# Ecological drivers of stability and instability in marine ecosystems

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## Abstract

A stability analysis of the steady state of marine ecosystems is described. The study was motivated by the approximate invariance of biomass in logarithmic size intervals, which is widely observed in marine ecosystems. This invariance is recovered as the steady state of dynamic models of size spectra, which, unlike traditional species-based models of food webs, explicitly account for the mass gained by an individual organism when it eats a prey item. Little is known about the ecological conditions affecting the stability of the steady state, and a new method is developed to examine this. The results show that stability is enhanced by: (a) decreasing the mean predator-to-prey mass ratio (PPMR), (b) increasing the diet breadth of predators, (c) increasing the strength of intrinsic mortality relative to predation mortality, (d) increasing the biomass conversion efficiency. When perturbed from steady state, size spectra develop a wave-like shape, with an average wavelength especially sensitive to the mean PPMR. These waves move from small to large body size at an average speed which depends on the rate of growth of organisms. In contrast to traditional food web models, stability is enhanced as connectance (diet breadth) increases and as food chain length is increased by reducing the PPMR.

## Keywords

Food web Jump-growth equation Power law Predator-to-prey mass ratio Size-dependent predation Size spectrum## Notes

### Acknowledgements

This research was supported by the RSNZ Marsden fund, grant number 08-UOC-034. We thank Gustav Delius for his contributions to the analytical calculations and derivation of the convolution kernel. We also thank David Wall for illuminating discussions and José Capitán and Julia Blanchard for comments on an earlier version of the manuscript.

