Analysis of steady states for classes of reaction-diffusion equations with hump-shaped density-dependent dispersal on the boundary

2020 ◽  
Vol 13 (1) ◽  
pp. 9-19
Author(s):  
Quinn Morris ◽  
Jessica Nash ◽  
Catherine Payne
Keyword(s):  
Reaction Diffusion ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Density Dependent ◽  
Density Dependent Dispersal

A mathematical framework for determining the stability of steady states of reaction–diffusion equations with periodic source terms

2019 ◽  
Vol 84 (4) ◽  
pp. 669-678
Author(s):  
Lennon Ó Náraigh ◽  
Khang Ee Pang
Keyword(s):  
Steady State ◽  
A Priori ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Steady States ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Mathematical Framework ◽  
Source Terms ◽  
The Stability

Abstract We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction–diffusion equations with periodic source terms in one spatial dimension. We formulate an a priori condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch’s theorem and Poincaré’s inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

scholarly journals Virus-host interactions shape viral dispersal giving rise to distinct classes of travelling waves in spatial expansions

2020 ◽  
Author(s):  
Michael Hunter ◽  
Tongfei Liu ◽  
Wolfram Möbius ◽  
Diana Fusco
Keyword(s):  
Cooperative Behavior ◽  
Test Organism ◽  
Host Density ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Effective Density ◽  
Density Dependent ◽  
Phage T7 ◽  
The Impact ◽  
Density Dependent Dispersal

Reaction-diffusion waves have long been used to describe the growth and spread of populations undergoing a spatial range expansion. Such waves are generally classed as either pulled, where the dynamics are driven by the very tip of the front and stochastic fluctuations are high, or pushed, where cooperation in growth or dispersal results in a bulk-driven wave in which fluctuations are suppressed. These concepts have been well studied experimentally in populations where the cooperation leads to a density-dependent growth rate. By contrast, relatively little is known about experimental populations that exhibit a density-dependent dispersal rate.Using bacteriophage T7 as a test organism, we present novel experimental measurements that demonstrate that the diffusion of phage T7, in a lawn of host E. coli, is hindered by steric interactions with host bacteria cells. The coupling between host density, phage dispersal and cell lysis caused by viral infection results in an effective density-dependent diffusion rate akin to cooperative behavior. Using a system of reaction-diffusion equations, we show that this effect can result in a transition from a pulled to pushed expansion. Moreover, we find that a second, independent density-dependent effect on phage dispersal spontaneously emerges as a result of the viral incubation period, during which phage is trapped inside the host unable to disperse. Our results indicate both that bacteriophage can be used as a controllable laboratory population to investigate the impact of density-dependent dispersal on evolution, and that the genetic diversity and adaptability of expanding viral populations could be much greater than is currently assumed.

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