On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity

Yunfeng Jia; Jianhua Wu; Hong-Kun Xu

doi:10.3934/cpaa.2013.12.1927

Article Contents

2013, Volume 12, Issue 5: 1927-1941. Doi: 10.3934/cpaa.2013.12.1927

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On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity

1.
College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062
2.
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119
3.
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424

Received: September 30, 2011

Revised: July 31, 2012

Published: August 2013

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Abstract

In this paper, a two competing fish species model with combined harvesting is concerned, both the species obey the law of logistic growth and release a toxic substance to the other. Use spectrum analysis and bifurcation theory, the stability of semi-trivial solution, positive constant solution and the bifurcation solutions of model are investigated. We discuss bifurcation solutions which emanate from positive constant solution and trivial solution by taking the growth rate as bifurcation parameter. By the monotonic method, the existence result of positive steady-state of the model is discussed. The possibility of existence of a bionomic equilibrium is also obtained by taking the economical factor into consideration. Finally, some numerical examples are given to illustrate the results.

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Mathematics Subject Classification: Primary: 92D25; 93C20; Secondary: 35K57.

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