Dottorato in Scienze dei Cambiamenti Climatici – XXIV ciclo

Raffaele Corrado

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Time fluctuations of vegetation patterns and early warning signals of desertification transition in semi-arid ecosystems

Introduction

The identification of early warning signals of regime shifts in ecosystems is a crucial issue since these transitions can cause severe losses of ecological and economic resources.

Environmental stresses induced either by an excess of anthropic load or by an increased frequency of meteorological extremes, can give rise in arid or semi-arid ecosystems to a desertification transition.

Many recent studies highlighted the interest in vegetation patchiness analyses as a tool to provide indicators of desertification risk [1-3].

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Outline

Model of desertification transition [1]: change of the cluster size distribution is taken as indicator of transition.

Shnerb-Manor [2]: explanation of the cluster size behavior reported by Kefi et al. in terms of birth-death process.

Instead of considering the spatial fluctuations of the vegetation pattern, we analyzed the time fluctuations of several global quantities as a function of the different model parameters. Here we focus on the role of the mortality m (external bias: the simplest way to account for habitat loss).

We found: a) Critical behavior of the system for m~mc

b) Sharp peak of the variance of the sizes of the largest cluster Smax at m* < mc

c) Strong non-Gaussianity of Smax distribution for m < m* d) Change of the skewness of Smax distribution at m ~ m*

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Kéfi model

w0, += [+1−q+∣0 ] b−c+

w0, -=d w- ,0=r f q+∣-w+ ,0=m

The model of Kéfi et al., based on a stochastic cellular automaton, simulates the evolution of vegetation in arid ecosystems. It evaluates the effects of several biological mechanisms modeled by parameters (δ, b, c, m, d, r, f). Each cell can be alive (state '+'), dead (state '0') or degraded (state '-'). Transitions between different states stochastically depend on the values of the parameters

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The analysis of both field data and numerical simulations show that far from transition, vegetation patch-size distribution follows a power law. However, at increasing m or at decreasing f, this distribution significantly deviates from a power law, showing an exponential cut-off.

Kefi et al. suggested that truncated power-law patch-size distributions could be a warning signal for the onset of desertification

Kéfi model

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Time series of densities

We performed numerical simulations (105 time steps) of a 100x100 CA. We analized time series of the living cell density, ρ+, for several values of mortality m. We found a critical value mc, i.e. the maximum value allowing a stationary state with ⟨ρ+⟩ non-zero.

⟨+⟩ vs m +/ ⟨+⟩ vs m

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m=C1mc−m1

m=C2 [ m

mc−m ]2

=C 2

C1

m2

mc−m12

C1=1.80±0.03, 1=0.390±0.005

C 2=0.0050±0.0003, 2=0.260±0.009

Time series of densities

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We also analyzed the ρ+ fluctuations distribution.

When m~mc, the PDF of fluctuations exhibits a strong non-Gaussianity.

Time series of densities

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Time series of largest cluster

Largest cluster size (Smax) of the living cells as a function of m. For low m there is a spanning cluster.

At increasing m it exists m*<mc such that the largest cluster breaks up into many small clusters (percolation threshold of the ρ+ phase).

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⟨S max ⟩ vs m

Time series of largest cluster

Standard deviation of the largest cluster size as a function of m.

The peak of this function is at m*~0.114.

Near this value of m the largest percolating cluster breaks up.

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S maxvs m

Time series of largest cluster

m=0.10

for ρ+: regular oscillations around an average value

for Smax: “intermittency” and large fluctuations under the average value

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Time series of largest cluster

m=0.16

m=0.16 is bigger than m* but the system is far from extinction

For this m we observe intermittency and large fluctuations above the average value

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Time series of largest cluster

m=0.16

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In figure: more than 300 cells in a large cluster, while mean value of Smax is 92.

This corresponds to a peak in time series.

Non-gaussianity of Smax fluctuations(Smax: size of largest cluster)

Time series of Smax at increasing values of m are clearly asymmetric. To quantify this aspect we computed the distribution density function of Smax fluctuations

m=0.06 m=0.08

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Non-gaussianity of Smax fluctuations

m=0.10

m=0.12Lecce, 2011/07/21 - 15 of 21

Non-gaussianity of Smax fluctuations

When m~mc the distribution becomes 'size-independent'.

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Non-gaussianity of Smax fluctuations

y=Kea [b y−s−eb y−s ] , y=S max−⟨S max ⟩

, y= y

Distributions are Gaussian only for very small m; for increasing values of m, a non-gaussian left or right tail appears. When m=m* the sign of the skewness changes, and the functional form of the distribution isn't evident.For high values of m, far from m*, the distribution is well described by the Bramwell-Holdsworth-Pinton distribution, analitically approximated by a Generalized Gumbel distribution

K=2.14, a=

2,b=0.938, s=0.374 ;

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Work in progress:

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We are studying the role of the other parameters:

For example, how mc depends on parameters?

For the meaning of b and f see references, at greater values corresponds better conditions.

Max mortality mc is practically linear in b and f .

Work in progress:

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Probability distribution of changes of the S&P500 index (left)

Probability distribution of changes of the largest cluster's size (right)

i−⟨⟩/

We are studying the time series of increments and we have found similarities with quite different phenomena:

Conclusions

Power law and truncated power law distribution of cluster sizes are common features in stochastic CA with underlying birth-death processes [2].

BHP distribution is frequently observed to describe the fluctuations of global quantities of systems with extended spatial and temporal correlations [3,4,5].

Here, the change of skewness observed for the distribution of the largest cluster size fluctuations is understood in term of percolation.

We propose this change of skewness and non-Gaussianity associated with negative skewness at m<m* as signals of a transition to desertification. Unlike [1], these signals arise when the ecosystems is apparently not a risk.

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References

[1] S. Kefi et al., Nature, 449, 213, 2007.[2] A. Manor, N. M. Shnerb, Phys. Rev. Let., 101,

268104, 2008; ibidem, 103, 030601, 2009. [3] M. Clusel, J.F. Fortin, P.C. Holdsworth, Phys.

Rev. E, 70, 046112, 2004. [4] S.T. Bramwell et al., Phys. Rev. Let., 84, 3744,

2000; S.T. Bramwell, Nature, 5, 443, 2009. [5] C. Pennetta, E. Alfinito. L. Reggiani and S. Ruffo, Physica

A, 340, 380 (2004).

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