Stability and Bifurcation of a Gordon–Schaefer Model with Additive Allee Effect

Abstract

The rarity of species increases its market price, consequently leading to the overexploitation of the species and even the extinction of the species. We study how the harvest intensity and the additive Allee effect impact on the Gordon–Schaefer model. In addition, by Sotomayor’s theorem and Poincaré–Andronov theorem, we prove the existence of Hopf bifurcation, saddle-node bifurcation and transcritical bifurcation, respectively. Finally, we illustrate our results by numerical simulations. We find that both the cost per unit of harvest and the additive Allee effect have a significant impact on human exploitation of the population. As the additive Allee effect reduces to the weak Allee effect, the lower harvest cost encourages humans to increase the exploitation of species. This threshold is a switch that controls the strong Allee effect. If it exceeds its threshold, then the motivation of humans to exploit the species increases.