Regimes of biological invasion in a predator prey system with the Allee effect

Transcription

1 Bulletin of Mathematical Biology 67 (2005) Regimes of biological invasion in a predator prey system with the Allee effect Sergei Petrovskii a,b,,andrew Morozov a,b,bai-lian Li b a Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prospekt 36, Moscow , Russia b Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California at Riverside, Riverside, CA , USA Received 28 April 2004; accepted 15 September 2004 Abstract Spatiotemporal dynamics of a predator prey system is considered under the assumption that prey growth is damped by the strong Allee effect. Mathematically, the model consists of two coupled diffusion-reaction equations. The initial conditions are described by functions of finite support which corresponds to invasion of exotic species. By means of extensive numerical simulations, we identify the main scenarios of the system dynamics as related to biological invasion. We construct the maps in the parameter space of the system with different domains corresponding to different invasion regimes and show that the impact of the Allee effect essentially increases the system spatiotemporal complexity. In particular, we show that, as a result of the interplay between the Allee effect and predation, successful establishment of exotic species may not necessarily lead to geographical spread and geographical spread does not always enhance regional persistence of invading species Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved. 1. Introduction Biological invasion has been attracting considerable attention recently due to its numerous adverse effects on ecosystem dynamics and biodiversity (Hengeveld, 1989; Corresponding author at: Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prospekt 36, Moscow , Russia. address: spetrovs@sio.rssi.ru (S. Petrovskii) /$ Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved. doi: /j.bulm

2 638 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Hastings, 1996; Shigesada and Kawasaki, 1997; Frantzen and van den Bosch, 2000; Keitt et al., 2001; Owen and Lewis, 2001; Wang and Kot, 2001). Although a considerable progress has been made during the last decade in understanding basic scenarios of species invasion, many important issues have not been properly addressed yet. Comprehensive identification of factors that affect rates of invasion and patterns of species spread and can potentially either enhance or hamper species invasion, is expected to open a possibility of biological control and to result in effective invasive species management (Sakai et al., 2001; Fagan et al., 2002). Biological invasion is known to have a few more or less clearly distinguishable stages [cf. Shigesada and Kawasaki (1997), Sakai et al. (2001)]. The first stage is introduction when a few organisms of an exotic species are brought, deliberately or unintentionally, into the given ecosystem. The second stage isestablishment when the introduced species is getting adapted to the new environmental conditions. The third stage is, in case the previous two have been successful and did not result in species extinction, the geographical spread when the exotic species invades new areas at thescale much larger compared to the domain where it was originally introduced. Later stages are related to the impact of the new species on the native ecological community and, possibly, on human health and society. Apparently, each stage has its own basic processes and specific problems. In this paper, we mainly focus on the spatiotemporal dynamics of the introduced species typical for the second and third stages of invasion. Thus, we are mainly interested in such issues as population growth, species extinction/persistence, patterns of species spread and related ecological pattern formation. Under what conditions the introduced species will fail to establish itself in the new environment, may it happen that successful introduction will not lead to geographical invasion, whether the spatial spread will take place through propagation of the population front or in a more complicated manner all these questions are highly relevant both from a theoretical point of view and from the point of immediate practical applications. From a theoretical perspective, it is well-known that many basic features of the species spread during biological invasion can be explained reasonably well by the interplay between local population growth and local dispersal due to self-motion of individuals (Fisher, 1937; Skellam, 1951; Okubo, 1980; Shigesada and Kawasaki, 1997). Mathematically, this model is described by a diffusion-reaction equation whose properties appear to depend essentially on the type of the population growth. While early studies tended to assume it to be logistic, more recently much attention has been paid to the impact of the Allee effect (Lewis and Kareiva, 1993; Owen and Lewis, 2001; Wang and Kot, 2001) because the Allee effect was shown to affect virtually all aspects of species interactions in space and time (Allee, 1938; Berryman, 1981; Dennis, 1989; Amarasekare, 1998; Courchamp et al., 1999; Gyllenberg et al., 1999). The Allee effect usually arises as a result of intraspecific interactions (Allee, 1938; Berryman, 1981). However, the impact of interspecific interactions on species invasion is important as well. In particular, it was shown that predation is likely to affect the rates of invasive species spread (Fagan and Bishop, 2000; Owenand Lewis, 2001; Petrovskii et al., in press). The spatiotemporal dynamics of a predator prey system relevant to biological invasion has been recently studied in much detail in the case that population growth is logistic (Petrovskii et al., 1998; Petrovskii and Malchow, 2000).However, the impact of the

