Skip to main content

Predator–prey models: an application for the plankton dynamics of lake Geneva

Abstract

In this paper we present a hierarchical Bayesian analysis for a predator–prey model applied to ecology considering the use of Markov Chain Monte Carlo methods. We consider the introduction of a random effect in the model and the presence of a covariate vector. An application to ecology is considered using a data set related to the plankton dynamics of lake Geneva for the year 1990. We also discuss some aspects of discrimination of the proposed models.

This is a preview of subscription content, access via your institution.

References

  • Arditi R, Abillon J, Vieirada Silva J (1978) A predator–prey model with satiation and intraspecific competition. Ecol Modell 5: 173–191

    Article  Google Scholar 

  • Bazykin AD (1985) Mathematical models in Biophysics. Nauka, Moscow

    Google Scholar 

  • Boyce WE, DiPrima RC (1977) Elementtary differential equations and boundary value problems. Wiley, New York

    Google Scholar 

  • Cangelosi AR, Hooten MB (2008) Models for bounded systems with continuous dynamics. Biometrics In Press

  • Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. J Am Stat Assoc 85(410): 398–409

    Article  Google Scholar 

  • Gelman A (2006) Prior distribution for variance parameters in hierarquical models (comment on article by browne and draper). Bayesian Anal 1: 515–534

    Article  Google Scholar 

  • Gelman A, Rubin BD (1992) Inference from iterative simulation using multiple sequences. Stat Sci 4: 457–511

    Article  Google Scholar 

  • Gianni G, Pasquali S, Ruggeri F (2008) Bayesian inference for functional response in a stochastic predator–prey system. Bull Math Biol 70: 358–381

    Article  Google Scholar 

  • Goel NS, Maitra SC, Montroll EW (1971) On the volterra and other nonlinear models of interacting populations. Rev Modern Phy 43: 231–276

    Article  Google Scholar 

  • Gompertz B (1825) On the nature of the function expressive of the law of human mortality. Philos Trans 115: 513–583

    Article  Google Scholar 

  • Gutierrez A (1992) The physiological basis of ratio-dependent predator–prey theory: a metabolic pool model of nicholson’s blowflies as an example. Ecology 73: 1552–1563

    Article  Google Scholar 

  • Harrison GW (1995) Comparing predator–prey models to luckinbilláas experiment with didinium and paramecium. Ecology 76: 157–174

    Article  Google Scholar 

  • Hassel MP, Varley GC (1969) New inductive population model for insect parasites and its bearing on biological control. Nature 223: 1133–1137

    Article  Google Scholar 

  • Jost C (1998) Comparing predator–prey models qualitatively and quantitatively with ecological time-series data. Ph.D. Thesis, Institut national agronomique Paris-Grignon

  • Lotka AJ (1924) Elements of physical biology. Willians and Wilkins, Baltimore, MD

    Google Scholar 

  • Malthus TR (1798) An essay on the principle of population and a summary view of the principle of population. England, Harmondsworth

    Google Scholar 

  • Philip JR (1957) Sociality and sparce populations. Ecology 38: 107–111

    Article  Google Scholar 

  • Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Ser B 55(1): 3–23

    Google Scholar 

  • Smith FE (1963) Population dynamics in daphnia magna and a new model for population growth. Ecology 44: 651–663

    Article  Google Scholar 

  • Spiegelhalter DJ, Thomas A, Best NG, R GW (1995) BUGS: Bayesian inference using Gibbs sampling, Version 0.50. Cambridge: MRC Biostatistics Unit

  • Spiegelhalter DJ, Best NG, Vander Linde A (2000) A bayesian measure of model complexity and fit (with discussion). J R Stat Soc Ser B 64: 583–639

    Article  Google Scholar 

  • Strebel DE, Goel NS (1973) On the isocline methods for analysing prey–predator interactions. J Theor Biol 39: 211–234

    Article  PubMed  CAS  Google Scholar 

  • Sutherland WJ (1983) Aggregation and the “ideal free” distribution. J Anim Ecol 52: 821–828

    Article  Google Scholar 

  • Szathmary E (1991) Simple growth laws and selection consequences. Trends Ecol Evol 6: 366–370

    Article  PubMed  CAS  Google Scholar 

  • Verhulst PFE (1838) Notice sur la loi que la population suit dans son accroissement. Correspondances Mathematiques et physiques 10: 113–121

    Google Scholar 

  • Volterra V (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118: 558–560

    Article  Google Scholar 

  • Volterra V (1938) Population growth, equilibria, and extinction under specified breeding conditions: a development and extension of the theory of the logistic curve. Hum Biol 10: 1–11

    Google Scholar 

  • Von Bertalanffy L (1951) Theoretische Biologie. 2. A. Frank, Bern

    Google Scholar 

  • Yeargers EK, Showkwiller RW, Herod JU (1996) An Introduction to mathematics of biology: with computer algebra models. Birkhauser, Boston

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Emílio Augusto Coelho-Barros.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Achcar, J.A., Mazucheli, J. & Coelho-Barros, E.A. Predator–prey models: an application for the plankton dynamics of lake Geneva. Environ Ecol Stat 18, 315–329 (2011). https://doi.org/10.1007/s10651-010-0134-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10651-010-0134-z

Keywords

  • Predator–prey models
  • Bayesian analysis
  • Random effects
  • Plankton dynamics
  • Markov Chain Monte Carlo