Derivations of animal movement models with accumulated memory

Animals continuously update their movement decisions using both real-time observations and historical information from experience, social interactions, or environmental cues, which we call accumulated memory (also called distributed delay memory). While memory is important for animals, how it influences movement strategies has received limited attention. We address this gap by integrating accumulated memory into three widely used models: advection-diffusion, Fickian-type diffusion, and Fokker-Planck type diffusion. These represent distinct strategies: (i) gradient-based movement, responding to environmental gradients; (ii) environment matching, symmetrically adjusting movement rates; and (iii) location-based movement, relying solely on local suitability. We derive each model from random walk models to compare how different memory-based movement strategies at the individual level give rise to distinct macroscopic population behaviors. Furthermore, we establish the local existence of solutions for a general model encompassing all three cases using fixed-point theory and provide a linear stability analysis. Numerical simulations show that the Fickian model always converges rapidly to a uniform state. Under memory-suppressed conditions, the advection-diffusion and Fokker-Planck models may exhibit aggregation, whereas under memory-enhanced conditions all models eventually reach uniformity, with the advection-diffusion and Fokker-Planck models sometimes displaying oscillatory wiggling pattern or periodic movement before stabilization.