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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References

[1] Dryland vegetation is known for its ability to exhibit high degrees of spatial organization with well-defined patterns of vegetated and unvegetated soil patches. Deterministic dynamics are usually invoked to explain the emergence of these patterns in arid ecosystems, while the effect of noise on the process of pattern formation has seldom been investigated. Relatively strong interannual rainfall fluctuations are a typical feature of dryland climates. Rainfall variability is commonly interpreted as a source of random disturbances for arid and semiarid vegetation, while its ability to play a more fundamental role in the process of vegetation pattern formation remains poorly understood. Here we show how random interannual climate fluctuations may indeed cause the emergence of vegetation patterns through the random alternation of stressed (i.e., water limited) and unstressed conditions. Thus, patterns emerge as a noise-induced phenomenon from a mechanism of climate-driven random switching of deterministic dynamics. Moreover, pattern formation is observed even when the deterministic dynamics of unstressed and stressed vegetation would - separately - lead to uniform spatial distributions of vegetation or bare soil.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References

[2] The emergence of self-organized vegetation patterns is a distinctive and recurrent feature of many water-limited ecosystems around the World [MacFayden, 1950; Greig-Smith, 1979; Ludwig and Tongway, 1995; Couteron and Kokou, 1997]. Typical aridland patterns include spotted, banded, and labyrinthine arrangements of vegetated patches separated by barren areas [e.g., Couteron and Lejeune, 2001; Tongway et al., 2001]. Until recently, the formation of these patterns has been explained mainly as the result of deterministic mechanisms of symmetry-breaking instability induced by resource redistribution and spatial interactions among neighboring plants [Lefever and Lejeune, 1997; Klausmeier, 1999; von Hardenberg et al., 2001; HilleRisLambers et al., 2001]. The decay of termite mounds [MacFayden, 1950] and local relief [Klausmeier, 1999] have been invoked as possible drivers responsible for the formation of spotted gaps and banded patterns, while the emergence of vegetation patterns in homogeneous and isotropic conditions was first investigated by Lefever and Lejeune [1997]. The effect of noise on the process of pattern formation in ecosystems has often been either ignored or at most included in cellular automata models [e.g., van Wijk and Rodriguez-Iturbe, 2002] to account for the uncertainty and unpredictability inherent to a number of environmental processes (e.g., disturbances, meteorological conditions, climate variability). In these models noise induces random fluctuations in the dynamics of vegetation but is not the cause for pattern formation. In fact, noise is usually believed to act on ecosystem dynamics as a source of disorganized fluctuations. However, in different contexts noise has been shown to be also able to induce ordered states in the dynamics of a system through noise-induced transitions [Horsthemke and Lefever, 1984], pattern formation [Van den Broeck et al., 1994; Garcia-Ojalvo and Sancho, 1999; Porporato and D'Odorico, 2004], noise-induced stability [D'Odorico et al., 2005], and stochastic resonance [Benzi et al., 1981].

[3] Here we show that the random interannual rainfall fluctuations existing in dryland ecosystems [e.g., Noy-Meir, 1973; Nicholson, 1980; D'Odorico et al., 2000] may trigger the formation of vegetation patterns. To this end, we capitalize on the theory of non-equilibrium stochastic alternation of global dynamics [Buceta et al., 2002], which shows how patterns may emerge from the random switching between two dynamical states. We interpret this random switching as an effect of interannual rainfall fluctuations: in each state the system undergoes different spatially-extended, deterministic dynamics, which - by themselves – would tend to spatially homogeneous conditions (e.g., completely bare or vegetated landscapes). The random alternation between these two dynamics induces the formation of heterogeneous, organized states with well-defined patterns [Buceta et al., 2002]. This theory provides a suitable mechanism to show how random interannual climate fluctuations may induce pattern formation in dryland vegetation through the externally-controlled switch between conditions favorable to vegetation growth or decay.

2. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References

[4] This study shows how vegetation patterns may emerge from the random alternation of stressed and unstressed conditions driven by interannual rainfall fluctuations. This mechanism of pattern formation is investigated through a model of vegetation dynamics. The spatial and temporal variability of vegetation biomass, V (normalized between 0 and 1), is modeled as a random sequence of two deterministic dynamics corresponding to (1) drought-induced vegetation decay, and (2) unstressed vegetation growth. In both cases, vegetation dynamics at any point, x(x, y), are expressed as the sum of two terms accounting for the local dynamics, f1,2(V(x, t)), and the spatial interactions, g(V(x, t), V(x′, t)), with the surrounding vegetation existing at all points, x′, in the neighborhood of x

  • equation image

with ζ being a dimensionless coefficient determining the relative importance of spatial vs. local dynamics. Two different functions, f1(V) and f2(V), are used to describe the local dynamics: the loss of vegetation occurring in water stressed conditions is assumed to be proportional to the existing biomass, while the unstressed growth of V is expressed by a logistic law, i.e., with rate proportional to the existing vegetation biomass, V, and to the available resources, 1 − V

  • equation image
  • equation image

with α1 being the mortality rate per unit V, and α2 the reproduction rate [e.g., Murray, 1989] of the logistic equation.

