Numerical simulations for the predator-prey model on surfaces with lumped mass method

Abstract

The predator-prey model is powerful mathematical tool to describe the dynamics of biological systems and promote research on biological populations. In this paper, we present a lumped mass finite element method for solving the predator-prey models on surfaces. The main purpose of the proposed method is to overcome the difficulty of the positivity preservation of the solutions. Based on positivity preservation results, we investigate the stabilities of semi-discrete and fully discrete approximations. Besides, numerical simulations are considered to illustrate the feasibility of the numerical method by convergence tests. Two classical phenomena of the predator-prey model are simulated on three different implicit surfaces.