Dynamic behaviors of two species amensalism model with a cover for the first species

Volume 16, Issue 3, pp 395--401http://dx.doi.org/10.22436/jmcs.016.03.09

Publication Date: September 15, 2016Submission Date: May 29, 2016

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Authors

Xiangdong Xie- Department of Mathematics, Ningde Normal University, Ningde, Fujian, 352300, P. R. China.Fengde Chen- College of Mathematics and Computer Sciences, Fuzhou University, Fuzhou, Fujian, 350116, P. R. China.Mengxin He- College of Mathematics and Computer Sciences, Fuzhou University, Fuzhou, Fujian, 350116, P. R. China.

Abstract

In this paper, a two species amensalism model with a cover for the first species takes the form\[\frac{dx}{dt}=a_1x(t)-b_1x^2(t)-c_1(1-k)x(t)y(t),\]\[\frac{dy}{dt}=a_2y(t)-b_2y^2(t),\]is investigated, where \(a_i, b_i, i = 1, 2\) and \(c_1\) are all positive constants, \(k\) is a cover provided for thespecies \(x\), and \(0 < k < 1\). Our study shows that if \(0 \leq k < 1-\frac{a_1b_2}{a_2c_1}\), then \(E_2(0, \frac{a_2}{b_2})\) is globally stable,and if \(1>k>1-\frac{a_1b_2}{a_2c_1}\), then \(E_3(x^*, y^*)\) is the unique globally stable positive equilibrium. Moreprecisely, the conditions which ensure the local stability of \(E_2(0, \frac{a_2}{b_2})\) is enough to ensure its globalstability, and once the positive equilibrium exists, it is globally stable. Some numerical simulationsare carried out to illustrate the feasibility of our findings.

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Keywords

• Amensalism model
• Lyapunov function
• stability.

MSC

•  34C25
•  92D25
•  34D20
•  34D40

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