[5] We use dichotomous Markov noise [Van den Broeck, 1983] as a mechanism driving the switching between the two local dynamics (equations (2) and (3)): with probability (1 − P) vegetation is water stressed and its dynamics are modeled by equations (1) and (2). With probability P plants are unstressed and vegetation growth is modeled by equations (1) and (3). Thus, the effect of (large scale) random, interannual rainfall fluctuations is modeled as the switching between the local dynamics (2) and (3). This alternation simultaneously occurs at all points in the 2D domain.

[6] Here we consider the case in which neither one of the dynamics of stressed and unstressed vegetation (equation (1)(3)) are – separately - able to generate patterns, while the random switch induces pattern formation. Because in the interval between switches the system has no time to reach the equilibrium point of the local potential, the dynamics can be investigated considering the deterministic equation obtained from (1) replacing the local term with its average value

  • equation image

[7] In fact, the response of (woody) vegetation to water stress conditions is relatively slow (a few decades) if compared to the year-to-year climate variability considered in this study [e.g., Archer, 1990; Barbier et al., 2006]. Thus, in the case of fast switching, the homogeneous stable state, V0, of the system is a solution of equation image (V0) + ζg(V0, V0) = 0. Patterns emerge when the spatial dynamics are able to destabilize V0. To this end, we investigate the conditions controlling the (linear) stability of the state V = V0 (see Appendix A) as a function of the dimensionless parameter ζ (equation (1)), and of other parameters determining the properties of the spatial dynamics. The results are shown in Figure 1: the solid line (equation (A9) in Appendix A) separates stable from unstable states in the parameter space. The V-shape is due to the discontinuous dependence of V0 on P (equation (A6) in Appendix A). For a given value of ζ (Figure 1, dotted line) there are two values, P1 and P2, of P (Figure 1) marking the transition between stable and unstable states. For P < P1 the unvegetated state is stable; for P1 < P < P2 the system tends to a spatially heterogeneous stable state with organized vegetated patches bordered by bare ground. For P > P2 the homogeneous state, V0, is stable.

image

Figure 1. Marginal stability curve ζ = ζ* as a function of the probability, P, of being in unstressed conditions (with χ = 2.0; ɛ = 0.25; and η = 1.454, see Appendix A). Patterns emerge for ζ > ζ*.

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[8] The linear stability analysis of the state V = V0 does not account for the existence of a bound for V at V0 = 0. The numerical simulations carried out to investigate the effect of this bound show that conditions of instability and pattern formation are reached for values of P1 (Figure 1, dashed line) that are slightly different from those predicted by the analytical methods. However, the existence of a bound at V = 0 does not qualitatively change the stability of the unvegetated state. Interestingly, Figure 1 (solid line) also shows the emergence of a completely non-deterministic limit behavior in the local dynamics for ζ = 0 (i.e., with no deterministic spatial interactions) and P ≈ 0.4, though this behavior disappears when the bound of the dynamics at V = 0 is accounted for (Figure 1).

[9] The dependence of vegetation patterns on the function used to model the spatial interactions has already been investigated elsewhere [Lefever and Lejeune, 1997; D'Odorico et al., 2006]. Here we focus on the role of climate fluctuations in the process of pattern formation. In particular, the parameter P (i.e., the probability of not being in water-stress conditions) increases along a rainfall gradient and can be considered as a surrogate variable for mean annual rainfall. Figure 2 shows the mean and standard deviation of V as a function of P. For relatively low values of P the system is in water stress conditions for most of the time and vegetation is unable to establish and grow. Thus, the system tends to a uniformly unvegetated state (i.e., V = 0; σv = 0). For relatively high values of P the system is unstressed for most of the time and vegetation is able to reach a uniformly vegetated state. In these conditions σv = 0 and no patterns emerge. In intermediate conditions neither one of the two uniform states (vegetated or unvegetated conditions) can be attained by the system because the repeated switching between the two dynamics does not allow the system to reach the steady states of the underlying deterministic dynamics. In this case - depending on the spatial interactions - vegetation may attain a spatially heterogeneous stable state with a sparse canopy separated by barren areas. The typical sequence of (noise-induced) patterns along the gradient in mean annual precipitation (i.e., in the P parameter) is shown in Figure 2 (insets). This sequence ranges from spotted vegetation, to labyrinthine patterns, and spotted gaps. The widespread occurrence of this type of patterns in dryland ecosystems is well documented [e.g., Ludwig and Tongway, 1995; Couteron and Lejeune, 2001; Barbier et al., 2006].

image

Figure 2. Dependence of mean and standard deviation of V on the probability, P, of not being in water-stressed conditions (ζ = 0.4 and same parameters as Figure 1). For 0.2 < P < 0.64 the spatial standard deviation is larger than zero, suggesting the emergence of spatially heterogeneous vegetation, i.e., pattern formation. The patterns generated by the model are shown in the insets and include spotted vegetation (inset on the left, P = 0.375), labyrinthine patterns (central inset, P = 0.50), and spotted bare ground gaps (inset on the right, P = 0.6).

