On the almost periodicity of nonautonomous evolution equations and application to Lotka-Volterra systems

TL;DR

This paper proves the existence and uniqueness of almost periodic solutions for nonautonomous evolution equations with Stepanov almost periodic coefficients, and applies these results to a Lotka-Volterra predator-prey model with time-dependent parameters.

Contribution

It establishes new conditions for almost periodic solutions in nonautonomous evolution equations and introduces a novel composition result for Stepanov almost periodic functions.

Findings

Existence and uniqueness of almost periodic solutions proven.

New composition theorem for Stepanov almost periodic functions.

Application to a generalized Lotka-Volterra predator-prey system.

Abstract

Consider the nonautonomous semilinear evolution equation of type: $(\star) \; u'(t)=A(t)u(t)+f(t,u(t)), \; t \in \mathbb{R},$ where $ A(t), \ t\in \mathbb{R} $ is a family of closed linear operators in a Banach space $X$, the nonlinear term $f$, acting on some real interpolation spaces, is assumed to be almost periodic just in a weak sense (i.e. in Stepanov sense) with respect to $t$ and Lipschitzian in bounded sets with respect to the second variable. We prove the existence and uniqueness of almost periodic solutions in the strong sense (Bohr sense) for equation $ (\star) $ using the exponential dichotomy approach. Then, we establish a new composition result of Stepanov almost periodic functions by assuming just the continuity of $f$ in the second variable. Moreover, we provide an application to a nonautonomous system of reaction-diffusion equations describing a Lotka-Volterra predator-prey model with diffusion and time-dependent parameters in a generalized almost periodic environment.