Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

Garvie, Marcus R.; Trenchea, Catalin

doi:10.1007/s00211-007-0106-x

Cited by 48 publications

(36 citation statements)

References 48 publications

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“…The aim of this paper is to present two stable finite-difference schemes for the numerical solution of (2) in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs in MATLAB. A detailed numerical analysis of the model equations was undertaken by Garvie and Trenchea (2005b), and the current work provides the computational and implementational details needed to study the key dynamical properties of these equations. We also give the results of some numerical experiments that have ecological and numerical implications.…”

Section: Model Equationsmentioning

confidence: 99%

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

2007

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We present two finite-difference algorithms for studying the dynamics of spatially extended predator-prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB: . Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator-prey interactions.

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Section: Model Equationsmentioning

confidence: 99%

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

2007

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“…It is reported that spatial inhomogeneities like the inhomogeneous distribution of nutrients as well as interactions on spatial scales like diffusion plays an important role on the dynamics of ecological populations [2][3][4][5][6][7]. Furthermore, Holling emphasized the influence of noise in the ecological dynamics [8].…”

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confidence: 99%

Pattern dynamics of a spatial predator–prey model with noise

Jin

2011

Nonlinear Dyn

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A spatial predator-prey model with colored noise is investigated in this paper. We find that the number of the spotted pattern is increased as the noise intensity is increased. When the noise intensity and temporal correlation are in appropriate levels, the model exhibits phase transition from spotted to stripe pattern. Moreover, we show the number of the spotted and stripe pattern, with respect to both noise intensity and temporal correlation. These studies raise important questions on the role of noise in the pattern formation of the populations, which may well explain some data obtained in the ecosystems.

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“…However, this breaks up at a certain distance from the invasion front (Figure 9(c)-(f)). The existence of a propagating band of unstable waves behind an invasion front has been documented previously in numerical simulations of models for cyclic predator-prey systems [45,46,31,30,23,19,20,24]. The wavetrain band becomes narrower as ω 1 is increased, and persists as one crosses the threshold for absolute instability of the wavetrain; the behavior behind the band then changes from source-sink type (Figure 9(c),(d)) to more disordered dynamics (Figure 9(e),(f)).…”

Section: Source-sink Solutionsmentioning

confidence: 58%

Absolute Stability of Wavetrains Can Explain Spatiotemporal Dynamics in Reaction-Diffusion Systems of Lambda-Omega Type

Smith¹,

Rademacher²,

Sherratt³

2009

SIAM J. Appl. Dyn. Syst.

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Abstract. The lambda-omega class of reaction-diffusion equations has received considerable attention because they are more amenable to mathematical investigation than other oscillatory reaction-diffusion systems and include the normal form of any reaction-diffusion system with scalar diffusion close to a standard supercritical Hopf bifurcation. Despite this, detailed studies of the dynamics predicted by numerical simulations have mostly been restricted to regions of parameter space in which stable wavetrains (periodic traveling waves) are selected by the initial or boundary conditions; we use the term "stability" to denote spectral stability on the real line. Here we consider the emergent spatiotemporal dynamics on large bounded domains, with Dirichlet conditions at one boundary and Neumann conditions at the other. Previous studies have established a parameter threshold below which stable wavetrains are generated by the Dirichlet boundary condition. We use numerical continuation techniques to analyze the spectral stability of wavetrain solutions, and we identify a second stability threshold, above which the selected wavetrain is absolutely unstable. In addition, we prove that the onset of absolute stability always occurs through a complex conjugate pair of branch points in the absolute spectrum, which greatly simplifies the detection of this threshold. In the parameter region in which the spectra of the selected waves indicate instability but absolute stability, our numerical simulations predict so-called "source-sink" dynamics: bands of visibly regular periodic traveling waves that are separated by localized defects. Beyond the absolute stability threshold our simulations predict irregular spatiotemporal behavior.

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Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

Cited by 48 publications

References 48 publications

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

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

Pattern dynamics of a spatial predator–prey model with noise

Absolute Stability of Wavetrains Can Explain Spatiotemporal Dynamics in Reaction-Diffusion Systems of Lambda-Omega Type

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