A Nonlinear Stability Analysis of Vegetative Turing Pattern Formation for an Interaction-Diffusion Plant-Surface Water Model System in an Arid Flat Environment

Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA.
Bulletin of Mathematical Biology (Impact Factor: 1.39). 04/2012; 74(4):803-33. DOI: 10.1007/s11538-011-9688-7
Source: PubMed


The development of spontaneous stationary vegetative patterns in an arid flat environment is investigated by means of a weakly nonlinear diffusive instability analysis applied to the appropriate model system for this phenomenon. In particular, that process can be modeled by a partial differential interaction-diffusion equation system for the plant biomass density and the surface water content defined on an unbounded flat spatial domain. The main results of this analysis can be represented by closed-form plots in the rate of precipitation versus the specific rate of plant density loss parameter space. From these plots, regions corresponding to bare ground and vegetative patterns consisting of parallel stripes, labyrinth-like mazes, hexagonal arrays of gaps, irregular mosaics, and homogeneous distributions of vegetation, respectively, may be identified in this parameter space. Then those theoretical predictions are compared with both relevant observational evidence involving tiger and pearled bush patterns and existing numerical simulations of similar model systems as well as placed in the context of the results from some recent nonlinear vegetative pattern formation studies.

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    • "It should be mentioned that apart from the model by Klausmeier (1999) and derivations thereof (Van der Stelt et al., 2013; Kealy and Wollkind, 2012) a large body of model studies have been published that dedicate pattern formation in arid ecosystems to a variety of mechanisms, including competition for surface water (Dunkerley, 1997; HilleRisLambers et al., 2001; Rietkerk et al., 2002), competition through soil water uptake by roots (Von Hardenberg et al., 2001; Meron et al., 2004), a combination of these mechanisms (Gilad et al., 2004) or plant– plant interactions only (Lefever and Lejeune, 1997; Lejeune and Tlidi, 1999; Lejeune et al., 1999, 2002). These models may be suitable depending on system characteristics such as climate, soil and plant properties and can be used to answer specific research questions. "
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    ABSTRACT: Spatially periodic patterns can be observed in a variety of ecosystems. Model studies revealed that patterned ecosystems may respond in a nonlinear way to environmental change, meaning that gradual changes result in rapid degradation. We analyze this response through stability analysis of patterned states of an arid ecosystem model. This analysis goes one step further than the frequently applied Turing analysis, which only considers stability of uniform states. We found that patterned arid ecosystems systematically respond in two ways to changes in rainfall: (1) by changing vegetation patch biomass or (2) by adapting pattern wavelength. Minor adaptations of pattern wavelength are constrained to conditions of slow change within a high rainfall regime, and high levels of stochastic variation in biomass (noise). Major changes in pattern wavelength occur under conditions of either low rainfall, rapid change or low levels of noise. Such conditions facilitate strong interactions between vegetation patches, which can trigger a sudden loss of half the patches or a transition to a degraded bare state. These results highlight that ecosystem responses may critically depend on rates, rather than magnitudes, of environmental change. Our study shows how models can increase our understanding of these dynamics, provided that analyses go beyond the conventional Turing analysis.
    Full-text · Article · Dec 2014 · Ecological Complexity
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    • "In our terminology, this means that the Gray-Scott model is considered in that paper. In this sense, [21] strongly relates to [32]. Finally, in [41] [40] spatially periodic patterns in Gray-Scott type equations with an additional advection term are considered: a situation that is similar to the lower-left side case of Figure 1 (γ = 1, C > 0). "
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    ABSTRACT: In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.
    Full-text · Article · Feb 2013 · Journal of Nonlinear Science
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    ABSTRACT: Self-organized vegetation patterns in space were found in arid and semi-arid areas. In this paper, we modelled a vegetation model in an arid flat environment using reaction-diffusion form and investigated the pattern formation. By using Hopf and Turing bifurcation theory, we obtain Turing region in parameters space. It is found that there are different types of stationary patterns including spotted, mixed, and stripe patterns by amplitude equation. Moreover, we discuss the changes of the wavelength with respect to biological parameters. Specifically, the wavelength becomes smaller as rainfall increases and larger as plant morality increases. The results may well explain the vegetation pattern observed in the real world and provide some new insights on preventing from desertification.
    No preview · Article · Sep 2013 · Nonlinear Dynamics
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