3 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Alleeeffect has not been properly addressed, although it was shown that it can significantly increase the complexity of the system dynamics (Petrovskii et al., 2002a,b; Morozov, 2003; Morozov et al., 2004). In this paper we consider a predator prey system where prey growth is damped by the Alleeeffect. By means of extensive computer simulations, we fulfil a thorough study of this system in connection to biological invasion and give a detailed classification of possible patterns of species spread. We show that the system dynamics is remarkably rich and that its complexity increases with an increase of the prey maximum growth rate. In particular, we show that, for sufficiently large prey growth rate, there is a parameter range where the pattern of species spread exhibits nonuniqueness subject to the initial conditions. We also show that, as a result of the interplay between predation and the Allee effect, successful establishment of an exotic species does not necessarily lead to its geographical spread and that geographical spread, if/when it takes place, does not guarantee species regional persistence. 2. Main equations We consider the following 1-D model of predator prey interaction in a homogeneous environment: H (X, T ) T P(X, T ) T 2 H = D 1 + F(H ) f (H, P), (1) X 2 2 P = D 2 + κ f (H, P) MP (2) X 2 [cf. Nisbet and Gurney (1982), Murray (1989), Holmes et al. (1994), Sherratt (2001)]. Here H and P are the densities of prey and predator, respectively, at moment T and position X. D 1 and D 2 are diffusivities and κ is the food utilization coefficient. The function F(H ) describes prey multiplication, f (H, P) describes predation, and the term MP stands for predator mortality. We consider Holling type II response for predator and use the following parametrization: f (H, P) = AHP (3) H + B where A describes predation intensity and B is the half-saturation prey density. We assume that prey population is damped by the Allee effect, its growth rate being parametrized as follows (Lewis and Kareiva, 1993): ( 4ω F(H ) = (K H 0 ) 2 ) H (H H 0 )(K H ) (4) where K is the prey carrying capacity, ω is the maximum per capita growth rate and H 0 quantifies the intensity of the Allee effect so that it is called strong if 0 < H 0 < K (when the growth rate becomes negative for H < H 0 )and weak if K < H 0 0[cf. Owen and Lewis (2001), Wang and Kot (2001)]. For H 0 K,theAllee effect is absent (Lewis and Kareiva, 1993).

4 640 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) For convenience, we introduce dimensionless variables u = H/K, v = P/(κ K ), t = at, x = X (a/d 1 ) 1/2 where a = Aκ K /B.Then, from Eqs. (1)and(2), we obtain: u(x, t) t v(x, t) t = 2 u + γ u(u β)(1 u) uv x αu, (5) uv δv. 1 + αu (6) = ɛ 2 v x 2 + Eqs. (5) and(6) contain five dimensionless parameters (against nine in the original equations), i.e., α = K /B, β = H 0 /K, γ = 4ωBK/(Aκ(K H 0 ) 2 ), δ = M/a and ɛ = D 2 /D 1.Thus,the behaviour of dimensionless solutions u and v appears to depend on five dimensionless combinations of the original parameters rather than on each of them separately. Invasion of an alien species is usually started when a number of individuals of an exotic species is locally brought into the given ecosystem. From the point of model (5)and(6), it means that the initial species distribution should be described by functions of finite support. Thus, we consider initial conditions of the following form: u(x, 0) = u 0 for u < x < u, otherwise u(x, 0) = 0, (7) v(x, 0) = v 0 for v < x < v, otherwise v(x, 0) = 0 (8) where u 0, v 0 are the initial population densities and u, v give the radius of the initially invaded domain. Initial conditions (7)and(8)alsocorrespond to the problem of biological control when, soon enough after introduction of an exotic species, a predatory species is introduced intentionally in an attempt to slow down or stop its spread [cf. Fagan and Bishop (2000), Owenand Lewis (2001), Petrovskii et al. (in press)]. Note that the initial conditions (7)and(8)are somewhat idealized and in reality the form of the species initial distribution can be much more complicated. However, the results of our computer simulations show that the type of the system dynamics depends more on the radius of the initially inhabited domain and on the population density inside rather than on the details of the population density profile. 3. Patterns of species spread From the point of ecological applications, it is very important to distinguish between the cases when invasion will likely be successful and the cases when it will likely fail. In practical ecology, invasion failure usually means that the introduced species fails to establish itself in the new environment. Thus, an unsuccessful species is expected to go extinct soon after its introduction. However, it remains unclear whether invasion failure may happen as well at later stages, i.e., whether invasive species can go extinct after having already spread over relatively large areas. The results that we present in this section show that, for an invasive species affected by the Allee effect, extinction may take place at a later stage after its geographical spread. Moreover, it seems reasonable to distinguish between geographical invasion and local invasion. We will call the invasion local in the case when the new species successfully establishes itself locally, i.e., around the place of original introduction, but

