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