On infinite dimensional Volterra type operators

Since Lotka and Volterra’s seminal and pioneering works (see [10]) many decades ago, modeling of interacting, competing species have received considerable attention in the fields of biology, ecology, mathematics (for example, see [3, 9]). In their remarkably simple deterministic model, Lotka and Volterra considered two coupled nonlinear differential equations that mimic the temporal evolution of a two-species system of competing predator and prey populations. They demonstrated that coexistence of both species was not only possible but inevitable in their model. Moreover, similar to observations in real populations, both predator and prey densities in this deterministic system display regular oscillations in time, with both the amplitude and the period determined by the prescribed initial conditions. Note that in [1, 2, 4, 5] finite dimensional Volterra and more general quadratic operators were studied. When a system is large enough, it is interesting to investigate quadratic Volterra operators define on an infinite dimensional simplex. First studies in this direction were considered in [6]. Iterations of such operators define more complicated nonlinear operators. To better understanding the dynamics of such operators, it is important to study such nonlinear operators. The aim of this paper is to study more general class of nonlinear operators which contains a particular case of that quadratic Volterra operators. It is provided a sufficient condition for Volterra type operators to be bijective