Multistability for system of three competing species

The study of the Volterra model describing the competition of three types is carried out. The corresponding system of first-order differential equations with a quadratic right-hand side, after a change of variables, reduces to a system with eight parameters. Two of them characterize the growth rates of populations; for the first species, this parameter is taken equal to one. The remaining six coefficients define the species interaction matrix. Previously, in the analytical study of the so-called symmetric model [May, Leonard, 1975] and the asymmetric model [Chi, Wu, Hsu, 1998] with growth factors equal to unity, relations were established for the interaction coefficients, under which the system has a one-parameter family of limit cycles. In this paper, we carried out a numerical-analytical study of the complete system based on a cosymmetric approach, which made it possible to determine the ratios for the parameters that correspond to families of equilibria. Various variants of one-parameter families are obtained and it is shown that they can consist of both stable and unstable equilibria. In the case of an interaction matrix with unit coefficients, a multicosymmetry of the system and a two-parameter family of equilibria are found that exist for any growth coefficients. For various interaction coefficients, the values of growth parameters are found at which periodic regimes are realized. Their belonging to the family of limit cycles is confirmed by the calculation of multipliers. In a wide range of values that violate the relationships under which the existence of cycles is ensured, a slow oscillatory establishment, typical of the destruction of cosymmetry, is obtained. Examples are given where a fixed value of one growth parameter corresponds to two values of another parameter, so that there are different families of periodic regimes. Thus, the variability of scenarios for the development of a three-species system has been established.