Numerical Analysis for Long Time Behaviour of Two Invasive Species with Free Boundaries

Abstract

<p>In this thesis, we examine the possible dynamical behaviour, of two invasive species when they are competing and invading the environment at the same time, based on a diffusive Lotka-Volterra competition system with free boundaries. In a recent work, Du and Wu [35] considered a weak-strong competition case of this model (with spherical symmetry) and theoretically proved the existence of a “chaseand-run coexistence” phenomenon, for certain parameter ranges when the initial functions are chosen properly. In this thesis, we use numerical approaches to extend the theoretical research of [35] in several directions.</p> <p>Firstly, we examine how the long-time dynamics of the model changes as the initial functions are varied, and the simulation results suggest that there are four possible long-time profiles of the dynamics, with the chase-and-run coexistence the only possible profile when both species can coexist.</p> <p>Secondly, we investigate the long term dynamics and the geometrical shape of the spreading fronts of the species in two space dimensions. Our numerical analysis suggests that 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.</p> <p>Thirdly, we investigate the effect of environmental heterogeneity on species spreading. We focus on the changes of long-time dynamics and spreading speeds of the species as the parameters describing the heterogeneity of the environment are varied. This is a very complicated issue and in order to gain a good understanding for the two competing species situation, it is also necessary to consider the one species case. The single species model is obtained by assuming one of the two species is identically 0. Such a model in time-periodic environment and in space-periodic environment has been theoretically treated in [25, 26], but more detailed properties are obtained in this thesis through numerical analysis. The biological interpretations of our results on the spreading speed, in the space-periodic case, mostly agree with those of Kinezaki, Kawasaki and Shigesada [55], but some differences exist. For the two-species model, our analysis here shows that the four types of long-time dynamical behaviour observed in homogeneous case are robust: they are retained under time-periodic as well as space-periodic perturbations of the environment. By varying the parameters in the time-periodic and space-periodic terms of the model, we have numerically examined their influence on the long-time dynamics and on the spreading speeds of the two species. Generally speaking, our results suggest that heterogeneity of the environment enhances the invasion of the two species, although there are subtle differences of the influences felt by the two</p> <p>The numerical methods here are based on that of Liu et al. [63, 62] and Chen et. al. [16]. In the two space dimensions 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 thesis can provide further insights to the biological and ecological 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.</p>