Consequences of Allee effect on the multiple limit cycles in a predator-prey model

In the present paper, we examine the effects of the Allee threshold on prey-predator interaction frameworks developed by utilising the Holling type-III response. Initially, we find the system's equilibrium and its local stability. According to our analysis, the system follows a transcritical bifurcation about the conversion rate and the saddle node, yet also Hopf bifurcation concerning the Allee factor. In the context of variations in the carrying capacity, Allee threshold and rate of consumption, numerically we investigate the system's stability scenario and bifurcation layout in the presence of numerous coexistence equilibria. The system exhibits a bistability as well as monostable nature as the Allee parameter progressively increases and three limit cycles arise for lesser and higher intensity of the Allee threshold in which an unstable limit cycle separates two stable limit cycles, as their basin. Depending on the value of the Allee threshold, the population either achieves a stable steady state or oscillates. Numerically, we notice that the system experiences fascinating dynamics followed by transcritical, saddle node and Hopf bifurcation and finally depicts the bistability scenario for various strengths of consumption rates.