Global dynamics of a tritrophic model for twopatches with cost of dispersal

To understand how cost of dispersal affects population dynamics, we study a threetrophiclevel 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, anda 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 themovement. The two competing consumers are identical in every aspect except their dispersal ratesbetween two patches. If two consumers have the same dispersal rate from patch 1 to patch 2, wecompletely determine the global dynamics of the model and show that there exists an optimaldispersal rate from patch 2 to patch 1 for the consumer such that, in terms of the theory of adaptivedynamics, it is a globally evolutionarily stable strategy and also a convergent stable strategy. Ifthere is a minimum dispersal speed from patch 1 to patch 2, we are able to completely determinethe evolutionarily stable strategy for dispersal between two patches. Our results offer insights intothe evolution of dispersal in multitrophic level food chains, e. g. , how the evolution of fast or slowdispersal 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 ratecan still evolve.