[Paper Review] Propagation Dynamics for Monotone Evolution Systems without Spatial Translation Invariance

Taishan Yi, Xiao‐Qiang Zhao|arXiv (Cornell University)|Jul 7, 2020

Mathematical and Theoretical Epidemiology and Ecology Models33 references4 citations

TL;DR

This paper establishes the existence of spatially inhomogeneous steady states and asymptotic propagation dynamics for monotone evolution systems lacking spatial translation invariance, using limiting systems and comparison principles. The key contribution is a general theory enabling analysis of traveling waves and spreading speeds in non-translation-invariant settings, validated through applications to time-delayed nonlocal equations, reaction-diffusion systems in cylinders, and KPP-type equations in inhomogeneous media.

ABSTRACT

In this paper,under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of monotone evolution systems without spatial translation invariance. Then we apply the developed theory to study traveling waves and spatio-temporal propagation patterns for time-delayed nonlocal equations, reaction-diffusion equations in a cylinder, and asymptotically homogeneous KPP-type equations. We also obtain the existence of steady state solutions and asymptotic spreading properties of solutions for a time-delayed reaction-diffusion equation subject to the Dirichlet boundary condition.

Motivation & Objective

• To develop a general framework for analyzing propagation dynamics in monotone evolution systems that lack spatial translation invariance.
• To establish the existence of spatially inhomogeneous steady states and asymptotic spreading properties in such systems.
• To extend existing spreading speed and traveling wave theories beyond translation-invariant settings, particularly for non-autonomous and spatially inhomogeneous systems.
• To apply the developed theory to time-delayed nonlocal equations, reaction-diffusion equations in cylindrical domains, and Dirichlet problems with shifting environments.
• To characterize the long-term behavior of solutions, including convergence to steady states and asymptotic spreading, in systems with spatio-temporal heterogeneity.

Proposed method

• Introduce two limiting systems with spatial translation invariance to approximate the original non-invariant system.
• Use comparison arguments to transfer orbital estimates from the limiting systems to the original system without translation invariance.
• Employ upward convergence and asymptotic annihilation properties of the limiting systems to characterize propagation dynamics.
• Apply Harnack-type inequalities and strict positivity of solutions to derive estimates in non-compact spatial domains.
• Utilize iterative properties of traveling wave maps and fixed point arguments to prove existence of steady states and convergence to spreading speeds.
• Leverage the comparison principle and sub/super-solution techniques to analyze solutions of time-delayed and nonlocal equations under Dirichlet conditions.

Experimental results

Research questions

• RQ1 How can propagation dynamics be characterized in monotone evolution systems that lack spatial translation invariance?
• RQ2 What conditions ensure the existence of spatially inhomogeneous steady states in non-translation-invariant systems?
• RQ3 Can asymptotic spreading speeds and traveling wave solutions be established for systems with spatio-temporal heterogeneity?
• RQ4 How do solutions behave asymptotically in time-delayed reaction-diffusion equations with Dirichlet boundary conditions?
• RQ5 To what extent can the theory be extended to KPP-type equations in locally inhomogeneous or asymptotically inhomogeneous media?

Key findings

• The existence of spatially inhomogeneous steady states is established for monotone semiflows without spatial translation invariance under appropriate limiting system assumptions.
• Asymptotic spreading speeds are characterized via comparison with limiting systems, even when the original system lacks translation invariance.
• For time-delayed nonlocal equations with a shifting habitat, the paper proves the existence of traveling wave solutions and asymptotic spreading properties.
• In the Dirichlet problem for time-delayed reaction-diffusion equations on the half-line, solutions exhibit asymptotic spreading with speed $ c^* $, and the solution converges to zero outside the spreading front.
• For KPP-type equations in spatially inhomogeneous media, the theory confirms the existence of a unique positive steady state with $ W( heta) o u_ heta^* $ as $ | heta| o ty $, and convergence of solutions to this state within the spreading region.
• The paper establishes that under monotonicity and homogeneity conditions, the spreading speed $ c^* $ satisfies $ c^* = 2 au_0 au_1 $, where $ au_0 $ and $ au_1 $ are derived from the linearized system at infinity.