Abstract
A two-species commensal symbiosis model involving Allee effect and one party can not survive independently is proposed and studied in this paper. Sufficient conditions which ensure the local and global stability of the boundary equilibrium and the positive equilibrium are obtained, respectively. Numeric simulations show that with the increasing of Alee effect, the system takes much longer time to reach its stable steady-state solution, though the Allee effect has no influence on the final density of the species. The Allee effect has instable effect on the system, however, such effect is controllable.
Highlights
IntroductionThe aim of this paper is to investigate the dynamic behaviors of the following two-species commensal symbiosis model involving Allee effect and one party can not survive independently, which takes the form dx =x dt b1x + c1y x+y (1.1)
The aim of this paper is to investigate the dynamic behaviors of the following two-species commensal symbiosis model involving Allee effect and one party can not survive independently, which takes the form dx =x dt b1x + c1y x+y (1.1)dy y dt = y(a2 – b2y) u + y, where a1, b1, c1, a2, b2, and u are all positive constants, x(t) and y(t) are the densities of the first and second species at time t, a1 is the death rate of the first species, a2 is the intrinsic growth rate of the second species, a2 b2 is the environment carrying capacity of the second species
1 Introduction The aim of this paper is to investigate the dynamic behaviors of the following two-species commensal symbiosis model involving Allee effect and one party can not survive independently, which takes the form dx =x dt dy y dt = y(a2 – b2y) u + y, where a1, b1, c1, a2, b2, and u are all positive constants, x(t) and y(t) are the densities of the first and second species at time t, a1 is the death rate of the first species, a2 is the intrinsic growth rate of the second species, a2 b2 is the environment carrying capacity of the second species
Summary
The aim of this paper is to investigate the dynamic behaviors of the following two-species commensal symbiosis model involving Allee effect and one party can not survive independently, which takes the form dx =x dt b1x + c1y x+y (1.1). Only recently did scholars pay attention to the commensal symbiosis model with one party can not survive independently [26,27,28,29,30,31]
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