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Derivation and Analysis of a Discrete Predator-Prey Model.

Sabrina H Streipert1, Gail S K Wolkowicz2, Martin Bohner3

  • 1Department of Mathematics and Statistics, McMaster University, 1280 Main St. W., Hamilton, ON, L8S4K1, Canada. streipes@mcmaster.ca.

Bulletin of Mathematical Biology
|May 21, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces a new discrete predator-prey model. The model predicts stable prey populations at low predator consumption rates and oscillating coexistence at higher rates, with analysis of bifurcations.

Keywords:
Difference equationsGlobal stabilityLyapunov functionNeimark–Sacker bifurcationPredator–prey

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Population Dynamics

Background:

  • Predator-prey models are fundamental to understanding population dynamics.
  • Existing models often use continuous differential equations, which may not capture discrete population changes.
  • A discrete model derived from first principles is needed to better reflect natural populations.

Purpose of the Study:

  • To derive a novel discrete predator-prey model from fundamental ecological assumptions.
  • To analyze the stability of the prey-only equilibrium and the conditions for coexistence.
  • To investigate the dynamics of the discrete model, including bifurcations and oscillations.

Main Methods:

  • Derivation of a discrete predator-prey model using an economic technique for continuous-discrete compounding.
  • Extension of phase plane analysis with a novel 'next iterate root-curve'.
  • Application of Lyapunov functions and bifurcation analysis (Neimark-Sacker bifurcation).

Main Results:

  • The prey-only equilibrium is globally asymptotically stable below a specific predator consumption-energy rate threshold.
  • Above this threshold, a stable coexistence equilibrium emerges, with solutions oscillating around it.
  • The study identifies conditions for local stability and destabilization via a supercritical Neimark-Sacker bifurcation, bounding oscillation amplitudes.

Conclusions:

  • The derived discrete predator-prey model offers a more realistic representation of population interactions.
  • The model accurately predicts transitions from stable prey populations to oscillating coexistence.
  • Bifurcation analysis reveals complex dynamics, including the emergence and stability of coexistence equilibria.