Global asymptotic stability of evolutionary periodic Ricker competition models

Open Problems and Conjectures in the Evolutionary Periodic Ricker Competition Model

“…In a recent paper [15], Elaydi et al studied the global dynamics of the following evolutionary Ricker model of a single species x (with a single mean trait u and individual trait v):…”

“…The global dynamics of the non-trivial fixed point (in the autonomous case) and of the non-trivial periodic cycle (in the non-autonomous periodic case) are accomplished by the idea of mixed monotone mappings introduced by Hal Smith in [18]. The following result is the main theorem in [15] concerning the global dynamics of the autonomous single-species Ricker model (1): 1) is globally asymptotically stable if it is locally asymptotically stable, where…”

“…with α > 0, σ 2 < 2 and c 1 ̸ = 0 as stated in Theorem 2.1 in [15]. Concerning the non-autonomous p−periodic case, i.e., when α(t + p) = α(t) for all t = 0, 1, .…”

“…Dealing with global stability in discrete models is a real challenge even for simple models. In the case of evolutionary models, Elaydi et al, in [15], were able to show global stability in a Ricker-type model, in both autonomous and periodic cases, by using the idea of mixed monotone maps, which were firstly studied by Kulenovic and Merino in [16] and later popularized by H. Smith in [17,18].…”

“…The study of global stability presented in [15] for the evolutionary Ricker-type model is far from complete since there are a large number of cases, depending on the parameters, where the authors were not able to show global stability.…”