UNIVERSITY of CALIFORNIA Santa Barbara. Modeling host-parasitoid dynamics. A Thesis submitted in partial satisfaction of the

Transcription

1 UNIVERSITY of CALIFORNIA Santa Barbara Modeling host-parasitoid dynamics A Thesis submitted in partial satisfaction of the requirements for the degree Master of Arts in Ecology, Evolution and Marine Biology by Abhyudai Singh Committee in charge: Professor Roger M. Nisbet, Chair Professor William W. Murdoch Professor Bruce Kendall December 2007

2 The dissertation of Abhyudai Singh is approved. Professor William W. Murdoch Professor Bruce Kendall Professor Roger M. Nisbet, Committee Chair December 2007

3 To my grandparents iii

4 Acknowledgements I would like to express my deepest appreciation to Roger Nisbet and Bill Murdoch for their guidance and insight. I would also like to thank Joao Hespanha for support without which this thesis would not have been possible. Last but not the least, I thank my parents (Yatindra and Neeta Singh) and my wife (Swati Singh). They have all been a great source of strength and inspiration all through this work. iv

5 Curriculum Vitæ Abhyudai Singh Abhyudai Singh was born in Allahabad, India, on December 3, Education 2006 M.S. Mechanical Engineering, Michigan State University, East Lansing M.S. Electrical and Computer Engineering, Michigan State University, East Lansing B.Tech. Mechanical Engineering, Indian Institute of Technology, Kanpur, India. Experience 2004 present Research Assistant, University of California, Santa Barbara Research Assistant, Michigan State University, East Lansing Teaching Assistant, Michigan State University, East Lansing. Selected Publications A. Singh, R. Mukherjee, K. Turner and S. Shaw. MEMS Implementation of Axial and Follower End Forces. Journal of Sound and Vibration, 286, , v

6 A. Singh and H. K. Khalil. Regulation of Nonlinear Systems Using Conditional Integrators. International Journal of Robust and Nonlinear Control, 15, , J. P. Hespanha and A. Singh. Stochastic Models for Chemically Reacting Systems Using Polynomial Stochastic Hybrid Systems. International Journal of Robust and Nonlinear Control, 15, , A. Singh and J. P. Hespanha. Lognormal Moment Closures for Chemically Reacting Systems. In Proc. of the 45th IEEE Conference on Decision and Control, San Diego, A. Singh and R. M. Nisbet. Semi-discrete Host-Parasitoid Models. Journal of Theoretical Biology, 247, , A. Singh and J. P. Hespanha. A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model. Bulletin of Math Biology, 69, , A. Singh and J. P. Hespanha. Stochastic Analysis of Gene Regulatory Networks Using Moment Closure. In Proc. of the 2007 American Control Conference, New York, vi

7 Abstract Modeling host-parasitoid dynamics by Abhyudai Singh Arthropod host-parasitoid interactions constitute a very important class of consumer resource dynamics. Discrete-time models, starting from the seminal work of Nicholson and Bailey are a tradition for such interactions. Although the classic Nicholson-Bailey model predicts an unstable equilibrium, host-parasitoid population are often observed to be stable in nature and a fundamental area of research in population ecology is to elucidate mechanisms that can account for this stability. This thesis investigates some of these mechanisms and is divided into two parts. The first part of the thesis introduces a semi-discrete approach to modeling host-parasitoid populations. This approach takes account of the various withingeneration dynamics by modeling them in continuous-time, while still modeling reproduction as a discrete event. Using this formalism, results connecting the stability of the host-parasitoid interaction with different forms of density dependence and the form of the functional response are derived. The latter results contradict previous conclusions from heuristically formulated models, and illustrate the need for such a semi-discrete approach in discrete-time host-parasitoid theory. The second part of the thesis investigates how variation in the risk of parasitism among individual hosts can stabilize the Nicholson-Bailey model equilibrium. The vii

8 famous CV 2 > 1 rule states that this equilibrium can be stabilized if there is sufficient variation in risk. Analysis in this thesis show that this is not a general result. Instead, unless risk is gamma-distributed across hosts, and except for host reproduction rates close to 1, it is the shape of the distribution of risk among hosts that determines stability. For unconditional stability, the distribution must be skewed, with a modal risk of zero. If the distribution of risk does not have this appropriate shape then even infinite coefficient of variation (CV ) cannot stabilize the equilibrium for certain values of host reproduction rates. viii

9 Contents Acknowledgements Curriculum Vitæ Abstract List of Figures iv v vii xi 1 Semi-discrete host-parasitoid dynamics Model Structure Effects of Functional Responses Quadratic Functional Responses Higher Order Functional Responses Type II and III Functional Responses Effects of Density Dependence Density Dependent Host Mortality Density Dependent Parasitoid Mortality Discussion Stability in host-parasitoid models due to variability in risk Nicholson-Bailey model with variability in host risk Stability, shape of the risk distribution and the CV 2 rule Risk has an inverse gaussian distribution ix

10 2.2.2 Risk is bounded from below Unexpected patterns among stability, R and CV Skewed risk, parasitoid efficiency, and host suppression Discussion What the field biologist should measure Stability and within-generation dynamics Bibliography 42 A Semi-discrete host-parasitoid models 47 A.1 Appendix : Stability Analysis for Quadratic functional responses. 47 A.2 Appendix : Stability Analysis for Higher Order Functional Responses 48 A.3 Appendix : Stability Analysis for Type II and III Functional Responses B Stability in host-parasitoid models due to variability in risk 51 B.1 Appendix : Stability analysis for host-parasitoid models B.2 Appendix : Stability condition in terms of distribution of risk.. 53 B.2.1 Stability for small values of R B.2.2 Stability for large values of R B.3 Appendix : Stability Analysis for host-parasitoid when risk is inverse gaussian distributed B.4 Appendix : Stability Analysis for host-parasitoid when risk is bounded from below B.5 Appendix : Parasitoid efficiency B.6 Appendix : Stability Analysis in terms of distribution of fraction parasitized B.6.1 Stability for small values of R B.6.2 Stability for large values of R x