## References

- Allesina S, Pascual M (2008) Network structure, predator-prey modules, and stability in large food webs. Theor Ecol 1:55–64CrossRefGoogle Scholar
- Andersen KH, Beyer JE (2006) Asymptotic size determines species abundance in the marine size spectrum. Am Nat 168:54–61CrossRefGoogle Scholar
- Andersen KH, Beyer JE, Pedersen M, Andersen NG, Gislason H (2008) Life-history constraints on the success of the many small eggs reproductive strategy. Theor Popul Biol 73:490–497PubMedCrossRefGoogle Scholar
- Anderson CNK, Hsieh C-h, Sandin SA, Hewitt R, Hollowed A, Beddington J, May RM, Sugihara G (2008) Why fishing magnifies fluctuations in fish abundance. Nature 452:835–839PubMedCrossRefGoogle Scholar
- Arino O, Shin Y-J, Mullon C (2004) A mathematical derivation of size spectra in fish populations. Comptes Rendus Biol 327:245–254CrossRefGoogle Scholar
- Barnes C, Bethea DM, Brodeur RD, Spitz J, Ridoux V, Pusineri C, Chase BC, Hunsicker ME, Juanes F, Kellermann A, Lancaster J, Ménard F, Bard F-X, Munk P, Pinnegar JK, Scharf FS, Rountree RA, Stergiou KI, Sassa C, Sabates A, Jennings S (2008) Predator and prey body sizes in marine food webs. Ecology 89:881CrossRefGoogle Scholar
- Benoît E, Rochet M-J (2004) A continuous model of biomass size spectra governed by predation and the effects of fishing on them. J Theor Biol 226:9–21PubMedCrossRefGoogle Scholar
- Blanchard JL (2008) The dynamics of size-structured ecosystems. Ph.D. thesis, University of YorkGoogle Scholar
- Blanchard JL, Jennings S, Law R, Castle MD, McCloghrie P, Rochet M-J, Benoît E (2009) How does abundance scale with body size in coupled size-structured food webs? J Anim Ecol 78:270–280PubMedCrossRefGoogle Scholar
- Blanchard JL, Law R, Castle MD, Jennings S (2010) Coupled energy pathways and the resilience of size-structured food webs. Theor Ecol 4:289–300CrossRefGoogle Scholar
- Blueweiss L, Fox H, Kudzma V, Nakashima D, Peters R, Sams S (1978) Relationships between body size and some life history parameters. Oecologia 37:257–272CrossRefGoogle Scholar
- Boudreau PR, Dickie LM (1992) Biomass spectra of aquatic ecosystems in relation to fisheries yield. Can J Fish Aquat Sci 49:1528–1538CrossRefGoogle Scholar
- Brose U, Williams RJ, Martinez ND (2006) Allometric scaling enhances stability in complex food webs. Ecol Lett 9:1228–1236PubMedCrossRefGoogle Scholar
- Camacho J, Solé RV (2001) Scaling in ecological size spectra. Europhys Lett 55:774–780CrossRefGoogle Scholar
- Capitán JA, Delius GW (2010) Scale-invariant model of marine population dynamics. Phys Rev E 81:061901CrossRefGoogle Scholar
- Cushing DH (1969) The regularity of the spawning season of some fishes. J Cons Int Explor Mer 33:81–92Google Scholar
- Datta S, Delius GW, Law R (2010) A jump-growth model for predator–prey dynamics: derivation and application to marine ecosystems. Bull Math Biol 72:1361–1382PubMedCrossRefGoogle Scholar
- Datta S, Delius GW, Law R, Plank MJ (2011) Stability analysis of marine ecosystems using the deterministic jump-growth equation. J Math Biol doi: 10.1007/s00285-010-0387-z PubMedGoogle Scholar
- Dunne JA, Williams RJ, Martinez ND (2002) Network structure and biodiversity loss in food webs: robustness increases with connectance. Ecol Lett 5:558–567CrossRefGoogle Scholar
- Emmerson MC, Raffaelli D (2004) Predator–prey body size, interaction strength and the stability of a real food web. J Anim Ecol 73:399–409CrossRefGoogle Scholar
- Fenchel T (1974) Intrinsic rate of natural increase: the relationship with body size. Oecologia 14:317–326CrossRefGoogle Scholar
- Frank KT, Petrie B, Choi JS, Leggett WC (2005) Trophic cascades in a formerly cod-dominated ecosystem. Science 308:1621–1623PubMedCrossRefGoogle Scholar
- Gasol JM, Guerro R, Pedrós-Alió C (1991) Seasonal variations in size structure and procaryotic dominance in sulfurous Lake Cisó. Limnol Oceanog 36:860–872CrossRefGoogle Scholar
- Gaedke U (1992) The size distribution of plankton biomass in a large lake and its seasonal variability. Limnol Oceanog 37:1202–1220CrossRefGoogle Scholar
- Guckenheimer J, and Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New YorkGoogle Scholar
- Hartvig M, Andersen KH, Beyer JE (2011) Food web framework for size-structured populations. J Theor Biol 272:113–122PubMedCrossRefGoogle Scholar
- Heath MR (1995) Size spectrum dynamics and the planktonic ecosystem of Loch Linnhe. ICES J Mar Sci 52:627–642CrossRefGoogle Scholar
- Hsieh C-h, Reiss CS, Hunter JR, Beddington JR, May RM, Sugihara G (2006) Fishing elevates variability in the abundance of exploited species. Nature 443:859–862PubMedCrossRefGoogle Scholar
- Huete–Ortega M, Marañón, E, Varela M, Bode A (2010) General patterns in the size scaling of phytoplankton abundance in coastal waters during a 10-year time series. J Plankton Res 32:1–14CrossRefGoogle Scholar
