Competing species model with behavioral adaptation
As said in the abstract, the author ''studies the properties of a modified Lotka-Volterra model for two competing species, in which the coefficients of the interaction terms are time-dependent averages of the level of interaction over the entire past.'' The main analytical effect of this seems to be an increased nonlinearity which is reflected by the existence of up to 6 stationary states in a certain range of the parameters. A noticeable fact is that none of the 3 permanent equilibria corresponding to partial or total extinction is stable. This is in contrast to the usual model. The other correspond to coexistence of both species, with a dominant one. The author concentrates on the case of three interior equilibria, of which one (intermediate) is always unstable and the other two (extreme) are separately stable or unstable depending on interaction parameters. Moreover, the author was able to find numerically a nonempty region of these parametes where no equilibrium is stable and numerical simulation suggests the existence of solutions passing successively near each of the extreme equilibria, and possibly of a limit cycle. We may observe finally that functionally the equation belongs to the class of infinite delay differential systems. But, treating as an independent variable each term bearing the dependence on the past, it can be reduced to a system of 4 ordinary differential equations, as the author does. This remark also aims at indicating that the study performed in this paper is far from being trivial.
scientific article; zbMATH DE number 3989164