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EFFECTS OF PREDATOR DIET BREADTH ON STABILITY OF SIZE SPECTRA

Published online by Cambridge University Press:  05 April 2012

M. J. PLANK
M. J. PLANK*
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand (email: michael.plank@canterbury.ac.nz)
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Abstract

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Many marine ecosystems have the remarkable property that the abundance of organisms of a given body size is approximately proportional to the inverse square of that size. Size-structured models have been developed for which this “invariance-of-biomass” state is an equilibrium solution. These models are built on the coupling of predator growth to prey abundance, where prey suitability is determined by a size-based function referred to as a feeding kernel. In this paper, the local stability of the equilibrium state is investigated in a limiting case where predators only consume prey of a preferred size. In this special case, it is shown analytically that the equilibrium state is always unstable. It is concluded that some degree of diet breadth, in terms of the range of prey sizes consumed by a predator, is an essential prerequisite for the invariance-of-biomass state to be stable, as widely observed in the field.

Keywords

body massecosystem dynamicsMcKendrick–von Foerster equationpredatorpreysize-structured model

MSC classification

Secondary: 92D40: Ecology
Type
Research Article
Information
The ANZIAM Journal , Volume 53 , Issue 1 , July 2011 , pp. 38 - 47
Copyright
Copyright © Australian Mathematical Society 2012

