Explicit Solutions for Recursive Integro-Difference Equations With Applications in Spatial Ecology

This paper presents a detailed analysis and derives closed-form solutions for sequences of functions defined by recursive integro-difference equations. These equations are fundamental to modeling systems in spatial ecology, such as population spread, gene flow, and biological control, where discrete-time dynamics interact with continuous spatial structure. The mathematical framework ingeniously combines nonsmoothing difference operators with smoothing integral operators, presenting unique analytical challenges. We build upon established theory and previous work on convolution-type operators to obtain explicit formulas through operational calculus and combinatorial methods. The resulting solutions provide powerful analytical tools for computing higher iterates of spatial ecological processes, offering advantages over purely numerical approaches. We demonstrate the broad applicability of these results through examples in population invasion, epidemiology, and neural field modeling, and discuss connections to existing ecological models. Comprehensive numerical validation confirms the accuracy of our derived solutions with machine precision.