Two-Species competition model with diffustion and harvesting: A numerical study

Predicting well observed states in ecology such as co-existence, competitive exclusion of one competitor, and bi-stability is vital during multi-species competitions. Models that examine these aspects of living systems have extensive applications in the overlapping areas of applied mathematics, population ecology, invasion science, evolutionary biology, and economics. The present study investigated a twin species Lotka-Volterra competition system accommodating diffusion and harvesting environments – a scenario widely anticipated in mathematical ecology. Our mathematical setup converts the diffusion and harvesting incorporated physical model into a system of second order nonlinear partial differential equations (PDES) describing the competition of the two species in closed domains. A finite difference numerical scheme is developed to solve the nonlinear boundary value problem (BVP) with Neumann boundary conditions. We applied our model results to the case of Brown and Pink shrimp in southeastern Gulf of Campeche that compete for resources in Marine Protected Areas (MPAs). For this specific example, we computed theoretical results for the sustainable biomass yield due to the competition and the mobility in the presence of multiple fishing zones. Our numerical solutions reveal that the speed or mobility of species is critical for the design of MPAs to attain a maximum sustainable biomass yield. Additionally, the results indicate that harvesting rate is adjustable for larger number of MPAs along the coastal line for efficient fishing. Further, it is observed that a sustainable biomass can be achieved for low mobile species such as the brown and pink shrimps by having smaller MPAs and both the species can co-exist in Gulf of Campeche.