Transition to spatiotemporal chaos can resolve the paradox of enrichment

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1 Ecological Complexity 1 (24) Transition to spatiotemporal chaos can resolve the paradox of enrichment Sergei Petrovskii a,, Bai-Lian Li b, Horst Malchow c a Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prosp. 36, Moscow, Russia b Ecological Complexity and Modeling Laboratory, Department of Botany and Plant Sciences, University of California, Riverside, CA , USA c Department of Mathematics and Computer Science, Institute of Environmental Systems Research, Osnabrück University, D-4969 Osnabrück, Germany Received 16 April 23; received in revised form 12 September 23; accepted 2 October 23 Abstract The dynamics and stability of interacting populations in connection to spatial phenomena such as pattern formation and spatiotemporal chaos have recently become a focus of intensive research in theoretical ecology. In this paper, we demonstrate a surprising relation between the long-standing enigma known as Rosenzweig s paradox of enrichment and the formation of chaotic spatiotemporal patterns in an ecological community. Using two different spatially explicit models (a standard diffusion reaction system and a diffusion reaction system with cutoff at low population densities), we show by means of computer simulations that transition to spatiotemporal chaos can prevent species extinction in a situation when it would be expected in the case of regular dynamics. The patterns arising in our models are self-organized, and not induced directly by pre-existing spatial heterogeneity of the environment. We also show that the type of the system s response to enrichment essentially depends on the system size and on the rates of eutrophication. 24 Elsevier B.V. All rights reserved. Keywords: Enrichment; Predator prey system; Species extinction; Spatiotemporal chaos 1. Introduction Ecosystem eutrophication is currently considered as a major upcoming threat to species biodiversity (Tilman et al., 21). A comprehensive theoretical framework aimed at understanding and predicting species responses to this destructive process is still lacking. In particular, the paradox of enrichment, when an increase in the nutrient input into a predator prey Corresponding author. Fax: address: spetrovs@sio.rssi.ru (S. Petrovskii). system can destabilize the community and even lead to extinction of the species, has been a challenge for a few generations of ecologists (Rosenzweig, 1971; Gilpin, 1972; May, 1972, 1974; Brauer and Soudack, 1978; Jansen, 1995; Abrams and Walters, 1996; Bohannan and Lenski, 1997; Nisbet et al., 1997; Genkai-Kato and Yamamura, 1999; Holyoak, 2; Jansen, 21). It was shown theoretically that an increase in the prey carrying capacity which arises as a prey response to system eutrophication (enrichment) leads to population oscillations of increasing amplitude. The minimum value of the population X/$ see front matter 24 Elsevier B.V. All rights reserved. doi:1.116/j.ecocom

2 38 S. Petrovskii et al. / Ecological Complexity 1 (24) density decreases and population extinction becomes more probable due to stochastic environmental perturbations. Although this kind of system response to eutrophication is not commonly seen in nature (McCauley and Murdoch, 199), the self-regulating mechanisms of the system are not always clear. Furthermore, extinction of a predator prey community following system eutrophication has been seen in some laboratory experiments (Luckinbill, 1974; Bohannan and Lenski, 1997). The apparent contradiction between the intuitively expected positive impact of increasing nutrient input and its actual destabilizing effect inspired a number of modifications of the original predator prey model. It was shown that enrichment of a predator prey community does not necessarily diminish the minimum value of oscillating population densities in the cases of either the existence of invulnerable individuals within the prey population (Abrams and Walters, 1996) or in the presence of an alternative unpalatable prey (Genkai-Kato and Yamamura, 1999). However, these modifications have left open the question whether the simplest one-predator one-prey system is intrinsically unstable with respect to eutrophication. The theoretical results mentioned above were obtained under assumption that the interacting populations were homogeneous in space. A crucial point has become the understanding that the dynamics of any biological community takes place not only in time but also in space (Hassell et al., 