High-order semi-implicit linear multistep LG scheme for a three species competition-diffusion system in two-dimensional spatial domain arising in ecology
In this paper, we consider a robust numerical approximation of a three-species fully-coupled competition-diffusion system of Lotka-Volterra-type in a two-dimensional spatial domain. The model is characterized by the presence of a very small diffusion parameter. If the diffusivity coefficient is sufficiently small, a spatially segregated pattern with very thin internal layers occur. For such problems, it is a challenging task to develop an efficient numerical method that is also capable of capturing the various transient regimes and fine spatial structures of the solutions. In this paper, we develop a high-order semi-implicit multistep scheme based on the Lagrange temporal interpolation coupled with a conforming finite element method for the nonlinear competition-diffusion problem in two spatial dimensions. A major advantage of the proposed method is that it is essentially linear in terms of the current time-step values (no need for nonlinear iterative treatment), while its order of convergence is higher. Moreover, the couplings of current step values of the unknowns are one sided, which is a very desirable property in terms of algorithmic efficiency since each unknown is solved sequentially. This avoids solving for all unknowns simultaneously. We also discuss the stability and convergence of the proposed schemes. Furthermore, various simulations are carried out to demonstrate the performance of the proposed method in simulating different type of interaction patterns such as the onset of spiral-like coexistence pattern, complex spatio-temporal patterns and competitive exclusion of the species.