Stochastic Structure and Nonlinear Dynamics of Food Webs: Qualitative Stability in a Lotka-Volterra Cascade Model

J. E. Cohen, T. Luczak, C. M. Newman, Z.-M. Zhou

Abstract

Two competing models currently offer to explain empirical regularities observed in food webs. The Lotka-Volterra model describes population dynamics; the cascade model describes trophic structure. In a real ecological community, both population dynamics and trophic structure are important. This paper proposes and analyses a new hybrid model that combines population dynamics and trophic structure: the Lotka-Volterra cascade model (LVCM). The LVCM assumes the population dynamics of the Lotka-Volterra model when the interactions between species are shaped by a refinement of the cascade model. A critical surface divides the three-dimensional parameter space of the LVCM into two regions. In one region, as the number of species becomes large, the limiting probability that the LVCM is qualitatively globally asymptotically stable is positive. In the region on the other side of the critical surface, and on the critical surface itself, this limiting probability is zero. Thus the LVCM displays an ecological phase transition: gradual changes in the probabilities of various kinds of population dynamical interactions related to feeding can have sharp effects on a community's qualitative stability. The LVCM shows that an inverse proportionality between connectance and the number of species, and a direct proportionality between the number of links and the number of species, as observed in data on food webs, need not be directly connected with the qualitative global asymptotic stability or instability of population dynamics. Empirical testing of the LVCM will require field data on the population dynamical effects of feeding relations.

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