11 List of Figures 1.1 Life cycle of the Host in year t Plots of the fraction of host larvae escaping parasitism f as a function of host density H t, as given by (1.14) (dashed line, functional response is a function of the current host larvae density) and (1.16) (solid line, functional response is a function of the initial host larvae density) for R = 2, T = 1, c =.01, m = 1 and P t = The stability region specified in (1.27), for the discrete-time model (1.23) as a function of strength of density-dependent mortality vs. parasitism (c 1 /kc) and R (the number of viable eggs produced by each adult host). The no-parasitoid equilibrium refers to (1.24) while the host-parasitoid equilibrium refers to (1.26) Plot of host and parasitoid equilibrium densities as a function of the strength of density-dependent mortality (c 1 ). Other parameter were taken as R = 2, T = 1, k = 1 and c = 1. H and P refer to the host-parasitoid equilibrium (1.26) and HNC refers to the noparasitoid equilibrium given by (1.24). The solid and dashed lines correspond to stable and unstable equilibrium densities, respectively Population densities of hosts and parasitoids in year t as given by the discrete-time model (1.23). Parameters taken as R = 2, c 1 =.01, k = 1, T = 1 and a) c =.05, b) c =.02 and c) c =.0125 which correspond to values below, in and above the stable region (1.27), respectively. Initial densities are 20 and 30 for the host and parasitoid respectively Gamma distribution with mean one and CV 2 =.5 (dotted line), CV 2 = 1 (dashed line), CV 2 = 3 (solid line) xi

12 2.2 Inverse gaussian distribution with mean one and CV 2 = 1 (dashed line), CV 2 = 3 (solid line) Gamma distribution with mean one and CV 2 = 1 shifted to the right by 0.1 (c = 0.1) Region of stability for the discrete-time model (2.9) when g(x) is a gamma distribution with mean 1 (c = 1) c = 0.05 and c = Parasitoid efficiency defined as ln(r) where P denotes the parasitoid equilibrium density as a function of host rate of increase for P different distributions of risk : 1) No variability in risk (Nicholson- Bailey model). All host have risk equal to 1. 2) Risk has a gamma distribution with mean 1 and variance ) Risk has an inverse gaussian distribution with mean 1 and variance 2. 4) Risk has a gamma distribution with mean 1 and variance 2. Solid lines and dashed lines represent stable and unstable host-parasitoid equilibrium, respectively Adult host equilibrium as a function of host rate of increase for different distributions of risk. See caption of Figure 2.5 for more details xii

13 Chapter 1 Semi-discrete host-parasitoid dynamics One of the central themes in ecology is the interaction between populations of consumers (e.g. predators, parasitoids) and resources (e.g. prey, host). There is a large body of literature that studies such interactions using two well developed approaches: continuous-time models, which are used to model populations with overlapping generations and all year round reproduction, and discrete-time models, which are more suited for populations which reproduce in a discrete pulse determined by season [27]. A large body of theory has focused on the dynamics of arthropod hostparasitoid systems, dating back to the seminal work of Nicholson and Bailey [33]. Reviews can be found in [16] and [27]. Discrete-time models are the traditional framework used, a choice that reflects the univoltine life histories of many temperate-region insects. In such systems the host species is usually vulnerable 1

14 to a particular parasitoid species at only one stage of its life cycle, commonly the larval stage. One common life cycle involves adult hosts that emerge during spring or summer, lay eggs and then die. The eggs mature into larvae and for a short interval of time are vulnerable to attack by the parasitoids. Surviving host larvae pupate, overwinter in the pupal stage, and emerge as adult hosts the following year. Adult parasitoids search for hosts during the window of time that host larvae are present, then die. Parasitized hosts mature into a juvenile parasitoid, pupate, overwinter, and emerge as adults the following year. Synchronized life cycles, little overlap of life stages, and no overlap of generations in both hosts and parasitoids suggest that discrete-time models are appropriate for these systems. The updating function in a discrete-time, host-parasitoid model relates the population densities at a fixed date in one year to those at the same date in the previous year, and describes the cumulative effect of all the processes that have occurred with in the year. However, life processes are continuous, and the form taken by this cumulative effect may be far from intuitive, especially where multiple processes operate simultaneously. A systematic way to formulate the updating function is presented in Box 4.1 of [27], and involves a hybrid approach, where the continuous processes involved in the within-year dynamics are described by a continuous-time model, and reproduction is modeled as a discrete event. We follow [35], and call such models semi-discrete. Semi-discrete models have previously been used to investigate interactions between adults and juveniles within a season with discrete between-season dynamics [11], to model systems where a host or resource has discrete generations and is attacked by a consumer with non-seasonal dynamics (e.g. [8, 3, 5]), and 2

