Abstract
The aim of this chapter is to provide interested outsiders with a brief (and therefore incomplete) overview of the kind of questions and insights concerning the dynamics of biological populations that can be formulated in mathematical language and derived by mathematical methods. Ideally the chapter should serve as an invitation to further reading and hence we give many pointers to the extensive literature. In order to highlight the ideas, we sacrifice the precise statement of assumptions (implying that some of our statements are sloppy from the point of view of the pedantic mathematician). Likewise we shall focus on the simplest examples that illustrate a key issue and not strive for generality. Our aim is to enlighten, not to impress. Moreover, we have not tried to hide our bias deriving from taste and experience, so the views we present are somewhat idiosyncratic.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Anderson, R.M. and R.M. May, Infectious diseases of humans. Oxford University Press, 2000.
AUTO continuation package, see URL: http://indy.cs.concordia.ca/auto/main.html
Behnke, H., Periodical cicades. J. Math. Biol., Vol. 40, p. 413–431 (2000).
Boer, M.P., B.W. Kooi and S.A.L.M. Kooijman, Multiple attractors and boundary crisis in a tri-trophic food chain. Math. Biosciences, Vol. 169, p. 109–128 (2001).
Boerlijst, M.C., Spirals and spots: novel evolutionary phenomena through spatial self-structuring. In: (eds), The geometry of ecological interaction. Simplifying spatial complexity. Cambridge University Press, 2000.
Bulmer, Periodical insects. Am. Nat., Vol. 3?, p. 1099–1117 (1977).
Butler, G.J., S.B. Hsu and P. Waltman, Coexistence of competing predators in a chemostat. J. Math. Biol., Vol. 17, p. 133–151 (1983).
Butler, G.J. and G.S.K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J. Appl.Math., Vol. 45, 1, p. 138–151 (1985).
Butler, G.J. and G.S.K. Wolkowicz, Predator-mediated competition in the chemostat. J. Math. Biol, Vol. 24, p. 167–191 (1986).
Butler, G.J., S.B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate. SIAM J.Appl.Math., Vol. 45, 3, p. 435–449 (1985).
Calsina, À. and J. Saldava, A model of physiologically structured population dynamics with nonlinear individual growth rate. J. Math. Biol., Vol. 33, p. 335–364 (1995).
Claessen, D., A.M. deRoos and L. Persson, Dwarfs and giants: cannibalism and competition in size-structured populations. Am. Nat., Vol. 155, p. 219–237 (2000).
CONTENT continuation package for different computing platforms, see URL: http://www.math.uu.nl/people/kuznet/index.html
Cushing, J.M., An introduction to structured population dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics 71. SIAM, 1998.
DeAngelis, D. and L. Gross, Individual-based models and approaches in ecology: Populations, communities and ecosystems. Chapman Hall, New York. 1992.
Dieckmann, U., R. Law and J.A.J. Metz (eds.), The geometry of ecological interactions: Cambridge University Press, (2000).
Dieckmann, U., J.A.J. Metz, M.W. Sabelis and K. Sigmund, Adaptive dynamics of infectious diseases: in pursuit of virulence management. Cambridge Studies in Adaptive Dynamics Cambridge University Press, 2002.
Diekmann, O. F.B. Christiansen and R. Law (eds.) Special Issue on Evolutionary Dynamics. J. Math. Biol., Vol. 34, Issues 5/6, (1996).
Diekmann, O., The many facets of evolutionary dynamics. J. Biol. Systems, Vol. 5, p. 325–339 (1997).
Diekmann, O., M. Gyllenberg, J.A.J. Metz and H.R. Thieme, On the formulation and analysis of general deterministic structured population models. I. Linear theory. J. Math. Biol., Vol. 36, 4, p. 349–388 (1998).
Diekmann, O. and H. Heesterbeek, Mathematical epidemiology of infectious diseases. Wiley, 2000.
Diekmann, O., M. Gyllenberg, H. Huang, M. Kirkilionis, J.A.J. Metz and H.R. Thieme, On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory. J. Math. Biol., Vol. 43, 2, p. 157–189 (2001).
Geritz, S.A.H.,. Kisdi, G. Meszna and J.A.J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree.. Evol. Ecol., Vol. 12, 1, p. 35–57 (1989).
Geritz, S.A.H., J.A. J. Metz,. Kisdia and G. Meszna, Dynamics of adaptation and evolutionary branching. Phys. Rev. Letters, Vol. 78, p. 2024–2027 (1997).
Geritz, S.A.H., M. Gyllenberg, F. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance. J. Math. Biol., Vol. 44, p. 548–560 (2002).
Hale, J.K. and A.S. Somolinos, Competition for fluctuating nutrient. J. Math. Biol., Vol. 18, p. 255–280 (1983).
Hanski, I.A. and M.E. Gilpin, Metapopulation biology. Ecology, genetics, and evolution. Academic Press, San Diego. 1997.
Hsu, S.B., A competition model for a seasonally fluctuating nutrient. J. Math. Biol., Vol. p. (1980).
Hsu, S.B. and P. Waltman, On a system of reaction-diffusion equations arising from competition in an unstirred chemostat. SIAM J. Appl. Math., Vol. 53, p. 1026–1044 (1993).
Huang, Y. and O. Diekmann, Predator migration in resone to prey density: what are the consequences?. J. Math. Biol., Vol. 43, p. 561–581 (2001).
Huisman, J. and F.J. Weissing, Oscillations and chaos generated by competition for interactively essential resources. Ecological Research, Vol. 17, p. 175–181 (2002).
Jäger, W., B. Tang and P. Waltman, Competition in the Gradostat. J. Math. Biol., Vol. 25, p. 23–42 (1987).
