On the nonautonomous Volterra-Lotka competition equations

Abstract

Abstract. A nonautonomous competitive Lotka-Volterra system of two equations is considered. It is shown that if the coefficients are continuous and satisfy certain inequalities, then any solution that is positive at some point has the property that one of its components vanishes while the other approaches a certain solution of the logistic equation.

Global attractor in autonomous competitive Lotka-Volterra systems

“…Montes de Oca and Zeeman [8,Lemma 3.1]). We also notice that these ideas, especially (1.2), are extended to nonautonomous (see [7,8], Ahmad [1], Ahmad and Lazer [3]) and retarded autonomous ( [5] and [4, §4.1- §4.3]) systems. No doubt the popularity of (1.2) is attributed to its conciseness.…”

On a non-autonomous stochastic Lotka-Volterra competitive system

“…For example, persistence and extinction of (1.1) were considered in [1,2,35,37]. Zhao et al [38] investigated permanence and global attractivity of model (1.1).…”

Extinction in nonautonomous Kolmogorov systems

“…supremum) of the function g, then system (LV) is persistent and globally attractive. To be more precise, Gopalsamy proved that in the almost periodic case the conditions (GAT) In [1] and [4] a nonautonomous logistic equation (L) u 0 (t) = u 0 (t)(a(t) − b(t)u 0 (t)), t ∈ R, is considered. It is well known that an autonomous logistic equation…”

“…Ahmad [1] and Coleman [4] showed that in the nonautonomous equation (L) the role of the globally attracting carrying capacity of the autonomous equation is played by a well defined canonical solution u * i (t) to which all other solutions converge.…”

“…Lemma 1 (Ahmad and Coleman [1], [4]). Assume that in (L) the functions a(t), b(t) are continuous and bounded above and below by positive constants.…”