Spatial movement with distributed memory.

Area of Science:

• Mathematical Biology
• Theoretical Ecology
• Animal Behavior Modeling

Background:

• Traditional Brownian motion models animal diffusion but often fails to capture complex, non-Brownian movements.
• Animal movement is influenced by perceptual judgment, spatial memory, and cognition, necessitating more sophisticated models.
• Existing models often overlook the dynamic interplay of memory growth, decay, and spatial dependence in animal navigation.

Purpose of the Study:

• To develop a novel mathematical model for animal movement incorporating explicit spatial memory.
• To investigate the impact of memory dynamics (growth, decay, spatial dependence) on movement patterns.
• To analyze the emergent spatiotemporal dynamics arising from the proposed memory-informed diffusion model.

Main Methods:

• Introduced a distributed delayed diffusion term to explicitly model spatial memory.
• Analyzed the model's behavior under different temporal delay kernel strengths (weak vs. strong).
• Investigated bifurcations (Hopf, Turing) and pattern formation in spatially non-homogeneous steady states.

Main Results:

• Under weak memory kernel conditions, the model simplifies to the Keller-Segel chemotaxis model.
• Strong memory kernels, simulating learning and decay, generate rich spatiotemporal dynamics.
• Observed emergent patterns include Turing patterns and checker-board structures, indicating complex collective behaviors.

Conclusions:

• Explicit spatial memory is crucial for accurately modeling non-Brownian animal movement.
• The proposed model captures complex dynamics, including pattern formation, driven by memory effects.
• This framework provides a foundation for understanding cognition-driven movement in ecological contexts.