15 systems with discrete consumer generations and very fast resource dynamics [12]. Semi-discrete models have also been used to address specific questions relating to host-parasitoid dynamics; for example the stabilizing effects of within-season movement of parasitoids (e.g. [37]), and the consequences of phenological asynchrony between parasitoids and their hosts [25, 13]. However, the main body of discrete-time, host-parasitoid theory does not take advantage of this formalism. In this paper we revisit discrete-time host-parasitoid theory using semi-discrete models. We derive new results relating stability to the form of the functional response, and to different forms of density dependence. Some previous conclusions from heuristically formulated models survive, but others are contradicted, notably those relating the form of the functional response to stability. Our approach can easily be extended to include more complex biological interactions in contexts where a number of processes can occur concurrently. 1.1 Model Structure by A general model describing host-parasitoid dynamics in discrete-time is given H t+1 = F (H t, P t ) (1.1a) P t+1 = G(H t, P t ) (1.1b) where H t and P t are the adult host and adult parasitoid densities, respectively, at a fixed date near the start of year t (where t is an integer). The host life cycle is illustrated in Figure 1.1 where H t adult hosts give rise to eggs which mature into RH t host larvae at the start of the vulnerable stage. Here R > 1 denotes 3

16 the number of viable eggs produced by each adult host. The time within the vulnerable larvae stage is denoted by τ which varies from 0 to T corresponding to the start and end of the the vulnerable stage, respectively. The update functions Start of year t H t R H t End of year t Pupae Adult Egg Larvae Pupae τ=0 τ=t Figure 1.1. Life cycle of the Host in year t. F, G are obtained by first writing a continuous-time model that describes the various continuous-mortality sources that occur during the vulnerable stage of the host and is given by the following system of ordinary differential equations dl(τ, t) dτ dp (τ, t) dτ di(τ, t) dτ = g 1 [L(τ, t), P (τ, t), I(τ, t)]p (τ, t)l(τ, t) (1.2a) g 2 [L(τ, t), P (τ, t), I(τ, t)]l(τ, t) (1.2b) = g 3 [L(τ, t), P (τ, t), I(τ, t)]p (τ, t) (1.2c) =g 1 [L(τ, t), P (τ, t), I(τ, t)]p (τ, t)l(τ, t) g 4 [L(τ, t), P (τ, t), I(τ, t)]i(τ, t), (1.2d) where P (τ, t), L(τ, t) and I(τ, t) are the density of parasitoids, un-parasitized and parasitized host larvae, respectively, at a time τ within the vulnerable stage in year t. For convenience of presentation, in the rest of the paper we suppress the dependence on τ and t and represent the above differential equations by dl dτ = g 1LP g 2 L, dp dτ = g 3P, di dτ = g 1LP g 4 I. (1.3) The functions g 2, g 3 and g 4 represent (potentially density-dependent) host mortality due to causes other than parasitism, parasitoid mortality, and mortality of 4

17 the parasitized larvae, respectively. The function g 1 is the (potentially densitydependent) attack rate of the parasitoids. The product g 1 L is frequently referred to as the functional response and represents the instantaneous rate at which hosts are attacked per parasitoid in the population. The above set of ordinary differential equations are then integrated starting from τ = 0 with initial conditions L(0, t) = RH t, P (0, t) = P t, I(0, t) = 0, (1.4) up to τ = T. Assuming that each parasitized host larvae gives rise to k adult parasitoids in the next generation, the update functions can be obtained as F (H t, P t ) = L(T, t) (1.5a) G(H t, P t ) = ki(t, t). (1.5b) The stability analysis for the resulting discrete-time model can then be investigated using the standard Jury conditions [10]. If g 2 = g 4 = 0, i.e. host larvae mortality is solely through parasitism and there is no parasitized host mortality, then from (1.2) we have that L(τ, t) + I(τ, t) is constant throughout the vulnerable period and is equal to RH t. In that case the model (1.1) takes the more familiar form where H t+1 = RH t f(h t, P t ) (1.6a) P t+1 = krh t [1 f(h t, P t )] (1.6b) f(h t, P t ) := F (H t, P t ) RH t (1.7) denotes the fraction of host larvae that escape parasitism and is calculated from the solution of the system (1.2) of differential equations. 5

18 1.2 Effects of Functional Responses If the attack rate g 1 is assumed to be a constant c, implying a linear functional response, and if g 2 = g 3 = g 4 = 0, then integrating the corresponding continuoustime model leads to the classic Nicholson-Bailey model with f(h t, P t ) = exp ( ct P t ), (1.8) in the discrete-time model (1.6) [Box 4.1, [27]]. A typical population time series of this model is as follows: at low densities of hosts and parasitoids both the populations grow, when the population of hosts becomes large enough the parasitoid begins to overexploit the host leading to a crash of the host population, followed by a crash of the parasitoids. Such cycles of over exploitation and crashes make this interaction unstable and both hosts and parasitoids show diverging oscillations. Various authors have investigated Type II [36] and Type III functional responses [17] with the conclusion that they do not stabilize such discrete generation host-parasitoid populations. They did this by assuming that the attack rate is a function of the initial host larvae density. For example, for a Type II functional response, it was taken as g 1 = c = 1 + L(0,t) L H c (1.9) 1 + RHt L H for some constants c and L H. By contrast, our semi-discrete model recognizes that in a real system, the number of hosts larvae declines throughout the vulnerable period due to attack by the parasitoids. Thus it is assumed that the attack rate is a function of L(τ, t), the current host larvae density instead of L(0, t). We illustrate next, using a simple class of functional responses which have the form cl m+1, that the within season variation in host density can change the qualitative 6

