Imitation strategy in the Rosenzweig-MacArthur model

Simulations of the dynamics of two consumers C_{1}, C_{2} and two resources R_{1}, R_{2} are presented within the Rosenzweig-MacArthur model (RMA). Consumers i=1,2 share their time/effort devoted to particular resources; for each i, the consumption rates of the two resources are β_{i} and 1-β_{i}. Here, the model dynamics are enriched by the evolution of these rates over time. Namely, at discrete time instants, the consumer with a smaller density C_{i} updates her rates to imitate the behavior of the more successful opponent, while the rates of the latter remain unchanged. These rules can be written as ''lose-shift'' and ''win-stay,'' respectively, where ''shift'' is directed to imitate the temporal winner. The phase diagram in the plane (β_{1},β_{2}) shows periodic orbits and chaos. The latter behavior is more prominent when resources are acquired unevenly. Compared with three alternative strategies, the ''win-stay, lose-shift'' strategy is an equilibrium in both the periodic and chaotic regimes.