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

Volume 16, Issue 3, pp 395--401
Publication Date: September 15, 2016 Submission 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 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.

### Share and Cite

##### ISRP Style

Xiangdong Xie, Fengde Chen, Mengxin He, Dynamic behaviors of two species amensalism model with a cover for the first species, Journal of Mathematics and Computer Science, 16 (2016), no. 3, 395--401

##### AMA Style

Xie Xiangdong, Chen Fengde, He Mengxin, Dynamic behaviors of two species amensalism model with a cover for the first species. J Math Comput SCI-JM. (2016); 16(3):395--401

##### Chicago/Turabian Style

Xie, Xiangdong, Chen, Fengde, He, Mengxin. "Dynamic behaviors of two species amensalism model with a cover for the first species." Journal of Mathematics and Computer Science, 16, no. 3 (2016): 395--401

### Keywords

• Amensalism model
• Lyapunov function
• stability.

•  34C25
•  92D25
•  34D20
•  34D40

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