Transition to spatiotemporal chaos can resolve the paradox of enrichment
Introduction
Ecosystem eutrophication is currently considered as a major upcoming threat to species biodiversity (Tilman et al., 2001). 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 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, May, 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, 2000, Jansen, 2001. 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 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, 1990), 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, 2000. 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, 2000) and theoretically Jansen, 1995, Jansen and Lloyd, 2000, Jansen, 2001 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, 2000, Jansen, 1995, Jansen, 2001, Jansen and Lloyd, 2000 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, 2000, Jansen, 2001) 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, 2001. 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, Sherratt et al., 1997, Petrovskii and Malchow, 1999, Petrovskii and Malchow, 2001, Sherratt, 2001, Medvinsky et al., 2002. 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., 2001.
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 the parameter values Petrovskii and Malchow, 1999, Petrovskii and Malchow, 2001, Sherratt, 2001. 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.
Section snippets
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, 2001: where X is the position (0<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
Results of computer simulations
The impact of eutrophication on the spatiotemporal dynamics of the predator–prey systems , was studied by means of computer simulations. The system , 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 , , cf. (
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, 2000, Jansen, 2001 where similar results were obtained for a spatially
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-00-80529 and DBI-98-20318), German Science Foundation (436 RUS 113/631), Russian Foundation for Basic
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