Optimal harvesting control and dynamics of two-species stochastic model with delays

Lidan Liu,Xinzhu Meng

doi:10.1186/s13662-017-1077-6

Abstract

Taking the stochastic effects on growth rate and harvesting effort into account, we propose a stochastic delay model of species in two habitats. The main aim of this paper is to investigate optimal harvesting and dynamics of the stochastic delay model. By using the stochastic analysis theory and differential inequality technology, we firstly obtain sufficient conditions for persistence in the mean and extinction. Furthermore, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are gained by using Hessian matrix, the ergodic method, and optimal harvesting theory of differential equations. To illustrate the performance of the theoretical results, we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients.

Highlights

1 Introduction The population dynamics could be affected by the process of migration among patches
When the intensity of the white noise is small, the species can still be persistent just as the deterministic model; see Figure (d). It shows that noise with small intensity can allow the species to preserve the prosperity, whereas noise with large intensity may be a cause of species extinction
( ) Most of the existing works [, ] considered the effects of white noise on the growth rate, whereas we have studied environment disturbance on that and harvesting effort affected by human and social factors

Summary

Introduction

The population dynamics could be affected by the process of migration among patches. Due to natural conditions, such as the geology, climate, and hydrology, and the human factors, which include the development of tourism and the locations of industries, the animal habitats have been divided into some small patches. Applying Itô’s formula to system ( ) yields d ln x (t) = k – (b + α D )x (t) + D e–d τ x (t – τ ) dt + σ dB (t) – σ dB (t), d ln x (t) = k – (b + α D )x (t) + D e–d τ x (t – τ ) dt + σ dB (t) – σ dB (t) Integrating both sides of these two differential equations, we get ln x (t) x ( ). We have x ∗ = a.s. We are in the position to prove that limt→+∞ x (t) = a.s. Since x ∗ = a.s., for sufficiently large t, we can derive from ( ) that t– ln x (t) ≤ k + – (b + α D ) x + t– σ B (t) – t– σ B (t).

Optimal harvesting

Numerical simulations and discussion