EXISTENCE AND STABILITY OF A PERIODIC SOLUTION FOR AN EQUATION GOVERNING DYNAMICS OF A RENEWABLE RESOURCE SUBJECTED TO ADDITIVE ALLEE EFFECTS

Abstract

Allee effect refers to reduction in individual fitness at low population densities. Among several Allee effects that are known to occur in species dynamics, we consider additive Allee effect which appear to exist in several vital systems in the real world. In this article we study the dynamics of a renewable resource that is subjected to additive Allee effects in a periodically varying environment. We derive conditions under which the considered model admits a positive periodic solution. Uniqueness and stability properties of this periodic solution are also investigated. Fixed point theory and comparison principle have been employed to establish the needed results. It is observed that the trivial solution of the considered model is unstable where as the positive periodic solution is asymptotically stable and attracts all other solutions with positive initial states. This study highlights the conditions under which the stock in a renewable resource that is influenced by additive Allee effect follows a stable periodic pattern eventually where the periodicity coincides with that of the environment.