Regularization of a backward problem for a Lotka–Volterra competition system

Abstract

Lotka–Volterra systems are used to describe the dynamics of biological systems. We study the backward problem for the Lotka–Volterra system to determine the population density of species at preceding times. The problem is ill-posed in the sense that if the solution exists it does not depend continuously on the given data. We propose two stable regularization methods to regularize the system. Furthermore, under some a priori assumptions on the sought solution, we show that the corresponding regularized solutions converge to the exact solutions in $L^2$-norm. Numerical simulations are presented to demonstrate the theoretical results.