey and predator populations Now, we examine how the adaptive defense by the shared prey affects long-term behavior of the shared prey, intermediate predator and omnivore populations. The equilibrium or mean population density of the shared prey tends to increase with the efficiency of the generalized defense against the intermediate predator (Fig. 1-4a, and c). However, when the specialized defense is ineffective (f sn = 0: circles in Fig. 1-4a) or moderately effective (f sn = 0.4: boxes in Fig. 1-4a) and efficiency of the generalized defense against intermediate predators is low, the system becomes unstable and oscillations of population densities and defensive efforts tend to decrease the mean population density below that at the equilibrium without any defense. For moderately effective specialized defense, the population density decreases even if the equilibrium is stabilized by the joint use of the generalized and specialized defenses until the specialized defense is abandoned. In contrast, increases in the efficiency of specialized defense against the intermediate predator always result in increases in the prey population density when the specialized defense is used jointly with the generalized one. However, when the generalized defense is ineffective (f gn = 0: circles in Fig. 1-4b) or moderately effective (f gn = 0.2: boxes in Fig. 1-4b) against the intermediate predator and efficiency of the specialized defense against the intermediate predator is low, the system is unstable and oscillations of population densities and defensive efforts tend to reduce the mean population density below that at the equilibrium without any defense. In two systems with a prey species and a single predator species (intermediate predator or omnivore), equilibrium prey density is R * = m i / (b i R D i a ir ) 18

22 (i {N, P}), and equilibrium predator density is i * = (rc R * / k) / (a ir D i ) (i {N, P}). Here, we consider only the generalized defense, because both defenses play the same role in the absence of one of the intermediate predator and omnivore. As is expected, adaptive defense increases the prey density but may not decrease the predator density. Although the numerator of the predator density i * decreases by prey defense, the denominator also decreases and it may compensate the decrease in the numerator, depending on the predation rate, conversion efficiency, cost coefficient and efficiency of the defense. Thus, densities of both predator and prey can increase counter-intuitively by virtue of prey defense (figure is not shown). When the direct energy path is more efficient than the indirect one and the intermediate predator is excluded in the absence of adaptive defense by the shared prey (Fig. 1-5a), two predators can coexist as a result of the generalized defense effective against the omnivore. In this case, population density of the intermediate predator and omnivore respectively decreases and increases with the efficiency of the specialized defense against the intermediate predator. However, the omnivore density can never exceed the one in absence of any defenses. In contrast, when the indirect path is more efficient than the direct one and the omnivore is excluded in the absence of adaptive defense (Fig. 1-5b), the intermediate predator density can be higher than the one in the case of no defense by the mechanisms explained above, although it decreases with the efficiency of the specialized defense. Furthermore, the omnivore is saved from extinction by the adaptive defense and the density increases. Thus, the adaptive defense can increase population densities of prey and two predators above those in the case of no defense (Fig. 1-4d and Fig. 1-5b), although it seems counterintuitive at a first glance. These results are qualitatively identical to those in the case where 19

23 population densities are plotted against the efficiency of the generalized defense (figure is not shown) Generalized defense and specialized defense against omnivore Now, we consider the case where the shared prey can use the generalized defense and the specialized defense against the omnivore Comparison between adaptive and fixed defenses First, we examine how adaptive defense affects persistence of intermediate predators and omnivores in comparison with the case of fixed defense. When the shared prey can adaptively vary defensive efforts (Fig. 1-6a), it employs the generalized defense if productivity is low since the intermediate predator is dominant and the specialized defense is not effective against the intermediate predator (s.-rn in Fig. 1-6a). As the productivity increases, the omnivore invades (left region s.-rnp in Fig. 1-6a) and the demand for the generalized defense decreases since the omnivore feeds on the intermediate predator and finally the intermediate predator becomes extinct (center region s.-rp in Fig. 1-6a). Then, the omnivore increases, and the specialized defense against the omnivore becomes necessary. By the specialized defense, the omnivore decreases and the intermediate predator can invade for a little higher productivity (right region s.-rnp in Fig. 1-6a). Then, the generalized defense is jointly used under the constraint on the total effort (0 e sp + e g 1). When the productivity increases further, the omnivore dominates the system even under the 20

24 maximum effort toward the specialized defense against the omnivore (right region s.-rp in Fig. 1-6a). Therefore, the intermediate predator and omnivore can coexist in a broader range of productivity when the shared prey adaptively varies effort toward each defense, in comparison with the case of fixed effort toward each defense (region s.-rnp in Fig. 1-6b, and c) How do the efficiencies of defenses affect the coexistence of predators? When the direct path is more efficient (b PR > b PN b NR ; upper panels in Fig. 1-7), stable three species coexistence is realized by the generalized defense (region s.-rnp-e g in Fig. 1-7b, and c) or joint use of the generalized and specialized defenses (region s.-rnp-e sp e g in Fig. 1-7b, and c) in wider regions as IGP becomes stronger (a PN becomes higher). Similarly as in Fig. 1-2, stable coexistence by the joint use of two types of defenses (s.-rnp-e sn e g ) is possible in regions where the specialized defense is more efficient than the generalized one. The equilibrium may become unstable if either or both of the generalized and specialized defenses are not sufficiently efficient. Similarly as in Fig. 1-2, the shared prey tends to switch two types of defenses temporally depending on the abundance of two predators and oscillations of both the population densities and defensive efforts occur (Fig. 1-8a, and b) because the system is likely to be bistable in the absence of any defense. However, either one of the generalized and specialized defenses may be discarded when it is inefficient (Fig. 1-8c, d, e, and f). When IGP is intermediate or strong, decreases in efficiency of the generalized defense tend to destabilize three species coexistence (especially on the vertical axes in Fig. 1-7b, and c) and decreases in efficiency of the specialized defense 21

25 drive the intermediate predator into extinction (on the horizontal axes in Fig. 1-7b, and c). Therefore, joint use of two types of defenses can stabilize the IGP system. When the indirect energy path is more efficient (b PR < b PN b NR ; lower panels in Fig. 1-7), the situation is a little complicated. The omnivore is excluded (s.-rn-e g in Fig. 1-7d) only when IGP is weak and generalized defense is effective against the omnivore. Exclusion of the intermediate predator is limited in regions where IGP is strong (s.-rp in Fig. 1-7g) or the shared prey uses only the generalized defense which is not sufficiently efficient against the omnivore (s.-rp-e g in Fig. 1-7d, e, f, and g). In all the other regions, three species equilibria exist but may be unstable. When efficiency of the generalized defense against the omnivore becomes high, oscillations of population densities occur for the maximum allocation of effort against the generalized defense (Fig. 1-8g, h) because of the oscillatory tendency inherent in the system where the indirect path is more efficient. However, out-of-phase moderate-amplitude oscillations of defensive efforts and irregular oscillations of population densities appear if IGP is weak and the generalized defense is inefficient against the omnivore (Fig. 1-7e and Fig. 1-8i, and j). In this case, oscillations in defensive efforts may be driven by population oscillations. For stronger IGP, the shared prey allocates the maximum effort toward the specialized defense (Fig. 1-7f and Fig. 1-8k, and l) when the generalized defense is not sufficiently efficient against the omnivore, but the population densities still oscillate. As in the case where the direct energy path is more efficient, stable three species coexistence realized by the joint use of generalized and specialized defenses may be destabilized by reducing the generalized defense to a specialized one against the intermediate predator (on the 22

26 vertical axes in Fig. 1-7e, f, and g) or removed by terminating the specialized defense against the omnivore (on the horizontal axes in Fig. 1-7d, e, f, and g). We have also found the disappearance of bistability observed in the case of no defense by transient allocation of defensive efforts; the intermediate predator goes extinct by transient allocation of the generalized defense, irrespective of initial values (Fig. 1-7b and Fig. 1-8m, and n) Long term behavior of shared prey and predator populations Now, we examine how the adaptive defense affects the population density of the shared prey, intermediate predator and omnivore. When the direct path is more efficient (b PR > b PN b NR ) and the specialized defense is ineffective (f sp = 0: circles in Fig. 1-9a), no defense is used and the prey density does not depend on efficiency of the generalized defense if it is low. As the efficiency increases and the generalized defense is employed, equilibrium population density of the shared prey monotonically increases. When specialized defense is moderately (f sp =0.4: boxes in Fig. 1-9a) or sufficiently effective (f sp =0.8: diamonds in Fig. 1-9a), equilibrium or mean density monotonically increases due to the sole or joint use of two types of defenses even if the system is unstable. However, when the generalized defense becomes sufficiently efficient against the omnivore and the direct path is weakened, the omnivore maintained by high conversion efficiency decreases. Then, by a top-down trophic cascade, the intermediate predator increases and the equilibrium population density of the shared prey monotonically decreases and finally becomes lower than that at the equilibrium without any defenses (Fig. 1-9a). In 23

