Early warning signals in the fully size-structured population model of de Roos and Persson [<b>17</b>].

<p>Independent white noise with σ = 0.002 is added to the death rates of all juvenile consumers. Bifurcation procedure and colors are identical to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g002" target="_blank">Figure 2</a>, with predator mortality staring at <i>µ_P</i> = 0.01 (note that the original article uses parameter δ instead of <i>µ_P</i>), and incremented with <i>Δµ_P</i> = 0.0002 after each 50,000 time units. The fold catastrophe in this model is located at approximately <i>µ_P</i> = 0.038. Coefficient of variation and lag-1 autocorrelation are computed for each value of <i>µ_P</i> over the last 40,000 time steps. (<b>A</b>) Coefficient of variation, and (<b>B</b>) Autocorrelation. All other parameters have default values as used by de Roos and Persson <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone.0062033-deRoos1" target="_blank">[17]</a>.</p

Bistability and catastrophic collapse in a structured predator-prey system.

<p>Bifurcation diagram as a function of predator death rate <i>µ_P</i>. (<b>A</b>) Equilibrium juvenile density J, and (<b>B</b>) Equilibrium predator density P. The equilibrium curves exhibit a so-called catastrophe fold. Between the bifurcation points T1 (<i>µ_P</i>≈0.553) and T2 (<i>µ_P</i>≈0.435) the system is bistable (indicated by solid lines), with an intermediate saddle-node equilibrium (indicated by the dashed line) which is unstable. Model parameters are b = 1, c = 1, <i>µ_J</i> = 0.05, <i>µ_A</i> = 0.1.</p

Early warning signals in coefficient of variation.

<p>For each value of <i>µ_P</i>, starting with <i>µ_P</i>  = 0.4, the model is simulated for 60,000 time units, of which the last 50,000 time units are used to compute population averages and variances. Hereafter, <i>µ_P</i> is incremented with <i>Δµ_P</i> = 0.001, towards the catastrophic collapse at <i>µ_P</i>≈0.553. Death rates are perturbed every time unit using white noise with standard deviation σ = 0.005. (<b>A</b>) Noise added to the juvenile population (<b>B</b>) Noise added to the adult population (<b>C</b>). Independent noise added to all three populations. (<b>D</b>) Identical, fully correlated, noise added to all three populations. Colors are blue for juveniles, green for adults, and red for the predators. For other model parameters see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g001" target="_blank">Figure 1</a>.</p

The predicted optimal proportion of <i>Loripes</i> in terms of dry shell mass in the diet of an energy intake maximizing red knot that has <i>ad libitum</i> access to both <i>Loripes</i> and <i>Dosinia</i>.

<p>Red knots with small gizzards are expected to feed exclusively on <i>Loripes</i>, whereas red knots with large gizzards are expected to have a large share of <i>Dosinia</i> in the diet. Grey area shows 95% prediction interval. See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0136144#pone.0136144.s005" target="_blank">S4 File</a> for more details.</p

Early warning signals in the linearized system.

<p>The model of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g001" target="_blank">Figure 1</a> is linearized using the Jacobian matrix. Predator death rate is set at <i>µ_P</i> = 0.5528 (very close to the bifurcation point T1 in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g001" target="_blank">Figure 1</a>). The death rate of either the juveniles or the adults is perturbed using white noise with standard deviation σ = 0.005. (<b>A</b>) When noise is added to the juvenile death rate, the juvenile population (indicated in blue) clearly shows critically slowing down, whereas the adult (green line) and predator (red line) populations do not show early warning signs. (<b>B</b>) When noise is added to the adult death rate, all three populations do not show any sign of early warning. Note that the fluctuations in the juvenile population in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g005" target="_blank">Figure 5A</a> are so large, that the full (i.c. not linearized) system would shift to the alternative steady state.</p

Early warning signals with correlated noise and discrete noise.

<p>Coefficient of variation and autocorrelation are monitored for increasing predator death rate towards the catastrophic collapse at <i>µ_P</i>≈0.553. Bifurcation procedure, parameters and colors are identical to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g002" target="_blank">Figure 2</a>. (<b>A</b>) Coefficient of variation when pink noise (1/f correlated noise) is added to the death rate of the adult population. (<b>B</b>) Coefficient of variation when discrete white noise is applied directly to the adult population numbers after each time step.</p

Early warning signals in autocorrelation.

<p>For the same simulation as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g002" target="_blank">Figure 2</a>, the lag-1 autocorrelation is computed over the last 50,000 time units. (<b>A</b>) Noise added to the death rate of the juvenile population (<b>B</b>). Noise added to the death rate of the adult population (<b>C</b>). Independent noise added to the death rates of all three populations. (<b>D</b>) Identical, fully correlated, noise added to the death rates of all three populations. For description of the simulation and for color index see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g002" target="_blank">Figure 2</a>.</p

Effect of the direction of perturbation on early warning signals.

<p>Predator death rate is fixed at <i>µ_P</i> = 0.55 (close to the catastrophe), and the death rate of either the juveniles or the adults is perturbed using white noise with standard deviation σ = 0.005. System trajectories are plotted in blue for 60 time units. The dominant eigenvector is indicated by the red arrow, and the second and third eigenvector are indicated by the black arrows. (<b>A</b>) When the juvenile death rate is perturbed, the system responds only in the direction of the dominant eigenvector, resulting in an early warning signal that is only apparent in juvenile population fluctuations. (<b>B</b>) When the adult death rate is perturbed, the system responds in the direction of the surface spanned by the second and third eigenvector (indicated in grey), resulting in damped oscillations and absence of early warning. For other model parameters see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone-0062033-g001" target="_blank">Figure 1</a>. For an animated rotation of these 3D figures, and for direction and scaling of axis see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone.0062033.s001" target="_blank">Movie S1</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062033#pone.0062033.s002" target="_blank">S2</a>.</p

Regression of DM_shell intake on non-toxic prey against gizzard mass.

<p>Data from this study on <i>Dosinia</i> was combined with data from van Gils <i>et al</i>. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0136144#pone.0136144.ref014" target="_blank">14</a>] on other non-toxic prey species. Adding the current data to the regression derived by van Gils <i>et al</i>. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0136144#pone.0136144.ref014" target="_blank">14</a>] slightly changes the regression line (though not significantly; from dashed to solid line), but greatly reduces standard error (from light to dark grey area). Parameter estimates are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0136144#pone.0136144.s006" target="_blank">S1 Table</a> (model 3.1). Regressions are linear regressions on log-transformed data. Note that van Gils <i>et al</i>. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0136144#pone.0136144.ref014" target="_blank">14</a>] averaged gizzard mass measurements per bird, whereas we estimated gizzard mass in each trial by interpolating measurements.</p

Mean gizzard mass of birds directly after catch, during the first and second series of trials.

<p>Directly after catch, the 6 red knots were randomly divided into two groups, group 1 (solid dots and line) and group 2 (open dots and dashed line). Both groups received different diets outside the experimental trials (soft or hard-shelled prey) to manipulate gizzard size. Initial differences in gizzard mass between groups were not significant (F_1,4 = 3.9,p = 0.12). After catch, all birds decreased gizzard mass, but group 1 had larger gizzards than group 2 during the first series of trials, and smaller gizzards during the second series (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0136144#pone.0136144.t002" target="_blank">Table 2</a>, models 1.1 to 1.5), showing that the manipulation of gizzard size was successful. Each group consisted of three birds. However, data collected on one bird from group 2 after it became sick during series 2 was omitted from the graphs and the analysis. Error bars show standard error.</p