Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit

Area of Science:

• Mathematical Biology
• Ecology
• Nonlinear Dynamics

Background:

• Lotka-Volterra models are foundational for studying species competition.
• Reaction-diffusion systems incorporate spatial dynamics.
• Cross-diffusion arises from interspecific interactions and can lead to complex spatial patterns.

Purpose of the Study:

• To rigorously derive a cross-diffusion system from a Lotka-Volterra reaction-diffusion model in the fast reaction limit.
• To analyze the linear stability of homogeneous equilibria in the derived cross-diffusion system.
• To investigate the potential for cross-diffusion induced instabilities (Turing instabilities).

Main Methods:

• Asymptotic analysis to obtain the cross-diffusion system in the fast reaction limit.
• Linear stability analysis of homogeneous equilibria.
• Numerical simulations to validate theoretical findings.

Main Results:

• A rigorous proof for the passage from a reaction-diffusion to a starvation-driven cross-diffusion system.
• Demonstration that cross-diffusion does not induce Turing instability in this specific model.
• Numerical simulations confirm the theoretical predictions regarding stability.

Conclusions:

• The fast reaction limit of the studied Lotka-Volterra system yields a starvation-driven cross-diffusion model.
• The model is stable with respect to Turing instability.
• This work provides theoretical and numerical insights into ecological dynamics governed by cross-diffusion.