- Jennings S, Pinnegar JK, Polunin NVC, Boon TW (2001) Weak cross-species relationships between body size and trophic level belie powerful size-based trophic structuring in fish communities. J Anim Ecol 70:934–944CrossRefGoogle Scholar
- Jennings S, Warr KJ (2003) Smaller predator–prey body size ratios in longer food chains. Proc Roy Soc B 270:1413–1417CrossRefGoogle Scholar
- Kerr SR, Dickie LM (2001) The biomass spectrum: a predator–prey theory of aquatic production. Columbia University Press, New YorkGoogle Scholar
- Knowlton N (2004) Multiple “stable” states and the conservation of marine ecosystems. Prog Oceanogr 60:387–396CrossRefGoogle Scholar
- Law R, Plank MJ, James A, Blanchard JL (2009) Size-spectra dynamics from stochastic predation and growth of individuals. Ecology 90:802–811PubMedCrossRefGoogle Scholar
- Maury O, Faugeras B, Shin Y-J, Poggiale J-C, Ari TB, Marsac F (2007) Modelling environmental effects on the size-structured energy flow through marine ecosystems. Part 1: the model. Prog Oceanogr 74:479–499CrossRefGoogle Scholar
- May RM (1972) Will a large complex system be stable. Nature 238:413–414PubMedCrossRefGoogle Scholar
- McKane AJ, Newman TJ (2005) Predator–prey cycles from resonant amplification of demographic stochasticity. Phys Rev Lett 94:218102PubMedCrossRefGoogle Scholar
- McKendrick AG (1926) Applications of mathematics to medical problems. Proc Edinb Math Soc 40:98–130Google Scholar
- Neutel A-M, Heesterbeek JAP, de Ruiter PC (2002) Stability in real food webs: weak links in long loops. Science 296:1120–1123PubMedCrossRefGoogle Scholar
- Pimm SL, Lawton JH (1977) Number of trophic levels in ecological communities. Nature 268:329–331CrossRefGoogle Scholar
- Platt T and Denman K (1978) The structure of pelagic marine ecosystems. Rapp P-v Réun Cons Int Explor Mer 173:60–65Google Scholar
- Platt T, Fuentes-Yaco C, Frank KT (2003) Spring algal bloom and larval fish survival. Nature 423:398–399PubMedCrossRefGoogle Scholar
- Pope JG, Shepherd JG and Webb J (1994) Successful surf-riding on size spectra: the secret of survival in the sea. Phil Trans R Soc B 343:41–49CrossRefGoogle Scholar
- Post DM, Pace ML, Hairston NG Jr (2000) Ecosystem size determines food-chain length in lakes. Nature 405:1047–1049PubMedCrossRefGoogle Scholar
- Post DM (2007) Testing the productive-space hypothesis: rational and power. Oecologia 153:973–984PubMedCrossRefGoogle Scholar
- Quinones RA, Platt T, Rodríguez J (2003) Patterns of biomass-size spectra from oligotrophic waters of the Northwest Atlantic. Prog Oceanogr 57:405–427CrossRefGoogle Scholar
- Rochet M-J and Benoît E (2011) Fishing destabilizes the biomass flow in the marine size spectrum. Proc R Soc B. doi: 10.1098/rspb.2011.0893 PubMedGoogle Scholar
- San Martin E, Irigoien X, Harris RP, López-Urrutia Á, Zubkov MZ, Heywood JL (2006) Variation in the transfer of energy in marine plankton along a productivity gradient in the Atlantic Ocean. Limnol Oceanogr 51:2084–2091CrossRefGoogle Scholar
- Sheldon RW, Parsons TR (1967) A continuous size spectrum for particulate matter in the sea. J Fish Res Board Can 24:909–915CrossRefGoogle Scholar
- Sheldon RW, Prakash A, Sutcliffe WH Jr (1972) The size distribution of particles in the ocean. Limnol Oceanogr 17:327–340CrossRefGoogle Scholar
- Sheldon RW, Sutcliffe WH Jr, Paranjape MA (1977) Structure of pelagic food chain and relationship between plankton and fish production. J Fish Res Board Can 34:2344–2353CrossRefGoogle Scholar
- Silvert W, Platt T (1978) Energy flux in the pelagic ecosystem: a time-dependent equation. Limnol Oceanogr 23:813–816CrossRefGoogle Scholar
- Silvert W, Platt T (1980) Dynamic energy-flow model of the particle size distribution in pelagic ecosystems. In: Kerfoot WC (ed) Evolution and ecology of zooplankton communities. University Press of New England, Hanover, pp 754–763Google Scholar
- Sterner RW, Bajpai A, Adams T (1997) The enigma of food chain length: absence of theoretical evidence for dynamic constraints. Ecol 78:2258–2262CrossRefGoogle Scholar
- Ursin E (1973) On the prey size preferences of cod and dab. Medd Dan Fisk Havunders 7:85–98Google Scholar
- von Foerster H (1959) Some remarks on changing populations. In: Stohlman JF (ed) The kinetics of cellular proliferation. Grune and Stratton, New York, pp 382–407Google Scholar
- Ware DM (1978) Bioenergetics of pelagic fish: theoretical change in swimming speed and ration with body size. J Fish Res Board Can 35:220–228CrossRefGoogle Scholar
- Zhou M, Huntley ME (1997) Population dynamics theory of plankton based on biomass spectra. Mar Ecol Prog Ser 159:61–73CrossRefGoogle Scholar
- Zhou M, Tande KS, Zhu Y, Basedow S (2009) Productivity, trophic levels and size spectra of zooplankton in northern Norwegian shelf regions. Deep Sea Res II 56:1934–1944CrossRefGoogle Scholar
- Zhou M, Carlotti F, Zhu Y (2010) A size-spectrum zooplankton closure model for ecosystem modelling J Plankton Res 32:1147–1165CrossRefGoogle Scholar