Abstract
A Lotka–Volterra commensal symbiosis model with first species subject to the Allee effect is proposed and studied in this paper. Local and global stability property of the equilibria are investigated. An amazing finding is that with increasing Allee effect, the final density of the species subject to the Allee effect is also increased. Such a phenomenon is different from the known results, and it is the first time to be observed. Numeric simulations are carried out to show the feasibility of the main results.
Highlights
1 Introduction The aim of this paper is to investigate the dynamic behaviors of the following two species commensal symbiosis model incorporating the Allee effect to the first species: dx x dt = x(b1 – a11x) β + x + a12xy, dy dt = y(b2 – a22y), (1.1)
It came to our attention that the Allee effect has different influence on systems (1.4) and (1.5)
Çelik and Duman [29] showed that Allee effects have a stabilizing role in the discrete-time predator–prey model, while Merdan [36] showed that the continuous predator–prey system subject to an Allee effect takes a much longer time to reach its stable steady-state solution
Summary
The aim of this paper is to investigate the dynamic behaviors of the following two species commensal symbiosis model incorporating the Allee effect to the first species: dx x dt = x(b1 – a11x) β + x + a12xy, dy dt = y(b2 – a22y), (1.1). Where bi, aii, i = 1, 2, β and a12 are all positive constants, bi, i = 1, 2, is the intrinsic growth rate of the species x and y, respectively; bi aii , i =. 2, is the carrying capacity of and y, respectively; a12 reflects the efficiency of every single population y that can contribute to population x. Allee effect first species, which has the following property: (1)
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