19 dynamics from instability to stability. These accelerating functional responses are mathematically easy to analyze and provide insight on host-parasitoid interactions for the more complicated Type III functional response which we investigate in Section We begin by considering quadratic functional responses Quadratic Functional Responses We consider the dynamical interaction dl dτ = cl2 P, di dτ = cl2 P, dp dτ = 0, (1.10) i.e. g 1 = cl, g 2 = 0, g 3 = 0 and g 4 = 0 which incorporates a quadratic functional response. The solution to the above continuous-time model is L(τ, t) = RH t 1 + crh t P t τ, P (τ, t) = P t, I(τ, t) = RH t L(τ, t) (1.11) which yields the following function f(h t, P t ) in the discrete-time model (1.6) f(h t, R t ) = L(T, t) RH t = ct RH t P t. (1.12) Stability analysis of the model (see Appendix A.1) reveals that the equilibrium point H = 1 kt Rc, P = k(r 1)H (1.13) is neutrally stable with a period of 2π/ arctan( R 2 1) for small amplitude oscillations. Thus, the cycle period is very long when R is only slightly greater than one (the minimum value for viable populations), drops to six for R = 2, and asymptotically approaches a four year period of oscillation with increasing R. The neutrally stability persists if we add a background density-independent parasitoid mortality, i.e, g 3 = c 1. We obtain the same form for the function f as 7

20 (1.12) with ct being replaced by c[1 exp( c 1 T )]/c 1. This does not affect the above neutral stability result which holds independent of c Higher Order Functional Responses We now consider a generalization of the above model by taking the functional response to be cl m+1 with m > 0, i.e. g 1 = cl m. Using similar analysis to that above, we obtain f(h t, R t ) = 1, (1.14) [1 + ct m(rh t ) m P t ] 1 m and the following equilibrium densities ( ) H R m 1/(m+1) 1 =, ct kmr m (R 1) P = k(r 1)H. (1.15) In Appendix A.2 we show that functional response of the form cl m+1 can stabilize the host-parasitoid interaction for m > 1. It is instructive to contrast this result with previous work [36, 17] where functional responses were incorporated by assuming that the attack rate g 1 is a constant and equal to cl(0, t) m = c(rh t ) m, i.e. a function of only the initial number of host larvae available. Such an assumption does not prevent the parasitoids from overexploiting the hosts as a large initial host density leads to a high attack rate throughout the vulnerable stage in spite of the fact that the host population has been reduced to low levels towards the end of the stage. Not surprisingly, the equilibrium of the corresponding discrete-time model given by f = exp[ ct (RH t ) m P t ] (1.16) is unstable for all m 0. The difference between this function and the one obtained from a semi discrete model (equation (1.14)) is illustrated in Figure 8

21 1.2. Note the larger values of f obtained from the semi discrete approach, i.e. more host larvae escaping parasitism when H t is large as compared to (1.16). This is simply because the semi-discrete approach takes into account that the attack rate is not a constant but decreases with the host larval densities throughout the vulnerable period. This leads to a larger host population escaping parasitism and prevents overexploitation, which contributes to the stability of the semi discrete model. Figure 1.2. Plots of the fraction of host larvae escaping parasitism f as a function of host density H t, as given by (1.14) (dashed line, functional response is a function of the current host larvae density) and (1.16) (solid line, functional response is a function of the initial host larvae density) for R = 2, T = 1, c =.01, m = 1 and P t = 1. 9

22 1.2.3 Type II and III Functional Responses Changing the attack rate to g 1 = ( ) m L c L H ( ) m+1 (1.17) L 1 + L H with m 1 incorporates a sigmoidal functional response with maximal value cl H. Analysis in Appendix A.3 shows that this yields the implicitly defined discrete-time model H t+1 Lm+1 H mht+1 m = RH t Lm+1 H m(rh t ) ct L HP m t (1.18a) P t+1 = k(rh t H t+1 ) (1.18b) with the following equilibrium densities. ( ) H R m 1 1 m+1 = L H, P = k(r 1)H, kct L mr m H > 1. (R 1)(kcT L H 1) (1.19) The product ct L H represents the number of hosts parasitized per parasitoid during the entire larval stage when the host larvae densities are large. Hence, the above equilibrium is only feasible if the population of parasitoids can grow, i.e. their reproductive rate (kct L H ) is larger than one, when a large number of hosts are available. The equilibrium in the above discrete-time model is stable for m < m where constant m is the solution to R m m R m 1 = kct L H R 1 (kct L H 1)(R 1) (1.20) and has a value greater than one. If kct L h >> 1 then m 1 and if R >> 1 then m kct L H /(kct L H 1). From the above analysis we conclude that a 10

23 Type III functional response can stabilize a host-parasitoid interaction as long as the initial portion of the functional response accelerates sufficiently strongly. Also, as m is always greater than one, a Type II functional response, which would correspond to m = 0, will lead to an unstable equilibrium. 1.3 Effects of Density Dependence It is well known that density-dependent effects, expressed through dependence of functions g 1, g 2, g 3 and g 4 on initial population density can stabilize the equilibria in the Nicholson-Bailey model. For example, [24] showed the stabilizing effect of density-dependent self-limitation in the host by introducing the discrete form of the logistic equation to the host equation and [20] provided an analysis of the stabilizing effects of density-dependent mutual interference between searching parasitoid adults. In this section we will investigate some density-dependent effects by incorporating them in the continuous-time portion of our hybrid model. A key advantage of using the hybrid approach here is that these density-dependent effects can act concurrently with host mortality from parasitism. This contrasts with previous studies where for ease of discrete-time model formulation, it was explicitly or implicitly assumed that the density-dependent mortality acts at a stage before or after the stage where hosts are attacked by parasitoids Density Dependent Host Mortality We first consider the dynamical interaction dl dτ = clp c 1L 2, di dτ = clp, dp dτ = 0, (1.21) 11

