Spatial Noise Stabilizes Periodic Wave Patterns in Oscillatory Systems on Finite Domains

Zhang

et al. 2010

Ecological Modelling

In this paper, we investigate the emergence of a predator-prey system with Ivlevtype functional response and reaction-diffusion. We study how diffusion affects the stability of predator-prey coexistence equilibrium and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Based on the bifurcation analysis, we give the spatial pattern formation, the evolution process of the system near the coexistence equilibrium point, via numerical simulation. We find that pure Hopf instability leads to the formation of spiral patterns and pure Turing instability destroys the spiral pattern and leads to the formation of chaotic spatial pattern. Furthermore, we perform three categories of initial perturbations which predators are introduced in a small domain to the coexistence equilibrium point to illustrate the emergence of spatiotemporal patterns, we also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations. This indicates that for prey-dependent type predator-prey model, pattern formations do depend on the initial conditions, while for predator-dependent type they do not. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pattern formation of a predator–prey system with Ivlev-type functional response

Zhang

et al. 2010

Ecological Modelling

IMA Journal of Applied Mathematics

“…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%

“…These may have the form of noise in parameter values (Hagan, 1981;Kopell, 1981;Kay & Sherratt, 2000) or forcing applied at a boundary of the domain. Both mechanisms have been studied extensively for chains of coupled oscillators (Ermentrout & Kopell, 1984, 1986Kopell et al, 1991;Ren & Ermentrout, 1998), but in oscillatory reaction-diffusion equations, there has been very little work on boundary-driven periodic travelling 760 J.…”

Section: Introductionmentioning

confidence: 99%

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

Sherratt

2008

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.

“…Other examples include PTWs and patterns of regional synchrony in geometrid moths in Northern Fennoscandia [5][6][7][8][9], PTWs and intermittent synchrony in red grouse [10,11], and PTWs in larch budmoth in the European Alps [12][13][14]; see Sherratt & Smith [15] for further examples. Modelling studies have demonstrated a number of potential causes for this spatial asynchrony, including invasions [16][17][18][19][20][21], heterogeneous habitats [13,[22][23][24], migration between subpopulations [25], migration driven by pursuit and evasion [26] and hostile habitat boundaries [27][28][29][30]. This paper concerns the last of these.…”

Section: Introductionmentioning

confidence: 99%

Generation of periodic travelling waves in cyclic populations by hostile boundaries

Sherratt

2013

Proc. R. Soc. A.

Self Cite

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.