Text S1: Simulation models and detailed method for early warning signal calculation

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1 1 Text S1: Simulation moels an etaile metho for early warning signal calculation Steven J. Lae, Thilo Gross Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, Dresen, Germany This Supporting Text etails the application of the generalize moeling-base early warning approach to the fishery simulation an the tri-trophic foo chain simulation. (The metho for the Allee effect application is escribe in the main text. The simulation moels, use to generate the ata to which the early warning signals were applie, are also escribe. Early warning signal calculation Fishery simulation of Biggs et al. We assume the following knowlege: The populations of ault piscivores A i an planktivores F i at the start of year i The number of ault piscivores harveste Q i = Q(A i uring year i That ault piscivores ie from other causes at a per-capita rate m That ault piscivores preate on the planktivores That there is a net movement of planktivores into (out of the foraging arenas from (to refugia [2] of R i in year i That per-capita, an in the absence of interactions with other moele populations, r piscivores woul be born an mature into ault piscivores the following year That however juvenile piscivores are preate on by planktivores [3] an are also controlle, for example by acciental preation, by the ault piscivores [4]. This knowlege is represente schematically in Fig. 1 of the main text. We constructe a generalize moel of this fishery by moeling only the year-en piscivore populations A i an planktivore populations F i with a iscrete map. As a consequence we coul avoi explicitly moeling the complications of intra-annual ynamics such as piscivore reprouction an maturation. We wrote: A i+1 = (1 m(a i Q i +ra i C i F i+1 = F i +R i D i. (1a (1b The first term of Eq. (1a gives the number of surviving ault piscivores after harvesting (Q i an other causes of mortality (m. The secon term ra i moels reprouction without losses an the thir C i = C(A i,f i gives the preatory or controlling effects of planktivores an ault piscivores on the number of new ault piscivores. Since C epens twice on the ault piscivore population, first through the number of juvenile piscivores born an again through the controlling effect of the aults, we assume C is quaratic in A i. In Eq. (1b, R i moels the net influx of planktivores into the foraging arenas. Since the planktivore population was initially small, we assume there was negligible outflow of planktivores, an that the inflow epene on a refuge size that was constant. Therefore R i = R was constant. [In the other

2 2 scenario consiere by Biggs et al. [5], where habitat estruction causes the refuge size to ecrease, R i woul not be constant.] The remaining term D i = D(A i,f i moels all other factors changing the planktivore population, most importantly preation by ault piscivores. Since the planktivore population was initially small, we assume D to be linear in F i. From the output of the simulation moel (escribe below we recore annual ault piscivore ensity, planktivore ensity, an piscivore catch. Since the ecosystem approache the critical transition more slowly than the moel with Allee effect above, to remove noise we foun it necessary to smooth estimates of the graients. For each new observation at time t i, we fitte a secon-orer polynomial (since the metho eventually results in ifferences of up to secon orer to the recor of each observe time series up to that time. Since it is the graients at the current time that are most important, in the fit we applie a weight to each ata point j of exp[(t j t i /τ] with τ = 10. Graients were then extracte irectly from the coefficients of the polynomial fit. This simulate the calculation of early warning signals on the fly, as new observations become available. Specifically, at each time i we fitte the three polynomials F j = p F0 +p F1 (j i 1+p F2 (j i 1 2, j = 0,...,i+1 Ã j = p A0 +p A1 (j i 1+p A2 (j i 1 2, j = 0,...,i+1 Q j = p Q0 +p Q1 (j i+p Q2 (j i 2, j = 0,...,i to our three input time series. We assume that at year i, as well as the fish harvest for that year (Q i we also have alreay available the piscivore an planktivore populations at the en of that year (A i+1 an F i+1, respectively. Changing this assumption an using only previous population observations oes not significantly change our results. These smoothe observations were sufficient to calculate the Jacobian of Eqs. (1, [ ] 1 m+r (1 mq J = (A C (A C (F, D (A 1 D (F where we use the notation H/ x H (x. The eigenvalues λ of the Jacobian, which may be complex numbers, are the solutions of et(j λi = 0 where I is the (2 2 ientity matrix. OurproceuretoestimatetheJacobianattimeiwasthenasfollows. Fromtheassumptionoflinearity of Q in A, Q (A = Q/Ã = p Q0/p A0. Estimates of C an D can be foun by solving Eqs. (1. Together with the assumptions that C is quaratic in A an D linear in F, it follows that C (A = 2 C/Ã = 2 ( p A0 +(1+r mãi (1 mp Q0 /p A0 D (F = D/ F = ( p F1 +R/p F0 Finally, by solving linear Taylor expansions of the form the erivatives C (F = C C (A A F D (A = D D (F F A were estimate. = = C = C (A A+C (F F, (2 ( p A0 +(r mãi (1+r mãi 1 (1 mp Q1 C (A p A1 /p F1 ( 2p F2 D (F p F1 /p A1