27 contrast, when the indirect path is more efficient (b PR < b PN b NR ), efficient generalized defense enhances the shared prey suppressed by high rate of omnivory. Thus, a positive bottom-up trophic cascade (Price et al. 1980; Ohgushi 2008) occurs and equilibrium population density of the shared prey monotonically increases with the efficiency of generalized defense against the omnivore (f sp ) as long as any defense is used (Fig. 1-9c). When specialized defense is not effective, the shared prey uses only the generalized defense and the equilibrium population density of the shared prey does not depend on the efficiency of specialized defense (Fig. 1-9b and d). In contrast, when the specialized defense becomes sufficiently efficient against the omnivore and the shared prey simultaneously uses two types of defenses in combination, the equilibrium population density of the shared prey decreases with the efficiency of specialized defense by a top-down trophic cascade irrespective of whether b PR > b PN b NP or b PR < b PN b NR except for the case of f gp = 0 (Fig. 1-9b and d). However, the density of the shared prey employing two types of defenses (diamonds and boxes in Fig. 1-9b, and d) does not fall below that in the case of no defense (Fig. 1-9b and d). Population densities of the intermediate predator and omnivore respectively increases and decreases with the efficiency of specialized defense against the omnivore. When the direct energy path is more efficient than the indirect one (Fig. 1-10a), the omnivore density is lower than that in the case of no defense. However, when the indirect path is more efficient, densities of both predators and prey can exceed those in the case with no defense when the generalized defense is employed and the efficiency of the specialized defense against the omnivore is intermediate (Fig. 1-9d and Fig. 24

28 1-10b). These results are qualitatively identical with those in the case where population densities are plotted against the efficiency of generalized defense (figure is not shown) Discussion In this article, we studied effects of adaptive generalized and specialized defenses by the shared prey on persistence of intermediate predators and omnivores and stability of an IGP system. We showed that adaptive defense could save the inferior predator from extinction and widen the region of coexistence of the omnivore and intermediate predator toward more productive environments in comparison with the case of fixed defense (Fig. 1-1 and 1-6). By concentrating defensive efforts toward a dominant predator, joint use of two types of adaptive defenses could promote three species coexistence in highly productive environments which was denied in the classical theory (Holt and Polis 1997), similarly as in a previous study on specialized defense (Nakazawa et al. 2010). Efficiencies of two types of defenses dramatically affected persistence of intermediate predators and omnivores predators and stability of the system (Fig. 1-2 and 1-7). Since, in our model, the shared prey could use both the generalized and specialized defense and the generalized one was reduced to a specialized one when it became inefficient against the intermediate predator or omnivore, our results included those in models with the sole use of generalized defense (Matsuda et al. 1993, 1996; Kimbrell et al. 2007) and those with a joint use of two kinds of specialized defenses (Nakazawa et al. 2010). When the generalized defense was solely used, increases in the efficiency against the intermediate predator were detrimental to the intermediate 25

29 predator and it became extinct (region s-rp-e g on the horizontal axes in Fig. 1-2). Decreases in the efficiency against the omnivore also make the defense relatively more efficient against the intermediate predator and the latter goes to extinction (region s-rp-e g on the horizontal axes in Fig. 1-7). However, increases in the efficiency against the omnivore reduced the relative efficiency of the direct energy path and often promoted three species coexistence when IGP (a PN ) was strong (on the horizontal axes in Fig. 1-7). However, previous theoretical studies on generalized defense tended to emphasize the former negative effect (Matsuda et al. 1993, 1996; Kimbrell et al. 2007).Since the effects of generalized defense largely depended on the efficiencies of generalized defense (Fig. 1-2, and 1-7), the effect of efficiencies of defenses should have been studied extensively (but see Appendix of Kimbrell et al. 2007). Addition of the specialized defense to generalized one had the effect to save the inferior predator from extinction forced by the sole use of the generalized defense, by suppressing the dominant predator. Thus, joint use of the generalized and specialized defenses could enhance stable coexistence of intermediate predators and omnivores in IGP systems compared with the sole use of the generalized defense. Efficient specialized defense enhanced three species coexistence, unless both the predation rate a NR (resp. a PR ) and conversion efficiency b NR (resp. b PR ) of the defended intermediate predator (resp. omnivore) were too low (figure not shown). However, not sufficiently efficient specialized defense often made the system unstable and caused oscillations of population densities and defensive efforts, especially when the generalized defense was inefficient against the target predator of the specialized defense (on and near the vertical axes in Fig. 1-2 and Fig. 1-7). Enhancement of species coexistence and oscillations of population densities by specialized defense was 26

30 also observed in previous theoretical studies (Kondoh 2007; Nakazawa et al. 2010). However, if one of the two specialized defense became general and effective against both the intermediate predator and omnivore, unstable equilibrium could be stabilized with the increasing efficiency of the generalized defense. Moreover, population density of the shared prey often increased when one of the two kinds of specialized defenses became effective against both the intermediate predator and omnivore. Thus, joint use of the generalized and specialized defense enhances stable coexistence of the intermediate predator and omnivore and increases population density of the shared prey more than the joint use of two kinds of specialized defenses. Instability of the system and oscillations of population densities did occur by adaptive defense, but previous works (Kondoh 2007; Nakazawa et al. 2010) did not fully examine how and why the instability appeared. On the contrary, we could find a variety of different oscillations in population densities and defensive efforts and reveal how and why different kinds of oscillations appeared by focusing on the relative efficiency between two energy paths from the shared prey to the omnivore. Population oscillations without accompanying oscillations of defensive efforts tended to appear when the indirect path was more efficient and the maximum allocation of effort toward the generalized defense (e g = 1; Fig. 1-3e, f and Fig. 1-8g, and h) or the specialized defense (e sp = 1; Fig. 1-8k, and l) was insufficient to stabilize the system. Temporal shifts of efforts toward the generalized and specialized defenses occurred when the direct path was more efficient if efficiency of the generalized defense was low and that of the specialized defense was intermediate or high (Fig. 1-3c, d and Fig. 1-8a, and b). Decreases in density of the target species of the specialized defense allowed non-target species to increase and the generalized defense became needed. Then, subsequent 27

31 alternation of dominance made reinforcement of the specialized defense necessary. However, only the effort toward the generalized defense (Fig. 1-3a, b, and Fig. 1-8c, and d) or that toward the specialized defense (Fig. 1-8e, and f) oscillated depending on the efficiencies of the generalized and specialized defenses. When the indirect path was more efficient, anti-phase oscillations of defensive efforts with moderate amplitudes and irregular oscillations of population densities appeared (Fig. 1-8i, and j). Sole use of the specialized defense could cause population oscillations only when it was targeted against the omnivore (Fig. 1-7f and Fig. 1-8k, and l). A wide possibility of combinations of multiple types of defenses may permit dynamic switches of efforts and promote persistence of the system. It is well-known that a variety of alternative stable states appear in IGP systems (Holt and Polis 1997; Takimoto et al. 2007; Verdy and Amarasekare 2010). However, for the values of parameters used above, no two equilibrium states were stable at the same time, except for the case of defenses with fixed effort similarly as in Nakazawa et al. (2010). When bistability appeared in the absence of any defense and efficiencies of both the generalized and specialized defenses were low, effort toward the generalized defense was transiently allocated before the omnivore or intermediate predator went extinct (Fig. 1-3g, h and Fig. 1-8m, and n). Thus, adaptive defense may have potential to eliminate the possibility of alternative stable states and bistability. However, when both coefficients of costs of the specialized (c sn or c sp ) and generalized (c g ) defenses were sufficiently high, both defenses monotonically decreased and either one of the intermediate predator and omnivore went extinct depending on the initial populations (figure is not shown). 28

32 Adaptive defense did not necessarily increase equilibrium or mean density of the shared prey in comparison with the case without any defense, although the effects were rarely studied in models assuming adaptive changes in traits to increase the fitness or instantaneous shifts of diets or phenotypic traits. Increases in efficiencies of the generalized and specialized defenses against the intermediate predator tended to increase the population density of the shared prey. However, instability or the cost of joint use of the generalized and specialized defenses sometimes lowered the density below that in the case of no defense when the efficiency was low (Fig. 1-4a). In contrast, increases in efficiencies of the generalized and specialized defenses against the omnivore often decreased the population density of the shared prey, especially when the direct path was more efficient (Fig. 1-9a and b). This might happen because of the indirect effects of increased predation pressure from the intermediate predator released from predation by the defense against the omnivore. Adaptive defense to increase the instantaneous fitness of the shared prey could increase not only the prey population density but also population densities of both intermediate predator and omnivore at the same time. When only the generalized defense was used because of the inefficiency of the specialized defense, both densities of the shared prey and intermediate predator (Fig. 1-4d and Fig. 1-5b) or the shared prey and omnivore (Fig. 1-9b and Fig. 1-10b) counterintuitively exceeded those in the case of no defense in systems with a prey and a single predator. As the specialized defense became efficient and jointly employed, the target species (intermediate predator in Fig. 1-5b and omnivore in Fig. 1-10b) decreased and the non-target species (omnivore in Fig. 1-5b and intermediate predator in Fig. 1-10b) increased and the intermediate predator and omnivore could coexist. Thus, densities of both the 29

33 intermediate predator and omnivore became higher than those in the case of no defense in intermediate ranges of efficiencies of the specialized defense. Since both the generalized and specialized defenses were effective, the share prey was also abundant in these ranges. However, when the direct path was more efficient but the predation rate of the omnivore on the shared prey was low, the omnivore density could never exceed that in the case of no defenses and mutual prosperity of the intermediate predator and omnivore could never occur. Thus, whether the joint use of adaptive generalized and specialized defenses can enhance all of prey, intermediate predator and omnivore depends not only on the efficiencies of defenses but also on the predation rates and conversion efficiencies of predators. Some parameters which were fixed throughout this study did not cause significant effects on our results. We assumed that the adaptive rates of the generalized and specialized defenses were the same and showed that the total defensive effort never exceeded 1 under the assumption. We confirmed that the property was preserved even if the assumption of equal adaptive rates were violated. However, when the adaptive rates of both or either one of the specialized (v sn or v sp ) and generalized (v g ) defenses became sufficiently low, three species coexistence tended to be destabilized and population densities and defensive efforts oscillated. Similar effects were recognized in a previous study on two kinds of adaptive specialized defenses (Nakazawa et al. 2010). The intrinsic growth rate of the shared prey (r) affected persistence of species similarly as productivity k; the intermediate predator and omnivore tended to be extinct when r was high and low, respectively, and coexisted when r was intermediate (Holt and Polis 1997). Higher values of mortality of the 30

