Resilience of Alternative States in Spatially Extended Ecosystems

Abstract

Alternative stable states in ecology have been well studied in isolated, well-mixed systems. However, in reality, most ecosystems exist on spatially extended landscapes. Applying existing theory from dynamic systems, we explore how such a spatial setting should be expected to affect ecological resilience. We focus on the effect of local disturbances, defining resilience as the size of the area of a strong local disturbance needed to trigger a shift. We show that in contrast to well-mixed systems, resilience in a homogeneous spatial setting does not decrease gradually as a bifurcation point is approached. Instead, as an environmental driver changes, the present dominant state remains virtually ‘indestructible’, until at a critical point (the Maxwell point) its resilience drops sharply in the sense that even a very local disturbance can cause a domino effect leading eventually to a landscape-wide shift to the alternative state. Close to this Maxwell point the travelling wave moves very slow. Under these conditions both states have a comparable resilience, allowing long transient co-occurrence of alternative states side-by-side, and also permanent co-existence if there are mild spatial barriers. Overall however, hysteresis may mostly disappear in a spatial context as one of both alternative states will always tend to be dominant. Our results imply that local restoration efforts on a homogeneous landscape will typically either fail or trigger a landscape-wide transition. For extensive biomes with alternative stable states, such as tundra, steppe and forest, our results imply that, as climatic change reduces the stability, the effect might be difficult to detect until a point where local disturbances inevitably induce a spatial cascade to the alternative state.

Simplifying complexity : reflections on ecosystems, psychiatric disorders and biogeochemical pathways

“…This is because the low biomass state is actually stronger than the high biomass state under these conditions. However, the perturbation is too small to create an actual travelling front of an expanding perturbation (see van de Leemput et al (2015)). While the system does recover eventually, such non-exponential recovery trajectory obviously hints at the proximity of the system to a runaway cascading collapse of the landscape.…”

“…Strong local disturbances do not always recover (Keitt et al 2001;van de Leemput et al 2015). There are two other possibilities.…”

“…Importantly, a finer-meshed lattice did not alter the resulting dynamics for the parameters we used. Note that one should always be aware of situations of no recovery due to the discretization method used (Keitt et al 2001;van de Leemput et al 2015). All differential equations were solved using an explicit Runge-Kutta (4,5) solver with adaptive step-size.…”

“…it has an infinite recovery time, so-called 'pinning' (Keitt et al 2001;van de Leemput et al 2015). Also, in heterogeneous landscapes we can have a partial transition of the landscape (van de Leemput et al 2015).…”

Resilience of tropical forest and savanna: bridging theory and observation

“…Moreover, theoretical studies of alternative stable states often use simple models that assume well-mixed systems without explicit consideration of spatial components (Scheffer, 2009). However, the behavior of such models can change considerably when local spatial interactions are taken into account (Wilson et al, 1996;Bel et al, 2012;Van de Leemput et al, 2015;Villa Martín et al, 2015).…”