Features of population dynamics synchronization in the system of three populations related by migration

The article is devoted to the study of oscillations (cycles) synchronization in a system of three populations coupled by migration in a circle. We consider a discrete time model of population dynamics, which is a system of three identical dissipatively connected logistic maps. One-dimensional bifurcation diagrams (trees) were constructed, supplemented with the capture index of the population numbers cycles phases at adjacent sites. We conducted numerical experiments that show phase multistability - the cycles coexistence with different phases. Using qualitative methods for studying dynamic systems, a complete phase portrait of the model is constructed, showing that in the phase space there are several periodic points corresponding to elements of synchronous (in-phase) and asynchronous (out-of-phase) cycles. The 2- and 3-cycle stability conditions are investigated. It is shown that these two cycles are represented by three options: 1) a fully synchronous regime, when the abundances in the three populations coincide at any point in time; 2) a partially synchronous regime, when abundances coincide only for two populations, 3) a non-synchronous (out-of-phase) regime, when all three numbers have different values. For a 2-cycle, the third regime is unstable, possible as part of a long transition process. We found that for the 3-cycle, besides to the synchronous and partially synchronous regime, it is possible a stable out-of-phase behavior of three populations. It is shown that stable and unstable periodic points lie on certain surfaces (invariant manifolds), which separate the areas of attraction for regimes with different degrees of phase synchronization.