Skip to main content
SpringerLink
Log in

Population Dynamics: A Mathematical Bird’s Eye View

  • Conference paper
Trends in Nonlinear Analysis

  • 588 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, R.M. and R.M. May, Infectious diseases of humans. Oxford University Press, 2000.

    Google Scholar 

  2. AUTO continuation package, see URL: http://indy.cs.concordia.ca/auto/main.html

    Google Scholar 

  3. Behnke, H., Periodical cicades. J. Math. Biol., Vol. 40, p. 413–431 (2000).

    Article  MathSciNet  Google Scholar 

  4. 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).

    Article  MathSciNet  MATH  Google Scholar 

  5. 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.

    Google Scholar 

  6. Bulmer, Periodical insects. Am. Nat., Vol. 3?, p. 1099–1117 (1977).

    Google Scholar 

  7. 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).

    Article  MathSciNet  MATH  Google Scholar 

  8. 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).

    MathSciNet  MATH  Google Scholar 

  9. Butler, G.J. and G.S.K. Wolkowicz, Predator-mediated competition in the chemostat. J. Math. Biol, Vol. 24, p. 167–191 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  10. 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).

    Article  MathSciNet  MATH  Google Scholar 

  11. 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).

    Article  MathSciNet  MATH  Google Scholar 

  12. 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).

    Article  Google Scholar 

  13. CONTENT continuation package for different computing platforms, see URL: http://www.math.uu.nl/people/kuznet/index.html

    Google Scholar 

  14. Cushing, J.M., An introduction to structured population dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics 71. SIAM, 1998.

    Google Scholar 

  15. DeAngelis, D. and L. Gross, Individual-based models and approaches in ecology: Populations, communities and ecosystems. Chapman Hall, New York. 1992.

    Google Scholar 

  16. Dieckmann, U., R. Law and J.A.J. Metz (eds.), The geometry of ecological interactions: Cambridge University Press, (2000).

    Google Scholar 

  17. 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.

    Google Scholar 

  18. Diekmann, O. F.B. Christiansen and R. Law (eds.) Special Issue on Evolutionary Dynamics. J. Math. Biol., Vol. 34, Issues 5/6, (1996).

    Google Scholar 

  19. Diekmann, O., The many facets of evolutionary dynamics. J. Biol. Systems, Vol. 5, p. 325–339 (1997).

    Article  MATH  Google Scholar 

  20. 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).

    Article  MathSciNet  MATH  Google Scholar 

  21. Diekmann, O. and H. Heesterbeek, Mathematical epidemiology of infectious diseases. Wiley, 2000.

    Google Scholar 

  22. 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).

    Article  MathSciNet  MATH  Google Scholar 

  23. 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).

    Google Scholar 

  24. 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).

    Article  Google Scholar 

  25. 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).

    Article  MathSciNet  MATH  Google Scholar 

  26. Hale, J.K. and A.S. Somolinos, Competition for fluctuating nutrient. J. Math. Biol., Vol. 18, p. 255–280 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  27. Hanski, I.A. and M.E. Gilpin, Metapopulation biology. Ecology, genetics, and evolution. Academic Press, San Diego. 1997.

    MATH  Google Scholar 

  28. Hsu, S.B., A competition model for a seasonally fluctuating nutrient. J. Math. Biol., Vol. p. (1980).

    Google Scholar 

  29. 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).

    MATH  Google Scholar 

  30. 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).

    Article  MathSciNet  MATH  Google Scholar 

  31. Huisman, J. and F.J. Weissing, Oscillations and chaos generated by competition for interactively essential resources. Ecological Research, Vol. 17, p. 175–181 (2002).

    Article  Google Scholar 

  32. Jäger, W., B. Tang and P. Waltman, Competition in the Gradostat. J. Math. Biol., Vol. 25, p. 23–42 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  33. 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.

    Google Scholar 

  34. Kendall, D.G., Mathematical models of the spread of infections. In: Mathematics and computer sciences in biology and medicine. Medical Research Council, London, 1965.

    Google Scholar 

  35. 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).

    Article  MathSciNet  Google Scholar 

  36. 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).

    Article  Google Scholar 

  37. Kooijman, S.A.L.M., Dynamic energy budgets in biological systems. Theory and applications in ecotoxicology.. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  38. 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).

    Google Scholar 

  39. Maynard Smith, J., Mathematical Ideas in Biology. Cambridge Univ. Press, 1968.

    Google Scholar 

  40. 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.

    Google Scholar 

  41. 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.

    Google Scholar 

  42. 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.

    Google Scholar 

  43. 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.

    Google Scholar 

  44. Monod, J., La technique de culture continue, theorie et application. Ann.Inst.Pasteur, Vol. 79, p. 390–410 (1950).

    Google Scholar 

  45. 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).

    Article  Google Scholar 

  46. 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).

    Article  Google Scholar 

  47. 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).

    Article  Google Scholar 

  48. Nisbet, R.M. and W.S.C. Gurney, Modelling fluctuating populations. John Wiley Sons, Singapore, 1982.

    MATH  Google Scholar 

  49. Nowak, M.A. and R.M. May, Virus dynamics. Oxford University Press, 2000.

    Google Scholar 

  50. Pugliese, A., On the evolutionary coexistence of parasites. Mathematical Bio-sciences, Vol. 177 /178, p. 355–375 (2002)

    Google Scholar 

  51. 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).

    Article  Google Scholar 

  52. 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).

    Google Scholar 

  53. de Roos, A.M. and L. Persson, Physiologically structured models–from versatile technique to ecological theory. Oikos, Vol. 94, p. 51–71 (2001).

    Article  Google Scholar 

  54. 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).

    Google Scholar 

  55. Segel, L.A., Modeling dynamic phenomena in molecular and cellular biology. Cambridge University Press, Cambridge, 1984.

    MATH  Google Scholar 

  56. Smith, H. and B. Tang, Competition in the gradostat: the role of the communication rate. J.Math.Biol., Vol. 27, p. 139–165 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  57. Smith, H. L., Competitive coexistence in an oscillating chemostat. SIAM J.Appl.Math., Vol. 40, 3, p. 498–522 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  58. Smith, H.L. and P. Waltman, The theory of the chemostat Cambridge Studies in Mathematical Biology 13 Cambridge University Press, Cambridge, 1995.

    Book  Google Scholar 

  59. Thieme, H.R., Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations. J. Math. Biol., Vol. 30, p. 755–763 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  60. 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).

    MathSciNet  MATH  Google Scholar 

  61. 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.

    Google Scholar 

  62. 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).

    Article  MATH  Google Scholar 

  63. Yodzis, P., Introduction to theoretical ecology. Harper Row, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Universiteit Utrecht, P.O. Box 80010, 3508 TA, Utrecht, The Netherlands

    Odo Diekmann

  2. Mathematics Department & Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, UK

    Markus Kirkilionis

Authors
  1. Odo Diekmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Kirkilionis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematics Department and Center for Scientific Computing, University of Warwick, CV4 7AL, Coventry, UK

    Markus Kirkilionis

  2. Institute of Scientific Computing, University of Heidelberg, Im Neuheimer Feld 368, 69120, Heidelberg, Germany

    Susanne Krömker

  3. Institute of Applied Mathematics, University of Heidelberg, Im Neuheimer Feld 293, 69120, Heidelberg, Germany

    Rolf Rannacher

  4. Department of Mathematics, University of Heidelberg, Im Neuheimer Feld 288, 69120, Heidelberg, Germany

    Friedrich Tomi

Rights and permissions

Reprints 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

Publish with us

Policies and ethics

Search

Navigation