5 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) does not spread over new areas due to the impact of certain factors. Correspondingly, we call the invasion geographical in the case that the invasive species succeeds to spread over large areas. In many cases, local invasion of exotic species is followed by geographical invasion, although the time lag between these two stages can be as long as a few decades. The question that still needs to be answered is what environmental or biological factors make this lag so long and whether local invasion must always be followed, sooner or later, by geographical invasion. The results that we present below show that an exotic species can invade locally but sometimes fails to invade geographically due to the interplay between predation and the Allee effect. Eqs. (5) and(6) with the initial conditions (7) and(8) were solved numerically in the domain L < x < L by finite-difference method. The steps of the numerical mesh were chosen as x = 0.2and t = and it was checked that a decrease of the mesh steps did not lead to any significant modification of the results. The no-flux condition was used at the boundaries and the radius L of the numerical domain was chosen large enough in order to make the impact of the boundaries as small as possible during the simulation time. Throughout this section we fix ɛ = 1, the effect of differential diffusivity will be addressed in Section 5. In order to identify different regimes and to reveal the corresponding structure of the parameter space, in total, over three thousand computer experiments were run for different parameter values. We obtain that all regimes observed in our simulations can be classified into threegroups,see Fig. 1.Thesegroupscorrespondto extinction,geographicalinvasion when the species keep spreading until they reach the domain boundaries, and regional persistence when the alien species invade locally and spread over a certain area but do not go farther. Before proceeding to regime description, the issue of regime dependence on the initial conditions should be clarified. It is well-known that in a predator prey system with logistic growth for prey, although the population density can fall to very small values, the species will never go extinct in the strict mathematical sense because the extinction state (0, 0) is unstable (Gilpin, 1972)and thus acts as a repeller. Evolution of the initial conditions (7) and (8)eventually leads, for any biologically reasonable parameter values, to formation of atravelling population front. Thus, the large-time asymptotical system dynamics does not depend on the initial conditions as long as they are described by finite functions (Volpert et al., 1994). Although the actual patterns of spread can be different for different parameter values, e.g., front propagation can be followed either by a steady spatially homogeneous species distribution or by spatiotemporal pattern formation in the wake (Sherratt et al., 1995; Petrovskii et al., 1998; Petrovskii and Malchow, 2000), any introduction of a new species will lead to its geographical invasion. This prediction of inevitable species spread does not seem realistic and was used as a justification for various modifications of the model, e.g., by implementing a threshold at low population densities (Brauer and Soudack, 1978; Wilson, 1998; Petrovskii and Shigesada, 2001). The situation becomes different when the invasive prey is affected by the Allee effect. In this case, already single-species models predict that the introduced species does go extinct when the population size is not large enough (Lewis and Kareiva, 1993; Petrovskii, 1994; Petrovskii and Shigesada, 2001). [Note that this phenomenon is often observed in nature as well, see Courchamp et al. (1999).] The impact of predation makes this threshold

6 642 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 1. Classification of invasionregimes. behaviour more prominent, cf. Section 3.2. Essentially, for any value of parameters in Eqs. (5) and(6), sufficiently small u 0 and/or u can turn any regime of species spread to extinction. In order to exclude this somewhat trivial case, in our computer experiments u 0 and u are always chosen sufficiently large Regimes of geographical invasion We begin with the regimes describing the unbounded spread of the invasive species which corresponds to the geographical stage of biological invasion. We found three different scenarios of the species spread, examples are shown in Figs According to the first scenario, the species is spreading over space through propagation of a travelling population front, see Figs Infront of the front the species is absent, behind the front it is present in considerable densities. Apparently, this type of species spread corresponds to successful invasion. Depending on parameter values, in the wake of the front there can arise either a stationary spatially homogeneous species distribution or irregular spatiotemporal population oscillations. Fig. 2 showsthe snapshots of the population density (solid curve stands for prey, dashed for predator) obtained for parameters α = 0.5, β = 0.27, γ = 3, δ = Here and below (except for Fig. 5), the initial conditions are u = 7, v = 2, u 0 = 1, v 0 = 0.1. Propagation of the population front is followed by a stationary homogeneous

