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Resilience of alternative states in spatially extended ecosystems.

van de Leemput IA, van Nes EH, Scheffer M - PLoS ONE (2015)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Environmental Sciences, Wageningen University, Wageningen, The Netherlands.

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.

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Critical size of a local disturbance and the speed of a travelling wave as a function of the maximal mortality rate c.(a) On an infinitely sized landscape, disturbances smaller than the critical size Δx (in m) are repaired, while larger disturbances will initiate a propagating wave that travels through the landscape with (b) a constant wave speed (in m d−1). The thick dashed line represents the Maxwell point. The thin dashed lines represent the two fold bifurcations in a non-spatial system. Left of the Maxwell point the entire landscape was initially set to the low biomass state, and the disturbance was set to the high biomass state. Right of the Maxwell point the landscape was initially set to the high biomass state, and the disturbance was set to the low biomass state (indicated by the small upper panels). In this model, an n-fold increase in diffusion rate leads to a -fold increase in both critical disturbance size and wave speed.
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pone.0116859.g003: Critical size of a local disturbance and the speed of a travelling wave as a function of the maximal mortality rate c.(a) On an infinitely sized landscape, disturbances smaller than the critical size Δx (in m) are repaired, while larger disturbances will initiate a propagating wave that travels through the landscape with (b) a constant wave speed (in m d−1). The thick dashed line represents the Maxwell point. The thin dashed lines represent the two fold bifurcations in a non-spatial system. Left of the Maxwell point the entire landscape was initially set to the low biomass state, and the disturbance was set to the high biomass state. Right of the Maxwell point the landscape was initially set to the high biomass state, and the disturbance was set to the low biomass state (indicated by the small upper panels). In this model, an n-fold increase in diffusion rate leads to a -fold increase in both critical disturbance size and wave speed.

Mentions: Moreover, for a travelling wave to develop, a critical area needs to be disturbed to the alternative state. The actual critical size of this disturbed area (Δx) increases towards the Maxwell point (Fig. 3A). The asymptotic speed of a travelling wave between alternative states is constant on a homogeneous landscape [39], and approaches zero towards the Maxwell point (Fig. 3B). A non-dimensional version of our model (see S1 Text) shows that an n-fold increase in diffusion rate D leads to a -fold increase in both x and wave speed (see S4 Fig.). Importantly, once the system has shifted to the alternative state, a wave travelling backwards cannot establish, so the new state is on its turn indestructible against local disturbances to the other state. Note that if a system is indestructible against local disturbances, such as fires or storms, it is not indestructible against disturbances that affect the entire landscape, such as periods of drought.


Resilience of alternative states in spatially extended ecosystems.

van de Leemput IA, van Nes EH, Scheffer M - PLoS ONE (2015)

Critical size of a local disturbance and the speed of a travelling wave as a function of the maximal mortality rate c.(a) On an infinitely sized landscape, disturbances smaller than the critical size Δx (in m) are repaired, while larger disturbances will initiate a propagating wave that travels through the landscape with (b) a constant wave speed (in m d−1). The thick dashed line represents the Maxwell point. The thin dashed lines represent the two fold bifurcations in a non-spatial system. Left of the Maxwell point the entire landscape was initially set to the low biomass state, and the disturbance was set to the high biomass state. Right of the Maxwell point the landscape was initially set to the high biomass state, and the disturbance was set to the low biomass state (indicated by the small upper panels). In this model, an n-fold increase in diffusion rate leads to a -fold increase in both critical disturbance size and wave speed.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4340810&req=5

pone.0116859.g003: Critical size of a local disturbance and the speed of a travelling wave as a function of the maximal mortality rate c.(a) On an infinitely sized landscape, disturbances smaller than the critical size Δx (in m) are repaired, while larger disturbances will initiate a propagating wave that travels through the landscape with (b) a constant wave speed (in m d−1). The thick dashed line represents the Maxwell point. The thin dashed lines represent the two fold bifurcations in a non-spatial system. Left of the Maxwell point the entire landscape was initially set to the low biomass state, and the disturbance was set to the high biomass state. Right of the Maxwell point the landscape was initially set to the high biomass state, and the disturbance was set to the low biomass state (indicated by the small upper panels). In this model, an n-fold increase in diffusion rate leads to a -fold increase in both critical disturbance size and wave speed.
Mentions: Moreover, for a travelling wave to develop, a critical area needs to be disturbed to the alternative state. The actual critical size of this disturbed area (Δx) increases towards the Maxwell point (Fig. 3A). The asymptotic speed of a travelling wave between alternative states is constant on a homogeneous landscape [39], and approaches zero towards the Maxwell point (Fig. 3B). A non-dimensional version of our model (see S1 Text) shows that an n-fold increase in diffusion rate D leads to a -fold increase in both x and wave speed (see S4 Fig.). Importantly, once the system has shifted to the alternative state, a wave travelling backwards cannot establish, so the new state is on its turn indestructible against local disturbances to the other state. Note that if a system is indestructible against local disturbances, such as fires or storms, it is not indestructible against disturbances that affect the entire landscape, such as periods of drought.

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Environmental Sciences, Wageningen University, Wageningen, The Netherlands.

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.

Show MeSH