1991; de Roos, 1991; Allen et al., 1993; Bascompte and Solé, 1994; Nisbet et al., 1997; Wilson, 1998; Satake and Iwasa, 2). The impact of space on the persistence of enriched predator prey systems was proved in laboratory experiments (Luckinbill, 1974). Recently, it has been shown both in laboratory experiments (Holyoak, 2) and theoretically (Jansen, 1995; Jansen and Lloyd, 2; Jansen, 21) that the existence of a patchy spatial structure makes a predator prey system less prone to extinction. In a spatially structured metapopulation, the temporal variations of the density of different sub-populations can become asynchronous and the events of local extinction can be compensated due to re-colonization from other sites (Allen et al., 1993). Although recent studies by Holyoak (2), Jansen (1995, 21), Jansen and Lloyd (2) provided a valuable insight into the role of space in resolving the paradox of enrichment, a few important issues have not yet been properly addressed. First, those studies focus more on the comparison between the dynamics of enriched and non-enriched systems rather than on the actual process of enrichment and the corresponding system response. Thus, the impact of the eutrophication rate has never been addressed. Meanwhile, there are a lot of examples where a system shows principally different behavior depending on the rates of external forcing. Thus, one can expect that fast and slow enrichment may have different impacts on the community functioning. Second, the above results have been obtained for a spatially structured metapopulation (i.e., for a system of coupled habitats) and it is not at all clear whether they can be immediately extended to the case of homogeneous environment and/or to a single isolated habitat. Third, the theoretical approach formulated in terms of a space-discrete metapopulation model (cf. Jansen, 1995; Jansen and Lloyd, 2; Jansen, 21) takes space into account in a rather implicit manner. The impact of the habitat size is not considered; however, it is well-known that the population dynamics in small and large habitats can exhibit different features (Hassell et al., 1991; de Roos, 1991; Bascompte and Solé, 1994; Petrovskii and Li, 21). Also, under the metapopulation approach the population densities inside each habitat are assumed to be homogeneous which is not often observed in real ecosystems. Particularly, it has been recently shown that spontaneous spatiotemporal pattern formation is an intrinsic property of a predator prey system (Pascual, 1993; Sherratt et al., 1995, 1997; Petrovskii and Malchow, 1999, 21; Sherratt, 21; Medvinsky et al., 22). As a result of the homogeneity break-up, the dynamics of a spatially extended predator prey system continuous in space and time can become chaotic, see the references above. It should be mentioned here that, although conclusive evidence of ecological chaos is still to be found, there is a growing number of indications of chaos in real ecosystems (Scheffer, 1991; Hanski et al., 1993; Ellner and Turchin, 1995; Dennis et al., 21). In this paper, we consider a model which allows us to address the issues outlined above. The dynamics of a predator prey system is described by the spatially explicit diffusion reaction equations, cf. (Holmes et al., 1994). This kind of model has been demonstrated to exhibit two types of dynamics, i.e. regular or chaotic, depending on the initial conditions and

3 S. Petrovskii et al. / Ecological Complexity 1 (24) the parameter values (Petrovskii and Malchow, 1999, 21; Sherratt, 21). We show that the system s response to a sufficiently large increase in the carrying capacity of the prey population (which is assumed to be a result of the system eutrophication) can be either populations extinction or transition to spatiotemporal chaos. We also show that the stability of the regular dynamics to the system enrichment increases with a decrease in the size of the system. Also, we show that the type of the system s response to enrichment essentially depends on the rate of eutrophication. 