Dynamics of a Birth-Pulse Single-Species Model with Restricted Toxin Input and Pulse Harvesting

Yi Ma,Bing Liu,Wei Feng

doi:10.1155/2010/142534

Abstract

We consider a birth-pulses single-species model with restricted toxin input and pulse harvesting in a polluted environment. Pollution accumulates as a slowly decaying stock and is assumed to affect the growth of the renewable resource population. Firstly, by using the discrete dynamical system determined by the stroboscopic map, we obtain an exact 1-period solution of system whose birth function is Ricker function or Beverton-Holt function and obtain the threshold conditions for their stability. Furthermore, we show that the timing of harvesting has a strong impact on the maximum annual sustainable yield. The best timing of harvesting is immediately after the birth pulses. Finally, we investigate the effect of the amount of toxin input on the stable resource population size. We find that when the birth rate is comparatively lower, the population size is decreasing with the increase of toxin input; that when the birth rate is high, the population size may begin to rise and then drop with the increase of toxin input.

Highlights

One of the most important results in the economics of natural resources explains why natural biological stocks are often overexploited
The main objective of the present paper is to study the dynamics of a birthpulse single-species model with the restricted toxin input and pulse harvesting in a closed polluted environment
We show the relationship both between pulse harvesting time and the maximum annual-sustainable yield and between the amount of toxin input and the stable equilibrium population size

Summary

Introduction

One of the most important results in the economics of natural resources explains why natural biological stocks are often overexploited. The economic importance of the pollution problem is compounded by the fact that, in contrast with overexploitation, it affects both common and private property resource stocks. Such environmental uncertainty affects the incentive to harvest a resource. In this paper, considering the results in 4 , we will study an optimal pulse harvesting problem of a single birth-pulse population in a polluted environment and see how the timing of pulse harvesting affects the maximum annual-sustainable yield?

Model Formulation

The Effects of Pulse Harvesting Time on the Maximum Annual-Sustainable Yield

The Effect of the Amount of Toxin Input on the Size of Population

Conclusion