El flujo sanguíneo

Fig. 4.1, 4.2 and 4.3, 4.4 respectively focus on identifying key thresholds for the modelled reef community to shift forwards (from dense seaweed cover to sea urchin barrens) and ‘backward’ s (from barren habitat to recovered seaweed bed). Tipping points are characterised in terms of the probability of long-term barren formation (e.g. Fig. 4.1a) or, conversely, the probability of long-term restoration of the seaweed bed (e.g. Fig. 4.3a), as a function of the final biomass densities of sea urchins (e.g. Fig. 4.1a), lobsters (e.g. Fig. 4.2b), and large lobsters of carapace length>140 mm (e.g. Fig. 4.3c). Monitoring monthly seaweed cover through all Monte-Carlo simulations against biomass densities of the three groups (Fig. 4.2 and 4.1) provides an alternative view of tipping points in the dynamics. Red sigmoid curves in Fig. 4.1, 4.2, 4.3 and 4.4 represent binomial logistic models fitted against the biomass densities of (a) sea urchins; (b) all size classes of rock lobster; and (c) large lobsters (carapace length>140 mm). Summary statistics (i.e. parameter estimates, variance explained by the model, Akaike Information Criterion) for the GLM associated with the establishment of barren habitat (Table 4.2) or the recovery of seaweeds (Table 4.4) show that sea urchin biomass density is the most reliable predictor of the shift to urchin barrens (81.6% of the total variance explained by the GLM; Fig. 4.1a), while rock lobster biomass density best relates to the model’s ability to return to a state of dense seaweed (62.7% of the total variance explained by the GLM; Fig.4.3b). The estimated thresholds clearly reveal the presence of a hysteresis in model dynamics, i.e. there are different tipping points associated with the ‘forward’ (Fig. 4.1 and 4.2; Table 4.2) and ‘backward’ (Fig. 4.3 and 4.4 ; Table 4.4) shifts. The community path and thresholds to restore the seaweed bed are different to the community trajectory of overgrazing driving seaweed beds to sea urchin barrens; sea urchin populations have to build to a biomass density of 10-40,000 g.200 m−2 _{for barren habitat to begin forming (Fig. 4.1a and 4.2a;}

Table 4.2a-b), while recovery of seaweeds on sea urchin barrens becomes highly likely only when there are virtually no sea urchins remaining on the reef (Fig. 4.3a and 4.4a; Table 4.4a-b). Note that, across all size classes combined, critical biomass densities of lobsters are of the same order, i.e. between 4000 and 6000 g.200m−2, for both the ‘forward’ (Fig. 4.1b and 4.2b; Table 4.2c-d) and the ‘backward’ shift (Fig. 4.3b and 4.4b; Table 4.4c-d). However, when considering the biomass density of large lobster (carapace length > 140 mm) only, the threshold biomass density for the seaweed bed to recover (Fig. 4.1c and

4.4. Results 91

4.2c; Table 4.4e-f) is _∼3-4 times higher than the critical biomass density at which sea urchin barrens can start forming (Fig. 4.3c and 4.4c; Table 4.2e-f).

4.4. Results 92

a

b

c

Figure 4.3: Probability of seaweed bed recovery through 8500 Monte- Carlo simulations initialised as sea urchin barrens (urchin culling as an extra parameter compared to ‘forward’ shift Monte-Carlo simulations). Black crosses show the final state for each simulation. The blue line with dots and standard error bars corresponds to data binned into 20 even intervals of biomass density (from 0 to the maximum value). Red sigmoid curves represent binomial logistic models fitted against the biomass densities of the different species modelled (a) sea urchins; (b) rock lobsters (all size classes combined); and (c) large lobster individuals (carapace length >140 mm). Threshold points at which the TRITON model shifts from sea urchin barren habitat back to the seaweed-dominated state are marked by green solid lines, with 95% confidence intervals shown as dashed lines.

4.4. Results 93

a

b

c

Figure 4.4: Proportion of seaweed bed cover through 8500 Monte-Carlo simulations initialised as sea urchin barrens (urchin culling as an extra parameter compared to ‘forward’ shift Monte-Carlo simulations). The blue line with dots and standard error bars corresponds to monthly model outputs binned into 20 even intervals of biomass density (from 0 to the maximum value). Red sigmoid curves represent binomial logistic models fitted against the biomass densities of the different species modelled (a) sea urchins; (b) rock lobsters (all size classes combined); and (c) large lobster individuals (carapace length >140 mm). Threshold points at which the TRITON model shifts from sea urchin barren habitat back to the seaweed- dominated state are marked by green solid lines, with 95% confidence intervals shown as dashed lines.

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Table 4.4: Summary statistics of different binomial logistic models of the probability of a shift in community structure from sea urchin barren habitat to the seaweed-dominated state. The different models consider, either the probability of a community shift at the end of 50-year simulations (a, c, e), or the relative seaweed bed cover (b, d, f) as metrics to define threshold points in terms of the biomass density of sea urchin (a, b), rock lobster (c, d) and large (carapace length superior to 140 mm) rock lobster (e, f).

a)Parameter Estimate Standard error z value P value

αCR 9.28 x 10−1 5.2 x 10−2 17.77 <2 x 10−16 βCR -1.237 x 10−4 3.5 x 10−6 -35.15 <2 x 10−16

Null deviance: 9670.1 on 8499 df; Residual deviance: 7316.6 on 8498 df

Variance explained: 24.3%; AIC: 7320.6;BCRthreshold95%= -16295 (-16381 - -16295)g.200m−2

b) Parameter Estimate Standard error z value P value

αCR -4.97 x 10−1 1.9 x 10−3 -259 <2 x 10−16 βCR -7.34 x 10−5 1.3 x 10−7 -582 <2 x 10−16

Null deviance: 4054306 on 5099999 df; Residual deviance: 3475451 on 5099998 df

Variance explained: 14.2%; AIC: 4082683;BCRthreshold75%= -21747 (-21769 - -21725)g.200m−2

c) Parameter Estimate Standard error z value P value

αRL -5.88 1.2 x 10−1 -47.27 <2 x 10−16 βRL 1.67 x 10−3 4 x 10−5 42.12 <2 x 10−16

Null deviance: 9670.1 on 8499 df; Residual deviance: 3604.9 on 8498 df

Variance explained: 62.7%; AIC: 3608.9;BRLthreshold95%= 5277 (4903 - 5687)g.200m−2

d) Parameter Estimate Standard error z value P value

αRL -4.349 3.4 x 10−3 -1263.3 <2 x 10−16 βRL 9.134 x 10−4 9.7 x 10−7 943.6 <2 x 10−16

Null deviance: 4054306 on 5099999 df; Residual deviance: 2252951 on 5099998 df

Variance explained: 44.4%; AIC: 2629804;BRLthreshold75%= 5964 (5984 - 5944)g.200m−2

e) Parameter Estimate Standard error z value P value

αRL140+ -1.54 3.1 x 10−2 -49.95 <2 x 10−16

βRL140+ 2.30 x 10−3 9 x 10−5 25.15 <2 x 10−16

Null deviance: 9670.1 on 8499 df; Residual deviance: 8307.0 on 8498 df

Variance explained: 14.0%; AIC: 8311;BRL140+threshold95%= 1954 (1788 - 2147)g.200m−2

f)Parameter Estimate Standard error z value P value

αRL140+ -1.928 1.4 x 10−3 -1386 <2 x 10−16

βRL140+ 8.17 x 10−4 1.7 x 10−6 469 <2 x 10−16

Null deviance: 4054306 on 5099999 df; Residual deviance: 3790144 on 5099998 df

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