A mathematical framework for memory and learning mechanisms In animals

Abstract

This thesis investigates traveling wave solutions in a class of partial di erential equation models describing knowledge-based animal movement via nonlocal perception and cognitive mapping. Such models, inspired by the work of Wang and Salmaniw (2023), account for how animals sense their environment, retain information in memory, and adjust movement strategies accordingly. Two types of perception kernels are considered: a Dirac delta kernel, representing instantaneous local sensing, and a Gaussian kernel, representing spatially distributed sensing with memory decay. The study applies the upper and lower solution method to establish the existence of traveling wave solutions for each kernel type. Proof outlines are presented for both cases, highlighting the key analytical steps. Numerical simulations in MATLAB are used to visualize the resulting wave pro les and propagation speeds, illustrating how sensing range and memory a ect movement dynamics. In addition, the thesis analyzes steady states of the system, performing stability analysis through linearization and phase portraits to determine long-term behavioral patterns. The results contribute to a deeper understanding of how sensory perception and memory can shape the spread and persistence of animal populations in heterogeneous environments.