34 intermediate predator (m N ) or omnivore (m P ) made the focal species extinct as was expected. In this study, we assumed that the shared prey could dynamically change the effort toward each defense to increase their fitness. However, some previous studies assumed that the focal organisms could instantaneously shift diets or change the rate of phenotypic variation (Křivan and Diehl 2005; Mougi and Kishida 2009). Such differences in assumptions on the way of trait variation may affect our results. Also, it may be an important future extension to consider adaptive foraging by predators as well as adaptive defense by prey (Yamauchi and Yamamura 2005) incorporating perfect (Křivan and Diehl 2005) or imperfect (Abrams and Fung 2010) decision, as analyzed in some previous studies (Matsuda et al. 1996; Mougi and Kishida 2009). If we incorporate type II functional responses into our model of adaptive defense as in the model of adaptive foraging by Abrams and Fung (2010), joint use of generalized and specialized defenses may further contribute to persistence and stability of the IGP system. We have studied adaptive defense assuming that defensive morphs of the shared prey are determined by phenotypic plasticity to increase the fitness without invoking a genetic basis. Recently, genetic basis of adaptive defense by anuran tadpoles (Rana pirica) expressing bulgy morph (specialized defense against salamander larvae) has started to be identified (Mori et al. 2005, 2009). To consider dynamics of genotypes as a basis of phenotypic plasticity for better understanding of the subject, we need to study adaptive defense in a longer time scale. 31

35 Figures (Chapter 1) Figure 1-1: Effects of adaptively varying and fixed efforts toward defense on persistence of the intermediate predator and omnivore and stability of the system. Horizontal axes indicate productivity of the system in terms of k. Vertical axes indicate (a) adaptively varying efforts toward two types of defenses (e sn and e g ), (b) fixed effort toward the specialized defense against the intermediate predator (e sn ), and (c) fixed effort toward the generalized defense (e g ), respectively. The other efforts toward the fixed defenses are (b) e g = 0.2 and (c) e sn = 0.2, respectively. Circle and diagonal cross symbols in Fig. 1-1a represent efforts toward the specialized defense against the intermediate predator and generalized defense, respectively. Equilibrium states are categorized by persistence of the intermediate predator and omnivore; R: only shared prey persists, RN: shared prey and intermediate predator persist, RNP: three species coexist, RP: shared prey and omnivore persist. If the system is unstable and population densities oscillate, u. is added as a prefix before RNP. Likewise, if an equilibrium state is stable, s. is added as a prefix before the state. Equilibrium states in the case of no defense are also shown as a function of productivity at the bottom of each figure. Other parameter values are f sn = 0.5, f gn = 0.2, f gp = 0.5, c sn = c g = 0.2, a NR = 1, a PR = 0.25, a PN = 0.9, b NR = b PR = b PN = 0.5, m N = m P = 0.5, r = 1, v sn = v g = 1. 32

36 Figure 1-2: Dependence of equilibrium states on efficiencies of the generalized and specialized defenses against the intermediate predator. In (a) to (c), the direct energy path from the shared prey to the omnivore is more efficient than the indirect one via the intermediate predator, b PR > b PN b NR, and vice versa, b PR < b PN b NR in (d) to (g). Horizontal and vertical axes indicate efficiencies of the generalized (f gn ) and specialized (f sn ) defenses against the intermediate predator, respectively. Equilibrium states are categorized similarly as in Fig In addition, if an equilibrium state is stable and the shared prey employs any defense, the defensive effort is added as a suffix after the state. Likewise, if equilibrium state is unstable and the shared prey employs at least either one of the defenses, we add e as a suffix after the state. Parameter values are k = 15, a PR = 0.25 for (a)-(c) and 6.5 for (d) -(g), b PR = 0.5 for (a) -(c) and 0.05 for (d) -(g), and a PN = 0.3 for (a), 0.6 for (b), 0.9 for (c), 0.2 for (d), 1 for (e), 1.8 for (f) and 2.6 for (g). The other parameter values are the same as those in Fig Diagonal crosses indicate efficiencies of the generalized and specialized defenses where we show temporal dynamics of population densities and defensive efforts in Fig

37 Figure 1-3: Examples of temporal dynamics of (a)(c)(e)(g) population densities, and (b)(d)(f)(h) defensive efforts. Black, red and blue lines in (a)(c)(e)(g) indicate population densities of the shared prey, intermediate predator and omnivore, respectively. Red and black lines in (b)(d)(f)(h) indicate defensive effort toward the specialized defense against the intermediate predator and that toward the generalized defense, respectively. Initial population densities and defensive efforts are R(0) = 2, N(0) = 2, P(0) = 2, e sn (0) = 0.1, e g (0) = 0.1. In (a) and (b), the efficiencies of the generalized and specialized defenses against the intermediate predator are f gn = 0.1 and f sn = 0.4, respectively. Other parameter values are identical with those in Fig. 1-2c. In (c) and (d) f gn = 0.1 and f sn = 0.4. Other parameter values are identical with those in Fig. 1-2b. In (e) and (f), f gn = 0.1 and f sn = 0.4. Other parameter values are identical with those in Fig. 1-2f. In (g) and (h) f gn = 0.1 and f sn = 0.1. Other parameter values are identical with those in Fig. 1-2b. 34

38 Figure 1-4: Equilibrium population density of the shared prey as a function of efficiency of each defense against the intermediate predator. In (a) and (b), the direct energy path from the shared prey to the omnivore is more efficient than the indirect one via the intermediate predator, b PR > b PN b NR, and vice versa, b PR < b PN b NR, in (c) and (d). We used mean population density over a period if the system is unstable and exhibits a periodic solution. In (a) and (c), circle, box and diamond symbols represent different values of efficiency of the specialized defense against the intermediate predator; f sn = 0 (shared prey uses only generalized defense), f sn = 0.4 and f sn = 0.8, respectively. In (b) and (d), circles denote that efficiency of the generalized defense against the intermediate predator, f gn = 0 (generalized defense is reduced to specialized one against omnivore). Boxes denote that f gn = 0.2 for (b), and 0.3 for (d). Diamonds denote that f gn = 0.3 for (b), and 0.55 for (d). Symbols are filled if the shared prey uses two types of defenses, whereas remained open if it uses a single type of defense or no defense. Dashed lines indicate the population density of the shared prey when it does not employ any defense. Parameter values are k = 15, a PR = 0.25 for (a)-(b), 6.5 for (c)-(d), b PR = 0.5 for (a)-(b), 0.05 for (c)-(d), a PN = 0.9 for (a)-(b), and 0.2 for (c)-(d), and the other parameter values are the same as those in Fig

39 Figure 1-5: Dependence of population densities of the intermediate predator (red symbols) and omnivore (blue symbols) on efficiency of the specialized defense against the intermediate predator. In (a), the direct energy path from the shared prey to the omnivore is more efficient than the indirect one via the intermediate predator, b PR > b PN b NR, and vice versa, b PR < b PN b NR in (b). Symbols are filled if the shared prey uses two types of defenses, whereas remained open if it uses a single type of defense or no defense. Dashed lines indicate the population densities of the intermediate predator (red lines) and omnivore (blue lines) when the shared prey does not employ any defense. Circles denote that efficiency of the generalized defense against the intermediate predator, f gn = 0 (generalized defense is reduced to specialized one against the omnivore). Boxes denote that f gn = 0.2 for (a), and 0.3 for (b). Diamonds denote that f gn = 0.3 for (a), and 0.55 for (b). Other parameter values in (a) and (b) are identical with Fig. 1-4(b) and (d), respectively. 36

40 Figure 1-6: Effects of adaptively varying and fixed efforts toward defense on persistence of the intermediate predator and omnivore and stability of the system. Horizontal axes indicate productivity of the system in terms of k. Vertical axes indicate (a) adaptively varying efforts toward two types of defenses (e sp and e g ), (b) fixed effort toward the specialized defense against the omnivore (e sp ), and (c) fixed effort toward the generalized defense (e g ), respectively. The other efforts toward the fixed defenses are (b) e g = 0.2 and (c) e sp = 0.2, respectively. Circle and diagonal cross symbols in Fig. 1-4a represent efforts toward the specialized defense against the omnivore and generalized defense, respectively. Equilibrium states are categorized similarly as in Fig Equilibrium states in the case of no defense are also shown as a function of productivity at the bottom of each figure. Other parameter values are f sp = 0.5, f gn = 0.5, f gp = 0.2, a PN = 1.2, and the other parameter values are the same as those in Fig