7 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 2. Snapshots of the population density showing species invasion through propagation of population front with steady homogeneous species distribution in the wake, parameters are given in the text. Here and below, solid curve stands for prey, dashed for predator. species distribution with the population density corresponding to the stable steady state of the homogeneous system. (For some other parameter values, the front can be followed by asuccession of a few promptly damping oscillations preceding the region of spatial homogeneity.) It is this pattern of species spread that is usually evoked in connection with biological invasion described by diffusion-reaction equations; moreover, for a long time it had been considered as the only possible regime that deterministic diffusive predator prey

8 644 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 3. Snapshots of the population density showing species invasion through propagation of population front with irregular spatiotemporal oscillations in the wake in the case that the amplitude of oscillation is not large. systems can provide [cf. Lewis (1996)]. For the parameters of Fig. 2,the population fronts of prey and predator propagate with the same speed. For other parameter values it may happen that the front of prey travels with a greater speed than the front of predator (see also Section 4). In that case, the predator invades into the space already inhabited by prey at its carrying capacity. However, for the parameter values when the homogenous steady state becomes unstable, the pattern of spread changes essentially. Fig. 3 shows the snapshots of the population

9 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 4. A regime of invasion similar to the one shown in Fig. 3 but in the case that the amplitude of spatiotemporal species oscillations is large. density obtained for parameters α = 0.5, β = 0.27, γ = 3, δ = In this case, propagation of the population front is followed by excitation of irregular spatiotemporal oscillations in population density. A similar phenomenon was observed earlier for the diffusive predator prey system with logistic growth [cf. Sherratt et al. (1995)]. Note that the domain with irregular oscillations is separated from the travelling front by a plateau, i.e., by a domain with nearly-homogeneous species spatial distribution. The values of the population density in this plateau correspond to the locally unstable equilibrium.

10 646 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 5. Species invasion through propagation of a periodic population wave which is generated by a spatially irregular wave-maker situated around the place of species introduction. This phenomenon of dynamical stabilization was considered in detail in Petrovskii and Malchow (2000), Petrovskii et al. (2001) and Malchow and Petrovskii (2002). The results of our computer experiments show that the pattern of spread when propagation of the population front is followed by excitation of irregular spatiotemporal oscillations is rather typical for the diffusive predator prey system with the Allee effect in the sense that it can be observed in a wide parameter range. Some features of the regime can vary with parameter values; for instance, the unstable plateau does not always exist.

11 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 6. Snapshots of the population density (only half ofthe domain is shown) showing species spread over space through propagation of a solitary moving population patch, or a pulse. Note that this pattern of spread does not lead to species invasion because the population density in the wake is zero. Also, for some parameters the amplitude of population oscillations becomes notably larger so that the pattern in the wake becomes more prominent, see Fig. 4 obtained for α = 0.5, β = 0.27, γ = 3, δ = Note that, although in this case the pattern as a whole looks like an ensemble of separated patches, the scenario of species invasion is still essentially the same: close inspection of the population density snapshots clearly reveals the travelling population front, cf. top, middle and bottom of Fig. 4.

12 648 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 7. A regime similar to that shown in Fig. 6 but inthe case that the shape of the wave exhibits oscillations. Only half of the domain is shown. It should be mentioned that for this regime, as well as for propagation of smooth population fronts, it is possible that for some parameter values the front of prey travels faster than the front of predator, see Section 4 for more details. In that case, first, invasion of prey takes place. The travelling population front of prey separates the domain where both species are absent (in front of the front) from the domain where prey is at its carrying capacity and predator is absent (behind the front). The population front of predator

13 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 8. A regime of invasion through propagation of separate patches and groups of patches. propagates into the region already inhabited by prey, in the wake of the predator front irregular population oscillations arise. In the cases shown in Figs. 3 and 4, thepatches arising behind the front are irregular. For other parameter values, however, the spatial structurecan be more regular. Fig. 5 shows thesnapshots of the population density obtained for parameters α = 0.5, β = 0.28, γ = 7, δ = 0.46 and the initial conditions u = 5, v = 3, u 0 = 1, v 0 = 1. In this case, species invasion takes place through propagation of a periodic travelling wave. The periodic wave is generated by an irregular wave-maker situated about the place of the initial species

