Derivation and Analysis of a Discrete Predator–Prey Model

Author

Sabrina H. Streipert
Gail S.K. Wolkowicz
Martin Bohner, Missouri University of Science and TechnologyFollow

Abstract

We derive a discrete predator–prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark–Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark–Sacker bifurcation as a function of the model parameters.

Recommended Citation

S. H. Streipert et al., "Derivation and Analysis of a Discrete Predator–Prey Model," Bulletin of Mathematical Biology, vol. 84, no. 7, article no. 67, Springer, Jul 2022.

The definitive version is available at https://doi.org/10.1007/s11538-022-01016-4

Department(s)

Mathematics and Statistics

Comments

Natural Sciences and Engineering Research Council of Canada, Grant None

Keywords and Phrases

Difference equations; Global stability; Lyapunov function; Neimark–Sacker bifurcation; Predator–prey

International Standard Serial Number (ISSN)

1522-9602; 0092-8240

Document Type

Article - Journal

Document Version

Final Version

File Type

text

Language(s)

English

Rights

© 2023 Springer, All rights reserved.

Publication Date

01 Jul 2022

PubMed ID

35596850