Supercritical and subcritical Turing pattern formation in a hyperbolic vegetation model for flat arid environments

Consolo, G.; Currò, Carmela; Valenti, Giovanna

doi:10.1016/j.physd.2019.03.006

Cited by 32 publications

(36 citation statements)

References 65 publications

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“…In Figure 3, we depict the dependence of the Landau coefficient L as a function of the critical plant loss B c for different values of coefficient p (p = 2, solid line, and p = 3, dashed line). As it can be noticed, in the case p = 2, subcritical patterns may form under the restriction dB ≲ 10, in accordance with previous works, 23,37 whereas, for p = 3, the subcritical region is slightly enlarged to dB ≲ 14. It means that a larger strength of the infiltration rate favors the formation of subcritical modes since it yields an increase of the critical plant loss value below which subcritical vegetation patterns can be observed.…”

Section: Numerical Resultssupporting

confidence: 91%

“…From 17, we argue that the onset of Turing bifurcation is affected by the strength of the rate of infiltration of water into the soil p, but is independent of inertial times, as expected by virtue of the stationary nature of the emerging patterns. 23…”

Section: Steady States and Linear Stability Analysismentioning

confidence: 99%

“…In this latter case, weakly nonlinear analysis need to be pushed at higher perturbative orders, as described in our previous work. 23 For simplicity, we here limit our analysis to the supercritical case. In this regime, GL equation admits the spatially homogeneous stationary solution √…”

Section: Multiple-scales Weakly Nonlinear Analysismentioning

confidence: 99%

“…To account for the natural presence of inertial effects in ecological systems (and reported by some experimental works [14][15][16][17], it is necessary to overcome the paradox of infinite speed of propapagation of disturbances-typical of parabolic models-and to consider that any realistic wave process evolves in space over a finite time. [18][19][20][21][22][23][24][25] This procedure generally leads to hyperbolic reaction-transport models that most properly account for the finite speed of propagation and, thus, are better suited to describe transient wave phenomena.…”

Section: Introductionmentioning

confidence: 99%

“…In our previous works, 22,23 we proposed a hyperbolic version of the Klausmeier model that, in the case of flat arid terrains and in dimensionless form, reads u t + J u x = (u, w) w t + J w…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

Consolo

Currò

Valenti

2020

Math Methods in App Sciences

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The formation of Turing vegetation patterns in flat arid environments is investigated in the framework of a generalized version of the hyperbolic Klausmeier model. The extensions here considered involve, on the one hand, the strength of the rate at which rainfall water enters into the soil and, on the other hand, the functional dependence of the inertial times on vegetation biomass and soil water. The study aims at elucidating how the inclusion of these generalized quantities affects the onset of bifurcation of supercritical Turing patterns as well as the transient dynamics observed from an uniformly vegetated state towards a patterned state. To achieve these goals, linear and multiple-scales weakly nonlinear stability analysis are addressed, this latter being inspected in both large and small spatial domains. Analytical results are then corroborated by numerical simulations, which also serve to describe more deeply the spatio-temporal evolution of the emerging patterns as well as to characterize the different timescales involved in vegetation dynamics.

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Section: Numerical Resultssupporting

confidence: 91%

Section: Steady States and Linear Stability Analysismentioning

confidence: 99%

Section: Multiple-scales Weakly Nonlinear Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

Consolo

Currò

Valenti

2020

Math Methods in App Sciences

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Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal

Eigentler

Sherratt

2018

J. Math. Biol.

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Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.

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An integrodifference model for vegetation patterns in semi-arid environments with seasonality

Eigentler

Sherratt

2020

J. Math. Biol.

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Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents except Antarctica. In some semi-arid regions, the climate is characterised by seasonality, which yields a synchronisation of seed dispersal with the dry season or the beginning of the wet season. We reformulate the Klausmeier model, a reaction–advection–diffusion system that describes the plant–water dynamics in semi-arid environments, as an integrodifference model to account for the temporal separation of plant growth processes during the wet season and seed dispersal processes during the dry season. The model further accounts for nonlocal processes involved in the dispersal of seeds. Our analysis focusses on the onset of spatial patterns. The Klausmeier partial differential equations (PDE) model is linked to the integrodifference model in an appropriate limit, which yields a control parameter for the temporal separation of seed dispersal events. We find that the conditions for pattern onset in the integrodifference model are equivalent to those for the continuous PDE model and hence independent of the time between seed dispersal events. We thus conclude that in the context of seed dispersal, a PDE model provides a sufficiently accurate description, even if the environment is seasonal. This emphasises the validity of results that have previously been obtained for the PDE model. Further, we numerically investigate the effects of changes to seed dispersal behaviour on the onset of patterns. We find that long-range seed dispersal inhibits the formation of spatial patterns and that the seed dispersal kernel’s decay at infinity is a significant regulator of patterning.

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Supercritical and subcritical Turing pattern formation in a hyperbolic vegetation model for flat arid environments

Cited by 32 publications

References 65 publications

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

Turing vegetation patterns in a generalized hyperbolic Klausmeier model

Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal

An integrodifference model for vegetation patterns in semi-arid environments with seasonality

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