24 i.e. g 1 = c, g 2 = c 1 L and g 3 = g 4 = 0. Here not only do both the host mortalities occur concurrently but also the density-dependent mortality rate is a function of the current density of hosts available. We assume that parasitized larvae immediately becomes juvenile parasitoids which do not face competition for resources with the host larvae, and hence, have no density-dependence mortality acting on them, i.e. g 4 = 0. In the absence of parasitoids, it is known that this system would lead to Beverton-Holt dynamics in the host ([14], P. 125). The solution to the above continuous-time model is L(τ, t) = I(τ, t) = cp t c 1 RH t exp(cp exp(cp t τ) + c 1 RH tτ) 1 t cp [ t 1 exp( cp t τ) ln 1 + c 1 RH t cp t (1.22a) ]. (1.22b) P (τ, t) = P t (1.22c) which gives the discrete-time host-parasitoid model as H t+1 = L(T, t) = RH t exp( ct P t ) 1 exp( ct P 1 + c 1 RH t) t P t+1 = I(T, t) = kcp t c 1 ln cp t [ 1 + c 1 RH t 1 exp( ct P t ) cp t (1.23a) ]. (1.23b) The above system has two non-trivial equilibrium points. The first is the noparasitoid equilibrium that is set by the strength of the host-density dependence H NC = R 1 c 1 T R, P = 0. (1.24) The Beverton-Holt-like host dynamics imply that the host population alone can never have overcompensation or exhibit cycles. Stability analysis using the Jury conditions show that the above equilibrium is stable in the full system for ln R < c 1 kc, (1.25) 12

25 a condition that precludes growth of a small parasitoid population when the hosts are at carrying capacity. Thus, sufficiently large values of c 1 /ck, which can be interpreted as the strength of density-dependent mortality vs. parasitism, stabilize the no-parasitoid equilibrium. Similar stability analysis on Mathematica reveals that the second equilibrium, where both host and parasitoid are present and given by H = ( exp( c 1 ) 1 ) kc 1 exp( c 1 kc ) R cp c 1 R, P = c1 ln R kc, (1.26) ct is stable for z < c 1 ck < ln R, (1.27) where z is the solution to z + 1 = R(ln R z ) R exp (z ). (1.28) The constant z is an increasing function of R, however increasing not as fast as ln R. Thus the size of the stability region specified in (1.27), increases with R as is illustrated in Figure 1.3. For c 1 /kc > ln R the no-parasitoid equilibrium (1.24) is stable and as we will see, for c 1 /kc < z one typically gets persistence of both host and parasitoid populations in the sense of bounded oscillations. Figure 1.4 plots the above equilibrium densities and illustrates the different transitions between stability and instability. Note that for each value of the parameters there is at most one stable equilibrium of the discrete-time model (1.23). This is in contrast to the model studied by [32] which reduces to the Ricker model for the host in the absence of parasitoids. Due to the overcompensation in the Ricker model, the Neubert-Kot model may exhibit two locally stable attractors, one where the consumer persists and other where it is extinct, for intermediate 13

26 Figure 1.3. The stability region specified in (1.27), for the discrete-time model (1.23) as a function of strength of density-dependent mortality vs. parasitism (c 1 /kc) and R (the number of viable eggs produced by each adult host). The no-parasitoid equilibrium refers to (1.24) while the host-parasitoid equilibrium refers to (1.26). values of c 1. Figure 1.5 illustrates a simulation of the discrete-time model (1.23), with attack rate c chosen such that c 1 /kc takes values below, in, and above the stable region (1.27). In summary, for a fixed parasitism rate, stability arises at intermediate levels of density-dependent mortality. At low levels of densitydependent mortality the host population becomes large enough for the parasitoids to overexploit it, which leads to familiar cycles of increase and crashes. On the other hand, strong density-dependent mortality causes a very fast decrease of 14

27 Figure 1.4. Plot of host and parasitoid equilibrium densities as a function of the strength of density-dependent mortality (c 1 ). Other parameter were taken as R = 2, T = 1, k = 1 and c = 1. H and P refer to the host-parasitoid equilibrium (1.26) and H NC refers to the no-parasitoid equilibrium given by (1.24). The solid and dashed lines correspond to stable and unstable equilibrium densities, respectively. the hosts throughout the vulnerable stage and prevents the parasitoids from exploiting the resource at all, which leads to their extinction. Stability occurs in the intermediate situation where parasitoids can get enough recruits for the next generation but also do not overexploit the hosts. The quantity q = R(exp c 1 kc 1)(ln R c 1 kc ) (R 1)(R exp c 1 kc ) (1.29) is of special interest to biological control, for it represent the ratio of the host 15

28 Figure 1.5. Population densities of hosts and parasitoids in year t as given by the discrete-time model (1.23). Parameters taken as R = 2, c 1 =.01, k = 1, T = 1 and a) c =.05, b) c =.02 and c) c =.0125 which correspond to values below, in and above the stable region (1.27), respectively. Initial densities are 20 and 30 for the host and parasitoid respectively. 16

