Cooperative hunting in a predator–prey system with Allee effects in the prey

Abstract

We propose a continuous‐time predator–prey model with Allee effects in the prey and cooperative hunting in predators to explore their effects on population interactions. If the predator’s reproductive number exceeds one when prey is at its Allee threshold, then although both populations eventually go extinct, the time that it takes to become extinct may be different between cooperation and no cooperation. If the predator’s maximum reproductive number is larger than one, then coexistence occurs for the model of no Allee effect, whereas dynamics of the other model also depend on Allee threshold. There always exists a region of initial population distributions, for which both populations become extinct. When the predator’s maximum reproductive number is smaller than one, then predators go extinct if predators do not engage in cooperative hunting or if the degree of cooperation is small. With intense cooperation, predators may persist. However, the region of initial populations for coexistence is smaller for the model with Allee effects than the model of no Allee effects. Recommendations for Resource Managers If predators are specialists and engage in cooperative hunting on a prey population that is subject to Allee effects, then estimate the degree of predator cooperation along with the predator’s reproductive numbers when the prey is at its carrying capacity (termed the predator’s maximum reproductive number) and at Allee threshold. If both reproductive numbers are larger than one or if the predator’s maximum reproductive number is larger than one, while the other reproductive number is smaller than one, then one should cull predators to avoid prey extinction. The number of predators to be removed will depend on the sizes of the two populations. If the maximum reproductive number of predators is smaller than one, then the recommended management plan will depend on the degree of hunting cooperation. If the degree is small, then culling predators can avoid prey extinction. If cooperation is intense, then culling a large number of predators will likely cause predator extinction while removing a small number of predators will yield coexistence.

A predator-prey model with cooperative hunting in the predator and group defense in the prey

“…Alves and Hilker [4] added a cooperation term cv to the predation rate λ to represent the increasing success of predation, which makes the functional response (λ + cv)u, where c > 0 describes the strength of the predator's cooperation in hunting. After their work, other phenomena, such as Allee effect, fear effect, and spatial diffusion and so on, were introduced to predator-prey model with cooperative hunting [19,25,30,31,38,41].…”

“…To investigate the existence and stability of the equilibria, we divide the discussion into two cases according to the size of c 1 : (I-i) K > 1, β < 1, and c > c 1 ; (I-ii) K > 1, β < 1, and c < c 1 (see Figure 6 d where c 1 is defined in (19).…”

Traveling waves in cooperative predation: relaxation of sublinearity

“…A two-dimensional Lotka-Volterra system was proposed in [1] which considers logistic growth of the prey and cooperative hunting on the predator. This model was extended in [14] to investigate Allee effects on the prey. In this paper, we will introduce spatial diffusion to the predator and study the following diffusive predator-prey model with cooperative predation.…”

Bifurcations of a predator-prey system with cooperative hunting and Holling III functional response

“…Similarly, the cooperative hunting among predators also is a common phenomenon in nature( [14]), which even is an effective mechanism to promote the evolution and diversity of species( [25]). A lot of recent studies pay attention to the cooperative hunting among predators( [2,5,10,11,20,26,28,33]). Alves and Hilker( [2]) considered the following four types of predator-dependent functional response to model the cooperative hunting among predators based on the above-mentioned four types of functional response.…”