Nonlinear Lotka-Volterra Competition Models

Abstract

The classical Lotka-Volterra equations that model the interactions between two species competing for a limited resource have many potential modifications to improve biological accuracy; this paper explores modifications to the exponent of the competition term.
After an introduction to the behavior of the classical Lotka-Volterra model is given, a nonlinear modification to this model by Taylor and Crizer is discussed. In section 2, an extension of this modification is proposed, in which the population variable of the competition term is raised first to the power of positive real numbers and, next, small integers. A proof is offered that at most 3 coexistent equilibrium points exist for any positive exponent values, and additional proofs further limit the number of equilibria for certain exponent and parameter values. In section 3, we prove that, in such models, the stability of the equilibria alternates between stable and unstable when considered in a northwest to southeast configuration. Combining these results allows us to describe the equilibrium behavior of a broad class of competition models.