Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions
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Forward and Pullback Dynamics of Nonautonomous Integrodifference Equations: Basic Constructions Huy Huynh1 · Peter E. Kloeden2 · Christian Pötzsche1 Dedicated to the memory of Russell Johnson Received: 8 June 2020 © The Author(s) 2020
Abstract In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an aperiodic influence on the models leading to nonautonomous integrodifference equations. In order to capture their long-term behaviour comprehensively, both pullback and forward attractors, as well as forward limit sets are constructed for general infinite-dimensional nonautonomous dynamical systems in discrete time. While the theory of pullback attractors, but not their application to integrodifference equations, is meanwhile well-established, the present novel approach is needed in order to understand their future behaviour. Keywords Forward limit set · Forward attractor · Pullback attractor · Asymptotically autonomous equation · Integrodifference equation · Urysohn operator Mathematics Subject Classification 37C70 · 37C60 · 45G15 · 92D40
The work of Huy Huynh has been supported by the Austrian Science Fund (FWF) under Grant Number P 30874-N35.
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Christian Pötzsche [email protected] Huy Huynh [email protected] Peter E. Kloeden [email protected]
1
Institut für Mathematik, Universität Klagenfurt, 9020 Klagenfurt, Austria
2
Mathematisches Institut, Universität Tübingen, 72076 Tübingen, Germany
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Journal of Dynamics and Differential Equations
1 Introduction Integrodifference equations not only occur as temporal discretisations of integrodifferential equations or as time-1-maps of evolutionary differential equations, but are of interest in themselves. First and foremost, they are a popular tool in theoretical ecology to describe the dispersal of species having non-overlapping generations (see, for instance, [18] or [9,17,25]). While the theory of Urysohn or Hammerstein integral equations is now rather classical [19], both numerically and analytically, our goal is here to study their iterates from a dynamical systems perspective. This means one is interested in the long term behaviour of recursions based on a fixed nonlinear integral operator. In applications, the iterates for instance represent the spatial distribution of interacting species over a habitat. One of the central questions in this context is the existence and structure of an attractor. These invariant and compact sets attract bounded subsets of an ambient state space X and fully capture the asymptotics of an autonomous dynamical system [8,23]. The dynamics inside the attractor can be very complicated and even chaotic [7]. Extending this situation, the main part of this paper is devoted to general nonautonomous difference equations in complete metric spaces. Their right-hand side can depend on time allowing to model the dispersal of species in temporal
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