Invasive behaviour under competition via a free boundary model: a numerical approach.

Abstract

What will happen when two invasive species are competing and invading the environment at the same time? In this paper, we try to find all the possible scenarios in such a situation based on the diffusive Lotka-Volterra competition system with free boundaries. In a recent work, Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) considered a weak-strong competition case of this model (with spherical symmetry) and theoretically proved the existence of a "chase-and-run coexistence" phenomenon, for certain parameter ranges when the initial functions are chosen properly. Here we use a numerical approach to extend the theoretical research of Du and Wu (Calc Var Partial Differ Equ, 57(2):52, 2018) in several directions. Firstly, we examine how the longtime dynamics of the model changes as the initial functions are varied, and the simulation results suggest that there are four possible longtime profiles of the dynamics, with the chase-and-run coexistence the only possible profile when both species invade successfully. Secondly, we show through numerical experiments that the basic features of the model appear to be retained when the environment is perturbed by periodic variation in time. Thirdly, our numerical analysis suggests that in two space dimensions the population range and the spatial population distribution of the successful invader tend to become more and more circular as time increases no matter what geometrical shape the initial population range possesses. Our numerical simulations cover the one space dimension case, and two space dimension case with or without spherical symmetry. The numerical methods here are based on that of Liu et al. (Mathematics, 6(5):72, 2018, Int J Comput Math, 97(5): 959-979, 2020). In the two space dimension case without radial symmetry, the level set method is used, while the front tracking method is used for the remaining cases. We hope the numerical observations in this paper can provide further insights to the biological invasion problem, and also to future theoretical investigations. More importantly, we hope the numerical analysis may reach more biologically oriented experts and inspire applications of some refined versions of the model tailored to specific real world biological invasion problems.