Abstract
A prey‐predator model with gestation delay, stage structure for predator, and selective harvesting effort on mature predator is proposed, where taxation is considered as a control instrument to protect the population resource in prey‐predator biosystem from overexploitation. It shows that interior equilibrium is locally asymptotically stable when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. An optimal harvesting policy can be obtained by virtue of Pontryagin′s Maximum Principle without considering gestation delay; on the other hand, the interior equilibrium of model system loses as gestation delay increases through critical certain threshold, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of model system is also observed. Finally, numerical simulations are carried out to show consistency with theoretical analysis.
Highlights
The dynamics of a class of stage-structured prey-predator models with gestation delay have been studied by several authors 1–9
The prey-predator models with stage structure and gestation delay play an important role in the modelling of multispecies population dynamics
The constant τ ≥ 0 denotes the gestation delay of mature predator, and τ ≥ 0 is based on the assumption that the reproduction of predator population after predating the prey population is not instantaneous but mediates by some discrete time lag required for gestation of mature predator population. d > 0 denotes the proportional transforming rate from immature predator population to mature predator population
Summary
The dynamics of a class of stage-structured prey-predator models with gestation delay have been studied by several authors 1–9. The prey-predator models with stage structure and gestation delay play an important role in the modelling of multispecies population dynamics. Optimal harvesting policies with taxation are discussed. Chaudhuri et al 15–17 , Pradhan and Chaudhuri 18 , and Kar et al 19–25 studied the optimal taxation policies for harvesting of the prey-predator system. From the above literature survey, it may be pointed out that no attempt has been made to study the optimal taxation policy of a stage-structured prey-predator system. Taxation instrument is discussed to control overharvesting from prey-predator system with gestation delay in 26. An optimal harvesting policy for mature predator is discussed in the absence of gestation delay.
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