# American Institute of Mathematical Sciences

September  2013, 12(5): 1927-1941. doi: 10.3934/cpaa.2013.12.1927

## 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, China 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, Taiwan

Received  October 2011 Revised  August 2012 Published  January 2013

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.
Citation: Yunfeng Jia, Jianhua Wu, Hong-Kun Xu. On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1927-1941. doi: 10.3934/cpaa.2013.12.1927
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