41 Figure 1-7: Dependence of equilibrium states on efficiencies of generalized and specialized defenses against the omnivore. In (a) to (c), the direct energy path from the shared prey to the omnivore is more efficient than the indirect one via the intermediate predator, b PR > b PN b NR, and vice versa, b PR < b PN b NR, in (d) to (g). Horizontal and vertical axes indicate efficiencies of the generalized (f gp ) and specialized (f sp ) defenses against the omnivore, respectively. Equilibrium states are categorized similarly as in Fig Other parameter values are f gn = 0.5, k = 15, a PR = 0.25 for (a)-(c), 6.5 for (d)-(g), b PR = 0.5 for (a)-(c), 0.05 (d)-(g), and a PN = 0.4 for (a), 0.6 for (b), 0.9 for (c), 0.2 for (d), 1 for (e), 1.8 for (f) and 2.6 for (g), and the other parameter values are the same as those in Fig Diagonal crosses indicate efficiencies of the generalized and specialized defenses where we show temporal dynamics of population densities and defensive efforts in Fig

42 Figure 1-8: Examples of temporal dynamics of (a)(c)(e)(g)(i)(k)(m) population densities, and (b)(d)(f)(h)(j)(l)(n) defensive efforts. Black, red and blue lines in (a)(c)(e)(g)(i)(k)(m) indicate population densities of the shared prey, intermediate predator and omnivore, respectively. Blue and black lines in (b)(d)(f)(h)(j)(l)(n) indicate defensive effort toward the specialized defense against the omnivore and that toward the generalized defense, respectively. Initial population densities of each species and defensive efforts are R(0) = 2, N(0) = 2, P(0) = 2, e sp (0) = 0.1, e g (0) = 0.1. In (a) and (b), the efficiencies of the generalized and specialized defenses against the omnivore are f gp = 0.05 and f sp = 0.5, respectively. Other parameter values are identical with those in Fig. 1-7b. In (c) and (d), f gp = 0.05, and f sp = Other parameter values are identical with those in Fig. 1-7a. In (e) and (f), f gp = 0.05, and f sp = 0.5. Other parameter values are identical with those in Fig. 1-7c. In (g) and (h), f gp = 0.8, and f sp = 0.5. Other parameter values are identical with those in Fig. 1-7f. In (i) and (j), f gp = 0.1 and f sp = 0.5. Other parameter values are identical with those in Fig. 1-7e. In (k) and (l), f gp = 0.1 and f sp = 0.5. Other parameter values are identical with those in Fig. 1-7f. In (m) and (n), f gp = 0.1, and f sp = 0.1. Other parameter values are identical with those in Fig. 1-7b. 39

43 Figure 1-9: Equilibrium population density of shared prey as a function of efficiency of each defense against the omnivore. In (a) and (b), the direct energy path from the shared prey to the omnivore is more efficient than the indirect one via the intermediate predator, b PR > b PN b NR, and vice versa, b PR < b PN b NR, in (c) and (d). We used mean population density over a period if the system is unstable and exhibits a periodic solution. In (a) and (c), circle, box and diamond symbols represent different values of efficiency of the specialized defense against the omnivore; f sp = 0 (shared prey uses only generalized defense), f sp = 0.4 and f sp = 0.8, respectively. In (b) and (d), circles denote that efficiency of the generalized defense against the omnivore, f gp = 0 (the generalized defense is reduced to the specialized one against the intermediate predator). Boxes denote that f gp = 0.4 for (b), and 0.3 for (d). Diamonds denote that f gp = 0.6. Symbols are filled if the shared prey uses two types of defenses, whereas remained open if it uses a single type of defense or no defense. Dashed lines indicate the population density of the shared prey when it does not employ any defense. Parameter values are k = 15, a PR = 0.25 for (a)-(b), 6.5 for (c)-(d), b PR = 0.5 for (a)-(b), 0.05 (c)-(d), and a PN = 0.9 for (a)-(b), 1 for (c)-(d), and the other parameter values are the same as those in Fig

44 Figure 1-10: Dependence of population densities of the intermediate predator (red symbols) and omnivore (blue symbols) on efficiency of the specialized defense against the omnivore. In (a), the direct energy path from the shared prey to the omnivore is more efficient than the indirect one via the intermediate predator, b PR > b PN b NR, and vice versa, b PR < b PN b NR in (b). Symbols are filled if the shared prey uses two types of defenses, whereas remained open if it uses a single type of defense or no defense. Dashed lines indicate the population densities of the intermediate predator (red lines) and omnivore (blue lines) when the shared prey does not employ any defense. Circles denote that efficiency of the generalized defense against the omnivore, f gp = 0 (generalized defense is reduced to specialized one against the intermediate predator). Boxes denote that f gp = 0.4 for (a), 0.3 for (b). Diamonds denote that f gp = 0.6. Other parameter values are identical with those of Fig. 1-7b and d. 41

45 Chapter 2 Effects of adaptive defense by pests and switching predation by omnivorous natural enemies on efficiency of biological control 42

46 Abstract One of the most important questions in biological control is whether multiple natural enemies can provide greater suppression of agricultural pests than a single best enemy. Intraguild predation (IGP) among natural enemies has often been invoked to explain failure of biological control by multiple enemies, and classical theoretical studies on IGP have supported this view. However, empirical studies are inconclusive and have yielded both positive and negative results. We extend classical models by considering anti-predator behavior of pests and diet switching of omnivorous natural enemies, and examine their effects on pest control. We assume that the pest can adaptively allocate effort toward the specific defense against each predator, and that the omnivorous natural enemy can consume disproportionately more of the relatively abundant prey (switching predation) by type III functional responses to prey items. The model predicts that adaptive defense augments pests but favors introduction of multiple natural enemies for controlling pests if IGP is weak. In contrast, switching predation does not make pest control by multiple natural enemies advantageous as in classical studies, in the absence of adaptive defense. However, switching predation reduces the necessity of defense by the pest against the omnivore and offsets the effect of adaptive defense. Thus, it makes the introduction of multiple natural enemies advantageous for pest control when the pest employs adaptive defense even if IGP is strong. These results suggest that types and combinations of behavior of prey and predators may greatly affect qualitative outcomes of biological control by multiple natural enemies Introduction 43

47 Outbreaks of agricultural pests cause serious damage to farm products. Therefore, management strategies for controlling pests have been studied for many years (Kogan 1998). Biological control suppresses pests by introducing their natural enemies (predators, parasitoids, and pathogens) and its control efficiency depends on the conditions of pests and natural enemies, location, temperature, humidity, and so on (Rosenheim 1998; Stiling and Cornelissen 2005; Janssen et al. 2006). It is one of the important components of integrated pest management, which adopts a combination of effective measures for suppressing pests, such as spreading chemical pesticides, introducing natural enemies, and structuring physical barriers (Kogan 1998; Yano 2004). In pest management, whether introduction of multiple natural enemies is more effective than that of a single enemy has been debated for a long time (Rosenheim et al. 1995). The effect of predator diversity on suppression of prey is variable depending on species composition or intra- and interspecific interactions (Sih et al. 1998; Stiling and Cornelissen 2005; Griffin et al. 2013). For example, Griffin et al. (2013) performed a meta-analysis and reported that higher predator diversity could enhance prey suppression relative to the case with a single predator species. Some empirical studies on biological control introducing multiple natural enemies supported this result of the meta-analysis (Eubanks 2001; Dinter 2002; Snyder and Ives 2003), while others did not (Rosenheim et al. 1993; Schausberger and Walzer 2001; Snyder and Ives 2001). One of the causes of the negative effect of introducing multiple natural enemies on efficiency of biological control is occurrence of intraguild predation (IGP) between them (Rosenheim et al. 1995; Rosenheim 1998; Janssen et al. 2006). 44

48 Classical theoretical studies investigating persistence and dynamics of IGP reveal that one of the necessary conditions for coexistence of the two predators is that the intermediate predator is competitively superior in consuming the shared prey to the omnivorous predator (Polis and Holt 1992; Holt and Polis 1997; Kuijper et al. 2003). Holt and Polis (1997) also showed that the equilibrium population density of the shared prey is lower when only the intermediate predator exploits it than when the two predators coexist. In other words, the classical theory of IGP implies that introduction of multiple natural enemies is not desirable for suppression of pests in comparison with that of a single efficient one. This prediction, however, does not match some observations that illustrate more efficient suppression of pests by multiple natural enemies engaging in IGP (Eubanks 2001; Dinter 2002; Snyder and Ives 2003). Janssen et al. (2006) proposed some ideas for modification of the classical models on pest control by multiple natural enemies including IGP to fill the gap between theory and observation. One of them is to take account of behavior of organisms. In nature, strength of interactions often dynamically varies by behavioral plasticity or rapid evolution of organisms, such as diet choice by predators or dynamically changing defense by prey. Optimal foraging or dynamical allocation of foraging effort to increase the fitness of predators is a form of diet choice. Another form is switching predation in which predators disproportionally consume more of the abundant prey (Murdoch et al. 1975; Hughes and Croy 1993). Some studies on biological control of pests in systems containing IGP have suggested that the rates of predation by the omnivorous natural enemy on the pest and another natural enemy varied depending on their relative abundance (Chow et al. 2008; Davey et al. 2013). Induced defense by prey was also observed in biological control systems. For example, herbivorous 45

