on Self-organization, Scale and Stability in a Spatial Predator–Prey Interaction

1.
I NTRODUCTION
Simple predator–prey models often predict extreme instability in interactions where the prey are depressed well below their carrying capacity—a phenomenon often called the paradox of enrichment (Rosenzweig, 1971; Gilpin, 1972) because such depression is more likely in nutrient rich systems. Instability in such models usually takes the form of limit-cycle oscillations, whose amplitude is often sufficient to imply the extinction of one or both players. Although the behaviour of some laboratory systems [e.g., Gause (1934); Huffaker (1958)] apparently conforms to this pattern, field and mesocosm studies generally show prolonged co-existence of prey and predator, accompanied by random fluctuations or small amplitude oscillations [e.g., McCauley and Murdoch (1987, 1990)]. Considerable effort has consequently been expended on constructing more elaborate models incorporating a variety of potential stabilizing mechanisms. Murdoch et al. (1998) give a wide-ranging survey of these possibilities, together with tests of some of them in the context of the interaction between Daphnia and its
Bulletin of Mathematical Biology (2000) 62, 61–86 Article No. bulm.1999.0130 Available online at on
Self-organization, Scale and Stability in a Spatial Predator–Prey Interaction
WILLIAM S. C. GURNEY AND A. ROY VEITCH Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, Scotland, U.K.
Simple predator–prey models often predict extreme instability in interactions where the prey are depressed well below their carrying capacity. Although the behaviour of some laboratory systems conforms to this pattern, field and mesocosm studies generally show prolonged co-existence of prey and predator. Prominent among the possible causes of this discrepancy are the effects of spatial heterogeneity. In this paper we show that both discrete and continuous representations of the spatial Rosenzweig–McArthur model with immobile prey can be stabilized by selforganized prey heterogeneity. This concordance of behaviour closely parallels that which we have previously established in the context of invasion waves. We use the continuous model variant to calculate the characteristic spatial scales of the selforganized structures. The discrete variant forms the basis of a simulation study demonstrating the variety of stable structures and elucidating their relation to the history of the system. We note that all stable prey distributions take the form of a network of occupied patches separated by prey-free regions, and liken the process which generates such assemblages to the formation of a landscape mozaic. c 2000 Society for Mathematical Biology
0092-8240/00/010061 + 26 $35.00/0 c 2000 Society for Mathematical Biology
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ቤተ መጻሕፍቲ ባይዱ
W. S. C. Gurney and A. R. Veitch
algal food. They conclude that many of the most straightforward mechanisms can readily be shown to be incompatible with currently available evidence, but prominent among those which cannot be so falsified is spatial heterogeneity. It has long been known that environmental heterogeneity can stabilize population processes. Gurney and Nisbet (1975) showed that non-linear dispersal can stabilize a population growing uncontrollably in a finite geographical region surrounded by territory where it suffers net mortality. Gurney and Nisbet (1976) demonstrated that donor-regulated populations exhibit spatial distributions in which some of the scales present in the resource distribution are amplified—an effect which McLaughlin and Roughgarden (1991, 1992) have shown to stabilize the spatial Lotka–Volterra predator–prey model, which is invariably neutrally stable in a homogeneous environment. The possibility that spatial predator–prey interactions might exhibit self-organization capable of producing stabilizing heterogeneities in prey distribution was first suggested by the work of Jansen (1994, 1995), who considered a model with two patches, each with unstable Rosenzweig–McArthur dynamics, linked by diffusive dispersal. This model has several equilibria, among which is one with identical prey (and predator) populations in the two patches. When this homogeneous equilibrium is unstable, the model can exhibit large amplitude limit cycles with the patch populations fluctuating in synchrony. Jansen (1994) showed that this homogeneous limit-cycle can become unstable, whereupon the system is attracted to a smaller amplitude limit-cycle in which the patch populations fluctuate about an equilibrium with low prey in one patch and high prey in its neighbour. Subsequent work (Jansen and de Roos, 1998) has shown that similar effects occur in arbitrary sized linear arrays of patches provided that the prey are significantly less mobile than the predator. A related effect has been reported by McCauley et al. (1996) in an individualbased simulation model with Rosenzweig–McArthur dynamics and a separate juvenile stage. They found that where prey mobility is low, systems with parameters implying large amplitude oscillations in a non-spatial model, can be stabilized by a mechanism intimately connected to the formation of a persistent, small-scale heterogeneous structure in the prey distribution. De Roos et al. (1998) have studied a related continuous time patch dynamic model with logistic local prey dynamics and predators with a type II functional response, whose searching covers an area which is large compared to the distance over which the prey can diffuse in a generation. They found that when the homogeneous equilibrium of this system is unstable, most trajectories other than those starting from an exactly homogeneous initial condition are attracted to a stable spatially heterogeneous steady state. The work of Jansen (1995), McCauley et al. (1996) and De Roos et al. (1998) suggests that a saturating predator functional response plays a key role in the selforganization of stabilizing structures in the spatial distribution of prey. Although such structures have, as yet, only been observed in individual-based and patch-