Numerical Investigation and Factor Analysis of the Spatial-Temporal Multi-Species Competition Problem

Abstract

This work considers the spatial-temporal multi­species competition model. A mathematical model is described by a coupled system of nonlinear diffusion reaction equations. We use a finite volume approximation with semi-implicit time approximation for the numerical solution of the model with corresponding boundary and initial conditions. To understand the effect of the diffusion to solution in one and two-dimensional formulations, we present numerical results for several cases of the parameters related to the survival scenarios. We control all non-diffusion parameters, including reproductive growth rate, competition rate, and initial condition of population density of competing species, and compare the dynamic and equilibrium under regular diffusion rate and small diffusion rate; we found that competing species with small diffusion rate can reach a higher equilibrium over the whole geographic domain, but requires more time steps. The random initial conditions' effect on the time to reach equilibrium is investigated. We control other parameters and examine the impact of the initial condition of the species population; we found that regardless of the values of initial conditions in the system, competing species populations will arrive at an equilibrium point. The influence of diffusion on the survival scenarios is presented. We control other parameters and examine the effect of diffusion of species; we found that when the ratio of diffusion rates passes some thresholds, the survival status will change. In real-world problems, values of the parameters are usually unknown yet vary in some range. To evaluate the impact of parameters on the system stability, we simulate a spatial­temporal model with random parameters and perform factor analysis for two and three­species competition models. From the perspective of the numerical experiment, we release control for all parameters and perform factor analysis on simulation results. We found that the initial population condition has a minimum effect on the final population, which aligns with the outcome of our controlled numerical experiment on the initial condition. Diffusion is the dominant factor when diffusion rates are on the same scale as other parameters. This dominant factor aligns with our controlled numerical experiment on diffusion rate, where the change in diffusion rate leads to different survival statuses of species. However, when diffusion rates are 1/10 on the scale of other parameters, reproductive growth rates and competition rates become the dominant factors.