A predator-prey model with different response functions to juvenile and adult prey in deterministic and stochastic environments

Abstract

In this paper, we formulate a stage-structured predator-prey model with Holling-I and Crowley-Martin functional responses in deterministic and stochastic environments, where Holling-I and Crowley-Martin functional responses conform respectively to predator feeds on juvenile and adult prey. In the deterministic case, by discussing the existence and stability of equilibria as well as equilibrium point bifurcations, we observe that the model can possess more than one positive equilibrium and exhibit rich dynamics such as bistability and complex bifurcations, meaning that its dynamics is easily affected by the environmental perturbations. In the stochastic case, by constructing appropriate Lyapunov functions we establish respectively the sufficient conditions for the ergodic stationary distribution and extinction of the model. Moreover, for the bistability scenario between a positive equilibrium and an interior limit cycle in the absence of noise, we can numerically observe the phenomenon of noise-induced state frequent switching between two stochastic attractors in the bistable zone. Biologically, our results can partially explain the phenomenon that in real world, for the inevitably small environmental noise intensities, the biological populations may remain at least two patterns of survival to switch.