Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics

We investigate a nonlinear integral equation derived through moment approximation from the individual-based representation of spatial logistic dynamics. The equation describes how the densities of pairs of individuals represented by points in continuous space are expected to equilibrate under spatially explicit birth–death processes characterized by constant fecundity with local natal dispersal and variable mortality determined by local competition. The equation is derived from a moment hierarchy truncated by a moment closure expressing the densities of triplets as a function of the densities of pairs. Focusing on results for individuals inhabiting two-dimensional habitats, we explore the solvability of the equation by introducing a dedicated space of functions that are integrable up to a constant. Using this function space, we establish sufficient conditions for the existence of solutions of the equation within a zero-centered ball. For illustration and further insights, we complement our analytical findings with numerical results.