49 two-spotted spider mites avoided plants inhabited by predaceous mites, detecting odors from the predator or the alarm pheromone emerging from the conspecifics exposed to the predators (Bernstein 1984; Grostal and Dicke 1999; Pallini et al. 1999). Some theoreticians reported that changes in behavior of organisms, such as optimal diet choice by predators (Křivan 2000; Křivan and Diehl 2005; Abrams and Fung 2010) or adaptive defense by prey (Kimbrell et al. 2007; Nakazawa et al. 2010), enhanced persistence and stability of the IGP system. However, effects of switching predation by omnivores in the IGP system on biological control have rarely been studied. In this article, we study the effects of IGP, adaptive defense by shared prey, and switching predation by omnivores on biological control by multiple natural enemies. We extend the classical IGP model of Holt and Polis (1997) by incorporating adaptive defense by the shared prey and switching predation by the omnivore. We assume adaptive and dynamical allocation of defensive efforts against predators by the shared prey. In addition, type III functional response is used to represent switching predation by the omnivore. This allows us to compare our results with those of the previous IGP modules that assumed optimal foraging (Křivan 2000; Křivan and Diehl 2005), providing an insight into how the way of modelling diet choice affects the system behavior. Incorporating either the adaptive defense or switching predation, we examine independent effects of them on persistence of each predator species and suppression of the shared prey. Next, we include both of them and examine their combined effects. Then, we try to elucidate when and why introduction of multiple predators can improve biological control by considering the predation pressures from the two predators. We also examine effects of IGP on biological control through the trophic cascade. Finally, we discuss how the success or failure of biological control by 46

50 multiple predators depends on types and combinations of behavior of prey and predators Model We extend the classical theoretical study on IGP (Holt and Polis 1997). We regard a pest species as a shared prey and two natural enemy species as an intermediate predator, and an omnivore which also consumes the intermediate predator. Population densities of the shared prey, intermediate predator, and omnivore are described as R, N, and P, respectively. Population dynamics of three species are assumed to be the same as in previous studies (Polis and Holt 1992; Holt and Polis 1997), except for the defense and switching predation terms, and are described as follows: dr dt = (r RC R k R ) R A RN N A RP P dn dt = (b RNA RN m N )N A NP P dp dt = (b RPA RP + b NP A NP m P )P (1-a) (1-b) (1-c) where r R is the intrinsic growth rate of the shared prey; k R is the inverse of density dependence of the shared prey; b ij is the conversion efficiency of predator species j consuming prey species i (i {R, N}, j {N, P}); m i is density-independent mortality of predator species i (i {N, P}); A ij is the per capita predation rate of predator species j consuming prey species i (i {R, N}, j {N, P}). C represents costs of defenses by the shared prey in terms of reduction in reproduction of its own (details are explained later). As we assume that the defensive cost affects only the reproductive rate of the shared prey, the intrinsic growth rate and the cost are not factored out in the first term 47

51 on the right-hand side of Eq.(1-a). Therefore, r R k R (not k R ) is the carrying capacity of the shared prey in the absence of defensive costs. With respect to functional responses of predators, we assume that the per capita predation rate of the omnivore on the prey species i (A ip ; i {R, N}) depends on the abundance of another prey species j (j {R, N}), and is described by the Holling type III functional response as follows: A RP = A NP = D P a RP R 2 1+D P a RP h RP R 2 +a NP h NP N 2 a NP N 2 1+D P a RP h RP R 2 +a NP h NP N 2 (2-a) (2-b) where a ip is the encounter rate of the omnivore and the prey species i (i {R, N}) in the absence of adaptive defense; h ip is the handling time of the omnivore consuming the prey species i (i {R, N}); D i is proportional reduction in the encounter rate of the shared prey and predator species i (i {R, N}) owing to defense by the shared prey (details are explained later). Equations(2-a) and (2-b) describe sigmoidal functions of population densities of the shared prey and intermediate predator, respectively, and because A RP A RP +A NP = D P a RP R 2 D P a RP R 2 +a NP N 2 = D P a RP D P a RP +a NP ( N R )2, (3-a) A NP A RP +A NP = a NP N 2 N = a NP ( R )2, D P a RP R 2 +a NP N 2 D P a RP +a NP ( N R )2 (3-b) they express disproportionately severe predation on more abundant prey (Matsuda et al. 1986; van Baalen et al. 2001). When the omnivore does not switch their diets, we assume that A NP and A RP are proportional to the prey density, i.e., A NP = a NP N, and A RP = D P a RP R, similarly as in the classical Lotka-Volterra model (Holt and Polis 1997). 48

52 The per capita predation rate on the shared prey by the intermediate predator is also assumed to be a linear function of the shared prey density: A RN = D N a RN R. For prey behavior, we assume that the shared prey can employ two kinds of predator-specific defenses against the intermediate predator and the omnivore, and that proportional reduction in the encounter rate with each predator owing to the defense is described as follows: D i = 1 f i e i (i {N, P}), (4) where e i represents the effort toward the defense against the predator species i (i {N, P}); f i represents the efficiency of the defense against the predator species i (0 f i 1, i {N, P}). We assume that the shared prey can allocate effort toward two kinds of defenses within the constraint, 0 e N + e P 1. Although the encounter rate decreases as the effort invested toward the defense increases, the shared prey incurs defensive costs in the form of proportional decrements in its own intrinsic growth rate as follows: C = 1 i c i e i (i {N, P}), (5) where c i represents the coefficient of costs of the defense against the predator species i (i {N, P}). Here, we assume that the shared prey adaptively varies effort toward two kinds of defenses to increase its own instantaneous fitness W described by the per capita growth rate of its own (W = (dr dt) R). The dynamics of effort allocation toward each defense is expressed by the replicator-like equation (Matsuda et al. 1996; Kondoh 2007; Nakazawa et al. 2010) and described as follows: de i dt = v ie i { W e i (e N W e N + e P W e P )} (i {N, P}), (6) 49

53 where v i is the rate of adaptation of defensive effort against the predator species i (i {N, P}). Shared prey can rapidly adjust the effort if v i is high. If both or either one of the two predators go extinct, the shared prey can abandon defense against the absent predators. If v N = v P = v, the total effort e T = e N + e P satisfies de T dt = v (e N W W + e e P ) (1 e N e T ). (7) P Because e N (t) > 0 and e P (t) >0 if e N (0) > 0 and e P (0) > 0, the constraint 0 e N + e P 1 is satisfied if 0 e N (0) + e P (0) 1. Hereafter, we assume v N = v P. This model includes two limiting cases. When the omnivore does not prey on the intermediate predator (no IGP; a NP = 0), configuration of this model is equivalent to exploitative competition (Stewart and Levin 1973), and some theoreticians have shown that three species can coexist only in non-equilibrium states (Hsu et al. 1978; Armstrong and McGehee 1980). However, predation on the intermediate predator by the omnivore (IGP) may enable stable coexistence of two predators and weaken control of the shared prey by the intermediate predator. When the top predator does not prey on the shared prey (no omnivory; a RP = 0), configuration of this model is equivalent to the tritrophic food chain. Some studies suggested that omnivory (predation on the shared basal prey) by the top predator weakens the effect of trophic cascades on basal prey, and population density of the prey may not be enhanced (Pace et al. 1999). We assume that, of the two energy paths from the basal prey to the omnivore, the direct path is more efficient than the indirect one through the intermediate predator, i.e. b RP > b RN b NP. Then, the omnivore can grow efficiently by consuming the shared prey if the predation rate is high, and alternative stable states may appear in the absence of 50

54 adaptive defense and switching predation (Holt and Polis 1997). To clarify effects of adaptive defense and switching predation, we assume that the intermediate predator is superior to the omnivore in exploiting the shared prey, which is a necessary condition for coexistence of two predators at equilibrium in classical IGP theory (Polis and Holt 1992; Holt and Polis 1997). The condition is met when the intermediate predator can more efficiently suppress the population density of the shared prey than the omnivore. If we do not consider adaptive defense by the shared prey and switching predation by the omnivore, the necessary condition for coexistence is m N b RN a RN < m P b RP a RP. (8) If we consider switching predation by the omnivore ignoring adaptive defense by the shared prey, the condition becomes more complex, and is described as follows: m N m P <, (9) b RN a RN (b RP m P h RP )a RP where (b RP m P h RP ) is positive if the omnivore can persist by consuming only the shared prey. Because analytical solutions of the system containing both or either one of adaptive defense and switching predation are too difficult to solve, we adopt numerical calculations to analyze the system. We derive equilibrium population densities and defensive efforts by setting the right-hand sides of Eqs.(1) and (6) to zero. Then, we evaluate their local asymptotic stability by the Routh-Hurwitz criterion. We numerically calculate mean population densities and defensive efforts over a period when the equilibria are unstable and the solution is periodic. A predator species is considered to be persistent if a positive equilibrium or steady state at which the species survives is stable or there exists a stable positive limit cycle. 51

55 Here, we can distinguish four cases depending on the absence or presence of adaptive defense and switching predation, respectively. First, we examine how persistence of each predator species and stability of the system depend on the encounter rates of the omnivore with the shared prey and intermediate predator in each case. Second, we compare the equilibrium population density of the shared prey among the case with two predators and two cases with a single predator, focusing on the strength of IGP (a NP ) and the encounter rate between the omnivore and the shared prey (a RP ). Finally, we compare predation pressure on the shared prey among the case with two predators and two cases with a single predator to explain why the introduction of multiple natural enemies can improve biological control by suppressing the shared prey Results Effects of encounter rates on equilibrium states We investigated effects of predation by the omnivore on equilibrium states in each of four cases described in the last paragraph of the model section Model with neither adaptive defense nor switching predation In the case without any defense and switching predation, coexistence of two predators was possible if and only if the omnivore consumed the shared prey slightly and the intermediate predator sufficiently (light grey region RNP in Fig. 2-1a). When the 52

