Abstract
This paper is devoted to study the permanence and periodic solution of a competitive system with infinite delay, feedback control, and the Allee effect. We derive sufficient conditions for the permanence and existence of a periodic solution in a competitive system with infinite delay, feedback control, and the Allee effect by using the differential inequality theory and constructing the Lyapunov function. We provide explicit estimates of the lower and upper bounds of the population density. This study reveals that the Allee effect plays an essential role in the permanence and increases the risk of population extinction.
Highlights
A basic question of theoretical and practical importance in population biology concerns the long-term survival of each species
In [14], it was shown that, for infinite delay ecosystem, feedback controls can avoid the extinction of the species without the Allee whereas the lower bound is a1L–b12M x2–c1M u 1 b11M
And we prove that the upper bound of x1(t) with Allee effect is invariant, and the lower
Summary
A basic question of theoretical and practical importance in population biology concerns the long-term survival of each species. Permanence, addressing the long-term survival of each component of a system, has emerged as the most important notion for the systems in ecology. Ahmad [1] studied the permanence of the following traditional two-species nonautonomous Lotka–Volterra system. The functions ak(t) and bkj(t) (1 ≤ k, j ≤ 2) defined on (–∞, +∞) are positive continuous upper bounded and have positive lower bounds. Letting gL(gM) = inf(sup) g(t) : t ∈ R for each bounded function g : R → R, he showed that one of them will be driven to extinction whereas the other will stabilize at a certain solution to the corresponding logistic. The permanence of the traditional two-species Lotka–
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