14 650 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) distribution, i.e., around x = 0. Depending on parameter values, the wave-maker either stays localized or gradually grows in size. In the latter case, the periodic oscillations are eventually displaced by irregular ones [cf. Petrovskii and Malchow (2001) for a similar phenomenon]. All the regimes described above correspond to species invasion through propagation of a population front. According to the second scenario, the species spread over the domain via propagation of a moving patch, or pulse, cf. Figs. 6 and 7 (only a half of the domain is shown). The dynamics is similar to predator prey pursuit [cf. Murray (1989)]. In this case, the invasive species is absent both in front of the pulse and in its wake which apparently means that invasion fails in spite of the fact that geographical spread has taken place. Depending on parameter values, the travelling population pulse can be either stationary when its shape does not change with time, or nonstationary when its shape oscillates with time; in both cases the pulse propagates with a constant speed. Fig. 6 shows the snapshots of the population density obtained for parameters α = 0.5, β = 0.28, γ = 3, δ = 0.44 when the species spread over the system via propagation of a stationary travelling pulse. Fig. 7 is obtained for α = 0.5, β = 0.28, γ = 3, δ = 0.425, it shows propagation of a nonstationary pulse. Finally, there is another, more exotic scenario of species geographical spread. Fig. 8 showsthe snapshots of the population density obtained for parameters α = 0.05, β = 0.28, γ = 3, δ = In this case, invasion takes place through formation and propagation of groups of moving patches. However, the patch motion is now much more complicated than the simple locomotion in the case of travelling pulses. The patches interact with each other, they merge and split, some of the patches or even groups of patches can disappear, new patches are formed, they can produce new groups of patches, etc. The inhabited area grows and eventually the groups of nonstationary patches occupy the whole domain. Although there is certain visual similarity between this regime of invasion and the regime shown in Fig. 4, comparison between the snapshots obtained for different times immediately showsthat in this case there is no stationary travelling population front. Another important distinction is that the size of the domain occupied by the moving patches does not grow monotonically, cf. top, middle and bottom of Fig. 8.Thathappens when the leading group of patches goes extinct in the course of the system dynamics Regimes of anomalous extinction The second group of regimes corresponds to species extinction. It is well-known that, in case the introduced species is affected by the Allee effect, it goes extinct if the initial population size is not large enough, i.e., either the radius of originally inhabited domain or the population density inside are less than certain critical values (Lewis and Kareiva, 1993; Petrovskii, 1994; Petrovskii and Shigesada, 2001). In this case, the population size decreases exponentially and the population stays localized in about the same domain where it had originally been introduced. We will refer to this type of population dynamics as ordinary extinction. When the introduced population is affected by predation, ordinary extinction can takes place as well. Although the critical values for the initial radius and the initial prey density appear somewhat larger in this case as a result of the pressure from

15 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) the predator, the qualitative features of ordinary extinction remain the same, cf. the previous paragraph. (Note that, in our model, extinction of prey inevitably leads to extinction of both species.) However, due to the impact of predation, species extinction can also follow other, rather unusual scenarios. Depending on parameter values, we observed two other regimes when species extinction is either preceded by formation of a distinct long-living spatiotemporal pattern or by long-distance population spread. Fig. 9 showsthe snapshots of the population density in the regime of species extinction through dynamical localization observed for parameters α = 0.1, β = 0.15, γ = 5, δ = At an early stage of the system dynamics, a moving patch is formed which propagates with approximately constant speed over distances much longer compared to the radius of the initial species distribution. This stage of the system dynamics is apparently similar to pulse propagation,see Fig. 6. Finally, however,the preyis caughtby thepredator and both species go extinct: starting from the moment shown in the bottom of Fig. 9 the population size exhibits fast decay. Fig. 10 shows the regime of patchy extinction (observed for α = 0.5, β = 0.2, γ = 1, δ = 0.365). In this case, the initial conditions eventually evolve into an ensemble of patches allocated over the domain. The patches interact with each other in a complicated manner somewhat similar to the patch dynamics shown in Fig. 8; finally, however, the species go extinct. We want to emphasize that, in both of these cases, the invasive population persists during aremarkably long time before the actual population decay takes place (for the parameters of Figs. 9 and 10, nearly one hundred times longer than it would be in the case of the ordinary extinction) and it can spread over large distances. During that period, the system dynamics is very similar to the regimes of geographical spread shown in Figs. 6 and 8, respectively. These results seem to reveal a new aspect of the extinction debt (Tilman et al., 1994; Loehle and Li, 1996)andalsoevoke a more general discussion regarding the ecological relevance of transient dynamics (Hastings, 2001): a population which is doomed to vanish, e.g., as a result of certain unfavourable environmental changes, can exhibit the dynamics which is, during a long time, virtually indistinguishable from the dynamics of persistent populations. The collapse comes unexpectedly and then it may be too late to apply a conservation strategy Regimes of local invasion Remarkably, the two groups of the regimes described above, i.e., species extinction and unbounded spatial spread, do not exhaust all possible types of the system dynamics. For certain parameter values, evolution of the initial species distribution leads to formation of quasistationary patches. Fig. 11 shows the snapshots obtained for α = 0.5, β = 0.32, γ = 3, δ = In this case, at an early stage of the system dynamics (for t 100), two symmetric dome-shaped patches are formed. At later stages, the position of their centres remains fixed and the shape of the patches changes with time in an oscillatory manner. A close inspection shows that, depending on the parameter values, the corresponding temporal fluctuations in the population density can be either periodic or chaotic (Morozov et al., 2004).

