Optimal harvesting policy of logistic population model in a randomly fluctuating environment

Abstract

In this paper, we investigate the optimal harvest policy of a stochastic logistic population model. The value of this study lies in two aspects: mathematically, we establish a stochastic threshold theorem to govern whether the population persists or not. In the case of population persistence, we prove the existence, uniqueness and global asymptotic stability of the invariant density of the Fokker–Planck equation associated with the SDE model, and we further show the relation between these two models. In addition, we give the unique optimal harvesting effort and the corresponding maximum of expectation of sustainable yield, which gives us the profile of the optimal harvesting policy of the SDE model. Ecologically, we find that big harvesting effort or big intensity of noise will lead the population to extinct risk almost surely. In addition, we find that under a fixed randomness strategy and proper harvesting, the maximum sustainable yield increases systematically as the harvesting effort increases, but overexploitation will reduce the level of maximize sustainable yield and eventually make the whole population extinct with probability one. Hence in order to obtain the optimal harvesting policy, we must decrease the harvesting effort and the intensity of noise. Furthermore, we find that our parameter perturbation method in this paper is more beneficial to the exploitation of renewable resources than the classical one given by Beddington and May (1977). The results show that different perturbation method can exhibit different stochastic population dynamics.