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3. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References

[10] The main point of this study is that interannual rainfall fluctuations may explain the emergence of vegetation patterns in dryland ecosystems. The proposed mechanism is noise induced, in that patterns emerge from the random switching between two deterministic dynamics. While the patterning mechanism remains deterministic (i.e., it results from deterministic spatial interactions), noise-driven switching between the two dynamics is needed to trigger pattern formation. The role of noise is to shift the system into the pattern-formation domain in the parameter space. This random switching is interpreted as the effect of interannual climate fluctuations, which drive the alternation between water stressed and unstressed conditions in vegetation dynamics.

[11] Vegetation patterns are commonly explained as the result of deterministic processes associated with (1) instability induced by diffusion-like mechanisms [Turing, 1952] of resource redistribution and vegetation encroachment [Klausmeier, 1999; von Hardenberg et al., 2001; HilleRisLambers et al., 2001; Rietkerk et al., 2002]; (2) fertility island formation caused by vegetation-deposition/erosion feedbacks [Schlesinger et al., 1990]; (3) near-neighbor facilitation and competition dynamics [Lefever and Lejeune, 2001]. Stochastic models [e.g., Jeltsch et al., 1996; van Wijk and Rodriguez-Iturbe, 2002] have generally used noise to account for the uncertainty inherent to external drivers and internal interactions in natural systems, thereby providing a framework for the generation of random patterns of vegetation. In this study we demonstrate that noise (i.e., interannual rainfall fluctuations) may play a more crucial role in the spatial dynamics of vegetation, i.e., the random forcing can indeed cause pattern formation through a mechanism of random switching between deterministic dynamics.

[12] A similar mechanism could be invoked to explain the emergence of vegetation patterns in mixed plant communities such as savannas. In fact, climate fluctuations may affect the relative competitive advantage between trees and grasses [e.g., Sankaran et al., 2004]. Thus, the model presented in this paper provides also a conceptual framework for testing recent non-equilibrium theories of savanna dynamics [e.g., Fernandez-Illescas and Rodriguez-Iturbe, 2003; Sankaran et al., 2004], i.e., that tree-grass coexistence emerge from the dynamical interaction of vegetation with environmental drivers and disturbance regime rather than from niche separation.

Appendix A

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References

[13] We follow Lefever and Lejeune [1997] and model spatial interactions as the combined effect of facilitation-competition mechanisms: near-neighbor interactions facilitate vegetation survival and growth, due to the favorable environment existing in the subcanopy soils, e.g., higher moisture contents due to lower evaporation losses and/or higher infiltration capacity [Walker et al., 1981; Scholes and Archer, 1997; Zeng et al., 2004; Greene, 1992; Greene et al., 1994]. Long-range interactions are dominated by competitions between lateral root systems, which expand beyond the vertical projection of the canopy [e.g., Casper et al., 2003; Caylor et al., 2006]. At any point, x(x, y), in a two-dimensional domain, Ω, the effect of interactions with vegetation at another point, x′(x′, y′), is assumed to be proportional to the biomass of the neighboring vegetation and to a weight function, w(r), of the distance, r = ∣r∣ = ∣xx′∣, between x′ and x. This weight function (Figure A1a) is positive for relatively small values of r (facilitation) and negative at larger distances (competition). In the absence of relief, the spatial interactions (hence, w(r)) depend on the length but not on the direction of the displacement vector, r, i.e., the process is isotropic. An integral formulation is used to account for the effect of the interactions with vegetation existing at all points, x′(x′, y′), in Ω [e.g., Murray, 1989]. For each of the two climate-controlled states (i.e., stressed and unstressed conditions) vegetation dynamics (1) can be expressed as

  • equation image

where the subscripts 1 and 2 denote the local dynamics of stressed and unstressed vegetation, respectively, while V represents vegetation biomass, which is normalized with respect to the ecosystem carrying capacity in state 2 (i.e., 0 ≤ V ≤ 1). The local dynamics are expressed by f1,2(V) as in equation (2) and (3). The weighting (or “kernel”) function, w(r), is here modeled as the difference between two Gaussian functions [e.g., Murray, 1989]

  • equation image

where b1 and b2 express the relative importance of facilitation and competition processes, while d1 and d2 are related to the radii of canopy and root footprints, respectively. The kernel function, w(r), has qualitatively the shape shown in Figure A1a when d1 < d2 and b1 > b2.

image

Figure A1. (a) Example of kernel function (equation (6)) used to model the spatial dynamics of facilitation and competition. (b) Relation between growth factor, γ, and wave number, k, in solutions of (A4) expressed as linear perturbations of V0 (ζ = 0.4; χ = 2.0; ɛ = 0.25; η = 1.454).

