Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Evolution of mobility in predator-prey systems

Pages: 3397 - 3432, Volume 19, Issue 10, December 2014      doi:10.3934/dcdsb.2014.19.3397

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Fei Xu - Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada (email)
Ross Cressman - Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada (email)
Vlastimil Křivan - Biology Centre ASCR, Institute of Entomology and Department of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia, Branišovská 31, 370 05 České Budějovice, Czech Republic (email)

Abstract: We investigate the dynamics of a predator-prey system with the assumption that both prey and predators use game theory-based strategies to maximize their per capita population growth rates. The predators adjust their strategies in order to catch more prey per unit time, while the prey, on the other hand, adjust their reactions to minimize the chances of being caught. We assume each individual is either mobile or sessile and investigate the evolution of mobility for each species in the predator-prey system. When the underlying population dynamics is of the Lotka-Volterra type, we show that strategies evolve to the equilibrium predicted by evolutionary game theory and that population sizes approach their corresponding stable equilibrium (i.e. strategy and population effects can be analyzed separately). This is no longer the case when population dynamics is based on the Holling II functional response, although the strategic analysis still provides a valuable intuition into the long term outcome. Numerical simulation results indicate that, for some parameter values, the system has chaotic behavior. Our investigation reveals the relationship between the game theory-based reactions of prey and predators, and their population changes.

Keywords:  Predator-prey system, game theory, dominant strategy, stability and chaos, Lotka-Volterra model, Rosenzweig-MacArthur model, population and strategy dynamics.
Mathematics Subject Classification:  Primary: 92D25, 92D15; Secondary: 91A22.

Received: June 2013;      Revised: October 2013;      Available Online: October 2014.