Abstract
ABSTRACTWe investigate a discrete-time predator–prey system with cooperative hunting in the predators proposed by Chow et al. by determining local stability of the interior steady states analytically in certain parameter regimes. The system can have either zero, one or two interior steady states. We provide criteria for the stability of interior steady states when the system has either one or two interior steady states. Numerical examples are presented to confirm our analytical findings. It is concluded that cooperative hunting of the predators can promote predator persistence but may also drive the predator to a sudden extinction.
Highlights
Cooperation is frequently observed and widespread among individuals of social animals in many biological systems
This investigation is motivated by the recent work of Alves and Hilker [2] who use continuous-time models of predator–prey interactions with cooperative hunting in predators to study the impacts of cooperation upon population interactions
The discrete model is based on the classical Nichelson–Bailey system with density dependent host growth rate and cooperative hunting of the predator is modelled via the attack rate of the predator
Summary
Cooperation is frequently observed and widespread among individuals of social animals in many biological systems. Earlier research incorporating cooperative hunting includes Berec [3] who uses ordinary differential equations to model predator–prey interactions with a Holling type II functional response Due to this functional response, Berec studies the effects of cooperative hunting relative to population oscillations. Chow et al [4] propose and investigate a discrete-time predator–prey model with cooperative hunting among predators to study the effects of cooperation upon predator–prey interactions. Their model derivation is built on the well-known Nicholson–Bailey system with density-dependent prey growth rate and it is proven that both populations coexist indefinitely if the maximal reproductive number of the predator is larger than one.
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