56 omnivore exploited the shared prey more often (a RP became higher), the intermediate predator went extinct because of severe predation and competitive pressure from the omnivore (region RP in Fig. 2-1a). In contrast, when the omnivore consumed the intermediate predator insufficiently (a NP was low), it was outcompeted by the competitively superior intermediate predator (region RN in Fig. 2-1a). Exclusion of either one of two predators occurred depending on initial population densities between regions RN and RP (bistable; region RN/RP in Fig. 2-1a). These results reconfirmed the classical theory (Holt and Polis 1997) Model with adaptive defense In the case with adaptive defense, three species coexistence was stable when a RP was low and a NP was high, similar to the case without adaptive defense (light grey region RNP in Fig. 2-1b). The shared prey abandoned any defense in this region because the intermediate predator was suppressed by predation from the omnivore, and the latter consumed the shared prey only slightly. When IGP became weaker, the shared prey employed defense against the competitively superior intermediate predator. Therefore, three species coexistence was still stable (light grey region RNP-e N in Fig. 2-1b). When a RP became higher and a NP was still low, the shared prey used defense against the omnivore, and three species coexistence was realized (light grey region RNP-e N e P in Fig. 2-1b). If a RP increased, three species coexistence became unstable and the population densities and defensive efforts oscillated temporally (grey region oscil.-rnp-e N e P in Fig. 2-1b). In contrast, when a NP was sufficiently high, the intermediate predator was excluded by strong IGP, and then the shared prey employed 53

57 defense against the omnivore (region RP-e P in Fig. 2-1b). As a NP increased, defensive effort e P increased and the intermediate predator was saved from extinction (light grey region RNP-e P ). However, further increases in a NP reinforced the omnivore and the maximum defensive effort by the shared prey against the omnivore was insufficient to save the intermediate predator from extinction (region RP-e P in Fig. 2-1b) Model with switching predation In the case of switching predation, the intermediate predator was hardly excluded by the omnivore because the latter switched its diet to the shared prey when the former became scarce. When IGP was weak, the omnivore was excluded by the intermediate predator because of competitive inferiority in consuming the shared prey (region RN in Fig. 2-1c). When a NP increased, the omnivore could persist and the two predators coexisted because predation on the intermediate predator compensated for competitive inferiority of the omnivore. Consequently, two predators could persist as long as the omnivore frequently attacked the intermediate predator (grey region RNP and oscil.-rnp in Fig. 2-1c). However, three species coexistence became unstable and oscillations of population densities occurred if a RP was large (grey region oscil.-rnp in Fig. 2-1c) Model with adaptive defense and switching predation In the case where both adaptive defense and switching predation were incorporated, the intermediate predator was rarely excluded, as in the case considering only 54

58 switching predation. The omnivore could persist unless both of the two encounter rates a NP and a RP were too low, and three species coexistence was realized in most parameter regions (Fig. 2-1d). When a RP was low, three species coexistence was stable if a NP was high (light grey region RNP in Fig. 2-1d) and defense against the intermediate predator was employed when a NP became lower (light grey region RNP-e N in Fig. 2-1d). However, the omnivore became extinct even if the intermediate predator was defended when a NP was too low (region RN-e N in Fig. 2-1d), as in the case with only adaptive defense. In contrast, when a RP was large, stability of the equilibrium states and presence or absence of defense by the shared prey, were highly variable depending on a NP. If a NP was sufficiently low, population densities oscillated and the shared prey temporally varied effort toward defense against the intermediate predator (region oscil.-rnp-e N in Fig. 2-1d). If a NP was higher, the shared prey abandoned effort toward defense against the intermediate predator because the predator decreased by IGP and population densities still oscillated (region oscil.-rnp in Fig. 2-1d). When a NP became slightly larger, defense against the omnivore became necessary, which indirectly favored the intermediate predator. Thus, the shared prey also employed defense against the intermediate predator and temporally switched defenses against two predators (region oscil.-rnp-e N e P in Fig. 2-1d). When a NP became even larger, constant effort toward defense against the omnivore became sufficient to suppress rebounds of both predators and the system converged to the stable positive equilibrium (region RNP-e P in Fig. 2-1d). The type III functional response reduced the necessity for defense against the omnivore because predation pressure was negligible at low prey density and saturated at high prey density. Thus, the specialized defense against the omnivore was 55

59 employed only when both encounter rates with the shared prey and intermediate predator, a RP and a NP, were high Performance of biological control We examined effects of adaptive defense and switching predation on suppression of the shared prey (pest) by predators (natural enemies). To evaluate the suppressive effects, * we considered the relative equilibrium population density of the shared prey R r defined as follows: R r = R r R k R (10) where r R k R is the population density of the shared prey in the absence of both predators (i.e., carrying capacity), and R * is that in the presence of both or either one of the predators Model with neither adaptive defense nor switching predation In the absence of any adaptive defense and switching predation, the competitively superior intermediate predator outcompeted the omnivore if a NP = 0 (no IGP), and R r * was identical with that in the system without the omnivore (circles in Fig. 2-2a). If a NP was sufficiently high and a RP was low, two predators coexisted but R * r could never fall below that in the system without the omnivore (open and closed boxes in Fig. 2-2a). In other words, introduction of multiple natural enemies could never improve 56

60 performance of biological control practices, and the classical analytical result by Holt and Polis (1997) was reconfirmed Model with adaptive defense When adaptive defense by the shared prey was incorporated in the absence of switching predation, introduction of multiple natural enemies could improve performance of biological control. If a RP was small, the omnivore was outcompeted by the intermediate predator and R * r became identical with that in the system without the omnivore (circles overlapping a dashed line in Fig. 2-2b). If a NP = 0 and a RP was sufficiently large, three species coexistence was realized because of adaptive defense, and R * r could be lower than when the omnivore was absent (circles below a dashed line in Fig. 2-2b). Similarly, R * r was minimized by introduction of multiple natural enemies if a NP became slightly larger and a RP was intermediate (open boxes below a dashed line in Fig. 2-2b). However, if a RP became sufficiently larger, the intermediate predator was excluded, and R * r became identical with that in the system without the intermediate predator (open boxes overlapping a chained line in Fig. 2-2b). When two predators were introduced simultaneously, and the threat of the omnivore (a RP ) was intermediate or high, the shared prey allocated more defensive effort toward the defense against the omnivore and less against the intermediate predator (Fig. 2-3b), and the shared prey decreased (Fig. 2-3a). If a NP was large, R * r lay between the dashed and chained lines and coalesced into the chained line when the intermediate predator became extinct (closed boxes in Fig. 2-2b). Consequently, if the shared prey could employ predator-specific defense against each predator, R * r could be lowered by introduction 57

61 of multiple natural enemies when IGP was weak (small a NP ) and the competitive ability of the omnivore (a RP ) was intermediate or high. This might happen because of difficulty defending two predators effectively under the constraint on the total effort (see section Predation pressure from two predators on the shared prey) Model with switching predation Even if switching predation by the omnivore was incorporated, the omnivore was competitively excluded by the intermediate predator if a NP = 0, and R * r was identical with that in the system without the omnivore (circles in Fig. 2-2c). When a NP became slightly high, R * r was never below that in the system without the omnivore although three species coexistence was stable (open boxes in Fig. 2-2c). When a NP was increased further, R * r became even larger than the value in the case without the intermediate predator if a RP was large (closed boxes in Fig. 2-2c). Because of the type III functional response, the omnivore reduced the predation rate on the shared prey, and increased that on the intermediate predator when the former was relatively rare, and this led to reduction in the predation pressure on the shared prey from the intermediate predator. Consequently, switching predation by the omnivore did not improve biological control of the shared prey by introduction of multiple natural enemies Model with adaptive defense and switching predation 58

62 Finally, we considered both adaptive defense and switching predation. Joint effects of them on R * r qualitatively differed from the sole effect of either one. Type III predation by the omnivore urged the shared prey to abandon employment of adaptive defense against the omnivore because the predation rate was negligible at low prey density and saturated at high prey density. Increases in the encounter rate between the shared prey and the omnivore (a RP ) also promoted the shared prey to reduce defensive effort against the intermediate predator (Fig. 2-3d) because the latter tended to be suppressed by the omnivore. Then the shared prey abundance approached the one in the absence of defense, which was lower than the level attained when the shared prey employed adaptive defense against an efficient single natural enemy (Fig. 2-3c). Therefore, a combination of switching predation by predators and adaptive defense by prey can improve biological control by introduction of multiple natural enemies even if IGP between multiple predators are included. When a RP was sufficiently high, R * r was minimized by multiple natural enemies even if IGP was strong (large a NP ) (Fig. 2-2d). Switching predation urged the shared prey to abandon defense against the omnivore and the adaptive defense was offset even if both encounter rates of the omnivore with the shared prey and intermediate predator were relatively high Predation pressure from two predators on the shared prey We examined contributions of two predators to suppression of the shared prey to elucidate why the introduction of multiple enemies could be successful when the shared prey employed adaptive defense. Figure 2-4 shows the sum of predation pressures from two predator species on the shared prey, A RN N + A RP P (Fig. 2-4a, c), 59

