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Spatial systems with heterogeneities are ubiquitous in nature, from precipitation, temperature, and soil gradients controlling vegetation growth to morphogen gradients controlling gene expression in embryos. Such systems, generally described by nonlinear dynamical systems, often display complex parameter dependence and exhibit bifurcations. The dynamics of heterogeneous spatially extended systems passing through bifurcations are still relatively poorly understood, yet recent theoretical studies and experimental data highlight the resulting complex behaviors and their relevance to real-world applications. We explore the consequences of spatial heterogeneities passing through bifurcations via two examples strongly motivated by applications. These model systems illustrate that studying heterogeneity-induced behaviors in spatial systems is crucial for a better understanding of ecological transitions and functional organization in brain development. [ABSTRACT FROM AUTHOR]

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-field limit is a spatially extended non-Markovian process characterized by nonlocal integro-differential equations describing the evolution of the probability for a patch of land to be in a given state (the generalized Kolmogorov equations (GKEs) of the process). We thus provide an accessible general framework for spatially extending many classical finite-state models from ecology and population dynamics. We demonstrate the practical effectiveness of our approach through a detailed comparison of our limiting spatial model and the finite-size version of a specific savanna-forest model, the so-called Staver--Levin model. There is remarkable dynamic consistency between the GKEs and the finite-size system in spite of almost sure forest extinction in the finite-size system. To resolve this apparent paradox, we show that the extinction rate drops sharply when nontrivial equilibria emerge in the GKEs, and that the finite-size system's quasi-stationary distribution (stationary distribution conditional on nonextinction) closely matches the bifurcation diagram of the GKEs. Furthermore, the limit process can support periodic oscillations of the probability distribution and thus provides an elementary example of a jump process that does not converge to a stationary distribution. In spatially extended settings, environmental heterogeneity can lead to waves of invasion and front-pinning phenomena. [ABSTRACT FROM AUTHOR]