Periodic dynamics for semilinear equation with biological applications

Abstract

This work investigates the existence of periodic solutions in a class of reaction diffusion systems arising in population dynamics. By leveraging the properties of semiFredholm operators and the contraction mapping principle, we establish sufficient conditions for the existence and uniqueness of periodic solutions for the following equation: ^d⁄_dtv(t) - Δv(t) = Bv(t) + N(t,v(t)) Specifically, we demonstrate that a small perturbation of the linear part and the Lipschitz constant of the nonlinear part guarantees the periodicity of solutions. As an application of our theoretical framework, we analyze a diffusive logistic model and establish the existence of biologically meaningful periodic oscillations.