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Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.
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Predicting the Effectiveness of Population Replacement Strategy Using Mathematical Modeling
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Predicting population extinction in lattice-based birth-death-movement models.

Stuart T Johnston1,2, Matthew J Simpson3, Edmund J Crampin1,2,4

  • 1Systems Biology Laboratory, School of Mathematics and Statistics, and Department of Biomedical Engineering, University of Melbourne, Parkville, Victoria, Australia.

Proceedings. Mathematical, Physical, and Engineering Sciences
|August 25, 2020
PubMed
Summary

This study introduces a new state-space diffusion approximation for population dynamics. This method accurately predicts population extinction, outperforming standard approximations and reducing computation time.

Keywords:
continuum limitdiffusionextinctionmathematical modellingpopulation dynamicsrandom walk

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Area of Science:

  • Ecology and population biology
  • Mathematical modeling

Background:

  • Population persistence and extinction are central ecological questions.
  • Lattice-based random walk models with exclusion simulate population dynamics but are computationally intensive.
  • Standard continuum approximations fail to accurately predict population extinction.

Purpose of the Study:

  • To develop a novel continuum approximation for population dynamics that accurately predicts extinction.
  • To compare the predictive power and computational efficiency of the new approximation against existing methods.

Main Methods:

  • Development of the state-space diffusion approximation, a novel continuum approximation.
  • Comparison with traditional lattice-based random walk models and standard continuum approximations.
  • Analysis of the influence of lattice size and initial population on long-term behavior.

Main Results:

  • The state-space diffusion approximation accurately captures population extinction dynamics.
  • The new approximation provides additional insights compared to standard methods.
  • Significant reduction in computation time demonstrated compared to the random walk model.

Conclusions:

  • The state-space diffusion approximation offers a computationally efficient and accurate tool for ecological modeling.
  • This method enhances the prediction of population persistence and extinction.
  • The approach is valuable for understanding long-term population behavior under various conditions.