# Simplification of structured population dynamics in complex ecological communities

## Abstract

Goldstone’s idea of slow dynamics resulting from spontaneously broken symmetries is applied to Hubbell’s neutral hypothesis of community dynamics, to efficiently simplify stage-structured multi-species models—introducing the quasi-neutral approximation (QNA). Rather than assuming population-dynamical neutrality in the QNA, deviations from ideal neutrality, thought to be small, drive dynamics. The QNA is systematically derived to first and second order in a two-scale singular perturbation expansion. The total reproductive value of species, as computed from the effective life-history parameters resulting from the non-linear interactions with the surrounding community, emerges as the new dynamic variables in this aggregated description. Using a simple stage-structured community-assembly model, the QNA is demonstrated to accurately reproduce population dynamics in large, complex communities. Further, the utility of the QNA in building intuition for management problems is illustrated by estimating the responses of a fish stock to harvesting and variations in fecundity.

## Keywords

Reproductive value Lotka–Volterra model Fisheries management Neutral theory of biodiversity Goldstone mode## Notes

### Acknowledgements

The authors acknowledge discussion and comments on this work by Ken Haste Andersen, Martin Pedersen and J.A.J. (Hans) Metz. This Beaufort Marine Research Award is carried out under the Sea Change Strategy and the Strategy for Science Technology and Innovation (2006–2013), with the support of the Marine Institute, funded under the Marine Research Sub-Programme of the Ireland’s National Development Plan 2007–2013.

