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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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The effect of spatially heterogeneous diffusion.A travelling wave of collapsing biomass triggered by a disturbance can come to a halt if it meets an area of increased diffusion rates. (a) The effect is illustrated in a simulated landscape with heterogeneous diffusion rates (c = 2.4 g m−1 d−1) (upper panel). The dashed line in the lower panel represents the initial disturbance and the solid lines depict the transient situation every 40 days. The shaded area depicts the final stable configuration. This configuration is stable, as long as the system does not suffer from other local disturbances. (b) In order to understand the conditions for pinning, we introduced a local disturbance in a landscape with a single spatial gradient in diffusion rate, representing a change from an area with low diffusion (D0) to an area with high diffusion (D0+ Dplus) (visualized in the small upper panels). The landscape was created by a sigmoidal function: (D0 = 1 m2d−1,p = 50, L = 100 m). The main panel represents the occurrence of pinning for different combinations of maximal mortality rate c and the level of increase in diffusion rate Dplus. Importantly, pinning only occurs if a traveling wave meets an area in which diffusion is higher. The thick black dashed line indicates the Maxwell point.
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pone.0116859.g005: The effect of spatially heterogeneous diffusion.A travelling wave of collapsing biomass triggered by a disturbance can come to a halt if it meets an area of increased diffusion rates. (a) The effect is illustrated in a simulated landscape with heterogeneous diffusion rates (c = 2.4 g m−1 d−1) (upper panel). The dashed line in the lower panel represents the initial disturbance and the solid lines depict the transient situation every 40 days. The shaded area depicts the final stable configuration. This configuration is stable, as long as the system does not suffer from other local disturbances. (b) In order to understand the conditions for pinning, we introduced a local disturbance in a landscape with a single spatial gradient in diffusion rate, representing a change from an area with low diffusion (D0) to an area with high diffusion (D0+ Dplus) (visualized in the small upper panels). The landscape was created by a sigmoidal function: (D0 = 1 m2d−1,p = 50, L = 100 m). The main panel represents the occurrence of pinning for different combinations of maximal mortality rate c and the level of increase in diffusion rate Dplus. Importantly, pinning only occurs if a traveling wave meets an area in which diffusion is higher. The thick black dashed line indicates the Maxwell point.

Mentions: We first simulated landscapes with spatially variable diffusion rates (Fig. 5). This can be seen as an intermediate between two simplified spatial models: a spatially discrete model with exchange between patches and a continuous model with homogeneous diffusion (see S3 Text and S5 Fig.). The results show that a travelling wave can slow down and can come to a halt, if it meets an area of increasing diffusion rates (Fig. 5A). Whether a travelling wave comes to a halt, so-called ‘pinning’, depends on three factors. Pinning is more likely if i) the actual level of increase in diffusion (Dplus) is high (Fig. 5B), ii) the scale on which the increase in diffusion occurs is low, i.e. the steepness of the diffusion function (set by p in the sigmoidal function described in the caption of Fig. 5) is high (S3 Text, S6 Fig.), and iii) there is little difference between the resilience of the alternative states, causing the system to be close to the Maxwell point (Fig. 5B). Thus even if growth and mortality rates are homogeneous, spatial variation in exchange or dispersal rates can allow spatial co-existence of alternative stable states.


Resilience of alternative states in spatially extended ecosystems.

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

The effect of spatially heterogeneous diffusion.A travelling wave of collapsing biomass triggered by a disturbance can come to a halt if it meets an area of increased diffusion rates. (a) The effect is illustrated in a simulated landscape with heterogeneous diffusion rates (c = 2.4 g m−1 d−1) (upper panel). The dashed line in the lower panel represents the initial disturbance and the solid lines depict the transient situation every 40 days. The shaded area depicts the final stable configuration. This configuration is stable, as long as the system does not suffer from other local disturbances. (b) In order to understand the conditions for pinning, we introduced a local disturbance in a landscape with a single spatial gradient in diffusion rate, representing a change from an area with low diffusion (D0) to an area with high diffusion (D0+ Dplus) (visualized in the small upper panels). The landscape was created by a sigmoidal function: (D0 = 1 m2d−1,p = 50, L = 100 m). The main panel represents the occurrence of pinning for different combinations of maximal mortality rate c and the level of increase in diffusion rate Dplus. Importantly, pinning only occurs if a traveling wave meets an area in which diffusion is higher. The thick black dashed line indicates the Maxwell point.
© Copyright Policy
Related In: Results  -  Collection

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

pone.0116859.g005: The effect of spatially heterogeneous diffusion.A travelling wave of collapsing biomass triggered by a disturbance can come to a halt if it meets an area of increased diffusion rates. (a) The effect is illustrated in a simulated landscape with heterogeneous diffusion rates (c = 2.4 g m−1 d−1) (upper panel). The dashed line in the lower panel represents the initial disturbance and the solid lines depict the transient situation every 40 days. The shaded area depicts the final stable configuration. This configuration is stable, as long as the system does not suffer from other local disturbances. (b) In order to understand the conditions for pinning, we introduced a local disturbance in a landscape with a single spatial gradient in diffusion rate, representing a change from an area with low diffusion (D0) to an area with high diffusion (D0+ Dplus) (visualized in the small upper panels). The landscape was created by a sigmoidal function: (D0 = 1 m2d−1,p = 50, L = 100 m). The main panel represents the occurrence of pinning for different combinations of maximal mortality rate c and the level of increase in diffusion rate Dplus. Importantly, pinning only occurs if a traveling wave meets an area in which diffusion is higher. The thick black dashed line indicates the Maxwell point.
Mentions: We first simulated landscapes with spatially variable diffusion rates (Fig. 5). This can be seen as an intermediate between two simplified spatial models: a spatially discrete model with exchange between patches and a continuous model with homogeneous diffusion (see S3 Text and S5 Fig.). The results show that a travelling wave can slow down and can come to a halt, if it meets an area of increasing diffusion rates (Fig. 5A). Whether a travelling wave comes to a halt, so-called ‘pinning’, depends on three factors. Pinning is more likely if i) the actual level of increase in diffusion (Dplus) is high (Fig. 5B), ii) the scale on which the increase in diffusion occurs is low, i.e. the steepness of the diffusion function (set by p in the sigmoidal function described in the caption of Fig. 5) is high (S3 Text, S6 Fig.), and iii) there is little difference between the resilience of the alternative states, causing the system to be close to the Maxwell point (Fig. 5B). Thus even if growth and mortality rates are homogeneous, spatial variation in exchange or dispersal rates can allow spatial co-existence of alternative stable states.

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