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


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 the species \(x\), 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. More precisely, the conditions which ensure the local stability of \(E_2(0, \frac{a_2}{b_2})\) is enough to ensure its global stability, and once the positive equilibrium exists, it is globally stable. Some numerical simulations are carried out to illustrate the feasibility of our findings.


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