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

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.