References

[1]Allesina, S. and Pascual, M., “Network structure, predator–prey modules, and stability in large food webs”, Theor. Ecol. 1 (2008) 55–64; doi:10.1007/s12080-007-0007-8.Google Scholar
[2]Andersen, K. H. and Beyer, J. E., “A symptotic size determines species abundance in the marine size spectrum”, Am. Nat. 168 (2006) 54–61; doi:10.1086/504849.CrossRefGoogle Scholar
[3]Andersen, K. H. and Pedersen, M., “Damped trophic cascades driven by fishing in model marine ecosystems”, Proc. Roy. Soc. Lond. B 277 (2010) 795–802; doi:10.1098/rspb.2009.1512.Google Scholar
[4]Benoît, E. and Rochet, M.-J., “A continuous model of biomass size spectra governed by predation and the effects of fishing on them”, J. Theor. Biol. 226 (2004) 9–21; doi:10.1016/S0022-5193(03)00290-X.CrossRefGoogle Scholar
[5]Blanchard, J. L., “The dynamics of size-structured ecosystems”, Ph. D. Thesis, University of York, 2008.Google Scholar
[6]Blanchard, J. L., Jennings, S., Law, R., Castle, M. D., McCloghrie, P., Rochet, M.-J. and Benoît, E., “How does abundance scale with body size in coupled size-structured food webs?”, J. Anim. Ecol. 78 (2009) 270–280; doi:10.1111/j.1365-2656.2008.01466.x.Google Scholar
[7]Blanchard, J. L., Law, R., Castle, M. D. and Jennings, S., “Coupled energy pathways and the resilience of size-structured food webs”, Theor. Ecol. 4 (2011) 289–300; doi:10.1007/s12080-010-0078-9.CrossRefGoogle Scholar
[8]Boudreau, P. R. and Dickie, L. M., “Biomass spectra of aquatic ecosystems in relation to fisheries yield”, Can. J. Fish. Aqua. Sci. 49 (1992) 1528–1538; doi:10.1139/f92-169.Google Scholar
[9]Camacho, J. and Solé, R. V., “Scaling in ecological size spectra”, Europhys. Lett. 55 (2001) 774–780; doi:0.1209/epl/i2001-00347-0.Google Scholar
[10]Capitán, J. A. and Delius, G. W., “Scale-invariant model of marine population dynamics”, Phys. Rev. E 81 (2010) 061901; doi:10.1103/PhysRevE.81.061901.Google Scholar
[11]Cohen, J. E., Pimm, S. L., Yodzis, P. and Saldaña, J., “Body sizes of animal predators and animal prey in food webs”, J. Anim. Ecol. 62 (1993) 67–78; doi:10.2307/5483.Google Scholar
[12]Datta, S., Delius, G. W. and Law, R., “A jump-growth model for predator–prey dynamics: derivation and application to marine ecosystems”, Bull. Math. Biol. 72 (2010) 1361–1382; doi:10.1007/s11538-009-9496-5. Corrected version: http://arxiv.org/abs/0812.4968.Google Scholar
[13]Datta, S., Delius, G. W., Law, R. and Plank, M. J., “A stability analysis of the power-law steady state of marine size spectra”, J. Math. Biol. 63 (2011) 779–799; doi:10.1007/s00285-010-0387-z.Google Scholar
[14]Fenchel, T., “Intrinsic rate of natural increase: the relationship with body size”, Oecolog. 14 (1974) 317–326; doi:10.1007/BF00384576.CrossRefGoogle ScholarPubMed
[15]Hartvig, M., Andersen, K. H. and Beyer, J. E., “Food web framework for size-structured populations”, J. Theor. Biol. 272 (2011) 113–122; doi:10.1016/j.jtbi.2010.12.006.CrossRefGoogle ScholarPubMed
[16]Heath, M. R., “Size spectrum dynamics and the planktonic ecosystem of Loch Linnhe”, ICES J. Mar. Sci. 52 (1995) 627–642; doi:10.1016/1054-3139(95)80077-8.Google Scholar
[17]Jennings, S. and Mackinson, S., “Abundance–body mass relationships in size-structured food webs”, Ecol. Lett. 6 (2003) 971–974; doi:10.1046/j.1461-0248.2003.00529.x.CrossRefGoogle Scholar
[18]Jennings, S., Pinnegar, J. K., Polunin, N. V.  C. and Boon, T. W., “Weak cross-species relationships between body size and trophic level belie powerful size-based trophic structuring in fish communities”, J. Anim. Ecol. 70 (2001) 934–944; doi:10.1046/j.0021-8790.2001.00552.x.Google Scholar
[19]Jennings, S., Warr, K. J. and Mackinson, S., “Use of size-based production and stable isotope analyses to predict trophic transfer efficiencies and predator-prey body mass ratios in food webs”, Mar. Ecol. Prog. Ser. 240 (2002) 11–20; doi:10.3354/meps240011.Google Scholar
[20]Kerr, S. R. and Dickie, L. M., The biomass spectrum: a predator-prey theory of aquatic production (Columbia University Press, New York, 2001).Google Scholar
[21]Law, R., Plank, M. J., James, A. and Blanchard, J. L., “Size-spectra dynamics from stochastic predation and growth of individuals”, Ecol. 90 (2009) 802–811; doi:10.1890/07-1900.1.Google Scholar
[22]Maury, O., Faugeras, B., Shin, Y.-J., Poggiale, C., Ari, T. B. and Marsac, F., “Modelling environmental effects on the size-structured energy flow through marine ecosystems. Part 1: The model”, Prog. Oceanog. 74 (2007) 479–499; doi:10.1016/j.pocean.2007.05.002.CrossRefGoogle Scholar
[23]McKendrick, A. G., “Applications of mathematics to medical problems”, Proc. Edinburgh Math. Soc. 40 (1926) 98–130.Google Scholar
[24]Plank, M. J. and Law, R., “Ecological drivers of stability and instability in marine ecosystems”. Theor. Ecol., to appear; doi:10.1007/s12080-01.Google Scholar
[25]Platt, T. and Denman, K., “The structure of pelagic marine ecosystems”, J. Conseil International pour l’Exploration de la Mer 173 (1978) 60–65.Google Scholar
[26]Pope, J. G., Rice, J. C., Daan, N., Jennings, S. and Gislason, H., “Modelling an exploited marine fish community with 15 parameters—results from a simple size-based model”, ICES J. Mar. Sci. 63 (2006) 1029–1044; doi:10.1016/j.icesjms.2006.04.015.Google Scholar
[27]Rochet, M.-J. and Benoît, E., “Fishing destabilizes the biomass flow in the marine size spectrum”, Proc. Roy. Soc. Lond. B 279 (2012) 284–292; doi:10.1098/rspb.2011.0893.Google Scholar
[28]San Martin, E., Irigoien, X., Harris, R. P., López-Urrutia, Á., Zubkov, M. Z. and Heywood, J. L., “Variation in the transfer of energy in marine plankton along a productivity gradient in the Atlantic Ocean”, Limnol. Oceanog. 51 (2006) 2084–3091; doi:10.4319/lo.2006.51.5.2084.Google Scholar
[29]Scharf, F. S., Juanes, F. and Rountree, R. A., “Predator size-prey size relationships of marine fish predators: interspecific variation and effects of ontogeny and body size on trophic-niche breadth”, Mar. Ecol. Prog. Ser. 208 (2000) 229–248; doi:10.3354/meps208229.Google Scholar
[30]Sheldon, R. W. and Parsons, T., “A continuous size spectrum for particulate matter in the sea”, J. Fish. Res. Board Canada 24 (1967) 909–915; doi:10.1139/f67-081.Google Scholar
[31]Sheldon, R. W., Prakash, A. and Sutcliffe, W. H., “The size distribution of particles in the ocean”, Limnol. Oceanog. 17 (1972) 327–340; doi:10.4319/lo.1972.17.3.0327.Google Scholar
[32]Sheldon, R. W., Sutcliffe, W. H. and Paranjape, M. A., “Structure of pelagic food chain and relationship between plankton and fish production”, J. Fish. Res. Board Canada 34 (1977) 2344–2353; doi:10.1139/f77-314.Google Scholar
[33]Silvert, W. and Platt, T., “Energy flux in the pelagic ecosystem: a time-dependent equation”, Limnol. Oceanog. 23 (1978) 813–816; doi:10.4319/lo.1978.23.4.0813.Google Scholar
[34]Stephens, D. W. and Krebs, J. R., Foraging theory (Princeton University Press, Princeton, NJ, 1986).Google Scholar
[35]Ursin, E., “On the prey size preferences of cod and dab”, Meddelelser fra Danmarks Fiskeri- og Havundersøgelser 7 (1973) 85–98.Google Scholar
[36]von Foerster, H., “Some remarks on changing populations”, in: The kinetics of cellular proliferation (ed. Stohlman, J. F.), (Grune and Stratton, New York, 1959) 382–407.Google Scholar
[37]Ware, D. M., “Bioenergetics of pelagic fish: theoretical change in swimming speed and ration with body size”, J. Fish. Res. Board Canada 35 (1978) 220–228; doi:10.1139/f78-036.Google Scholar