A limit set trichotomy for order-preserving systems on time scales. (English) Zbl 1059.37016
Let \(T\) be a time scale and \(X\) a metric space. A mapping \(\varphi:\{(t,\tau)\in T^2:\,\tau\leq t\}\times X\to X\) is said to be a \(2\)-parameter semiflow on \(X\) (or evolutionary process), if the mappings \(\varphi(t,\tau,\cdot)=\varphi(t,\tau) : X \to X\), \(\tau\leq t\), satisfy the following properties: (i) \(\varphi(\tau,\tau)x=x\) for all \(\tau\in T\), \(x\in X\), (ii) \(\varphi(t,s)\varphi(s,\tau)=\varphi(t,\tau)\) for all \(\tau,s,t\in T\), \(\tau\leq s\leq t\), (iii) \(\varphi(\,\cdot,\cdot)x:\{(t,\tau)\in T^2:\,\tau\leq t\} \to X\) is continuous for all \(x\in X\). The authors consider such systems that are also order preserving in normal cones of strongly ordered Banach spaces. Then, it appears that there are only three possible asymptotic behaviors and they are analyzed in the paper. This, in particular, generalizes an old theorem of Müller and is applied to a specific model from population dynamics.
Reviewer: Jerzy Ombach (Kraków)
MSC:
37C65 | Monotone flows as dynamical systems |
37B55 | Topological dynamics of nonautonomous systems |
92D25 | Population dynamics (general) |
34D05 | Asymptotic properties of solutions to ordinary differential equations |