Influence of Allee effect in prey and hunting cooperation in predator with Holling type-III functional response

Abstract

Cooperative hunting is a widespread phenomenon in the predator population which promotes the predation and the coexistence of the prey-predator system. On the other hand, the Allee effect among prey may drive the system to instability. In this work, we consider a prey-predator model with Type-III functional response involving the hunting cooperation in predator and Allee effect in the growth rate of the prey population. Here our aim mainly is to demonstrate the impact of both the Allee effect and hunting cooperation on the system dynamics. Mathematically our analysis primarily focuses on the stability of coexisting equilibrium points and all possible bifurcations that the system may exhibit. We have observed transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and SN-TC bifurcation point respectively in the course of studying the global dynamics.

Appendices

Appendix A1.

corresponding to the bifurcation diagram given in Fig. 2

Fig. 6

Phase portrait diagrams corresponding to Fig.(2) in different region from ${R_1\rightarrow R_7}$ respectively with the fixed parameter values are $\beta =0.2, \eta =1, h=0.5$ for the various choice of $\sigma$ and $\alpha$ is a$\sigma =1.1, \alpha =5.0$,b $\sigma =0.716, \alpha =6.23$, c $\sigma =0.57, \alpha =2.38$, d $\sigma =0.75, \alpha =9.44$, e $\sigma =0.72, \alpha =16.91$, f $\sigma =0.56, \alpha =14.18$, g $\sigma =0.02, \alpha =8.48$

Appendix A2.

corresponding to the bifurcation diagram given in Fig. 3

Fig. 7

Phase portrait diagrams corresponding to Fig.(3) in different region from ${R_1\rightarrow R_6}$ respectively with the fixed parameter values are $\beta =-0.05, \eta =1, h=0.5$ for the various choice of $\sigma$ and $\alpha$ are a $\sigma =0.80, \alpha =2.16$, b $\sigma =0.719, \alpha =5.23$, c $\sigma =0.544, \alpha =2.344$, d $\sigma =0.76, \alpha =9.47$, e $\sigma =0.74, \alpha =12.57$, f $\sigma =0.587, \alpha =11.96$