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

Mikhail Nikolaev (), Alexey Nikitin () and Ulf Dieckmann
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Mikhail Nikolaev: Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
Alexey Nikitin: Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
Ulf Dieckmann: Complexity Science and Evolution Unit, Okinawa Institute of Science and Technology Graduate University, Onna 904-0412, Japan

Mathematics, 2024, vol. 12, issue 24, 1-20

Abstract: 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.

Keywords: nonlinear integral equations; spatial logistic dynamics; individual-based models; fixed-point (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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