2. Mathematical model The dynamics of a predator prey system can be qualitatively described by the following equations (Nisbet and Gurney, 1982; Murray, 1989; Pascual, 1993; Holmes et al., 1994; Sherratt et al., 1995; Sherratt, 21): U(X, T) T V(X, T) T ( = D 2 U X 2 + αu 1 U ) b ( = D 2 V X 2 + κγ U γ U + h V, (1) ) U U + h µ V, (2) where X is the position ( < X < l where l is the length of the domain or the system size ), T is the time, U, V are the densities of prey and predator, respectively, α stands for the maximum per capita growth rate of prey, b is the carrying capacity, h is the half-saturation prey density, coefficient γ describes the intensity of predation, κ is the coefficient of food utilization, and µ is the mortality rate of predator. Coefficient D describes the intensity of dispersion due to the mobility of individuals (Skellam, 1951; Okubo, 198); for simplicity we assume that it is the same for both species. Without diffusion terms (i.e. assuming spatially homogeneous species distribution), the system Eqs. (1) and (2) is equivalent to the family of predator prey models considered by Rosenzweig (1971) in his original paper. Environmental properties are assumed to be homogeneous, so that none of the coefficients in Eqs. (1) and (2) depends on space. At the boundaries of the domain, the zero-flux boundary conditions are used. By rescaling of the variables, u = U/b, v = Vγ/(αb), t = αt, x = X(α/D) 1/2, we arrive at the system which contains only dimensionless variables and parameters: u t = 2 u + u(1 u) u x2 v t = 2 v x 2 + k ( u u + H v rv v, (3) u + H ) (4) where H = h/b, k = κγ/α, r = µ/κγ and <x< L, L = l(α/d) 1/2. Enrichment of the system likely leads to an increase in the prey growth rate and/or to an increase in the prey carrying capacity and thus to a decrease in either k or H. In terms of a homogeneous predator prey model, the dynamics that may lead to species extinction corresponds to the case when the trajectory in the phase space of the system comes close to the boundary of the biologically meaningful domain u,v. It should be noted that in the model Eqs. (3) and (4) the population actually never goes extinct because the boundary of the domain u, v acts as a repeller. However, the situation when the population density falls to a very small value and then grows again does not seem realistic. Referring to the dynamics of a real ecological community, the smaller the population density is, the higher is the probability of species extinction due to stochastic environmental fluctuations (Goeland Richter Dyn, 1974; Lande, 1993). Thus, when the population density becomes sufficiently small it can be treated as species extinction (Gilpin, 1972; May, 1972, 1974). This approach to virtually incorporate environmental stochasticity into fully deterministic model (3) and (4) is known as the practical stability concept (Brauer and Soudack, 1978; Legovic, 1987). Let us note that the system (3) and (4) has only one homogeneous steady co-existence state, (ū, v), which is situated in the domain u, v under constraints r<1 and H (1 r)/r (for H>(1 r)/r, the only attracting steady state is the prey only state (1,); thus, for these parameter values the population of predator cannot persist). Under the condition H< H c (r) = (1 r)/(1 + r), the steady state (ū, v) becomes unstable; in this case, the only attractor in the phase plane of the corresponding homogeneous system is the stable limit cycle which appears via the

4 4 S. Petrovskii et al. / Ecological Complexity 1 (24) Hopf bifurcation. Computer simulations show that, for increasing distance from H c (r), particularly, for decreasing H the local oscillatory kinetics of the system (3) and (4) undergoes gradual quantitative changes, namely, the size of the limit cycle increases and it comes closer and closer to the axes u = and v =. As a result, the community comes in danger of extinction. On the other hand, computer simulations show that a decrease in k does not lead to any significant decrease in the minimum value of population density. Correspondingly, we restrict our consideration to the case when enrichment results only in a decrease in H. When space is taken into account the dynamics of the system (3) and (4) becomes essentially different. Even a small inhomogeneous perturbation of a homogeneous state of the system (3) and (4) can lead to formation of spatiotemporal patterns (Petrovskii and Malchow, 1999). Note that, since the dispersal rates are assumed to be the same for prey and predator, the patterns cannot appear due to the Turing instability (cf. Segel and Jackson, 1972) and should be ascribed to another mechanism (Petrovskii and Malchow, 21). Depending on the details of the species distribution, there can be two different patterns corresponding to two basically different regimes of the system dynamics, i.e. regular or chaotic. A sufficiently small perturbation drives the system to the regular regime with a smooth inhomogeneous spatial distribution of species as a typical pattern (see Fig. 1a). The temporal variations of species are periodic (see Fig. 