Compatibility of local and global stability conditions for some discrete population models

Abstract

In this paper, we investigate the stability of the model that has been proposed to study the growth of Bobwhite Quail Populations of Northern Wisconsin and prove that local stability implies global stability. We will prove the main results by using a suitable Lyapunov function and for illustration we apply the results on Hassell and Smith models. Also, we will show that for different values of the parameters, the population will exhibit variations in dynamics. Some illustrative examples and graphs are included to demonstrate the validity and applicability of the results.

Rebuttal of “Compatibility of local and global stability conditions for some discrete population models,” by S.H. Saker

“…Secondly, it should be noticed that the techniques used in the proofs of Theorems 2.1 and 2.3 in [1] are different from the ones employed in [2,3]. However, as I show below, Saker's proofs are wrong.…”

“…Paper [1] contains two main results (Theorems 2.1 and 2.3), and a corollary (labelled as Theorem 2.2). The author claims that these results are new, and establishes a comparison with some previous papers.…”

“…The author claims that these results are new, and establishes a comparison with some previous papers. The aim of this rebuttal is not only to clarify that this claim is false (that is, these results are not new), but also to show that the proofs in [1] Amazingly, paper [3] is included in the list of references of [1], but the author does not recognize in his discussion that Theorem 2.3 was already proven there; rather than this, he choose to compare his result with other papers, that were already cited in [3].…”

“…The note [2] is not included in the bibliography of paper [1], but I suspect that Dr. Saker knows the paper. For instance, he uses exactly the same motivation and references as in [2].…”

“…Theorem 2.1 in [1] states that condition (3) ensures the global asymptotic stability of the positive equilibriumx, that is, all solutions of (1) with positive initial condition converge tox. For it, the author claims that the map V (x) = (x−x) 2 is a (strict) Lyapunov function for (1).…”