Discrete monotone dynamics and time-periodic competition between two species

Abstract

Convergence to a fixed point for every positive semi-orbit of a monotone discrete-time dynamical system in a strongly ordered Banach space is investigated. The dynamical system is generated by a compact continuous self-mapping $T\colon [a,b]\to [a,b]$ of a closed order interval $[a,b]\overset{def}{=}(a + V_+)\cap (b - V_+)$ in an ordered Banach space $V$, where the positive cone $V_+$ of $V$ has nonempty interior $\overset{o}{V}_+ = \rm{Int} (V_+)$, and \hbox{$a,b\in V$} with $b-a\in \overset{o}{V}_+$. The mapping $T$ is strongly monotone on the open order interval $[[a,b]]\overset{def}{=} (a + \overset{o}{V}_+)\cap (b - \overset{o}{V}_+)$. Finally, assume that (i) the fixed points of $T$ contained in $[[a,b]]$ form a totally ordered set; (ii)for every nonempty compact set ${\cal K}\subset [[a,b]]$ of fixed points of $T$, $\min {\cal K}$ is lower Ljapunov stable or $\max {\cal K}$ is upper Ljapunov stable; (iii) for $n=1,2,\dots$, each fixed point $p\in [a,b]\setminus [[a,b]]$ of $T^n$, $a\neq p\neq b$, is ejective; and (iv) if dim $(V) < \infty$, then each fixed point $p\in [a,b]\setminus [[a,b]]$ of $T$ is extreme for $[a,b]$. Then, for the dynamical system generated by $T$, each positive semi-orbit starting in $[[a,b]]$ converges to a fixed point of $T$ in $[[a,b]]\cup \{ a,b \}$. The proof of this result combines ejective fixed-point theory with some geometric properties of maximal unordered subsets of $[[a,b]]$. Applications include large-time asymptotic behavior of competition between two species modeled by a time-periodic competitive system of two weakly coupled reaction-diffusion equations. The questions of extinction of one of the two species and unstable or stable coexistence of both species as well as the compressive case are discussed.