Almost automorphic solutions for Lotka-Volterra systems with diffusion and time-dependent parameters

Abstract

In this work we study the response for a class of Lotka-Volterra prey- predator systems with diffusion and time-dependent parameters to a large class of oscillatory type functions, namely the pseudo almost automorphic type oscillations. To this end, using the exponential dichotomy approach and a fixed point argument, we propose to analyze a class of nonau- tonomous semilinear abstract evolution equation of the form (⋆)z′(h) = A(h)z(h) + g(h, z(h)), h ∈ R, where A(h), h ∈ R is a family of closed linear operators acting in a Banach space T, the nonlinear term g is μ- pseudo-almost automorphic in a weak sense (Stepanov sense) with re- spect to h and Lipschitzian in T with respect to the second variable. Therefore, according to the results obtained for equation (⋆) we establish the existence and uniqueness of μ-pseudo-almost automorphic solutions in the strong sense (Bohr sense) to a nonautonomous system of reaction- diffusion equations describing a Lotka-Volterra prey-predator model with diffusion and time-dependent parameters in a generalized almost auto- morphic environment.