Jansen, V.A.A. and A.M. de Roos, The role of space in reducing prey-predator cycles. In: U. Dieckmann, R. Law and M. a. J. Metz (eds), The geometry of ecological interaction. Simplifying spatial complexity. Cambridge University Press, 2000.
Kendall, D.G., Mathematical models of the spread of infections. In: Mathematics and computer sciences in biology and medicine. Medical Research Council, London, 1965.
Kirkilionis, M., O. Diekmann, B. Lisser, M. Nool, A. deRoos and B. Sommeijer, Numerical continuation of equilibria of physiologically structured population models. I. Theory. Mathematical Models and Methods in Applied Sciences, Vol. 11, 6, p. 1–27 (2001).
de Koeijer, A., O. Diekmann and P. Reijnders, Modelling the spread of Phocine Distemper Virus among harbour seals.. Bull. Math. Biol., Vol. 60, 3, p. 585–596 (1998).
Kooijman, S.A.L.M., Dynamic energy budgets in biological systems. Theory and applications in ecotoxicology.. Cambridge University Press, Cambridge, 1993.
Lovitt, R.W. and J.W.T. Wimpenny, The gradostat: a bidirectional compound chemostat and its application in microbiological research. J. Gen. Microbiol., Vol. 127, p. 261–268 (1981).
Maynard Smith, J., Mathematical Ideas in Biology. Cambridge Univ. Press, 1968.
Metz, J.A.J., O. Diekmann and (Editors), The dynamics of physiologically structured populations. Lecture Notes in Biomathematics 68, Springer-Verlag, Berlin and Heidelberg 1986.
Metz, J.A.J., S.A.H. Geritz, G. Meszna, F.J.A. Jacobs and J.S. v. Heerwaarden, Adaptive dynamics: A geometrical study of the consequences of nearly faithfull reproduction. In: S. J. v. Strien and S. M. Verduyn-Lunel (eds), Stochastic and spatial structures of dynamical systems. North Holland, Elsevier, 1996.
Metz, J.A.J., D. Mollison and F. van den Bosch, The dynamics of invasion waves. In: U. Dieckmann, R. Law and M. A. J. Metz (eds), The geometry of ecological interaction. Simplifying spatial complexity. Cambridge University Press, 2000.
Metz, J.A.J., S. Mylius and O. Diekmann, When does evolution optimise? On the relation between types of density dependence and evolutionary stable life history parameters. Report from: IIASA, Nr.: WP-96–04, 1996.
Monod, J., La technique de culture continue, theorie et application. Ann.Inst.Pasteur, Vol. 79, p. 390–410 (1950).
Mylius, S. and O. Diekmann, On evolutionary stable life histories, optimization and the need to be specific about density dependence. Oikos, Vol. 74, p. 218–284 (1995).
Mylius, S.D. and O. Diekmann, The resident strikes back: invasion induced switching of resident attractor. J. theor. Biol., Vol. 211, p. 297–311 (2001).
Neutel, A.-M., J.A.P. Heesterbeek and P.C. de Ruiter, Stability in real food webs: weak links in long loops. Science, Vol. 296, p. 1120–1123 (2002).
Nisbet, R.M. and W.S.C. Gurney, Modelling fluctuating populations. John Wiley Sons, Singapore, 1982.
Nowak, M.A. and R.M. May, Virus dynamics. Oxford University Press, 2000.
Pugliese, A., On the evolutionary coexistence of parasites. Mathematical Bio-sciences, Vol. 177 /178, p. 355–375 (2002)
Rand, D. A., H. B. Wilson and J. M. McClade, Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Phil. Trans. R. Soc. Lond. B, Vol. 343, p. 261–283 (1994).
de Roos, A.M. Numerical methods for structured population models: the escalator boxcar train. Num. Meth. Part. Diff. Equations, Vol. 4, p. 173–195 (1988).
de Roos, A.M. and L. Persson, Physiologically structured models–from versatile technique to ecological theory. Oikos, Vol. 94, p. 51–71 (2001).
de Roos, A.M. and L. Persson, Competition in size-structured populations: mechanisms inducing cohort formation and population cycles. Theor. Pop. Biol., (2002, in press).
Segel, L.A., Modeling dynamic phenomena in molecular and cellular biology. Cambridge University Press, Cambridge, 1984.
Smith, H. and B. Tang, Competition in the gradostat: the role of the communication rate. J.Math.Biol., Vol. 27, p. 139–165 (1989).
Smith, H. L., Competitive coexistence in an oscillating chemostat. SIAM J.Appl.Math., Vol. 40, 3, p. 498–522 (1981).
Smith, H.L. and P. Waltman, The theory of the chemostat Cambridge Studies in Mathematical Biology 13 Cambridge University Press, Cambridge, 1995.
Thieme, H.R., Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol., Vol. 30, p. 755–763 (1992).
Tucker, S.L. and S.O. Zimmermann, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables. SIAM J. Appl. Math., Vol. 48, 3, p. 549–591 (1988).
Tuljapurkar, S. and H. Caswell, Structured-population models in marine, terrestrial, and freshwater systems. Population and Community Biology Series 18. Chapman Hall, New York, 1997.
Yang, K. and H.I. Freedman, Uniqueness of limit cycles in Gause type models of predator-prey systems. Math. Biosciences, Vol. 88, p. 67–84 (1988).
Yodzis, P., Introduction to theoretical ecology. Harper Row, 1989.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Diekmann, O., Kirkilionis, M. (2003). Population Dynamics: A Mathematical Bird’s Eye View. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds) Trends in Nonlinear Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05281-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-662-05281-5_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07916-0
Online ISBN: 978-3-662-05281-5
eBook Packages: Springer Book Archive