29 equilibrium with parasitoids and without parasitoids. As one varies c 1 /kc in the stability region, q increases monotonically between q and one, where q = R(exp (z ) 1)(ln R z ). (1.30) (R 1)(R exp (z )) The quantity q represent the maximum level of host depression that is consistent with a stable dynamics. We calculated q =.34 and.29 for R = 2 and 10 respectively. As 10 appears to be an upper limit on observed values of R [27], this shows that strong host depression is accompanied by instability for biologically appropriate parameters Density Dependent Parasitoid Mortality We first consider the scenario of density-independent parasitoid mortality which is incorporated in the following dynamical interaction dl dτ = clp, di dτ = clp, dp dτ = c 2P. (1.31) This leads to a discrete-time model identical to the Nicholson-Bailey model given by (1.8) with ct replaced by c[1 exp( c 1 T )]/c 1. Thus density-independent parasitoid mortality has no effect on stability. We now investigate if density-dependent parasitoid mortality could stabilize the host-parasitoid interaction. Such densitydependent mortality could arise due to limitation of adult parasitoids by a resource other than the hosts or by a specialist predator that responds to density of parasitoids. Towards this end, we change the above interaction to dl dτ = clp, di dτ = clp, dp dτ = c 2P 2, (1.32) 17

30 i.e. g 1 = c, g 2 = g 4 = 0 and g 3 = c 2 P. Analysis of this system leads us to the discrete-time model (1.6) with the function f given by f(h t, P t ) = 1 (1.33) (1 + c 2 T P t ) c c 2 which has equilibrium densities H = R c2 c 1 kc 2 T (R 1), P = R c 2c 1 c 2 T. (1.34) Note that this form of function f is identical to the popular version of f given by the zero term of the negative binomial distribution [23]. This similarity is interesting given that May s model was motivated by a different situation density dependence in the parasitoid attack rate that is induced by assuming that hosts vary in their susceptibility to attack. Stability arises in this model if c 2 > c [23]. Here, we have shown that strong density-dependent parasitoid mortality may similarly stabilize the host-parasitoid interaction. As with May s model, stability is associated with an increased host equilibrium, i.e. the parasitoids have reduced capacity to overexploit their host. 1.4 Discussion The preceding analyses revisit two classic questions in host-parasitoid theory: how is the stability of interacting populations affected by the form of the functional response and by density dependence. Our primary findings regarding density dependence are unsurprising, but the results on the effects of the functional response contrast starkly with previous work that used heuristically formulated models. 18

31 The key previous result on the effects of the functional response was that of [17] who argued that accelerating functional response do not compensate for the instability in the Nicholson-Bailey model. This implies very different dynamics in discrete-time models than in their continuous-time counterparts where accelerating functional responses are stabilizing [29]. Figure 1.2 shows the reason for the different outcomes; essentially, our model recognizes that parasitism slows down when the host population becomes small, thereby precluding the smallest host populations. This leads to an increase in the equilibrium host density and to stability. The ecological importance of our findings on the stabilizing effects of accelerating functional responses will depend on the mechanism responsible for the acceleration. The model in this paper assumes that the parasitoid attack rate depends on the instantaneous host density. This is obviously an idealization, as some time delay is inevitably involved. For type II functional responses, there are mechanisms (e.g. those involving handling time) for which this delay is likely to be short, but it is harder to defend this assumption with accelerating functional responses caused by changes in parasitoid behavior. Reference [26] argued that for aquatic predators, non-delayed, type III functional responses are seldom observed, and we suspect that a similar situation holds for parasitoids. Yet the delays involved in parasitoid response may plausibly be much shorter than the duration of the vulnerable stage of the host, and in such circumstances our model remains a more reasonable caricature of reality than its discrete-time counterpart. The take-home message for applications is the need for careful consideration of underlying mechanisms and any associated time delays when constructing models of any particular system that includes accelerating functional responses. Our conclusion that density dependence in either host or parasitoid is stabilizing is, 19

32 by itself, unsurprising. But the dynamics of our model differs in important ways from previous work, not least in the absence of multiple attractors [32, 22]. The work reported in this paper illustrates the limitations of models with the form of equations (1.6), namely H t+1 = RH t f(h t, P t ) (1.35a) P t+1 = krh t [1 f(h t, P t )]. (1.35b) Much theory on the periods of population cycles in consumer-resource systems assumes this form or generalizations that allow adult individuals to survive from one year to the next [28, 27], with one important conclusion being that in a broad range of situations, consumer-resource interactions lead to a lower bound on cycle periods close to 6 time units. Shorter periods typically require very large values of the geometric rate of increase, R, which is expected to be less than 10 in natural systems [27]. By contrast, the model in this paper with the quadratic functional response has a cycle period of 5 when R = 3.23, well within the ecologically feasible range. Yet the data analyzed by [28] was broadly consistent with expectations from the simpler model. Further work on cycle periods using the semi-discrete formalism is clearly called for. Finally, we note the generality of our approach. Here we focused on two aspects of consumer-resource interactions that have for decades been considered important for stability of consumer-resource interactions. Many other processes can be considered with semi-discrete models. For example, we have work in progress on the consequences of host-feeding where parasitoids face a choice between eating or ovipositing in a host. Previous work (e.g. [38, 6, 21]) using continuous-time models has identified situations where host feeding has no effect 20

33 on stability, and situations where it is stabilizing. With partially synchronized life cycles, we lack intuition and systematic investigation is required. 21

