Spatio-temporal dynamics for a class of monotone evolution systems
Taishan Yi, Xiao-Qiang Zhao
TL;DR
This work addresses propagation and steady-state behavior in monotone evolution systems lacking translational invariance by developing an abstract framework governed by limiting systems with leftward/rightward spreading speeds ($c_-^*$, $c_+^*$) and uniform asymptotic annihilation (UAA). It extends the theory from discrete-time to continuous-time semiflows and to nonautonomous/periodic settings, proving upward convergence to a positive profile $r^*$, asymptotic annihilation outside growth bands, and the existence or nonexistence of nontrivial fixed points and traveling waves under (UC) and (UAA) combinations (including GB). The framework is then applied to an integro-difference equation, yielding fixed-point connectivity results and highlighting a counterexample where nonlinearities at finite locations create nontrivial fixed points despite limiting behavior. Overall, the results broaden the analysis of spatio-temporal spreading in heterogeneous media and shifting habitats, with implications for ecology and epidemiology.
Abstract
In this paper, under an abstract setting we establish the spreading properties and the existence, non-existence and global attractivity of spatially heterogeneous steady states for a large class of monotone evolution systems without the translational monotonicity under the assumption that one limiting system has both leftward and rightward spreading speeds and the other one has the uniform asymptotic annihilation. Then we apply the developed theory to study the global dynamics of asymptotically homogeneous integro-difference equations, and provide a counter-example to show that the value of the nonlinear function at the finite range of location may give rise to nontrivial fixed points.