[Paper Review] Propagation Phenomena for A Reaction-Advection-Diffusion Competition Model in A Periodic Habitat

Xiao Yu, Xiao‐Qiang Zhao|arXiv (Cornell University)|Oct 16, 2014

Mathematical and Theoretical Epidemiology and Ecology Models20 references3 citations

TL;DR

This paper investigates spreading speeds and spatially periodic traveling waves in a two-species Lotka-Volterra reaction-advection-diffusion model within a periodic habitat. By combining abstract semiflow theory with upper solution methods, it establishes the existence of a single rightward spreading speed and proves its coincidence with the minimal wave speed for periodic traveling waves, under linear determinacy conditions.

ABSTRACT

This paper is devoted to the study of propagation phenomena for a Lotka-Volterra reaction-advection-diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trival steady state for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. We also obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction-diffusion model.

Motivation & Objective

• To analyze propagation phenomena in a two-species Lotka-Volterra competition model with reaction, advection, and diffusion in a periodic spatial habitat.
• To establish the existence and global attractivity of semi-trivial periodic steady states for the system with periodic initial data.
• To prove the existence of a single rightward spreading speed and its equivalence to the minimal wave speed for spatially periodic traveling waves.
• To derive sufficient conditions under which the spreading speed is linearly determinate, extending classical results to periodic heterogeneous environments.
• To apply the theoretical framework to a prototypical reaction-diffusion model, validating the general results in a concrete setting.

Proposed method

• Utilizes a continuous-time semiflow framework on a space of periodic functions to model the dynamics of two competing species in a periodic habitat.
• Applies abstract theory from Fang and Zhao (2012) and Liang and Zhao (2007) on monotone semiflows with weak compactness to handle systems with boundary fixed points.
• Employs the method of upper solutions to prove the existence of a single rightward spreading speed for the system.
• Establishes the existence of spatially periodic traveling waves connecting the semi-trivial state $(u_1^*, 0)$ to $(0, u_2^*)$ via comparison and fixed-point arguments.
• Adapts the concept of linear determinacy to periodic habitats by analyzing the principal eigenvalue of the linearized system at the origin.
• Uses variational and spectral techniques to relate the spreading speed to the minimal wave speed of periodic traveling waves.

Experimental results

Research questions

• RQ1 Does a single rightward spreading speed exist for the reaction-advection-diffusion competition model in a periodic habitat?
• RQ2 Is the spreading speed coincident with the minimal wave speed of spatially periodic traveling waves in this periodic environment?
• RQ3 Under what conditions is the spreading speed linearly determinate in a periodic habitat?
• RQ4 How do advection and spatial heterogeneity affect the invasion dynamics of competing species in a periodic environment?
• RQ5 Can the abstract theory of semiflows with weak compactness be adapted to prove existence of periodic traveling waves in this context?

Key findings

• The system admits a unique rightward spreading speed $c^*_+$, which is shown to be equal to the minimal wave speed of spatially periodic traveling waves.
• The spreading speed is linearly determinate under a set of sufficient conditions involving the principal eigenvalue of the linearized system at the origin.
• The semi-trivial steady state $(u_1^*(x), 0)$ is globally attractive for periodic initial data, establishing long-term dominance of the first species under certain conditions.
• Existence of $L$-periodic rightward traveling waves connecting $(u_1^*(x), 0)$ to $(0, u_2^*(x))$ is proven via upper solution construction and semiflow theory.
• The theory is extended to handle the case where the origin is an isolated equilibrium, allowing for the existence of waves connecting the dominant species to extinction.
• The results are applied to a prototypical reaction-diffusion model, confirming the theoretical predictions in a concrete biological setting.