Stochastic Spatial Models in Ecology: A Statistical Physics Approach

Abstract

Ecosystems display a complex spatial organization. Ecologists have long tried to characterize them by looking at how different measures of biodiversity change across spatial scales. Ecological neutral theory has provided simple predictions accounting for general empirical patterns in communities of competing species. However, while neutral theory in well-mixed ecosystems is mathematically well understood, spatial models still present several open problems, limiting the quantitative understanding of spatial biodiversity. In this review, we discuss the state of the art in spatial neutral theory. We emphasize the connection between spatial ecological models and the physics of non-equilibrium phase transitions and how concepts developed in statistical physics translate in population dynamics, and vice versa. We focus on non-trivial scaling laws arising at the critical dimension \(D = 2\) of spatial neutral models, and their relevance for biological populations inhabiting two-dimensional environments. We conclude by discussing models incorporating non-neutral effects in the form of spatial and temporal disorder, and analyze how their predictions deviate from those of purely neutral theories.

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Acknowledgements

MAM is grateful to the Spanish-MINECO for financial support (under Grant FIS2013-43201-P; FEDER funds), as well as to J. Hidalgo, S. Suweis, A. Maritan, C. Borile for a long term collaboration on topics related to the content of this paper.

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Appendix: General Scaling Relationships

Appendix: General Scaling Relationships

In this brief Appendix, we discuss general condition imposed on the functions f and g by the properties of the function \(\varPsi \), depending on the exponent \(\varDelta \), see Eq. (13), Eq. (14) and [105]. Let us write the normalization condition for P(nA)

$$\begin{aligned} \sum _n P(n;A) \approx g(A) f(A) \int _{n_0/f(A)}^\varLambda dx ~x^{-\varDelta } =1 . \end{aligned}$$
(30)

The infrared cutoff \(\varLambda \) is related to the fact that the function \(\psi (x)\) is a power-law for small x only and rapidly decays for larger arguments, see e.g. Fig. 5. The integral is singular for small x and \(\varDelta >1\) and thus

$$\begin{aligned} 1 \sim g(A) f(A) f(A)^{\varDelta -1} = g(A)f(A)^{\varDelta }\ . \end{aligned}$$
(31)

On the other hand, if \(\varDelta <1\), the integral is weakly dependent on f(A), so that

$$\begin{aligned} 1 \sim g(A) f(A)\ . \end{aligned}$$
(32)

Similarly, the first moment of \(\varPsi \) is

$$\begin{aligned} \langle n \rangle \sim g(A) f^2(A) f(A)^{\varDelta -1} = g(A) f(A)^{\varDelta +1} \end{aligned}$$
(33)

if \(1<\varDelta <2\) and

$$\begin{aligned} \langle n \rangle \sim g(A) f^2(A) \end{aligned}$$
(34)

for \( \varDelta >2\). Combining the expressions above, different regimes emerge as a function of \(\varDelta \): if \(\varDelta <1\), \( f(A) = \langle n \rangle \), while for \(1< \varDelta <2\), \( f(A) = \langle n \rangle ^{1/(2-\varDelta )} \), while no specific prediction for f(A) can be made in the case \(\varDelta \ge 2\). In particular, for \(\varDelta <1\) one has a simple scaling form \(f(A) =\langle n \rangle \) and \(g(A)= 1/\langle n \rangle \) which applies, for example, to the 1D case as described in the main text. The marginal case \(\varDelta =1\) is treated in detail in Sect. 2.5.

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Pigolotti, S., Cencini, M., Molina, D. et al. Stochastic Spatial Models in Ecology: A Statistical Physics Approach. J Stat Phys 172, 44–73 (2018). https://doi.org/10.1007/s10955-017-1926-4

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Keywords

  • Neutral theory
  • Voter model
  • Community ecology
  • Non-equilibrium phase transitions