A finite element discretization with semi-implicit nonlinear multistep scheme for a two-dimensional competition-diffusion system of three competing species with different mobility rates

In ecological studies of pattern formation, models of the competitive-diffusion type are generally singularly perturbed, and the numerical approximation of such models is challenging. In this paper, we present finite element discretization combined with a second-order semi-implicit nonlinear multistep scheme for a two-dimensional three-species competition-diffusion system with distinct mobility rates. The method employs a $C^0$-conforming Galerkin finite element approximation in space and a Crank-Nicolson/Adams-Bashforth-type time integration that treats the diffusion terms implicitly while linearizing the nonlinear reaction terms in a stage-by-stage manner. The resulting scheme is linear at each time step and avoids iterative nonlinear solvers. Rigorous stability analysis shows that the discrete method inherits the asymptotic stability properties of the continuous model without restrictions on the time step size. Numerical simulations for various mobility regimes demonstrate the ability of the proposed method to capture complex spatio-temporal patterns, including droplet-like, banded, spiral, and glider-type structures.