Lattice-based stochastic models motivate non-linear diffusion descriptions of memory-based dispersal

Area of Science:

• Ecology and Evolutionary Biology
• Mathematical Biology
• Computational Neuroscience

Background:

• Population dispersal is significantly influenced by individual movement, cognition, and memory.
• Classical partial differential equation (PDE) models for animal dispersal often lack detailed spatial memory effects.
• The precise mathematical formulation of memory effects within PDE transport terms (diffusion, advection) remains unclear.

Purpose of the Study:

• To develop a more biologically intuitive model for incorporating spatial memory into population dispersal.
• To derive a continuum limit description that captures memory-dependent movement mechanisms.
• To investigate the impact of memory effects on population dispersal patterns.

Main Methods:

• Development of a straightforward lattice-based individual-based model (IBM) accounting for crowding and historical distribution.
• Derivation of the continuum limit of the stochastic IBM to obtain a nonlinear diffusion equation.
• Stochastic simulations and numerical exploration of the derived mean-field PDE model.

Main Results:

• A novel nonlinear diffusion equation with memory-based diffusion terms was derived from the IBM.
• The study elucidates the relationship between memory-based diffusion and individual movement mechanisms influenced by memory.
• Numerical simulations confirmed that the new PDE model accurately represents the behavior of the stochastic IBM.

Conclusions:

• Individual-based models offer a more intuitive approach to implementing memory effects in dispersal simulations compared to heuristic PDE extensions.
• The derived nonlinear diffusion equation provides a new framework for studying memory-influenced population dispersal.
• Spatial memory significantly impacts population dispersal dynamics, as demonstrated by the model's predictions.