1b) and strongly correlated in space; the trajectory in the local phase plane coincides with the limit cycle of the spatially homogeneous system. For a larger perturbation, the species distribution evolves to the formation of chaotic spatiotemporal pattern (Petrovskii and Malchow, 1999, 21; Medvinsky et al., 22), see Fig. 1c. In this case, the temporal variation of the species density becomes remarkably irregular (see Fig. 1d), and Population Density (a) Prey Density (b) Space 2 Population Density (c) Prey Density (d) Space 2 Fig. 1. Spatial (a, c) and temporal (b, d) variations of the population densities in case of regular (a, b) and chaotic (c, d) dynamics. Snapshots of the species distribution in (a) and (c) (solid line for prey, dotted for predator) are taken at moment t = 2 and correspond to the initial distribution chosen in the form of allocated constant-gradient perturbation of the co-existence steady state, i.e., u(x, ) =ū, v(x, ) = v+νx+δ with parameter values ν = 1 5 and (a) δ =.1 and (c) δ =.5. Other parameters are k = 2., r =.3, H =.43.

5 S. Petrovskii et al. / Ecological Complexity 1 (24) uncorrelated in space since the spatial autocorrelation function promptly decays with distance (more rigorously, temporal variations at two positions can be regarded as uncorrelated when the distance between the positions exceeds the correlation length L corr ) (Petrovskii and Malchow, 21; Petrovskii et al., 23). Besides, the spatial distribution of the species shows a very high sensitivity to variations of the initial conditions (cf. Sherratt et al., 1995; Medvinsky et al., 22): the property which is characteristic for chaotic dynamics. We want to stress that the patterns shown in Fig. 1a and c are self-organized since the model (3) and (4) does not have any prescribed spatial structure. Our goal is to investigate what can be the impact of enrichment on persistence/extinction of the species taking into account the existence of two different regimes of dynamics. Obviously, the system response to enrichment depends on the properties of the system s spatiotemporal dynamics. In the case that the temporal variations of the species are synchronized or strongly correlated over the whole space, as it takes place in the regular regime, a fall of the species density to a small value takes place almost simultaneously throughout the system and thus the population may go extinct without any chance of subsequent re-colonization (Allen et al., 1993). Conversely, if the temporal variations are desynchronized, as it takes place for the predator prey system (3) and (4) in the chaotic regime, a local extinction does not necessarily lead to global extinction because the empty sites can be re-colonized via migration of the individuals from other sites, i.e. via diffusion in terms of the model (3) and (4). 3. Results of computer simulations The impact of eutrophication on the spatiotemporal dynamics of the predator prey systems (3) and (4) was studied by means of computer simulations. The system (3) and (4) was solved numerically by finite differences. Sensitivity of the results with respect to the values of the grid steps was checked and they were chosen reasonably small to avoid any essential numerical artifact. Besides, we tested the numerical code by comparison numerical results with some analytical predictions known for the system (3) and (4), cf. (Murray, 1989, p. 279; Petrovskii and Malchow, 21). We begin with the case when the system before eutrophication is in the regime of regular spatiotemporal oscillations, cf. Fig. 1a and b. In order to distinguish between different types of community dynamics, we consider the temporal variations of the spatially averaged prey density. We use the spatially averaged density as a convenient measure to describe the correlation between the temporal variations in population density at different positions in space. Evidently, in the case that the temporal variations are strongly correlated throughout the domain (as in the case of regular spatiotemporal oscillations, see Fig. 1a), the amplitude of the variation of the average density is close to the amplitude of the local variations which tends to grow with the system eutrophication. If the population oscillations are not correlated throughout the domain, the amplitude of the variation of the average density is significantly smaller than that of the local density (cf. Bascompte and Solé, 1994); particularly, it takes place in the case of spatiotemporal chaos (Medvinsky et al., 22). We consider separately two cases: instantaneous