Fig 1.
Example systems with alternative stable states in space.
(a) Shallow lake: clear water with Chara vegetation vs. turbid water (photo by Ruurd Noordhuis). (b) Salt marsh: vegetation vs. bare marshland (photo by Johan van de Koppel). (c) Musselbed: mussels vs. bare soil (photo by Andre Meijboom).
Fig 2.
Simulations in a homogeneous landscape with local alternative states and diffusion of the modeled species.
Initially, the left side of the landscape is set to the high biomass state, and the right side to the low biomass state. With these initial conditions a moving front establishes, shifting the entire landscape to the state with the highest resilience (a) c = 2.2, (b) c = 2.3487, (c) c = 2.5 (g m−1 d−1). The four figures in each panel represent: 1) snapshots of the moving front with the grey arrows indicating the shifting direction (scale = 30 m), 2) the local change in biomass per day due to growth and mortality (fN), 3) the local change in biomass per day due to diffusion (fD), and 4) the net local change in biomass per day. Note that local dynamics and diffusion precisely cancel out if conditions are such that the modeled system is at the Maxwell point (panel b). The scale of all change-in-biomass plots ranges from -0.5 to 0.5 (g m−1 d−1).
Fig 3.
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.
Fig 4.
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.
Fig 5.
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.
Fig 6.
The effect of spatially heterogeneous conditions.
A gradual increase in growth conditions in space (e.g. north-south gradient in temperature) results in a distinct shift in space from the low biomass state to the high biomass state (i.e. a stable standing wave) on the location where conditions cross the Maxwell point. Upper panel: The solid black line represents the local growth rate r on the landscape. The solid grey line indicates where the conditions cross the Maxwell point (MP), and the two dashed grey lines indicate where the two fold bifurcations (F) are crossed. Lower panel: The shaded area represents the stable end configuration of biomass for any initial configuration (c = 2.35 g m−1 d−1).
Fig 7.
Travelling wave-type of spread of aquatic vegetation (Chara spec).
Lake Veluwe, the Netherlands, from 1993 to 1999 (from: Monitoring of aquatic vegetation of the IJsselmeer Area by Rijkswaterstaat, an Agency of the Ministry of Infrastructure and the Environment, The Netherlands).