Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting

Dongmei  Xiao

doi:10.3934/dcdsb.2016.21.699

Article Contents

2016, Volume 21, Issue 2: 699-719. Doi: 10.3934/dcdsb.2016.21.699

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Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting

Dongmei Xiao^1,

1.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received: December 31, 2014

Revised: May 31, 2015

Published: February 2016

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Abstract

In this paper, we investigate a class of population model with seasonal constant-yield harvesting and discuss the effect of the seasonal harvesting on the survival of the population. It is shown that the population can be survival if and only if the model has at least a periodic solution and the initial amount of population is not lower than the minimum periodic solution. And if the population goes to extinction then it must be in the finite time. As an application of the conclusion, we systemically study the global dynamics of a logstic equation with seasonal constant-yield harvesting, and prove that there exists a threshold value $h_{MSY}$ of the intensity of harvesting, called the maximum sustainable yield, which is strictly greater than the maximum sustainable yield of logstic equation with constant-yield harvesting, such that the model has exactly two periodic solutions: one is attracting and the other is repelling if $0 < h < h_{MSY}$, a unique periodic solution which is semi-stable if $h=h_{MSY}$ and all solutions which go down to zero in the finite time if $h>h_{MSY}$. Hence, the logstic equation with seasonal constant-yield harvesting undergoes saddle-node bifurcation of the periodic solution as $h$ passes through $h_{MSY}$. Biologically, these theoretic results reveal that the seasonal constant-yield harvesting can increase the maximum sustainable yield such that the ecological system persists comparing to the constant-yield harvesting.

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Mathematics Subject Classification: Primary: 34C25, 92D25; Secondary: 58F14.

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