enrichment and gradual enrichment. Firstly, we consider the case when eutrophication occurs suddenly at a certain moment t. Correspondingly, the value of H changes instantly from an initial value H toanew value H 1 = H H. We assume that the local kinetics of the system is already oscillatory before the parameter changes, i.e. the point (r, H ) in the parameter plane lies below the Hopf bifurcation curve. In order to investigate the impact of enrichment on the community functioning, we have accomplished numerical simulations for values of r, H and H from a wide range. Typical results are presented in Fig. 2. Only temporal variations of the prey density are shown; the density of predator exhibits qualitatively similar behavior. Fig. 2a shows the response of a spatially homogeneous predator prey system to enrichment after H changed from H =.43 to H 1 =.15 at the moment t = 2. The amplitude of the population oscillations increases so that the prey density periodically falls to a very small value. According to the above arguments, it likely leads to species extinction. This is in full coincidence with the results of earlier studies (Rosenzweig, 1971; Gilpin, 1972; May, 1972). The situation remains qualitatively the same when prior to eutrophication the species distribution was spatially inhomogeneous (see Fig. 1a as an example)

6 42 S. Petrovskii et al. / Ecological Complexity 1 (24) (a) (b) (c) (d) Fig. 2. Temporal variation of spatially averaged prey density (semilogarithmic plot) in case of instantaneous enrichment: (a) spatially homogeneous oscillations obtained for H 1 =.15; (b) transition to chaos obtained for H 1 =.15; (c) transition to chaos obtained for H 1 =.13; (d) transition to chaos obtained for H 1 =.15 in the predator prey model with cutoff at ɛ = 1 4. In all cases, H =.43, other parameters are the same as in Fig. 1. For the initial conditions we used the distribution of species shown in Fig. 1a. provided that the magnitude of enrichment H is sufficiently small, cf. Table 1. The regular regime persists and the temporal variations of the population densities remain correlated throughout the domain. Table 1 Predator prey system response to enrichment L, dimensionless H, dimensionless The types of the predator prey system response to enrichment for different length L of the domain and different magnitude H of enrichment observed for parameters k = 2.,r =.3,H =.43. Minus sign stands for the cases when the dynamics of the system remains regular, plus stands for the cases when the enrichment drives the system into spatiotemporal chaos. However, in the inhomogeneous case the dynamics of the system undergoes principal changes for larger values of H (correspondingly, for smaller H 1 ). Fig. 2b shows the species temporal variations when at t = 2 the value of H decreases from H =.43 to H 1 =.15, i.e. for the same parameters as in Fig. 2a. The type of the system dynamics now changes from regular to chaotic. The properties of the spatial distribution of species undergo corresponding changes, cf. Fig. 1a and c. Although the local density may still fall to a small value, the amplitude of the temporal variation of the average density decreases significantly; this reflects the fact that the oscillations of the species density at different positions in space become desynchronized. Desynchronized dynamics excludes the situation when the species density falls to a small value simultaneously at each position in space. Thus, following the above arguments, it decreases the probability of species extinction. For somewhat

7 S. Petrovskii et al. / Ecological Complexity 1 (24) larger eutrophication, i.e. for larger H, the impact of transition to chaos on temporal variations of the population density becomes even more prominent, see Fig. 2c. We want to stress that the results shown in Fig. 2b and c are typical and have been selected from many other qualitatively similar results. An issue of considerable interest is to understand how the type of system s response to eutrophication can be modified by variation in the length L of the spatial domain. Results of our numerical simulations show that the properties of the system dynamics do depend on L. Particularly, in the spatiotemporal chaos regime, the amplitude of temporal variations of the spatially averaged density decreases monotonously with L. This result immediately follows from the fact that a larger domain includes a larger number of dynamically independent sub-domains, the length of each sub-domain being on the order of L corr (Petrovskii and Malchow, 21). Thus, one