Example systems with alternative stable states in space.

<p>(<i>a</i>) Shallow lake: clear water with Chara vegetation vs. turbid water (photo by Ruurd Noordhuis). (<i>b</i>) Salt marsh: vegetation vs. bare marshland (photo by Johan van de Koppel). (<i>c</i>) Musselbed: mussels vs. bare soil (photo by Andre Meijboom).</p

Critical size of a local disturbance and the speed of a travelling wave as a function of the maximal mortality rate <i>c</i>.

<p><i>(a)</i> On an infinitely sized landscape, disturbances smaller than the critical size <i>Δx</i> (in m) are repaired, while larger disturbances will initiate a propagating wave that travels through the landscape with <i>(b)</i> 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 <i>n</i>-fold increase in diffusion rate leads to a </p><p></p><p></p><p></p><p><mi>n</mi></p><p></p><p></p><p></p>-fold increase in both critical disturbance size and wave speed.<p></p

Travelling wave-type of spread of aquatic vegetation (<i>Chara spec</i>).

<p>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).</p

Body size data of Australian diving beetles

Excel file containing 1) body sizes of individuals of diving beetle species collected in the field, 2) body size ranges for additional beetle species, taken from Watts and Humphreys 200

The effect of spatially heterogeneous diffusion.

<p>A travelling wave of collapsing biomass triggered by a disturbance can come to a halt if it meets an area of increased diffusion rates. (<i>a</i>) The effect is illustrated in a simulated landscape with heterogeneous diffusion rates (<i>c</i> = 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. (<i>b</i>) 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 (<i>D</i>_<i>0</i>) to an area with high diffusion (<i>D</i>_<i>0</i>+ <i>D</i>_<i>plus</i>) (visualized in the small upper panels). The landscape was created by a sigmoidal function: </p><p></p><p></p><p></p><p><mi>f</mi><mi>D</mi></p><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><p><mo>∂</mo></p><p><mo>∂</mo><mi>x</mi></p><p></p><p><mo>(</mo></p><p></p><p><mo>(</mo></p><p></p><p><mi>D</mi><mn>0</mn></p><mo>+</mo><p><mi>D</mi></p><p><mi>p</mi><mi>l</mi><mi>u</mi><mi>s</mi></p><p></p><p></p><p></p><p><mi>x</mi><mi>p</mi></p><p></p><p></p><p><mi>x</mi><mi>p</mi></p><mo>+</mo><p></p><p></p><p><mo>(</mo></p><p><mi>L</mi><mo>/</mo><mn>2</mn></p><mo>)</mo><p></p><p></p><mi>p</mi><p></p><p></p><p></p><p></p><mo>)</mo><p></p><p></p><p><mo>∂</mo><mi>N</mi></p><p><mo>∂</mo><mi>x</mi></p><p></p><p></p><mo>)</mo><p></p><p></p><p></p><p></p>(<i>D</i>_<i>0</i> = 1 m²d⁻¹_,<i>p</i> = 50, <i>L</i> = 100 m). The main panel represents the occurrence of pinning for different combinations of maximal mortality rate <i>c</i> and the level of increase in diffusion rate <i>D</i>_<i>plus</i>. 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.<p></p

Appendix C. Reliability in identification of increasing trends in variance and autocorrelation.

Reliability in identification of increasing trends in variance and autocorrelation

Appendix A. Examples from past climate transitions.

Examples from past climate transitions

The effects of diversity and diversity loss on the outcome of the introduction of a predator into a diverse native community.

<p>Initially, native species prevent the introduced predator from invading by reducing the predator to a low biomass. After species extinctions (shaded areas) or an increase in species-specific mortality m_F,I (arrow), at low diversity, the feedback mechanism fails and the introduced invades very suddenly. Low diversity communities have a lower initial biomass and the effect of diversity loss has a larger effect on the total biomass of less diverse systems – see supporting information S2. For clarity and ease of comparison between simulations, we here use a fixed rather than random interspecific competition coefficient (<i>α_i,j</i>) (p = 0.0015; e = 0.6; r = 1; g = 0.7; H = 20; m = 0.22; <i>α_i,j</i>₌0.3; K_i =50, m_F,i = [0,0.5], I = 5).</p

Traps and opportunities for increasing workforce diversity.

<p>Traps and opportunities for increasing workforce diversity.</p

Low diversity can be reinforced by feedbacks with three measures of inclusion: applicant diversity, appointment bias and departure bias.

<p>Low diversity can be reinforced by feedbacks with three measures of inclusion: applicant diversity, appointment bias and departure bias.</p