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Predicting population extinction from early observations of the Lotka–Volterra system

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  • Skvortsov, Alex
  • Ristic, Branko
  • Kamenev, Alex

Abstract

Population extinction is one of the central themes in population biology. We propose a statistical algorithm for long-term prediction of an extinction event in the paradigmatic predator–prey model. The algorithm is based on noisy and sporadic observations of the Lotka–Volterra (LV) system at the early stages of its evolution, when the system is still very far from extinction. There are two stages in the algorithm: first, the unknown parameters (reaction rates) of the LV system are estimated using the Approximate Bayesian Computation method; then an analytical expression for the time-scale of extinction (which involves the estimated parameters) is applied to compute the probability density function of extinction time. The proposed algorithm is validated by numerical simulations for the case of a stochastic LV system specified by the birth–death rate equations. The algorithm can be seen as an initial step in the quest for long-term prediction of rare “catastrophic” events in complex stochastic dynamic systems (epidemics, host-parasite dynamics, enzyme kinetics, dynamic trading, etc.).

Suggested Citation

  • Skvortsov, Alex & Ristic, Branko & Kamenev, Alex, 2018. "Predicting population extinction from early observations of the Lotka–Volterra system," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 371-379.
    • Handle: RePEc:eee:apmaco:v:320:y:2018:i:c:p:371-379
    DOI: 10.1016/j.amc.2017.09.029
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    References listed on IDEAS

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    1. Barry W. Brook & Julian J. O'Grady & Andrew P. Chapman & Mark A. Burgman & H. Resit Akçakaya & Richard Frankham, 2000. "Predictive accuracy of population viability analysis in conservation biology," Nature, Nature, vol. 404(6776), pages 385-387, March.
    2. Christophe Andrieu & Arnaud Doucet & Roman Holenstein, 2010. "Particle Markov chain Monte Carlo methods," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(3), pages 269-342, June.
    3. Richard Robinson, 2013. "Bursting with Randomness: A Simple Model for Stochastic Control of Gene Expression," PLOS Biology, Public Library of Science, vol. 11(8), pages 1-1, August.
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    Cited by:

    1. Vadillo, Fernando, 2019. "Comparing stochastic Lotka–Volterra predator-prey models," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 181-189.
    2. Ahmed Nafidi & Meriem Bahij & Ramón Gutiérrez-Sánchez & Boujemâa Achchab, 2020. "Two-Parameter Stochastic Weibull Diffusion Model: Statistical Inference and Application to Real Modeling Example," Mathematics, MDPI, vol. 8(2), pages 1-11, January.
    3. Nafidi, A. & Bahij, M. & Achchab, B. & Gutiérrez-Sanchez, R., 2019. "The stochastic Weibull diffusion process: Computational aspects and simulation," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 575-587.

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