can expect that the type of the system response to enrichment may depend on the length L of the domain. In order to address this issue, we have repeated our simulations for various L. The results are summarized in Table 1. We obtain that the regular dynamics in small habitats tends to be more stable to enrichment than it is in large ones. Since transition to chaos is shown to decrease the probability of species extinction, it means that a large domain is more favorable for community functioning than a small one. This result seems to be in a good agreement with intuitive expectations. However, it also leads to a somewhat paradoxical inference that an instantaneous enrichment of smaller magnitude may appear to be more dangerous than that of larger magnitude, especially in a small domain. The matter is that the minimal values of the oscillating population densities decrease when H increases; thus it can happen that, for some values of H, the average population densities fall to dangerously small values while the dynamics of the system still remains regular. For larger H the population oscillations at different sites become desynchronized, that makes the populations less prone to extinction. The above conclusions about the role of space in resolving the paradox of enrichment are sketched in Fig. 3. Here, curves 1 and 2 show the probability of 1 1 Probability of Global Extinction 2 H 1 H 2 Magnitude of Eutrophication Fig. 3. A sketch of the extinction probability dependence on system s enrichment for spatially homogeneous (curve 1) and spatially inhomogeneous (curve 2) systems. The risk of extinction becomes dangerously high when the magnitude of eutrophication exceeds certain critical value H 1. However, when the magnitude of eutrophication reaches another critical value H 2 the probability of population extinction in the spatially inhomogeneous system drops down significantly due to transition to spatiotemporal chaos.

8 44 S. Petrovskii et al. / Ecological Complexity 1 (24) population extinction for spatially homogeneous and inhomogeneous cases, correspondingly. An increase in the probability of global extinction at small H appears from the fact that the amplitude of local population oscillations grows with H while the oscillations remain strongly correlated throughout the domain (assuming that prior to eutrophication the dynamics was regular, cf. Fig. 1a and b) so that the populations face a high risk of extinction for H> H 1, where H 1 is a certain critical value depending on various environmental and demographic factors. In the case of spatially homogeneous system, the risk continues growing monotonously with H until extinction becomes inevitable. For the spatially inhomogeneous system, however, when the magnitude of eutrophication exceeds certain critical value H 2 transition to spatiotemporal chaos takes place. As a result, the local population oscillations become desynchronized and the probability of global extinction decreases significantly. Besides instantaneous changes, we also consider the case when eutrophication takes place gradually during the time interval from t to t 1 = t + t. Typical results are shown in Fig. 4. For simplicity, we assume that the value of H changes from H to H 1 linearly with time. In this case, the type of the system response depends also on the value of t. The results of our computer simulations show that, while for small t the type of the system response is the same as for instantaneous enrichment (i.e. transition to chaos for sufficiently large H, cf. Figs. 2b and 4a), for sufficiently large t the dynamics of the system remains regular. That leads to a rather unexpected conclusion that a fast eutrophication may appear to be less dangerous for the community functioning than a slow one. The above results relate to the case when, prior to eutrophication, the dynamics of the system is regular. Conversely, the results of our computer simulations indicate that the regime of spatiotemporal chaos is stable with respect to a decrease in H. In this case, 1-1 (a) 1-5 (b) (c) 1-5 (d) 1-2 Fig. 4. Temporal variation of spatially averaged prey density (semilogarithmic plot) in case of gradual enrichment for parameters H =.43 and H =.28 and (a, c) t = 4 and (b, d) t = 2. Other parameters and the initial conditions are the same as in Fig. 2. The plots in (c, d) are obtained in the model with cutoff at (c) ɛ = 1 5 and (d) ɛ =

