# Probabilistic Foundations of Spatial Mean-Field Models in Ecology and Applications

@article{Patterson2020ProbabilisticFO, title={Probabilistic Foundations of Spatial Mean-Field Models in Ecology and Applications}, author={Denis D. Patterson and Simon A. Levin and Carla Staver and Jonathan Touboul}, journal={SIAM J. Appl. Dyn. Syst.}, year={2020}, volume={19}, pages={2682-2719} }

Deterministic models of vegetation often summarize, at a macroscopic scale, a multitude of intrinsically random events occurring at a microscopic scale. We bridge the gap between these scales by demonstrating convergence to a mean-field limit for a general class of stochastic models representing each individual ecological event in the limit of large system size. The proof relies on classical stochastic coupling techniques that we generalize to cover spatially extended interactions. The mean…

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## References

SHOWING 1-10 OF 64 REFERENCES

### On the complex dynamics of savanna landscapes

- Environmental ScienceProceedings of the National Academy of Sciences
- 2018

Analysis of a set of simple, yet realistic, models for savanna–forest vegetation dynamics based on minimal ecological assumptions reveals that diverse cyclic behaviors occur for broad ranges of parameter space and that large shifts in landscape structure can result from endogenous dynamics and not just from external drivers or from noise.

### Tropical tree cover in a heterogeneous environment: A reaction-diffusion model

- Environmental SciencePloS one
- 2019

The Maxwell point of the homogeneous reaction-diffusion equation without savanna trees is derived and it is shown that the front between forest and nonforest settles at this point as long as savanna tree cover near the front remains sufficiently low.

### Mean field limit for disordered diffusions with singular interactions

- Mathematics
- 2013

Motivated by considerations from neuroscience (macroscopic behavior of large ensembles of interacting neurons), we consider a population of mean field interacting diffusions in $\mathbf {R}^m$ in the…

### Propagation of chaos in neural fields

- Mathematics
- 2014

The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which the authors can resolve quite fine structure of the neuron's activity in space and where averaging effects occur.

### Allee Effects, Invasion Pinning, and Species’ Borders

- Environmental ScienceThe American Naturalist
- 2001

The properties of invasion models when a species cannot persist below a critical population density known as an “Allee threshold” are studied, suggesting caution when interpreting abrupt range limits as stemming either from competition between species or a hard environmental limit that cannot be crossed.

### Spatial feedbacks and the dynamics of savanna and forest

- Environmental ScienceTheoretical Ecology
- 2019

Theoretical models and empirical analyses have combined to suggest that, for some ranges of climate, alternative stable states in vegetation structure may be possible: positive feedbacks with fire…

### Transient phenomena in ecology

- Environmental ScienceScience
- 2018

Current knowledge of transient dynamics is reviewed, showing that hitherto idiosyncratic and individual patterns can be classified into a coherent framework, with important general lessons and directions for future study.

### Dispersal increases the resilience of tropical savanna and forest distributions

- Environmental SciencebioRxiv
- 2018

This work examines the contributions of dispersal to determining savanna and forest distributions using a reaction-diffusion model, and suggests that both models are limited in that they ignore spatial processes, such as dispersal.

### Seed Dispersal Near and Far: Patterns Across Temperate and Tropical Forests

- Environmental Science
- 1999

Dispersal affects community dynamics and vegetation response to global change. Understanding these effects requires descriptions of dispersal at local and regional scales and statistical models that…

### Spatial aspects of interspecific competition.

- Economics, MathematicsTheoretical population biology
- 1998

It is proved rigorously that if one species has a competitive advantage over each of the others, then eventually it takes over all the sites in the system and produces a stationary distribution with an interesting spatial structure.