Differential and Integral Equations

Discrete monotone dynamics and time-periodic competition between two species

Peter Takáč

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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.

Article information

Differential Integral Equations, Volume 10, Number 3 (1997), 547-576.

First available in Project Euclid: 2 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37L05: General theory, nonlinear semigroups, evolution equations
Secondary: 35K57: Reaction-diffusion equations 37N25: Dynamical systems in biology [See mainly 92-XX, but also 91-XX] 47H05: Monotone operators and generalizations 92D25: Population dynamics (general)


Takáč, Peter. Discrete monotone dynamics and time-periodic competition between two species. Differential Integral Equations 10 (1997), no. 3, 547--576. https://projecteuclid.org/euclid.die/1367525667

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