On a Predator–Prey System with Digestion Delay and Anti-predation Strategy

Abstract

Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both direct effect (through predation) and indirect effect (e.g., fear effect). In the literature, the indirect effect has been largely missing in predator–prey models, until some recent works. Based on the recent work (Wang et al. in J Math Biol 73:1179–1204, 2016) where a fear effect is considered in an ODE model as a cost, in this paper, we also consider a benefit from the anti-predation response in addition to the cost, as well as a time delay in the transfer of biomass from the prey to the predator after predation. This results in a system of delay differential equations (DDEs). By analyzing this nonlinear DDE system, we obtain some insights on how the anti-predation response level (indirect effect) and the biomass transfer delay jointly affect the population dynamics; particularly we show how the nonlinearity in the predation term mediated by the fear effect affects the long term dynamics of the model system. We also perform some numerical computations and simulations to demonstrate our results. These results seem to suggest a need to revisit existing predator–prey models in the literature by incorporating the indirect effect and biomass transfer delay.