63 and defensive efforts allocated by the shared prey (Fig. 2-4b, d). Predation pressure from each predator was calculated as a product of the predation rate of each predator including reduction by adaptive defense and the population density of each predator. When we included only the defense but not switching predation, the sum of the predation pressures from two predators (grey and light grey region in Fig. 2-4a) exceeded those in two cases with a single predator (dashed and chained lines in Fig. 2-4a) if the encounter rate between the shared prey and the omnivore was low or intermediate. It should be noted that in spite of the increasing a RP, predation pressure from the omnivore showed a slightly decreasing trend due to the increasing defensive effort against it (e P in Fig. 2-4b), while that from the intermediate predator increased owing to the decreasing defensive effort against it (e N in Fig. 2-4b) in the latter range. However, in the former range, no defensive effort was employed and the defensive cost was low. The higher predation pressure from the two predators and the high total defensive cost reduced the prey population density below that in the case with only the intermediate predator in the latter range. Similarly, when we incorporated both the adaptive defense and switching predation, the sum of the predation pressures from two predators was higher than those in two cases with a single predator if a RP was intermediate to high (Fig. 2-4c). It should be noted that the shared prey employed no defense against the omnivore which adopted the type III functional response and gradually diminished defense against the intermediate predator (Fig. 2-4d) although both predators persisted and frequently attacked the shared prey. Although the defensive costs were low, much higher predation pressure from the two predators could explain the lower prey population density than that in the case with only the intermediate predator. 60

64 Effects of IGP and trophic cascade We also examined the effects of IGP (predation on the intermediate predator) as an agent to promote trophic cascades on the equilibrium population density of the shared prey (Fig. 2-5). In general, the prey population density increased with the strength of IGP (a NP ) and trophic cascades tended to invalidate biological control. However, two counterintuitive results are shown in Fig In the absence of adaptive defense and switching predation, the population density in the case of weak omnivory (a RP = 0.05; open diamonds in Fig. 2-5a) exceeded that in the case of no omnivory (a RP = 0; open circles in Fig. 2-5a). The increasing strength of omnivory has been considered to weaken trophic cascades and favor biological control (Shurin et al. 2002). However, in this case, direct effects of weak predation by the omnivore might be weakly negative to the shared prey but enhance the omnivore and promote indirect positive effects of trophic cascades on the shared prey through strong predation on the intermediate predator. When both adaptive defense and switching predation were incorporated, population density of the shared prey could decrease with the strength of IGP (a RP = 0.8; closed boxes in Fig. 2-5d). This is also a counterintuitive result because it means that IGP weakens trophic cascades and improves biological control. Increases in the strength of IGP with the type III functional response of the omnivore reduced the predation pressure on the shared prey and stabilized the unstable system of exploitative competition. This might suppress sharp increases in the shared prey population. If the strength of omnivory was intermediate to high (a RP = 0.4; open boxes and a RP = 0.8; closed boxes in Fig. 2-5d) 61

65 and the strength of IGP was low to intermediate, the prey density was still lower than that in the case with only the intermediate predator. Thus, multiple predators and IGP could improve biological control Discussion In this study, we investigated effects of IGP, adaptive defense by pests, and switching predation by omnivorous predators on the efficiency of biological control by introduction of multiple natural enemies. We extended classical theoretical studies on IGP (Polis and Holt 1992; Holt and Polis 1997) by including two kinds of adaptive defenses by the shared prey and switching predation by the omnivore, and examined persistence of each predator species and degree of suppression of the shared prey by each behavior. Adaptive defenses and switching predation enhanced three species coexistence in communities with weak and strong IGP respectively. Thus, separate or joint effects of adaptive behavior by prey and predators may be one of the factors to explain ubiquity of IGP in nature. It was also found that introduction of multiple natural enemies could improve biological control even if IGP was included, and that the effects depended on types and combinations of prey and predator behavior. IGP could even enhance biological control by multiple natural enemies in the presence of adaptive defense and the type III functional response Effects of behaviors on efficiency of pest control 62

66 When the shared prey employed predator-specific adaptive defenses to increase its own fitness, introduction of multiple natural enemies could improve biological control. If the defense was predator-nonspecific and effective against both predators, introduction of multiple natural enemies was less effective than that of a single more efficient intermediate predator because the negative effects of IGP could never be counteracted by the nonspecific defense (Ikegawa et al. 2014). When only the efficient intermediate predator was introduced, the shared prey could effectively employ adaptive defense against the predator and increase its population density. However, when the omnivore with a high rate of omnivory was also introduced, the shared prey must have allocated more defensive effort against the omnivore, and less against the intermediate predator. Then, by the indirect effect of reduced predation on the intermediate predator by the omnivore, and the direct effect of reduction in defensive effort against the intermediate predator, predation pressure on the shared prey increased and the shared prey decreased (Fig. 2-4a, b). This effect was observed even if no cost accompanied the defenses because the shared prey switched defenses with the increasing rate of omnivory due to the constraint on the total effort (0 e N + e P 1). Type III predation by the omnivore contributed to improvement of biological control by multiple natural enemies when the shared prey employed predator-specific adaptive defenses. It urged the shared prey to abandon adaptive defense against the omnivore because the predation rate was negligible at low prey density and saturated at high prey density. Increases in the strength of omnivory also prompted the shared prey to reduce defensive effort against the intermediate predator because the latter tended to be suppressed by the omnivore. Then, the shared prey abundance approached that in the absence of defense, which was lower than the level attained when the shared prey 63

67 employed adaptive defense against the more efficient intermediate predator. Therefore, switching predation by predators can invalidate adaptive defense by prey and improve biological control by introduction of multiple natural enemies even if IGP between multiple predators is included and relatively severe. We examined robustness of our results by varying some parameters fixed until now. The indicator of productivity k R (or r R ) did not affect qualitative results greatly. However, when k R was large (resp. small), the shared prey employing adaptive defense was less (resp. more) effectively suppressed by two predators. Further increases in k R destabilized the system. The effects were observed regardless of the omnivore s functional response (type I or type III). This suggests that positive effects of multiple natural enemies on efficiency of biological control tend to appear in relatively poor environments. Similarly, efficiencies (f N, and f P ) or costs (c N, and c P ) of defenses, mortality rates of predators (m N, and m P ), conversion efficiencies (b RN, b RP, and b NP ), and handling times (h RP, and h NP ) did not strongly affect qualitative outcomes as long as necessary conditions for three species coexistence, Eq.(8) or (9), were met. Adaptation rates of the shared prey v N and v P did not affect the trend in the equilibrium population densities as well but influenced stability of the system as in Nakazawa et al. (2010) Effects of IGP and trophic cascade We found that IGP can weaken trophic cascades on the population density of the shared prey and improve biological control, and that omnivory on the shared prey can enhance the target prey and depress biological control. These findings are 64

68 counterintuitive and inconsistent with traditional theory (Shurin et al. 2002). When the encounter rate between the omnivore and the shared prey is high and the system of exploitative competition with adaptive defense by the shared prey is unstable, weak IGP by the omnivore with the type III functional response can stabilize the system. Then, as the amplitude of the oscillations decreases, the mean population density of the shared prey also decreases. We have found that omnivory on the shared prey can enhance the target prey and depress biological control. However, this effect appeared only when the functional response of the omnivore is type I. If suppression of the shared prey by omnivory is weak and enhancement of the omnivore is substantial, the trophic cascade can be reinforced and the shared prey may increase. However, if the functional response of the omnivore is type III to the two prey items, omnivory has the effect of decreasing the predation rate on the intermediate predator and does not promote a trophic cascade. Thus, the shared prey density tends to be smaller than in the case of a type I functional response in which the predation rates on the two prey are independent. Therefore, effects of omnivory on the shared prey depend on the type of functional responses Comparison with previous theoretical studies Although empirical experiments of introduction of multiple natural enemies have shown positive (Eubanks 2001; Dinter 2002; Snyder and Ives 2003), negative (Rosenheim et al. 1993; Schausberger and Walzer 2001; Snyder and Ives 2001), and neutral (Rosenheim 2001; Venzon et al. 2001) outcomes for efficiencies of pest control by multiple natural enemies, previous theoretical studies tended to derive negative 65

69 results. Kakehashi et al. (1984) studied effects of the introduction of two parasitoid species and showed that aggregative responses of two parasitoid species to the shared host might disrupt biological control by multiple natural enemies owing to strong niche overlap between two enemies. A classical theoretical study on IGP also showed negative effects of introduction of multiple natural enemies on biological control (Holt and Polis 1997), and there is a gap between the theory and the observation showing positive results. In fact, some theoretical studies did show positive effects of introduction of multiple natural enemies on biological control, but the mechanism we presented here is different from those presented in earlier studies. In a paper to show that short-term experimental studies may cause erroneous decisions about the introduction of natural enemies (Briggs and Borer 2005), they studied a transient behavior of a simple mathematical model and demonstrated that introduction of an omnivore could enhance biological control in a short-term. Additional prey or predators from outside of an IGP module may change dynamics and persistence of species in the module (Kondoh 2008). Some studies reported that an alternative prey for both or either one of the two predators enhances coexistence of the predators and suppression of the shared prey (Daugherty et al. 2007; Okuyama 2009). In contrast, we have demonstrated that introduction of multiple natural enemies can improve long-term suppression of pests in an equilibrium or in a limit cycle without incorporating an additional prey or predator when adaptive defense is employed by the shared prey and the functional response of the omnivore is type III. In the absence of switching predation, our model of predator-specific adaptive defenses by dynamical changes in defensive effort is essentially the same as the one in 66