16 652 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 9. An example of dynamical localization : the population densities form a solitary moving patch which propagates a long distance before the species go extinct. Only half of the domain is shown. From the point of prospective ecological applications, this regime of the system dynamics is probably the most interesting. Field observations give many examples when the invasive species, after introduction, remains localized inside a certain area during a long time. Thus, their local invasion and subsequent regional persistence is not followed by geographical spread. There exist different explanations of this phenomenon such as the impact of environmental borders, time-lag related to mutations and evolutionary changes caused by adaptation in the new environment, etc. Our results provide another explanation

17 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 10. An example of long-living transients corresponding to the patchy extinction : species extinction is preceded by spatiotemporal pattern formation going during a relatively long time. and show that invasive species can be held localized purely due to certain inter- and intraspecific interactions such as the interplay between the Allee effect and predation. 4. Parameter space structure In the previous section, we demonstrated that the predator prey system with the Allee effect for prey exhibits very rich dynamics and predicts a wide variety of patterns/regimes

18 654 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 11. The regime of local invasion when the initial species introduction leads to formation of a few standing patches. The position of the patches does not change with time while their shape is either stationary or oscillates with time depending on parameter values. of species spread. A natural question arising here is about the relation between the regimes and possible transition between them that may take place in response to parameter variation. One way to address this issue is to study the structure of the system parameter space in order to locate the domains corresponding to different regimes of invasion. For that purpose, we fulfil a detailed numerical study of the system dynamics. In total, about three thousand computer experiments were run for different parameter values. The

19 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 12. The map in the parameter plane of the Eqs. (5) and(6) forγ = 1. Different domains correspond to different regimes, see details in the text. results are shown in Figs Notethat, since the model depends on four parameters (assuming here that ɛ = 1isfixed,the case ɛ 1will be addressed separately in the next section), it looks virtually impossible toaccomplish a detailed study of the whole R 4 + parameter space. It is readily seen that in this case even as many as 104 computer experiments, if run for parameter values spread homogeneously over the parameter space, would provide only meagre information about the position of different domains. In order to overcome this difficulty, we apply a certain strategy in choosing parameter values. Since the main goal of this paper is to study the dynamics of invasive species subject to the interplay between the impact of the Allee effect (quantified by β)andpredation (quantified by δ), it looks more relevant to focus on the detailed structure of the (δ, β) plane. Correspondingly, we firstly choose a certain hypothetical value for the half-saturation density, i.e., α = 0.5. Then we select a few values of the maximum per capita prey growth rate, i.e., in dimensionless units, γ = 1(Fig. 12), γ = 3(Fig. 13) andγ = 7(Fig. 14). Then, for each of these values, the (δ, β) parameter plane was studied thoroughly. In order to make the search in the (δ, β) plane more effective, one should also take into account that nontrivial dynamics can only take place inside the rectangle {0 <δ Ω = (1 +α) 1, 0 <β 0.5}.Hereδ is positive due to its biological meaning and β is assumed to be positive because we are concerned with consequences of the strong Allee effect. The values δ>ω = (1 + α) 1 correspond to predator extinction because the phase plane of Eqs. (5) and(6) inthespatially homogeneous case does not possess a co-existence steady state in the biologically meaningful domain {u 0,v 0} and the prey-only steady

