Impact of diffusion mechanisms on persistence and spreading

Area of Science:

• Mathematical Biology
• Nonlinear Dynamics
• Statistical Physics

Background:

• The Kolmogorov-Petrovsky-Piskunov (KPP) equation is a fundamental model in population dynamics and reaction-diffusion systems.
• Standard diffusion models like Fickian, Stratonovich, and Fokker-Planck describe different physical phenomena but lack a unified framework.
• Understanding how diffusion characteristics influence population persistence and spread is crucial for ecological and epidemiological applications.

Purpose of the Study:

• To introduce and analyze a generalized q-diffusion framework for the KPP equation.
• To investigate the dependence of persistence and asymptotic spreading speed on the diffusion parameter q and the spatial phase shift between growth rate and diffusion coefficient.
• To compare the effects of q-diffusion with traditional Fickian diffusion in ecological and epidemiological contexts.

Main Methods:

• Analytical methods were employed to derive theoretical results.
• Numerical simulations were conducted to explore parameter space and validate analytical findings.
• The study focused on analyzing the principal eigenvalue for persistence and the asymptotic spreading speed.

Main Results:

• Persistence and spreading properties are generally dependent on the q-diffusion parameter.
• Q-diffusion can either enhance or diminish persistence and spreading speed compared to Fickian diffusion, depending on the configuration of growth rate r(x) and diffusion coefficient D(x).
• The spatial arrangement of r(x) relative to D(x) has distinct effects for q > 0, q = 0, and q < 0.

Conclusions:

• The choice of diffusion model significantly impacts predictions of persistence and spreading in ecological and epidemiological systems.
• The generalized q-diffusion framework provides a more comprehensive understanding of reaction-diffusion dynamics.
• This research highlights the importance of considering generalized diffusion models for accurate modeling and effective control strategies in applied fields.