Local existence of solutions to a nonlinear autonomous PDE model for population dynamics with nonlocal transport and competition

TL;DR

This paper proves the local existence of classical solutions for a nonlinear, nonlocal PDE model describing population dynamics with long-range transport and competition, using smoothing and energy methods.

Contribution

It establishes local existence results for a novel nonlinear PDE model incorporating nonlocal transport in population dynamics.

Findings

Existence of classical solutions under regular initial conditions

Use of smoothing and energy estimates to prove convergence

Model captures long-range travel and competition effects

Abstract

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with a nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence of approximators converging to a classical solution.