PSEUDO ALMOST PERIODIC MILD SOLUTIONS OF QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH APPLICATION TO MATHEMATICAL BIOLOGY

摘要

Using the theory of semigroup of bounded linear operators in a complex Banach space, we establish the existence and uniqueness of a pseudo almost periodic mild solution of a quasilinear functional differential equation. The main result is applied to a partial integro-differential equation modeling nonlocal reaction-diffusion problem in biology. Then we develop a numerical method based on the nonstandard finite difference discretization. We use a non-local approximation to approximate a nonlinear term to preserve the positivity of the solution. We then provide a detailed stability analysis and establish the convergence of this numerical method. We prove that this method is unconditionally stable. Finally, we present several numerical results which show the existence of positive asymptotically stable solutions of the model.