70 Nakazawa et al. (2010). Nakazawa et al. (2010) showed that the adaptive defense enhanced coexistence of two predators but destabilized the IGP system when environmental productivity and the rate of omnivory (r R k R and a RP in our model) increased and IGP (a NP in our model) was weak. Because it can be easily confirmed by normalizing the prey population by r R k R that increasing productivity effectively enhances interactions between the shared prey and two predators, our results are qualitatively the same as theirs. When IGP is weak and omnivory is strong, the system without adaptive defense tends to be bistable because of exploitative competition. Thus, adaptive defense and shifting effort toward the more threatening predator cause cycles of efforts and population densities between two boundary states where one predator is absent and the other coexists with the basal prey, which was called compositional cycles by Nakazawa et al. (2010). In contrast, switching predation tended to enable stable or unstable coexistence of two predators unless IGP was too weak, and adaptive defense against the intermediate predator permitted persistence of the omnivore and coexistence of two predators when IGP was weak (compare Fig. 1c and 1d). However, strong omnivory caused oscillations of population densities similarly as in the case of no defense and stronger IGP. In this case, defensive effort against the intermediate predator followed the population oscillations and delayed oscillations of defensive effort appeared. Therefore, the role of adaptive defense on stability of the system was different depending on the types of functional responses. Effects of another form of diet choice by the omnivore on IGP systems were also studied by some theoreticians (Křivan 2000; Křivan and Diehl 2005). Křivan and Diehl (2005) assumed optimal foraging by the omnivore through phenotypic plasticity, in which the omnivore preyed on the less profitable prey only when the abundance of the 67

71 more profitable one decreased below a threshold density. They suggested that three species coexistence was enhanced by optimal foraging, but it was limited to the case with weak omnivory (small a RP in our model) and strong IGP (large a NP in our model) (Křivan and Diehl 2005). When the more profitable intermediate predator decreased, the omnivore additionally preyed on the less profitable shared prey, and then the intermediate predator might go extinct if omnivory was severe because of competitive pressure in addition to direct predation by the omnivore. In contrast, in our model, the omnivore preyed on the shared prey more frequently when the intermediate predator decreased and the predation rate on the intermediate predator became negligible when the intermediate predator was scarce because of the type III functional response. Therefore, switching predation could promote coexistence by decreasing direct predation from the omnivore on the intermediate predator, even if competitive pressure from the omnivore increased through consumption of the shared prey Conclusions and perspectives Now we can discuss effective control measures of pests by the introduction of natural enemies. When the pest does not employ adaptive defense against natural enemies, the most effective single enemy should be introduced. However, rapid evolution of prey defense seems pervasive as shown by Hiltunen et al. (2014). When the pest employs adaptive defense, introduction of multiple natural enemies can be more effective than that of a single one if IGP between the enemies is absent or weak. When IGP by one of the natural enemies is severe, introduction of multiple natural enemies may still be 68

72 effective if the omnivore switches its diet depending on the relative abundance of their prey and the pest employs adaptive defenses. In this article, we considered a combination of dynamical changes in defensive effort of prey to increase its fitness and plastic changes in predation rates directly dependent on prey densities. However, we did not examine the effects of plastic shifts between defensive and non-defensive morphs of the prey or dynamical changes in predator s traits to maximize their fitness. Vos et al. (2004) assumed herbivore-induced plastic defense by plant species and predator-induced one by herbivores in a tritrophic food chain system. They showed that the induced defenses enhanced stable three species coexistence whereas the permanent fixed ones did not (Vos et al. 2004). Kondoh (2003) considered a food web model and examined effects of adaptive food choice that maximizes energy gain by increasing foraging efforts allocated toward resources with higher profitability than the average. He showed that adaptive foraging could enhance long term persistence and stability of complex communities. Recently, some theoreticians suggested that phenotypic plasticity and rapid evolution might result in qualitatively different dynamics in predator-prey systems (Cortez 2011; Yamamichi et al. 2011). Yamamichi et al. (2011) showed that phenotypic plasticity tends to stabilize population dynamics more strongly than rapid evolution because of time delay inevitable in the latter. Since none of these studies have examined effects of dynamical changes in predation rates or plastic changes in defensive traits on success of biological control by multiple natural enemies, theoretical works incorporating such factors will be needed in future. We have found that adaptive behavior of pests and natural enemies profoundly influences the successful introduction of multiple natural enemies for biological 69

73 control. Although some empirical studies examined effects of the behavior of organisms, such as adaptive defense by pests (Bernstein 1984; Grostal and Dicke 1999; Pallini et al. 1999), or diet choice by natural enemies (Chow et al. 2008; Davey et al. 2013) in agricultural systems, their effects on suppression of the pests were rarely explored (but see Chow et al. 2008). One of the reasons for this is that the effects of behavior on some ecological properties of relevant species (e.g. fecundity, mortality, activity, etc.) are too difficult to quantitatively examine even if the behavior itself can be observed. Our results suggest that it is important to measure the strength of IGP, record the absence or presence of adaptive defense, and examine the type of functional response of the omnivore. Since some empirical works revealed the importance of modes of dispersal of pests (Bernstein 1984; Pallini et al. 1999) and natural enemies (Saavedra et al. 1997; Magalhães et al. 2004), introduction of spatial heterogeneity and dispersal of pests or natural enemies into our model may be useful to uncover effects of multiple natural enemies. Because diet choice or prey preference of an omnivorous natural enemy need not be switching predation (Foglar et al. 1990; Walzer and Schausberger 1999), it is also necessary to study effects of other types of diet choice, e.g., optimal diet choice, on persistence, stability, and suppression of the shared prey in combination with the effects of adaptive defense by shared prey in an IGP system. By considering such factors observed in natural systems, our model can contribute more to development of effective biological control in actual farms. 70

74 Figures (Chapter 2) Figure 2-1: Dependence of equilibrium states on the encounter rates between the omnivore and two prey species in systems (a) including neither adaptive defense nor switching predation, (b) including only adaptive defense, (c) including only switching predation, and (d) including both adaptive defense and switching predation. Horizontal and vertical axes indicate the encounter rate between the omnivore and the shared prey (a RP ), and that between the omnivore and the intermediate predator (a NP ), respectively. Regions in the parameter space are first categorized by persistence of predators; in RNP both predators persist (shaded regions), in RN only the intermediate predator persists, in RP only the omnivore persists, and in RN/RP either one of two predators persists (bistability dependent on initial values). If the system is unstable and population densities and defensive efforts oscillate, we add oscil. as a prefix of each state (dark shaded regions). In addition, if the shared prey employs each defense, we add it as a suffix of each state. The other parameter values are r R = 5, k R = 3, b RN = b RP = b NP = 0.5, a RN = 1, m N = m P = 0.5, h RP = h NP = 0.75, f N = f P = 0.75, c N = c P = 0.25, v N = v P = 1. 71

75 Figure 2-2: Dependence of the equilibrium population density of the shared prey on the encounter rate between the omnivore and the shared prey (a RP ) in systems (a) including neither adaptive defense nor switching predation, (b) including only adaptive defense, (c) including only switching predation, and (d) including both adaptive defense and switching predation. We used mean population density over a period if the system is unstable and the asymptotic solutions are periodic. Horizontal axes are identical with those in Fig Vertical axes indicate the relative equilibrium population density of the shared prey in the presence of either one or both predators (R * ) to that in the absence of both predators (carrying capacity of the shared prey; r R k R ). Dashed lines indicate the equilibrium population density of the shared prey in the absence of the omnivore as a function of a RP and chained lines indicate that in the absence of the intermediate predator. Each symbol represents different values of the encounter rate between the omnivore and the intermediate predator (circles: a NP = 0, open boxes: a NP = 0.2, closed boxes: a NP = 0.4). Large symbols indicate three species coexistence and small ones indicate extinction of one of two predators. The other parameter values are identical with those in Fig

76 Figure 2-3: Temporal variations of population densities and defensive efforts in systems (a-b) including only adaptive defense, and (c-d) including both adaptive defense and switching predation. Solid, dashed, and chained lines in (a) and (c) indicate population densities of the shared prey, intermediate predator, and omnivore, respectively. Similarly, dashed, and chained lines in (b) and (d) indicate efforts toward the defense against the intermediate predator, and omnivore, respectively. Thick lines indicate the presence of both predators whereas thin ones indicate the absence of the omnivore. Parameter values are a RP = 0.5, a NP = 0.15 for (a-b), and a RP = 0.8, a NP = 0.25 for (c-d). The other parameter values are identical with those in Fig

77 Figure 2-4: Dependence of predation pressure on the shared prey (Fig. 2-4a, c), and defensive efforts by the shared prey (Fig. 2-4b, d) on the encounter rate between the omnivore and the shared prey (a RP ). Switching predation by the omnivore is included in Fig. 2-3c and d, while not in Fig. 2-3a and b. Horizontal axes are identical with those in Fig Dashed lines indicate predation pressure and defensive effort in the absence of the omnivore and chained lines indicate those in the absence of the intermediate predator. (a and c) Vertical axes indicate the sum of predation pressure from two predators. Predation pressure from the omnivore is expressed by light grey regions, and that from the intermediate predator is expressed by dark grey regions. (b and d) Vertical axes indicate defensive efforts toward the intermediate predator (e N ), and the omnivore (e P ). e N is expressed by dark grey symbols and e P is by light grey symbols. We set the strength of IGP at 0.2 (a NP = 0.2) and the other parameter values are identical with those in Fig