Optimal behavior dynamics of the two-species community with intraspecific competition and migration

Some problems of the theory of optimal foraging are considered, namely, the problem of predator's choice of the most suitable patch and finding conditions for leaving it. The dynamics of the interaction between the predator and the prey is determined by the Lotka-Volterra system, which takes into account the intraspecific competition of the prey and the possibility of migration of the predator and the prey. Some fractions of populations participate, in the processes of interaction and migration. The problem of finding optimal shares from the point of view of Nash equilibrium is solved. In this case, a partition of the phase space of the system into domains with different behavior of the populations was obtained. We study the optimal trajectories of the corresponding dynamical system with a variable structure, their behavior on the boundaries of the phase space partition. The equilibrium positions are found and their global stability is proved under certain restrictions on the system parameters. In one of the cases of the relationship between the parameters, the study of the qualitative behavior of the optimal trajectories gives rise to the problem of the existence of limit cycles. In this case, an estimate of the corresponding domain of attraction of equilibrium is given.