20 656 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 13. The map in the parameter plane of the Eqs. (5) and(6) forγ = 3. state (1, 0) is a stable node. The values β>0.5 are readily seen to always correspond to the ordinary extinction and thus are not of much interest either. Fig. 12 shows the map in the(δ, β) plane obtained for γ = 1. Here domain 1 corresponds to species extinction (including both ordinary and anomalous extinction). Domain 3 corresponds to geographical spread of invasive species either through propagation of population fronts with irregular spatiotemporal oscillation in the wake (Figs. 3 and 4) orthrough the spread of moving patches (Fig. 8). Since it is not always easy to distinguish between these two regimes in numerical experiments, we are not going into the details of the fine structure of domain 3. Domain 4 corresponds to geographical invasion through propagation of smooth population fronts with stationary homogeneous species distribution in the wake (Fig. 2). Sub-domain 4 corresponds to the case when the front of invasive prey travels faster than the front of invasive predator, here the dashed curve separating sub-domain 4 can be obtained analytically [cf. Petrovskii and Malchow (2000)]. Domain 5 corresponds to local invasion through formation of quasi-stationary patches, see Fig. 11. The structure of the (δ, β) parameter plane changes for larger values of γ. Fig. 13 shows the map obtained for γ = 3, notations are the same as above. Now a new domain appears, i.e., domain 2 corresponding to propagation of solitary population pulses. Sub-domain 3 corresponds to the case when the front of prey travels faster than the front of predator, cf. Section 3.1 for details. A higher value of γ corresponds to a stronger prey; thus, it is not surprising that the domain where the front of prey outruns the front of predator (below the dashed curve) has grown in size compared to the case γ = 1showninFig. 12.

21 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) Fig. 14. The map in the parameter plane of the Eqs. (5)and(6)forγ = 7. Domain 6 corresponds to nonuniqueness of invasion regime, see details in the text. Fig. 14 shows the map in the (δ, β) parameter plane for γ = 7. Here a new domain 6 appears with peculiar properties. (More precisely, this domain appears already for γ = 3butinthat case its size is very small and thus it is not seen in Fig. 13.) For parameters from domain 6, the pattern of species spread exhibits nonuniqueness subject to the initial conditions. Depending on u 0, v 0, u, v,species invasion takes place either through propagation of solitary pulses (Figs. 6 and 7)orthrough propagation of periodical waves with irregular wave-makers (Fig. 5). Asterisk indicates the sub-domains where the population wave of prey travels faster than the wave of predator. In the case of propagating population pulses, it means that the pulse width grows with time. In the case of periodic waves, it means that they actually propagate into the region already invaded by prey. One can clearly see the tendency that an increase in γ makes prey more invasive compared to predator: the larger γ is, the higher the dashed curve lays. In order to estimate threshold values of γ for which the bifurcations of the (δ, β) plane take place, we accomplished an additional series of computer experiments. We obtained that domain 2 of solitary pulses arises for γ 1.3. As for domain 6, we observe that it arises at about the same value of γ 1.3; however, for γ<5 its width is very small and it can hardly be seen in the parameter plane. The maps in the (δ, β) plane give important information about possible transitions between different regimes that may occur as a result of system response to parameter changes. For the sake of simplicity, we consider the situation when only δ can change and all other parameters are fixed. Let us start with the case when δ is small. Since δ is (dimensionless) predator mortality, small δ likely means that prey is under strong

22 658 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) pressure from the predator. Thus, it is not surprising that small δ typically corresponds to species extinction (unless γ is sufficiently large and β is sufficiently small, cf. the left-hand bottom corner of Fig. 14). In order to avoid multivalence, we restrict further analysis to the case γ = 3. An increase in δ makes this pressure smaller and extinction changes, depending on other parameter values, either to local invasion (domain 5) or to geographical spread through pulse propagation (domain 2). Note that, since the impact of predation is still too strong, none of these two regimes lead to global persistence of the invading species. However, further increase in δ changes these regimes first to patchy invasion (either preceded by population front propagation, cf. Figs. 3 and 4,ornot, cf.fig. 8) and then to invasion through propagation of smooth population fronts,see Fig. 2. Similar analysis can be made varying other parameters. In particular, it is straightforward to see that similar succession of regimes takes place when β is varied from a large value to asmallone. 5. Concluding remarks The impact of predation on the spread of invasive species has long been an issue of significant interest (Murray, 1989; Sherratt et al., 1995; Shigesada and Kawasaki, 1997; Fagan and Bishop, 2000; Owen and Lewis, 2001; Petrovskii et al., in press). In most cases, however, these studies were reduced to the case of populations with logistic growth. Meanwhile, for many ecological populations their growth rate is believed to be damped by the Allee effect (Allee, 1938; Berryman, 1981). Although it was shown that the Allee effect can change population dynamics significantly (Dennis, 1989; Lewis and Kareiva, 1993; Courchampet al., 1999; Gyllenberget al., 1999; Petrovskii et al., 2002a),its impact on species invasion has not been investigated in detail [but see Morozov (2003)]. In this paper we have shown, using a predator prey system as a paradigm, that the spatiotemporal dynamics of invading species can become much more complex under the influence of the Allee effect and exhibit regimes of invasion that have not been studied theoretically before. Apredator prey system with the Allee effect for prey can exhibit such patterns of spread as a patchy invasion (Fig. 8), geographical invasion without regional persistence (Figs. 6 and 7) and local invasion without geographical spread (Fig. 11). Remarkably, similar patterns are often observed in nature. In contrast, a predator prey system without Allee effect only predicts species spread through either smooth population waves or travelling fronts with population oscillations in the wake (similar to what is shown in Figs. 2, 3 and 4 respectively) [cf. Sherratt et al. (1995), Petrovskii et al. (1998), Petrovskii and Malchow (2000)]. By means of extensive numerical simulations, werevealed the structure of the parameter space of the system. That structure gives important information about possible transitions between different regimes of invasion. Here we consider one example how it can enhance our understanding of various aspects of biological invasion. It is well-known that, between the stage of exotic species establishment in the new environment and the stage of its geographical spread, there often exists a time-lag that can be sometimes as long as years, or even decades. The nature of this time-lag iswidely seen in species adaptation to the new conditions. However, no specific dynamical mechanism has ever been proposed to explain