34 Chapter 2 Stability in host-parasitoid models due to variability in risk It has been well known that enough variation in the risk of parasitism among individual hosts can, if risk is gamma-distributed, stabilize the otherwise unstable equilibrium of the Nicholson-Bailey host-parasitoid model [2]. Reference [19] showed that the condition for stability is CV 2 > 1, where CV is the coefficient of variation of the distribution of risk. An important assumption is that risk is independent of local host density if the host is non-uniformly distributed in space [7]. The CV 2 rule, and earlier results stressing the role of spatially aggregated parasitism (e.g. [23]), stimulated hundreds of studies investigating parasitism patterns in the field. Pacala and Hassell surveyed the studies and showed that several data sets met the criterion [34]. We show here that it is not the amount of variation, measured for example 22

35 by CV 2, but the shape of the distribution of risk that is crucial in stabilizing the host-parasitoid interaction. We further show that the CV 2 rule is restricted to the gamma distribution or to situations when the host rate of increase, R, is close to one. In particular, to get stability robust to high rates of host increase, the distribution needs to have its mode at zero and thereafter to decline; that is the distribution needs to approximate a situation where a substantial fraction of the host population is effectively in a refuge. We also reinforce the previous observation of reference [30] that skewed risk of the type needed for stability increases the host equilibrium above the Nicholson- Bailey model value. Indeed, we develop a new, simple, and general criterion for stability: adult host equilibrium density must increase as host reproductive rate R increases. 2.1 Nicholson-Bailey model with variability in host risk We here formulate the Nicholson-Bailey parasitoid-host model using adult host density as a state variable (e.g. [14]), rather than larvae as is frequently the case (e.g. [27]). Consider a univoltine insect with H t being the adult density at a fixed date near the start of year t (where t is an integer). These adults give rise to eggs that mature into RH t larvae. Here R > 1 denotes the number of viable eggs produced by each adult. The larvae turn into pupae and metamorphose into adults the next year. Assume that this insect is a host species, which becomes vulnerable to attacks from a parasitoid during the larval stage. Adult (female) 23

36 parasitoids search for hosts during the window of time that host larvae are present, then die. Parasitized hosts mature into a juvenile parasitoids, pupate, overwinter, and emerge as adults the following year. If the parasitoids have a constant attack rate c and the duration of the larval stage is T, then the host-parasitoid dynamics is given by the Nicholson-Bailey model H t+1 = RH t exp( ct P t ) (2.1a) P t+1 = k(rh t H t+1 ) (2.1b) where H t and P t are the adult host and adult parasitoid densities, respectively, in year t, and k denotes the number of parasitoids that emerge per parasitized host larva. Typically we set k = 1. It is well known that the equilibrium of this model H no variability = ln(r) kct (R 1), P no variability = ln(r) ct (2.2) is unstable for all R > 1 and both hosts and parasitoids show diverging oscillations (e.g. [27]). Variability is introduced by assuming that for each host, the parasitoids have a different attack rate, which we can think of as the risk the individual host larva faces from the average parasitoid [7]. This risk is assumed to be independent of local host density and does not change as the larvae develop. Let the distribution of risk across the larval population be p(x) at the beginning of the larval stage. Thus the probability that a larva has a c value (risk) in the infinitesimal interval [x, x + dx] is p(x)dx. The distribution is defined only for non-negative values of and is assumed to remain the same from year to year. The fraction of larvae that have risk in the interval [x, x + dx] is p(x)dx. This fraction faces a mortality rate of xp t, hence, its fraction surviving would be 24

37 p(x) exp( xt P t )dx. The total fraction of larvae surviving would be x=0 p(x) exp( xt P t )dx, (2.3) which leads to the discrete-time model H t+1 = RH t p(x) exp( xt P t )dx (2.4a) x=0 P t+1 = k(rh t H t+1 ). (2.4b) This is a general form of the variable-risk model first proposed by [2] in which risk was assumed to be gamma-distributed. Before proceeding with the equilibrium and stability analysis of this model we discuss different ways that host risk can be interpreted. We introduced the above model by assuming host larvae face different mortality rates depending upon the attack rate of the parasitoids. An alternative way to introduce variability is to assume that all larvae have same mortality rate, i.e. same value of c, but are exposed to the parasitoids for different durations. Hence, risk is now defined in terms of T, the length of time a larva is exposed to attack. In such a case we have the discrete-time model H t+1 = RH t p(x) exp( xcp t )dx (2.5a) x=0 P t+1 = k(rh t H t+1 ) (2.5b) where now p(x) represents the distribution of time exposed to parasitoids T across the larvae population. Another way to define risk would be through the product ct. Hence not only do the host larvae face different mortality rates but also this mortality rate acts for different durations T. In this scenario we have the 25

38 following discrete-time model H t+1 = RH t p(x) exp( xp t )dx (2.6a) x=0 P t+1 = k(rh t H t+1 ) (2.6b) where now p(x) denotes the distribution of ct values across the host larvae population. A general way to define risk, which will also lead to the discrete-time model (2.6) would be by T c(τ)dτ where now c(τ) represents the attack ate that a host τ=0 larva faces at a time instant τ in the larval stage and τ = 0, τ = T correspond to start and end of the larval stage, respectively. All the above ways of defining risk are actually special cases of this. As in the Nicholson-Bailey model both parameters c and T appear together, all these different ways to define risk lead to the same form of the discrete-time model, give and take some constants. 2.2 Stability, shape of the risk distribution and the CV 2 rule The discrete-time model (2.6) defines a general class of parasitoid-host models in which risk varies among hosts, without specifying the particular probability distribution that describes how risk varies among individuals. Stability analysis of the model (Appendix B.1) provides a remarkably simple and elegant stability condition: the host-parasitoid equilibrium is stable, if and only if, the adult host equilibrium H* satisfies dh dr > 0. (2.7) 26

