Global Dynamics of a Tritrophic Model for Two Patches with Cost of Dispersal

Abstract

To understand how cost of dispersal affects population dynamics, we study a three-trophic level food chain model, proposed by DeAngelis et al. [D. DeAngelis, G. S. K. Wolkowicz, Y. Lou, Y. X. Jiang, M. Novak, R. Svanback, M. Araujo, Y. S. Jo, and E. A. Cleary, Am. Nat., 178 (2011), pp. 15–29], in two patches. The system consists of one resource species, two consumers, and a top predator. The top predator feeds on two consumers and both consumers feed on the resource. Only consumers move between the patches, possibly with a fraction of loss in population during the movement. The two competing consumers are identical in every aspect except their dispersal rates between two patches. If two consumers have the same dispersal rate from patch 1 to patch 2, we completely determine the global dynamics of the model and show that there exists an “optimal” dispersal rate from patch 2 to patch 1 for the consumer such that, in terms of the theory of adaptive dynamics, it is a globally evolutionarily stable strategy and also a convergent stable strategy. If there is a minimum dispersal speed from patch 1 to patch 2, we are able to completely determine the evolutionarily stable strategy for dispersal between two patches. Our results offer insights into the evolution of dispersal in multitrophic level food chains, e.g., how the evolution of fast or slow dispersal for the consumer species depends upon the variation of the predation risk in the habitat. Our result suggests that even if most individuals die during the movement, a positive dispersal rate can still evolve.