23 S. Petrovskii et al. / Bulletin of Mathematical Biology 67 (2005) how species regional persistence actually changes to its geographical invasion. Our results seem to suggest such a mechanism. In an introduced species, the individual fitness under new environmental conditions is likely to be low which means that the species is more prone to predation and to environmental stochasticity, so that the species is more likely to go extinct when it is at low density. Virtually, it means large β and/or small δ. Intermsof the diagrams shown in Figs ,itmeans that it falls either to domain 1 corresponding to species extinction or, in case fitness is not very low, to domain 5 corresponding to regional persistence. As a result of species adaptation to the new environment, the individual fitness is likely to increase so that theparameters move from domain 5 either to domain 3 or to domain 4; in both cases the regime of regional persistence gives way to species geographical spread. The results shown in Figs were obtained for ɛ = D 2 /D 1 = 1. However, we want to emphasize that this not a principal limitation and the particular value ɛ = 1wasmainly chosen in order to exclude another parameter. Our tentative numerical simulations made for 0.5 <ɛ<2show that all the regimes described above exist also in that case, although the position of the domains in the (δ, β) plane is somewhat different. The initial conditions that we used in numerical simulations correspond to the problem of biological control, see the lines after Eqs. (7) and(8). (Note that, in our computer experiments, the radius of the domain initially inhabited by predator is always smaller than that for prey.) Another biologically interesting case could be given by the situation when either prey is introduced into an ecosystem where predator is established, or predator is introduced into an ecosystem where prey is established. Mathematically, it means that only one of the functions u(x, 0) and v(x, 0) is finite. Although this problem still remains to be investigated in detail, we fulfilled some tentative simulations to make an early insight into thecorrespondingsystemdynamics. Ourresultsindicatethat, in this case, thenumber of possible invasion regimes is likely to be less than it was for the problem (5) (8); in particular we failed to observe the regimes shown in Figs In this paper, our study of biological invasion has been restricted to the 1-D case. That was done mainly for practical reasons because the time needed to fulfil necessary computer simulations increases essentially in the case of two spatial dimensions. However, we want to mention that, although a regular investigation of this issue is still lacking, our tentative computer experiments indicate that the invasion patterns described above exist in the 2-D predator prey system as well [cf. also Petrovskii et al. (2002a,b), Morozov (2003)]. In conclusion, we want to mention that the structure of the (δ, β) plane tends to become more complex with an increase in γ.thistendency can probably be easier understood if placed into a more general contextofself-organization in dynamical systems. It is well-known from other branches of natural science that the dynamics of an open system becomes the more complicated the more open it is, i.e., the higher is the energy/mass input into a given system (Prigogine, 1980; Haken, 1983). In particular, a higher energy/mass input drives the system further from its equilibrium state and enhances formation of complex spatiotemporal patterns. [A classical example of this increasing complexity is given by the transition between laminar and turbulent flows in a pipe which takes place when the speed of the fluid (and thus the mass/energy input) becomes sufficiently high.] In the predator prey model (1)and(2)the role of energy input that keeps the system away from the trivial extinction equilibrium is played by the biomass increase

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