3 3 This generalize moeling approach require estimates of the ault piscivore mortality m an reprouction rate r, an planktivore influx R from refugia. We assume the fishery manager coul estimate m = 0.4 an r = 1.1 from knowlege of the fish species an the fishery. Effective values m = 0.5 an r = 1 can be erive from the actual simulation s equations [5]. The planktivore influx R woul be harer to estimate in practice. In the absence of such knowlege, the ecosystem manager coul make use of the fact that the fish populations were initially stable. This knowlege constrains R to the range 0.1 to 1.5, as values outsie these ranges prouce an eigenvalue with magnitue greater than one between times t = 0 to 10. We use R = 0.9. Inee, this knowlege coul also be use to constrain m an r to the ranges 0.1 to 0.7 an 0.95 to 1.5, respectively. We foun that any combination of m, r an R that gave an initially stable system robustly preicte the later critical transition. Tri-trophic foo chain To obtain an early warning signal for this transition, we constructe a generalize moel of the tri-trophic foo chain t X 1 = A(X 1 G(X 1,X 2 t X 2 = G(X 1,X 2 H(X 2,X 3 t X 3 = H(X 2,X 3 M(X 3,µ, where X 1, X 2 an X 3 are the biomasses of the three species in the foo chain, A(X 1 is the primary prouction rate of X 1, G(X 1,X 2 an H(X 2,X 3 are the rates at which X 2 an X 3 consume X 1 an X 2, respectively, an M(X 3,µ is the total mortality per unit time of the top-level preator X 3, with the argument µ enoting it is subject to a changing external influence. The approach coul also easily be extene to the case of incomplete biomass conversion efficiencies, if estimates of those efficiencies were available. We assume: access to observations of all three biomasses X 1, X 2 an X 3, an the top-preator mortality M(X 3,µ; that this mortality is linear in X 3 so that M = m(µx 3 ; an that the preation rates G(X 1,X 2 an H(X 2,X 3 are linear in their respective preator biomasses. The observe time series were first smoothe by the same filter as in the fishery simulation, but with τ = 20. The Jacobian matrix of Eqs. (3 is A (1 G (1 G (2 0 J = G (1 G (2 H (2 H (3, 0 H (2 H (3 M (3 whereweenotef (a F/ X a. WecomputethequantitiesintheJacobianasfollows. Byrearranging Eqs. (3, first H then G then A can be foun. With the assumptions of linearity liste above, it follows that M (3 = M/ X 3 = p M0 /p X3,0 H (3 = H/ X 3 = (p X3,1 +p M0 /p X3,0 G (2 = G/ X 2 = (p X2,1 +p X3,1 +p M0 /p X2,0. Through a Taylor expansion as in Eq. (2, H (2 = ( H H ( (3 X 3 / X 2 = 2p X3,2 +p M1 H (3 p X3,1 /p X2,1 G (1 = ( G G ( (2 X 2 / X 1 = 2p X2,2 +2p X3,2 +p M1 G (2 p X2,1 /p X1,1 (3a (3b (3c

4 4 The last unknown in the Jacobian to be etermine is Simulation moels Single population with Allee effect à (1 = (2p X1,2 +2p X2,2 +2p X3,2 +p M1 /p X1,1. To test the early warning signal escribe in the main text we simulate a simple moel t X = AX2 k 2 mx +σξ(t, (4 +X2 base on Yeakel et al. [1]. We inclue a small aitive noise term σξ(t, with stanar eviation σ an autocorrelation ξ(tξ(t = δ(t t. Throughout the simulation we slowly change the mortality rate m accoring to m = t. Fishery simulation of Biggs et al. We generate ata for this early warning signal test with the previously publishe fishery moel of Biggs et al. [5]. Their moel is a hybri iscrete-continuous system an explicitly moels the ault piscivore, juvenile piscivore an planktivore populations. The intra-annual ynamics are continuous, moeling: the harvest of ault piscivores; the preation an control of planktivores an juvenile piscivores, respectively, by ault piscivores; preation on juvenile piscivores by planktivores; an the movement of both planktivores an juvenile piscivores between the foraging arenas an refugia. At the en of each year iscrete upate rules were applie that controlle the mortality of ault piscivores, maturation of the surviving juvenile piscivores into ault piscivores, an birth of more juvenile piscivores. There was an aitive noise term on the planktivore population ynamics. For further etails of the moel incluing its mathematical formulation see Biggs et al. [5]. Tri-trophic foo chain To generate example time series ata we use a conventional equation-base moel in which a proucer biomass, X 1, preator biomass, X 2, an top preator biomass, X 3, follow t X 1 = A n X 1 (K n X 1 BX2 1X 2 K 3 +X1 2 +σ 1 ξ 1 (t (5a t X 2 = BX2 1X 2 K 3 +X1 2 A px2x 2 3 K p +X2 2 +σ 2 ξ 2 (t (5b t X 3 = A px2x 2 3 K p +X2 2 mx 3 +σ 3 ξ 3 (t. (5c In these equations we assume Holling type-3 preator-prey interaction, logistic growth of the proucer an linear mortality of the top preator an ξ i (t are aitive noise terms with ξ i (tξ j (t = δ ij δ(t t. If any biomass ecrease to zero, we suppresse the noise term so that the corresponing population remaine extinct. References 1. Yeakel JD, Stiefs D, Novak M, Gross T (2011 Generalize moeling of ecological population ynamics. Theor Ecol 4:

5 5 2. Walters CJ, Martell SJD (2004 Fisheries Ecology an Management. Princeton University Press. 3. Walters C, Kitchell JF (2001 Cultivation/epensation effects on juvenile survival an recruitment: Implications for the theory of fishing. Can J Fish Aquat Sci 58: Carpenter SR, Brock WA, Cole JJ, Kitchell JF, Pace ML (2008 Leaing inicators of trophic cascaes. Ecol Lett 11: Biggs R, Carpenter SR, Brock WA (2009 Turning back from the brink: Detecting an impening regime shift in time to avert it. Proc Natl Aca Sci USA 106: