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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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Resilience to local disturbances on a small and a large landscape.(a) Local disturbances to the alternative stable state (i.e. the low biomass state) were performed on one side of the landscape, in order to have a symmetrical landscape. (b) Mean biomass on the landscape in equilibrium (Ntotal/L) as a function of the maximal mortality rate for a system on a relatively small landscape (L = 5 m). The solid parts of the curve represent the two stable landscape-wide equilibria. The dashed part of the curve represents the disturbance threshold, i.e. the size of the disturbed patch needed to induce a systemic shift to the alternative stable landscape-wide state. (c) The same as panel b but for a system on a large landscape (L = 50 m). Note that the disturbance threshold remains very close to the less resilient of the two stable equilibria implying that only a small disturbance is needed to induce a shift to the more resilient landscape-wide state. (d) Resilience of the high biomass state, in terms of the fraction of the landscape that needs to be perturbed to the alternative state to trigger a shift (Δx/L), for a system on a small landscape. (e) The same as panel d but for a system on a large landscape. Note that resilience shows a steep drop. (f) Engineering resilience of the high biomass state, in terms of the recovery rate of a local disturbance to the low biomass state for a system on a small landscape. (g) The same as panel f, but for a system on a large landscape.
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pone.0116859.g004: Resilience to local disturbances on a small and a large landscape.(a) Local disturbances to the alternative stable state (i.e. the low biomass state) were performed on one side of the landscape, in order to have a symmetrical landscape. (b) Mean biomass on the landscape in equilibrium (Ntotal/L) as a function of the maximal mortality rate for a system on a relatively small landscape (L = 5 m). The solid parts of the curve represent the two stable landscape-wide equilibria. The dashed part of the curve represents the disturbance threshold, i.e. the size of the disturbed patch needed to induce a systemic shift to the alternative stable landscape-wide state. (c) The same as panel b but for a system on a large landscape (L = 50 m). Note that the disturbance threshold remains very close to the less resilient of the two stable equilibria implying that only a small disturbance is needed to induce a shift to the more resilient landscape-wide state. (d) Resilience of the high biomass state, in terms of the fraction of the landscape that needs to be perturbed to the alternative state to trigger a shift (Δx/L), for a system on a small landscape. (e) The same as panel d but for a system on a large landscape. Note that resilience shows a steep drop. (f) Engineering resilience of the high biomass state, in terms of the recovery rate of a local disturbance to the low biomass state for a system on a small landscape. (g) The same as panel f, but for a system on a large landscape.

Mentions: So far, we considered a landscape without borders, to rule out any edge effects. We now turn to the more realistic case of finite landscapes. In case the landscape is small (or diffusion is high, which is mathematically the same) the system does in practice behave like a well-mixed system (Fig. 4B). Consequently, shifts will occur almost simultaneously across the landscape. In contrast, if the landscape is large, a local disturbance can cause a travelling wave that propagates through space, always in the direction of the least stable state. A relatively small disturbance can therefore already lead to systemic collapse (Fig. 4C). As a result, resilience, here defined as the capacity to recover upon a local disturbance, of the alternative states changes abruptly around the Maxwell point (Fig. 4E, see S2 Fig. for other models). As stress on the dominant state increases (e.g. through the mortality rate in our model), resilience to local disturbances remains unaltered in the sense that the system will recover, even from large disturbances, until a critical point is reached (the Maxwell point) where resilience sharply drops to a point where even a local disturbance can induce a traveling wave that will eventually bring the entire landscape in the alternative state. This is quite unlike the gradual decrease of resilience on a small landscape (Fig. 4D), corresponding to the classical well-mixed situation.


Resilience of alternative states in spatially extended ecosystems.

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

Resilience to local disturbances on a small and a large landscape.(a) Local disturbances to the alternative stable state (i.e. the low biomass state) were performed on one side of the landscape, in order to have a symmetrical landscape. (b) Mean biomass on the landscape in equilibrium (Ntotal/L) as a function of the maximal mortality rate for a system on a relatively small landscape (L = 5 m). The solid parts of the curve represent the two stable landscape-wide equilibria. The dashed part of the curve represents the disturbance threshold, i.e. the size of the disturbed patch needed to induce a systemic shift to the alternative stable landscape-wide state. (c) The same as panel b but for a system on a large landscape (L = 50 m). Note that the disturbance threshold remains very close to the less resilient of the two stable equilibria implying that only a small disturbance is needed to induce a shift to the more resilient landscape-wide state. (d) Resilience of the high biomass state, in terms of the fraction of the landscape that needs to be perturbed to the alternative state to trigger a shift (Δx/L), for a system on a small landscape. (e) The same as panel d but for a system on a large landscape. Note that resilience shows a steep drop. (f) Engineering resilience of the high biomass state, in terms of the recovery rate of a local disturbance to the low biomass state for a system on a small landscape. (g) The same as panel f, but for a system on a large landscape.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4340810&req=5

pone.0116859.g004: Resilience to local disturbances on a small and a large landscape.(a) Local disturbances to the alternative stable state (i.e. the low biomass state) were performed on one side of the landscape, in order to have a symmetrical landscape. (b) Mean biomass on the landscape in equilibrium (Ntotal/L) as a function of the maximal mortality rate for a system on a relatively small landscape (L = 5 m). The solid parts of the curve represent the two stable landscape-wide equilibria. The dashed part of the curve represents the disturbance threshold, i.e. the size of the disturbed patch needed to induce a systemic shift to the alternative stable landscape-wide state. (c) The same as panel b but for a system on a large landscape (L = 50 m). Note that the disturbance threshold remains very close to the less resilient of the two stable equilibria implying that only a small disturbance is needed to induce a shift to the more resilient landscape-wide state. (d) Resilience of the high biomass state, in terms of the fraction of the landscape that needs to be perturbed to the alternative state to trigger a shift (Δx/L), for a system on a small landscape. (e) The same as panel d but for a system on a large landscape. Note that resilience shows a steep drop. (f) Engineering resilience of the high biomass state, in terms of the recovery rate of a local disturbance to the low biomass state for a system on a small landscape. (g) The same as panel f, but for a system on a large landscape.
Mentions: So far, we considered a landscape without borders, to rule out any edge effects. We now turn to the more realistic case of finite landscapes. In case the landscape is small (or diffusion is high, which is mathematically the same) the system does in practice behave like a well-mixed system (Fig. 4B). Consequently, shifts will occur almost simultaneously across the landscape. In contrast, if the landscape is large, a local disturbance can cause a travelling wave that propagates through space, always in the direction of the least stable state. A relatively small disturbance can therefore already lead to systemic collapse (Fig. 4C). As a result, resilience, here defined as the capacity to recover upon a local disturbance, of the alternative states changes abruptly around the Maxwell point (Fig. 4E, see S2 Fig. for other models). As stress on the dominant state increases (e.g. through the mortality rate in our model), resilience to local disturbances remains unaltered in the sense that the system will recover, even from large disturbances, until a critical point is reached (the Maxwell point) where resilience sharply drops to a point where even a local disturbance can induce a traveling wave that will eventually bring the entire landscape in the alternative state. This is quite unlike the gradual decrease of resilience on a small landscape (Fig. 4D), corresponding to the classical well-mixed situation.

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