## References

- Andersen KH, Farnsworth KD, Pedersen M, Gislason H, Beyer JE (2009) How community ecology links natural mortality, growth, and production of fish populations. ICES J Mar Sci 66(9):1978–1984. doi: 10.1093/icesjms/fsp161 CrossRefGoogle Scholar
- Auger P (1983) Hierarchically organized populations: interactions between individual, population, and ecosystem levels. Math Biosci 65(2):269–289CrossRefGoogle Scholar
- Auger P, Poggiale JC (1998) Aggregation and emergence in systems of ordinary differential equations. Math Comput Model 27(4):1–21CrossRefGoogle Scholar
- Auger P, de la Parra RB, Poggiale JC, Sanchez E, Nguyen-Huu T (2008) Aggregation of variables and applications to population dynamics. In: Structured population models in biology and epidemiology. Lecture notes in mathematics, vol 1936, Springer, Berlin, pp 209–263CrossRefGoogle Scholar
- Azaele S, Pigolotti S, Banavar JR, Maritan A (2006) Dynamical evolution of ecosystems. Nature 444:926–928PubMedCrossRefGoogle Scholar
- Berman A, Plemmons RJ (eds) (1994) Nonnegative Matrices in the mathematical sciences. SIAM, PhiladelphiaGoogle Scholar
- Bodenschatz E, Zimmermann W, Kramer L (1988) On electrically driven pattern-forming instabilities in planar nematics. J Phys (Paris) 49:1875–1899CrossRefGoogle Scholar
- van den Bosch F, Metz JAJ, Diekmann O (1990) The velocity of spatial population expansion. J Math Biol 28(5):529–565CrossRefGoogle Scholar
- Caldarelli G, Higgs PG, McKane AJ (1998) Modelling coevolution in multispecies communities. J Theor Biol 193:345PubMedCrossRefGoogle Scholar
- Caswell H (2000) Matrix population models: construction, analysis and interpretation, 2nd edn. Sinauer Associates, SunderlandGoogle Scholar
- CBD (1992) Multilateral convention on biological diversity (will annexes). http://www.cbd.int/convention/convention.shtml
- Chow CC (2007) Multiple scale analysis. Scholarpedia 2(10):1617CrossRefGoogle Scholar
- Claessen D, de Roos AM (2003) Bistability in a size-structured population model of cannibalistic fish—a continuation study. Theor Popul Biol 64(1):49–65PubMedCrossRefGoogle Scholar
- De Roos AM, Schellekens T, Van Kooten T, Persson L (2008a) Stage-specific predator species help each other to persist while competing for a single prey. Proc Natl Acad Sci U S A 105(37):13,930–13,935. doi: 10.1073/pnas.0803834105 Google Scholar
- De Roos AM, Schellekens T, Van Kooten T, Van De Wolfshaar K, Claessen D, Persson L (2008b) Simplifying a physiologically structured population model to a stage-structured biomass model. Theor Popul Biol 73:47–62PubMedCrossRefGoogle Scholar
- Doncaster CP (2009) Ecological equivalence: a realistic assumption for niche theory as a testable alternative to neutral theory. PLoS ONE 4(10):e7460. doi: 10.1371/journal.pone.0007460 CrossRefGoogle Scholar
- Drake JA (1990) The mechanics of community assembly. J Theor Biol 147:213–233CrossRefGoogle Scholar
- Engen S, Lande R, Sæther BE, Dobson FS (2009) Reproductive value and the stochastic demography of age-structured populations. Am Nat 174(6):795–804PubMedCrossRefGoogle Scholar
- Fisher RA (1930) The genetical theory of natural selection. Oxford University Press, OxfordGoogle Scholar
- Froese R, Pauly D (2009) FishBase. World Wide Web electronic publication, www.fishbase.org, version 07/2009
- Gilbert CH (1912) Age at maturity of the Pacific Coast salmon of the genus Oncorhynchus. Scripps Institution of Oceanography Library Paper 10. http://repositories.cdlib.org/sio/lib/10
- Goldstone J (1961) Field theories with superconductor solutions. Nuovo Cim 19:154–164CrossRefGoogle Scholar
- Goldstone J, Salam A, Weinberg S (1962) Broken symmetries. Phys Rev 127(3):965–970CrossRefGoogle Scholar
- Goodman LA (1968) An elementary approach to the population projection-matrix, to the population reproductive value, and to related topics in the mathematical theory of population growth. Demography 5:382–409Google Scholar
- Grafen A (2006) A theory of Fisher’s reproductive value. J Math Biol 53(1):15–60PubMedCrossRefGoogle Scholar
- Greiner G, Heesterbeek JAP, Metz JAJ (1994) A singular perturbation theorem for evolution equations and time-scale arguments for structured population models. Canadian Appl Math Quart 2(4):435Google Scholar
- Hofbauer J, Schreiber SJ (2010) Robust permanence for interacting structured populations. J Diff Equ 248(8):1955–1971CrossRefGoogle Scholar
- Hoppensteadt F (1969) Asymptotic series solutions of some nonlinear parabolic equations with a small parameter. Arch Ration Mech Anal 35:284–298CrossRefGoogle Scholar
- Hubbell SP (2001) The unified neutral theory of biodiversity and biogeography. Princeton University Press, PrincetonGoogle Scholar
- ICES (2001) Report of the ICES advisory committee on ecosystems. ICES Cooperative Research Report 249, International Council for the Exploration of the Sea, Copenhagen, DenmarkGoogle Scholar
- ICES (2008) Report of the ICES Advisory Committee 2008. Tech Rep Book 6, ICES Advice 2008Google Scholar
- Kaschner K, Ready JS, Agbayani E, Rius J, Kesner-Reyes K, Eastwood PD, South AB, Kullander SO, Rees T, Close CH, Watson R, Pauly D, Froese R (2008) Aquamaps: predicted range maps for aquatic species. World Wide Web electronic publication, www.aquamaps.org, version 05/2008. Accessed 28 Feb 2009
- Kevorkian J, Cole JD (1996) Multiple scale and singular perturbation methods. Springer, New YorkCrossRefGoogle Scholar
- Law R, Morton RD (1996) Permanence and the assembly of ecological communities. Ecology 77(3):762–775CrossRefGoogle Scholar
- Lefkovitch LP (1965) The study of population growth in organisms grouped by stages. Biometrics 21:1–18CrossRefGoogle Scholar
- Leslie P (1945) The use of matrices in certain population mathematics. Biometrika 33(3):183–212PubMedCrossRefGoogle Scholar
- MacArthur R (1970) Species packing and competitive equilibrium for many species. Theor Popul Biol 1:1–11PubMedCrossRefGoogle Scholar
- MacArthur RH (1972) Geographical ecology. Harper and Row, New YorkGoogle Scholar
- May RM (1972) Will a large complex system be stable? Nature 238:413–414PubMedCrossRefGoogle Scholar
- Metz J, Nisbet R, Geritz S (1992) How should we define ‘fitness’ for general ecological scenarios? Trends Ecol Evol 7(6):198–202PubMedCrossRefGoogle Scholar
- Metz J, Mollison D, van den Bosch F (2000) The dynamics of invasion waves. In: Dieckmann U, Law R, Metz J (eds) The geometry of ecological interactions: simplifying spatial complexity. Cambridge University Press, Cambridge, pp 482–512CrossRefGoogle Scholar
- Metz JAJ (2008) Fitness. In: Jørgensen SE, Fath BD (eds) Encyclopedia of Ecology, vol 2. Elsevier, Oxford, pp 1599–1612CrossRefGoogle Scholar
- Plagány ÉE (2007) Models for an ecosystem approach to fisheries. FAO Fisheries technical paper 477, FAO, RomeGoogle Scholar
- Post WM, Pimm SL (1983) Community assembly and food web stability. Math Biosci 64:169–192CrossRefGoogle Scholar
- Samuelson PA (1978) Generalizing Fisher’s “reproductive value”: “incipient” and “penultimate” reproductive-value functions when environment limits growth; linear approximants for nonlinear Mendelian mating models. Proc Natl Acad Sci U S A 75(12):6327–6331PubMedCrossRefGoogle Scholar
- Sethna JP (2006) Statistical mechanics entropy, order parameters and complexity. Oxford University Press, OxfordGoogle Scholar
- Szilágyi A, Meszéna G (2009) Limiting similarity and niche theory for structured populations. J Theor Biol 258:27–37PubMedCrossRefGoogle Scholar
- Van Leeuwen A, De Roos AM, Persson L (2008) How cod shapes its world. J Sea Res 60(1–2, Sp. Iss. SI):89–104. doi: 10.1016/j.seares.2008.02.008 (Niels Daan Symposium on Sustainable Management of Marine Living Resources, Wageningen, 19 Apr 2007)Google Scholar