39 Although elegant, this result on its own does not relate stability to the distribution of risk. We now investigate what forms of the distribution of risk can lead to stability. The details are in Appendix B.2, where we show that a necessary and sufficient condition for stability for values of R close to one (the minimum host reproduction rate needed for a viable host population) is CV 2 > 1. However, this condition does not necessarily induce stability at large values of R, which instead requires the modal probability of parasitism to be zero x Figure 2.1. Gamma distribution with mean one and CV 2 =.5 (dotted line), CV 2 = 1 (dashed line), CV 2 = 3 (solid line) A necessary and sufficient condition for stability for large values of R, is that this mode at zero should be characterized by a risk distribution that either increases unboundedly as risk approaches zero (p(0) is not finite) or when p(0) is 27

40 finite, the distribution falls off rapidly (p (0) is sufficiently negative, more precisely p (0) < p(0) 2 ). For example, if p(x) is a gamma distribution, CV 2 > 1 implies this particular shape near x = 0 (see Figure 2.1, which plots gamma distributions for different values of CV ), and in this case, happens to be necessary and sufficient for stability for all values of R > 1, hence the CV 2 > 1 rule. We must emphasize, however, that the CV 2 > 1 rule is true only for a specific distribution of risk, namely a gamma distribution, and is not general. To reinforce that point, we next show two examples of distributions of risk where even an infinite cannot stabilize the host-parasitoid equilibrium for certain values of R Risk has an inverse gaussian distribution The inverse gaussian distribution (Figure 2.2) has the property that with increasing CV, the inverse gaussian distribution becomes more and more skewed towards x = 0 but unlike the gamma distribution it always has a non-zero modal probability of parasitism. This violates the condition for stability for large values of R, provided in the previous section. When the distribution of risk is an inverse gaussian distribution one can show that the equilibrium cannot be stabilized for R > 4.92 irrespective of how large R is (see Appendix B.3). This value of R is well within the ecologically feasible range, which is expected to be less than 10 in natural systems [18, 27]. 28

41 Figure 2.2. Inverse gaussian distribution with mean one and CV 2 = 1 (dashed line), CV 2 = 3 (solid line) Risk is bounded from below It is well known that the presence of a (fractional) refuge can stabilize a Nicholson-Bailey-type model over some range of parameter values [15]. To investigate what happens when we explicitly exclude a refuge, we next analyze the effects of variability when all the hosts have a risk greater than an arbitrary low nonzero risk, c. In this case, stability arises at low values of R and intermediate values of CV 2, but not at lower or higher values of CV 2 (Figure 2.4). Let the distribution of risk be given by 0 if x c p(x) = g(x c ) if x > c, (2.8) which one can think of as a function g(x) shifted to the right by c (Figure 2.3). 29

42 Figure 2.3. Gamma distribution with mean one and CV 2 = 1 shifted to the right by 0.1 (c = 0.1). The host-parasitoid model is now given by H t+1 =RH t p(x) exp( xp t )dx = RH t exp( c P t ) x=0 y=0 g(y) exp( yp t )dy (2.9a) P t+1 =k(rh t H t+1 ). (2.9b) When all the host larvae have a minimum risk c, the host equilibrium in now constrained by the condition (Appendix B.4) ln(r) kc (R 1) > H, (2.10) and as the left-hand-side of this inequality is a decreasing function of R, we have that at some large enough critical value R, the host equilibrium will be a decreasing function of R, and hence unstable (from (2.7)). A formal stability 30

43 analysis of the above model (see Appendix B.4) shows that the equilibrium of this model will always be unstable at some large enough value of R, irrespective of the form taken by the function g(x). Figure 2.4. Region of stability for the discrete-time model (2.9) when g(x) is a gamma distribution with mean 1 (c = 1) c = 0.05 and c = 0.1. This model also shows that a distribution that has a minimum level of risk leads to an unstable equilibrium for some values of R, even when it is combined with a distribution that, on its own and with modal risk of 0, induces stability for all values of R. This point can be illustrated by letting the function g(x) in 31

44 (2.9) have the form as that of a gamma distribution with mean c and CV 2 > 1 (Figure 2.3). As noted above, increasing in the gamma distribution makes it more skewed towards its minimum value (as in Figure 2.1). However, in this example, when CV 2 is large, the fraction surviving parasitism closely approximates the Nicholson-Bailey value: exp( c P t ) g(y) exp( yp t )dy = y=0 exp( c P t ) (1 + ccv 2 P t ) 1/CV 2 exp( c P t ), (2.11) and hence, from (2.9) we recover the Nicholson-Bailey model with attack rate c. Thus, if g(x) is highly skewed towards 0 (hence p(x) becomes skewed towards c ) then the host-parasitoid equilibrium is no longer stable. In summary, for such distributions of risks where large CV corresponds to the distribution being skewed towards c, stability arises at low values of R and the corresponding stability region is maximized at intermediate levels of variability (as shown in Figure 2.4). Note that the stability region becomes smaller with increasing c. This result serves to reinforce the earlier intuition that, when variability in host risk is stabilizing for large values of R, it is because the model approximates a refuge Unexpected patterns among stability, R and CV 2 In both the above examples, stability arose at low values of R due to sufficient variability in host risk, but was lost at larger values as these distributions did not have the appropriate shape near x = 0. It is quite possible to have the opposite scenario where the distribution does not have apparently-sufficient variability (CV < 1) but can still induce stability at large values of R, if it has a mode at zero followed by a sharp decline. For example, when p(x) = (1 + 2x 2 ) exp( 2x) 32