9 S. Petrovskii et al. / Ecological Complexity 1 (24) although eutrophication of the system increases the amplitude of local population oscillations, it does not likely increase the danger of species extinction significantly because the dynamics of the system remains chaotic and the population oscillations are not correlated throughout the domain. 4. Concluding remarks In this paper, we have shown that enrichment of a predator prey system can drive it to spatiotemporal chaos. Since chaos leads to desynchronization of population oscillations at different positions in space, it can prevent population extinction in a situation where it would be inevitable in the case of regular dynamics. Our results are in a good agreement with the considerations of previous authors (Jansen, 1995; Holyoak, 2; Jansen, 21) where similar results were obtained for a spatially structured patchy system. Moreover, we have shown that the actual response of the system depends essentially on the rate of eutrophication and on the length of the domain. Results of our computer simulations (only a small part of them is shown in Figs. 2 and 4 and Table 1, while in total computer experiments were run for a few hundreds different values of r, H, H,L and t) prove that, for an enrichment of given magnitude, transition to chaos is more likely to be observed in a large domain and for higher rates of eutrophication. Correspondingly, it means that the species extinction resulting from eutrophication of a given system is more likely to happen in a small domain and for low eutrophication rates. In order to arrive at the above conclusions, we used a model based on standard diffusion reaction equations. The question may arise, however, whether this model is valid when the populations are at the edge of extinction, i.e., at very low densities. Conceptual applicability of diffusion terms to describe re-distribution of species in space due to random motion of the individuals for any value of population density (ultimately, even for a single individual) was shown by Skellam (1951) and Okubo (198). Meanwhile, applicability of reaction terms may seem far less apparent or even doubtful, especially for a logistic population which, in a strict mathematical sense, never goes extinct. To check the robustness of our results to the choice of the model, we consider a modified approach: a diffusion reaction system with cutoff at low densities when the species extinction is taken into account explicitly. According to the modified model, at each position in space, whenever the population densities fall below certain prescribed value ɛ they are set to zero (cf. Wilson, 1998). For such a model, we do not need the practical stability concept any more. Now species extinction means extinction in a strict sense, i.e. u(x, t) = v(x, t) = for all x inside the given area. Note that we are not much concerned here with the exact value of ɛ. Moreover, an attempt to estimate the exact value would hardly make any ecological sense in terms of very schematic model (3) and (4). The main goal of this approach is to demonstrate that our results are not an artifact of the continuous-population approximation, and are at least partially robust to the fundamental discreteness of the biological populations. It seems obvious, however, that, since the conditions u<ɛand/or v<ɛ mean immediate extinction of corresponding species, relevant values of ɛ are likely to be small. Rigorous mathematical investigation of the model with cutoff may appear to be difficult and certainly lies beyond the scope of this paper. However, the way to implement cutoff into computer simulations is straightforward. The results obtained under the modified approach (some of them are shown in Figs. 2d, 4c and d) are in agreement with the results obtained above in the standard models (3) and (4). It should be mentioned that the cutoff remarkably enhance chaotic dynamics. We have not found a single parameter set when the regular regime of the system dynamics would persist (unless ɛ is smaller than the minimal value of oscillating population densities which is trivial). Instead, the community either goes extinct or the regular dynamics changes to chaos. Fig. 2d gives an example of transition to chaos obtained in the case of instantaneous enrichment for the same parameters and initial conditions as in Fig. 2b but with cutoff at ɛ = 1 4. Fig. 4c and d shows the temporal variation of the average prey density calculated in the model with cutoff in the case of gradual enrichment for the same parameter values as in Fig. 4a and b, respectively. While for the high rates of enrichment the existence of cutoff does not impose any significant alterations to the system response, cf. Fig. 4a and c, in the case of lower