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[14] When dichotomous Markov noise is used to model the random switching, vegetation dynamics can be expressed by the stochastic integral-differential equation

  • equation image

with ξDM being a dichotomous Markov noise assuming values −1 and 1 with probability P and (1 − P), respectively, and f± (V(x, t)) = (f1 (V(x, t)) ± f2 (V(x, t)))/2.

[15] Buceta et al. [2002] have shown how systems with fast switching between two processes having the same Swift-Hohenberg spatial coupling but different local dynamics may be able to exhibit pattern formation even though – separately - neither of the two processes is capable of generating patterns. We apply the same approach to the case in which spatial interactions are expressed by the integral term in (A1).

[16] If the random switching is fast with respect to the relaxation times of the deterministic dynamics, pattern emergence can be investigated replacing the (stochastic) local term in (A3) with its average (equation (4)). In dimensionless form equation (A3) becomes

  • equation image

with

  • equation image

where ζ and ɛ < 1 represent the relative importance of local/spatial dynamics (see equation (1)), and of competition/facilitation processes, respectively; χ > 1 depends on the ratio between the radii of root system and canopy footprints, while η expresses the relative importance of logistic growth and stress-induced mortality. To simplify the notation, in what follows we drop the header “∼” and indicate with x(x, y) the dimensionless coordinate vector.

[17] The homogeneous steady states, are obtained as solutions of (A4) for V = V0 = const.

  • equation image

[18] In particular, we study the linear stability of the homogeneous stable state

  • equation image

by seeking for solutions of (A4) in the form of a sum of V0,with a perturbation term, δV,

  • equation image

where equation imageV is the amplitude of the perturbation, γ is its growth factor, k the wave-number vector, i = equation image the imaginary unit, and “·” the scalar product operator. V = V0 is unstable when γ > 0, because any disturbance, δV, of V0 would indefinitely grow with time. To determine the relation between growth factor and wave number in (A7), we insert equation (A7) in (A4), and obtain (after a Taylor expansion for small values of equation imageV)

  • equation image

with k = ∣k∣ and W(k) = equation imagew(∣x∣)eik · xdx. Figure A1b shows that, for given values of ζ, ɛ, and η, γ is negative both for P = 0 and for P = 1, indicating that the corresponding homogeneous solutions (i.e., V0 = 0 and V0 = 1, respectively) are stable. Thus, no patterns emerge when, in the absence of climate fluctuations (i.e., with no switching), the system evolves towards homogeneous unvegetated (P = 0; Figure A1b, dotted line) or vegetated (P = 1; Figure A1b, dashed line) states. γ may be positive for intermediate values of P, indicating how climate-driven random switching between deterministic dynamics may indeed trigger instability. The instability of the state, V0, is associated with the emergence of spatial patterns when the most unstable mode, kmax, is different from zero (uniform solution), i.e., when W(0) < 0 [see Murray, 1989, p. 488]. kmax can be determined from equation (A8), kmax = equation image. The condition of marginal stability (γmax = γ(kmax) = 0) becomes

  • equation image

with V0 in (A9) being a function of P (equation 10). Because in Figures 2 and A1 the most unstable mode, kmax, is greater than zero (i.e., χ > 1/equation image), the instability (i.e., ζ > ζ*) of the homogeneous stable state, V0, is associated with the emergence of spatial patterns.

[19] Numerical simulations were carried out to simulate these patterns and to investigate the effects of the bound V = 0 on the stability of the state V0 = 0. To this end, equation (A4) was solved by finite differences (in time) over a square lattice (128 pixels × 128 pixels, with pixels having about the same size as an individual plant canopy), using a simulation length that allowed for 10,000 switches between the stressed and unstressed deterministic dynamics. In these simulations the deterministic vegetation decay was faster than vegetation encroachment and the ratio between the dynamical timescale and the characteristic time of the noise was equal to 25. The numerical results (Figure 1) were –qualitatively - in good agreement with the analytical model (equation (A8)).

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References

[20] This research was funded by the Fondazione CRT, Cassa di Risparmio di Torino, CESMO (through Politecnico di Torino), and by NASA (grant NNG-04-GM71G).

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Results
  5. 3. Discussion and Conclusions
  6. Appendix A
  7. Acknowledgments
  8. References