Defining a stability boundary for three species competition models

Abstract

A periodic steady state is a familiar phenomenon in many areas of theoretical biology and provides a satisfying explanation for those animal communities in which populations are observed to oscillate in a reproducible periodic manner. In this paper we explore models of three competing species described by symmetric and asymmetric May–Leonard models, and specifically investigate criteria for the existence of periodic steady states for an adapted May–Leonard model: x ˙ = r ( 1 − x − α y − β z ) x y ˙ = ( 1 − β x − y − α z ) y z ˙ = ( 1 − α x − β y − z ) z . Using the Routh–Hurwitz conditions, six inequalities that ensure the stability of the system are identified. These inequalities are solved simultaneously, using numerical methods in order to generate three-dimensional phase portraits to illustrate the steady states. Then the “stability boundary” is defined as the almost linear boundary between stability and instability. All the mathematics discussed is suitable for advanced undergraduate mathematics or applied mathematics students, offering them the opportunity to incorporate a computer algebra system such as Mathematica, DERIVE or Matlab in their investigations. The adapted May–Leonard model provides a practical application of steady states, stability and possible limit cycles of a nonlinear system.