tipping points in ecological dynamics

1. tipping points in ecological dynamics Richard Law University of York

2. background larger basin faster return rate lower temporal autocorrelation less asymmetry in basin no flickering smaller basin slower return rate greater temporal autocorrelation more asymmetry in basin flickering environmental parameter tipping point Scheffer et al 2009, Nature 461

3. lake eutrophication (fold bifurcation) water clarity little affected by increasing nutrient load clear nutrients + fish can push lake over basin boundary to turbidity turbid nutrient load reducing nutrient load not enough to restore clear water Lake Paul and Peter: Carpenter et al 2011, Science 332:1079 Scheffer et al 2001, Nature 413:591

4. spatial pattern in arid areas savanna, tree patches: Rietkerk et al 2004 Rietkerk et al 2004 Science 305: 1927 savanna, tree patches: Rietkerk et al 2004 www.kcet.org/updaily/the_back_forty/botany/desert-nurses.html

5. desertification under grazing spatial pattern as a warning signal of collapse: Kefi et al 2007 Nature 449: 213 away from power law close to power law grazing pressure increases

6. bifurcation at high grazing low grazing high grazing vegetation goes smoothly to zero threshold at which vegetation goes to zero departure from power law comes before the crash Kefi et al 2007

7. regime shifts in fisheries fold bifurcation stable steady state with fish present stable steady state with fish absent sudden collapse of fish stock McCall 2009

8. reproduction feeding preference risk of death from predation Log abundance growth Log body mass model for dynamics ecosystem approach internalises: • growth: eating other organisms • death: being eaten by other organisms food supply for smallest sizes log abundance vs log body mass: size spectrum

9. steady-state: numerical solution log u(x) x stability matters in size spectra Datta, Delius and Law 2010, Bull Math Biol

10. travelling wave narrow diet x not a surprise: this is a predator-prey system

11. dynamical regimes found under exploitation steady-state attractor hysteresis: interior attractor extinction attactor extinction attractor cyclic attractor Plank unpublished

12. fluctuations under harvesting time series of larval fish abundance, California Current System, 1951 to 2002 exploited stocks unexploited stocks age truncation effect fishing more variation in abundance forces the exploited population towards semelparity Hsieh et al 2006, Nature 443:859

13. path to chaos: insect life histories logistic map Nt+1 = rNt(1-Nt) Hassell et al. 1976 J Anim Ecol 45: 471 bifurcation diagram equilibrium 2-point cycle 4-point cycle chaos Colorado potato beetle

14. bifurcations in flour beetles: model larvae Lt+1 = bAt exp(-cel Lt – cea At + E1t) pupae Pt+1 = Lt (1 - mul) exp(E2t) adults At+1 = [Pt exp(-cpa At) + At (1- mua)] exp(E3t) cpa cpa recruitment from pupal to adult stage Costantino et al 1997 275:389

15. bifurcations in flour beetles: experiments Costantino et al 1997

16. take-home messages a lot of interest in tipping points in ecology some are potentially important evidence is debatable warning signals are hard to detect and then there is evolution