10 46 S. Petrovskii et al. / Ecological Complexity 1 (24) rates the impact of cutoff renders regular dynamics to chaos, cf. Fig. 4b and d. It should be also mentioned that chaotic dynamics is remarkably less sensitive to the value of ɛ and the community exhibiting spatiotemporal chaos goes extinct only if ɛ is of the same order as the stationary value ū of prey density. Thus, our computer simulations using the modified model with cutoff at low densities confirm the main result of this paper that in the case of large eutrophication of a spatially extended predator prey system only chaotic populations can survive. A regular population either goes extinct or regular dynamics gives a way to spatiotemporal chaos. Acknowledgements Professor A.B. Medvinsky, an Editorial Board Member, acted as the Chief Editor handling the review processes for this manuscript. The authors are thankful to Tom Over and Will Wilson for their helpful comments on the earlier version of the manuscript, and to Vincent Jansen for useful discussion of our results. This work was partially supported by the U.S. National Science Foundation (DEB and DBI ), German Science Foundation (436 RUS 113/631), Russian Foundation for Basic Research ( and ), by the University of California Agricultural Experiment Station and by UCR Center for Conservation Biology. References Abrams, P.A., Walters, C.J., Invulnerable prey and the paradox of enrichment. Ecology 77, Allen, J.C., Schaffer, W.M., Rosko, D., Chaos reduces species extinction by amplifying local population noise. Nature 364, Bascompte, J., Solé, R.V., Spatially induced bifurcations in single-species population dynamics. J. Anim. Ecol. 63, Bohannan, B.J.M., Lenski, R.E., Effect of resource enrichment on a chemostat community of bacteria and bacteriophage. Ecology 78, Brauer, F., Soudack, A.C., Response of predator prey nutrient enrichment and proportional harvesting. Int. J. Control 27, Dennis, B., Desharnais, R.A., Cushing, J.M., Henson, S.M., Costantino, R.F., 21. Estimating chaos and complex dynamics in an insect population. Ecol. Monogr. 71, de Roos, A.M., McCauley, E., Wilson, W.G., Mobility versus density-limited predator prey dynamics on different spatial scale. Proc. R. Soc. Lond. B 246, Ellner, S., Turchin, P., Chaos in a noisy world: new methods and evidence from time-series analysis. Am. Nat. 145, Genkai-Kato, M., Yamamura, N., Unpalatable prey resolves the paradox of enrichment. Proc. R. Soc. Lond. B 266, Gilpin, M.E., Enriched predator prey systems: theoretical stability. Science 177, Goel, N.S., Richter Dyn, N., Stochastic Models in Biology. Academic Press, New York, 198 pp. Hanski, I., Turchin, P., Korplmakl, E., Henttonen, H., Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos. Nature 364, Hassell, M.P., Comins, H.N., May, R.M., Spatial structure and chaos in insect population dynamics. Nature 353, Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R., Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, Holyoak, M., 2. Effects of nutrient enrichment on predator prey metapopulation dynamics. J. Anim. Ecol. 69, Jansen, V.A.A., Regulation of predator prey systems through spatial interactions: a possible solution to the paradox of enrichment. Oikos 74, Jansen, V.A.A., 21. The dynamics of two diffusively coupled predator prey populations. Theor. Popul. Biol. 59, Jansen, V.A.A., Lloyd, A.L., 2. Local stability analysis of spatially homogeneous solutions of multi-patch systems. J. Math. Biol. 41, Lande, R., Risks of population extinction from demographic and environmental stochasticity and random catastrophes. Am. Nat. 142, Legovic, T., A recent increase in jellyfish populations: a predator prey model and its implications. Ecol. Modell. 38, Luckinbill, L.S., The effects of space and enrichment on a predator prey system. Ecology 55, May, R.M., Limit cycles in predator prey communities. Science 177, May, R.M., Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, 212 pp. McCauley, E., Murdoch, W.W., 199. Predator prey dynamics in environments rich and poor in nutrients. Nature 343, Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., Li, B.-L., 22. Spatiotemporal complexity of plankton and fish dynamics. SIAM Review 44, Murray, J.D., Mathematical Biology. Springer, Berlin, 718 pp. Nisbet, R.M., Gurney, W.S.C., Modelling Fluctuating Populations. Chichester, Wiley, 34 pp. Nisbet, R.M., Diehl, S., Wilson, W.G., Cooper, S.D., Donalson, D.D., Kratz, K., Primary-productivity gradients and

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