scholarly journals Generation of periodic travelling waves in cyclic populations by hostile boundaries

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120756 ◽  
Author(s):  
Jonathan A. Sherratt
Keyword(s):  
Spatial Organization ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Diffusion Type ◽  
Predator Prey Model ◽  
Periodic Travelling Wave ◽  
Steady State Levels ◽  
Spatio Temporal ◽  
Cyclic Populations

Many recent datasets on cyclic populations reveal spatial patterns with the form of periodic travelling waves (wavetrains). Mathematical modelling has identified a number of potential causes of this spatial organization, one of which is a hostile habitat boundary. In this paper, the author investigates the member of the periodic travelling wave family selected by such a boundary in models of reaction–diffusion type. Using a predator–prey model as a case study, the author presents numerical evidence that the wave generated by a hostile (zero-Dirichlet) boundary condition is the same as that generated by fixing the population densities at their coexistence steady-state levels. The author then presents analysis showing that the two waves are the same, in general, for oscillatory reaction–diffusion models with scalar diffusion close to Hopf bifurcation. This calculation yields a general formula for the amplitude, speed and wavelength of these waves. By combining this formula with established results on periodic travelling wave stability, the author presents a division of parameter space into regions in which a hostile boundary will generate periodic travelling waves, spatio-temporal disorder or a mixture of the two.

scholarly journals Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

2008 ◽  
Vol 5 (22) ◽  
pp. 483-505 ◽  
Author(s):  
Jonathan A Sherratt ◽  
Matthew J Smith
Keyword(s):  
Travelling Waves ◽  
Empirical Studies ◽  
Reaction Diffusion ◽  
Field Studies ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Oscillatory Reaction ◽  
Relevant Field ◽  
Spatio Temporal ◽  
Cyclic Populations

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction–diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction–diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

scholarly journals Nonuniform Continuity of the Osmosis K(2, 2) Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Aiyong Chen ◽  
Yong Ding ◽  
Wentao Huang
Keyword(s):  
Differential Equations ◽  
Small Amplitude ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Qualitative Theory ◽  
Travelling Wave Solutions ◽  
Solution Map ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. II. Abruptly vanishing diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 361-378 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Wave Form ◽  
Initial Development ◽  
Finite Concentration ◽  
Distinct Process ◽  
And Diffusion

In this paper we continue our study of some of the qualitative features of chemical polymerization processes by considering a reaction-diffusion equation for the chemical concentration in which the diffusivity vanishes abruptly at a finite concentration. The effect of this diffusivity cut-off is to create two distinct process zones; in one there is both reaction and diffusion and in the other there is only reaction. These zones are separated by an interface across which there is a jump in concentration gradient. Our analysis is focused on both the initial development of this interface and the large time evolution of the system into a travelling wave form. Some distinct differences from our previous analysis of smoothly vanishing diffusivity are found.

Travelling waves for a reaction–diffusion system in population dynamics and epidemiology

2005 ◽  
Vol 135 (4) ◽  
pp. 663-675 ◽  
Author(s):  
Shangbing Ai ◽  
Wenzhang Huang
Keyword(s):  
Population Dynamics ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Diffusion Constant ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Dimensional System ◽  
Travelling Wave Solutions ◽  
Wave Solutions

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction–diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u′, ν), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b:= b(a) ∈ (γ, ∞), and furthermore, b(·): (0, γ) → (γ, ∞) is continuous and bijective

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

Travelling waves for delayed reaction–diffusion equations with global response

2005 ◽  
Vol 462 (2065) ◽  
pp. 229-261 ◽  
Author(s):  
Teresa Faria ◽  
Wenzhang Huang ◽  
Jianhong Wu
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Index Formula ◽  
Delayed Response ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Non Local

We develop a new approach to obtain the existence of travelling wave solutions for reaction–diffusion equations with delayed non-local response. The approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space, coupled with an index formula of the associated Fredholm operator and some careful estimation of the nonlinear perturbation. The general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation, and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non-local and delayed response for the matured population.

The development of travelling waves in a simple isothermal chemical system II. Cubic autocatalysis with quadratic and linear decay

1990 ◽  
Vol 430 (1879) ◽  
pp. 315-345 ◽  
Keyword(s):  
Numerical Solutions ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Chemical System ◽  
Necessary Condition ◽  
Travelling Wave Solution ◽  
Diffusion Front ◽  
Initial Value ◽  
Linear Decay

The possibility of travelling reaction-diffusion waves developing in the isothermal chemical system governed by the cubic autocatalytic reaction A + 2B → 3B (rate k 3 ab 2 ) coupled with either the linear decay step B → C (rate k 2 b ) or the quadratic decay step B + B → C (rate k 4 b 2 ) is examined. Two simple solutions are obtained,namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the autocatalyst B. Both of these suggest that, for the quadratic decay case, a wave will develop only if the non-dimensional parameter k ═ k 4 / k 3 a 0 < 1 (where a 0 is the initial concentration of the reactant A), with there being no restriction on the initial input of the autocatalyst B. However, for the linear decay case the initiation of a travelling wave depends on the parameter v ═ k 2 / k 3 a 2 0 and that, in addition, there is an input threshold on B before the formation of a wave will occur. The equations governing the fully developed travelling waves are then considered and it is shown that for the quadratic decay case the situation is similar to previous work in quadratic autocatalysis with linear decay, with a necessary condition for the existence of a travelling-wave solution being that K < 1. However, the case of linear decay is quite different, with a necessary condition for the existence of a travelling wave solution now found to be v < 1/4 Numerical solutions of the equations governing this case reveal further that a solution exists only for v < v c , with v c ≈ 0.0465, and that there are two branches of solution for 0 < v < v c . The behaviour of these lower branch solutions as v → 0 is discussed. The initial-value problem is then considered. For the quadratic decay case it is shown that the uniform state a ═ a 0 , b ═ 0 is globally asymptotically stable (i. e. a → a 0 , b → 0 uniformly for large times) for all k > 1. For the linear decay case it is shown that the development of a travelling wave requires β 0 > v (where β 0 is a measure of the initial input of B) for v < v c . These theoretical results are then complemented by numerical solutions of the initial-value problem for both cases, which confirm the various predictions of the theory. The behaviour of the solution of the equations governing the travelling waves is then discussed in the limits K → 0, v → 0 and K → 1. In the first case the solution approaches the solution for K ═ 0 (or v =0) on the length scale of the reaction-diffusion front, with there being a long tail region of length scale O ( K -1 ) (or O ( v -1 )) in which the autocatalyst B decays to zero. In the latter case we find that the concentration of reactant A is 1 + O [(1 - k )] and autocatalyst B is O[(1 - k 2 ] with the thickness of the reaction-diffusion front becoming large, of thickness O [(1- k ) -3/2 ].

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