Solvability of a System of Nonlinear Integral Equationswith Piecewise Constant Kernels

The model of population dynamics of a single-species biological community proposed by W. Dieckmann and R. Law is considered. Changes in the population are described by a system of integro-differential equations, which characterizes the dynamics of spatial moments and in the state of equilibrium is reduced to a nonlinear integral equation. The solvability of this equation is studied, according to which the solution of the original system is written out, for which a nonlinear integral operator is constructed and the problem of finding its fixed point is solved. Sufficient conditions for the existence of a nontrivial solution are established. An analytical example of the values of biological parameters that satisfy these conditions is given.