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Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model

1 Department of Mathematics, Visva-Bharati, Santiniketan 731235, India
2 Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Rabigh-25732, Saudi Arabia
3 Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India

Special Issues: Mathematical Modeling to Solve the Problems in Life Sciences

The predation strategy for predators and the avoidance strategy of prey are important topics in ecology and evolutionary biology. Both prey and predators adjust their behaviours in order to gain the maximal benefits and to increase their biomass for each. In the present paper, we consider a modified Leslie-Gower predator-prey model where predators cooperate during hunting and due to fear of predation risk, prey populations show anti-predator behaviour. We investigate step by step the impact of hunting cooperation and fear effect on the dynamics of the system. We observe that in the absence of fear effect, hunting cooperation can induce both supercritical and subcritical Hopf- bifurcations. It is also observed that fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. Moreover, the system shows two different types of bi-stabilities behaviour: one is between coexisting equilibrium and limit cycle oscillation, and another is between prey-free equilibrium and coexisting equilibrium. We also observe generalized Hopf-bifurcation and Bogdanov-Takens bifurcation in two parameter bifurcation analysis. We perform extensive numerical simulations for supporting evidence of our analytical findings.
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Keywords Leslie-Gower model; fear effect; cooperation; multiple limit cycles; bifurcation; bi-stability

Citation: Saheb Pal, Nikhil Pal, Sudip Samanta, Joydev Chattopadhyay. Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model. Mathematical Biosciences and Engineering, 2019, 16(5): 5146-5179. doi: 10.3934/mbe.2019258


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