Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB

Garvie, Marcus R.

doi:10.1007/s11538-006-9062-3

Cited by 280 publications

(187 citation statements)

References 51 publications

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“…Efficient computational methods are discussed in detail by Garvie (2007), and corresponding computer codes are available at www.uoguelph.ca/wmgarvie. Figure 6.…”

Section: Periodic Travelling Wave Generation By Invasionmentioning

confidence: 99%

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

Sherratt

Smith

2008

J. R. Soc. Interface.

158

123

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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 reactiondiffusion 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 reactiondiffusion 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.

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“…Efficient computational methods are discussed in detail by Garvie (2007), and corresponding computer codes are available at www.uoguelph.ca/wmgarvie. Figure 6.…”

Section: Periodic Travelling Wave Generation By Invasionmentioning

confidence: 99%

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

Sherratt

Smith

2008

J. R. Soc. Interface.

158

123

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“…Over the last three decades, the simple form of this periodic wave family has provided an invaluable reference point for the study of periodic travelling waves in more general reaction-diffusion systems. This work has focussed in particular on the existence and stability of periodic travelling waves (Ermentrout, 1981;Maginu, 1979Maginu, , 1981Kapitula, 1994), other cases with exact solutions (Cope, 1979;Romero et al, 2000) and the generation of periodic travelling waves by environmental heterogeneities (Auchmuty & Nicolis, 1976;Hagan, 1981;Kopell, 1981;Kay & Sherratt, 2000;Sherratt, 2003) and behind invasive wavefronts (Sherratt, 1994(Sherratt, , 1996Ermentrout et al, 1997;Petrovskii et al, 1998;Petrovskii & Malchow, 2000, 2001Webb & Sherratt, 2004;Garvie, 2007).…”

Section: λ λ λ-ω ω ω Systemsmentioning

confidence: 99%

“…Oscillatory reaction-diffusion equations are a class of model with important applications in many areas of biology, ranging from calcium signalling in cell biology (Sneyd & Sherratt, 1997;Timofeeva & Coombes, 2003) to population dynamics in ecology (Pascual, 1993;Medvinsky et al, 2002;Garvie, 2007). In one space dimension, the most important solution type for such equations is periodic travelling waves.…”

Section: Introductionmentioning

confidence: 99%

A comparison of periodic travelling wave generation by Robin and Dirichlet boundary conditions in oscillatory reaction-diffusion equations

Sherratt

2008

IMA Journal of Applied Mathematics

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Periodic travelling waves are an important solution form in oscillatory reaction-diffusion equations. I have shown previously that such waves arise naturally near a boundary at which a Dirichlet condition is applied. This result has applications in ecology, providing a potential explanation for the periodic waves seen in a number of natural populations. However, in ecological applications the Dirichlet boundary condition typically arises as a simple approximation to a more realistic Robin condition. In this paper, I consider the generation of periodic travelling waves by Robin boundary conditions and how the wave amplitude compares with that arising from Dirichlet conditions. I study a 'λ-ω' system of equations, which is the normal form of an oscillatory reaction-diffusion system with scalar diffusion matrix close to a Hopf bifurcation. I consider a Robin boundary condition close to the Dirichlet limit, with proximity measured by a small parameter , and I study the equations as a perturbation problem in this small parameter. I show that the perturbation is singular and that although the solution itself changes at O( ), the amplitude of the periodic travelling wave which this solution approaches far from the boundary is unchanged at both O( ) and O( 2 ). This provides strong justification for the use of the Dirichlet approximation to the Robin condition when studying periodic travelling wave generation in equations of λ-ω type. Finally, I discuss the ecological applications of the results.

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“…A number of recent contributions have begun to suggest that it is critical to begin to fully consider the implications of spatial flows on the dynamics of ecological communities [7]. In the literatures [11,12], the spatial component of ecological interactions (trophic-interactions) is identified as an important factor in shaping ecological communities. However, there is lack of recognition about the role of spatial considerations in food chain dynamics [7].…”

Section: Introductionmentioning

confidence: 99%

Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model

Melese¹,

Gakkhar²

2011

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In this paper, a spatial tri-trophic food chain model with ratio-dependent Michaelis-Menten type functional response under homogeneous Neumann boundary conditions is studied. Conditions for Hopf and Turing bifurcation are derived. Sufficient conditions for the emergence of spatial patterns are obtained. The results of numerical simulations reveal the formation of labyrinth patterns and the coexistence of spotted and stripelike patterns.

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Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB

Cited by 280 publications

References 51 publications

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

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

A comparison of periodic travelling wave generation by Robin and Dirichlet boundary conditions in oscillatory reaction-diffusion equations

Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model

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