Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction

Wang, Xiaoli; Shi, Junping; Zhang, Guohong

doi:10.1016/j.jmaa.2020.124860

Cited by 16 publications

(8 citation statements)

References 54 publications

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“…For the models without delay, we have given clear conditions for symmetry-breaking pattern formation by diffusion-driven instability to occur (Lemma 3) for the whole range of model parameters. In previous works, where other aspects of the models were the focus of the analysis, similar conditions were not as clearly defined and were usually restricted to a certain range of the mortality parameter (e.g., Theorem 2.5 in [36] for m < 2, and Theorem 2.2 in [37] for m > 2).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In this paper, we focused on the conditions for Turing instability to occur. Once a patterned vegetation arises, it may evolve through different non-uniform steady-state solutions, helping to maintain vegetation beyond the limit point where a uniform vegetated state would experience a critical transition to the bare soil, desert state [12,13,35,36]. Future work will address the effect of different types of realistic delays on the stability and evolution of patterned vegetation in Klausmeier-type models.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…These two vegetated equilibria coincide when a = 2m at the point P 12 = (m, 1), which is unstable and where the system undergoes a fold bifurcation. Detailed stability analyses of these equilibria can be found in previous works (e.g., [36,37,41]). We focus on the conditions for stability of P 2 , the uniformly vegetated state, summarised in the next lemma.…”

Section: Non-spatial Modelmentioning

confidence: 99%

“…However, the problem in a general domain Ω, usually in R or R 2 , can be dealt with similarly by considering the corresponding spatial eigenvalue problem [25]. A stability analysis of (2) in a general domain for m < 2 was presented in [36]. In a general interval [0, lπ] for m > 2, with a slightly different nondimensionalized form of (1), a stability analysis was presented in [37].…”

Section: Spatial Modelmentioning

confidence: 99%

“…where M represents density-independent mortality and J is the yield of plant biomass from uptaken water. This classical model was later analysed, modified, and extended in different aspects (e.g., [28][29][30][31][32][33][34][35][36][37]). In the case of flat terrains, the gradient term in (1) can be replaced with a second diffusion term with a different diffusion constant.…”

Section: Introductionmentioning

confidence: 99%

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Delay Effects on Plant Stability and Symmetry-Breaking Pattern Formation in a Klausmeier-Gray-Scott Model of Semiarid Vegetation

Medjahdi,

Lachachi,

Castro

et al. 2024

Symmetry

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The Klausmeier–Gray–Scott model of vegetation dynamics consists of a system of two partial differential equations relating plant growth and soil water. It is capable of reproducing the characteristic spatial patterns of vegetation found in plant ecosystems under water limitations. Recently, a discrete delay was incorporated into this model to account for the lag between water infiltration into the soil and the following water uptake by plants. In this work, we consider a more ecologically realistic distributed delay to relate plant growth and soil water availability and analyse the effects of different delay types on the dynamics of both mean-field and spatial Klausmeier–Gray–Scott models. We consider distributed delays based on Gamma kernels and use the so-called linear chain trick to analyse the stability of the uniformly vegetated equilibrium. It is shown that the presence of delays can lead to the loss of stability in the constant equilibrium and to a reduction of the parameter region where steady-state vegetation patterns can arise through symmetry-breaking by diffusion-driven instability. However, these effects depend on the type of delay, and they are absent for distributed delays with weak kernels when vegetation mortality is low.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Non-spatial Modelmentioning

confidence: 99%

Section: Spatial Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Delay Effects on Plant Stability and Symmetry-Breaking Pattern Formation in a Klausmeier-Gray-Scott Model of Semiarid Vegetation

Medjahdi,

Lachachi,

Castro

et al. 2024

Symmetry

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Predicting the Emergence of Localised Dihedral Patterns in Models for Dryland Vegetation

Hill

2024

J Nonlinear Sci

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Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as ‘fairy circles’. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction–diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction–diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. We consider four well-established vegetation models and compute their key predictive quantities, observing that models which possess similar values exhibit qualitatively similar localised patterns; we then complement our results with numerical simulations of various localised states in each model. Here, localised vegetation patches emerge generically from Turing instabilities and act as transient states between uniform and patterned environments, displaying complex dynamics as they evolve over time.

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Effects of Degenerate Bifurcations and their Applications to a Holling-Type II Predator–Prey System

Wei,

Guo,

Guo

2024

J Dyn Diff Equat

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Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction

Cited by 16 publications

References 54 publications

Delay Effects on Plant Stability and Symmetry-Breaking Pattern Formation in a Klausmeier-Gray-Scott Model of Semiarid Vegetation

Delay Effects on Plant Stability and Symmetry-Breaking Pattern Formation in a Klausmeier-Gray-Scott Model of Semiarid Vegetation

Predicting the Emergence of Localised Dihedral Patterns in Models for Dryland Vegetation

Effects of Degenerate Bifurcations and their Applications to a Holling-Type II Predator–Prey System

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