Numerical Analysis of Crowding Effects in Competing Species

Abstract

. In recent decades, scientists have observed that the mortality rate of some competing species increases superlinearly as populations grow to unsustainable levels. This is modeled by terms representing crowding effects in a system of nonlinear differential equations that describes population growth of two species competing for resources under the effects of crowding. After applying nondimensionalization to reduce parameters in the system, the stability of the steady state solutions of the system is examined. A semi-implicit numerical scheme is proposed which guarantees the positivity of the solutions. The long term behavior of the numerical solutions is studied. The error estimate between the numerical solution and the true solution is given.