Simple models of cannibalism: a critique and a new approach

O. Diekmann, R.M. Nisbet, William Gurney, F. van den Bosch

Research output: Contribution to journalArticle

64 Citations (Scopus)

Abstract

We show how to incorporate a functional response in recent models of Gurtin, Levine, and others for egg cannibalism. Starting from a relatively complicated model with vulnerability spread over an age interval of finite duration ϵ, we arrive at a much simpler model by passing to the limit ϵ ↓ 0. It turns out that survivorship through the vulnerable stage is implicitly determined by the solution of a scalar equation. Subsequently we study the existence and stability of steady states, and we find (analytically in a simple case, numerically in more general situations) curves in a two-dimensional parameter space where a nontrivial steady state loses its stability and a periodic solution arises through a Hopf bifurcation.
LanguageEnglish
Pages21-46
Number of pages26
JournalMathematical Biosciences
Volume78
Issue number1
DOIs
Publication statusPublished - Mar 1986

Fingerprint

Cannibalism
cannibalism
Ovum
functional response models
Hopf bifurcation
Functional Response
survival rate
Vulnerability
Hopf Bifurcation
duration
Parameter Space
Periodic Solution
Scalar
Model
Curve
Interval

Keywords

  • cannibalism
  • egg cannibalism
  • Hopf bifurcation
  • scalar equation

Cite this

Diekmann, O. ; Nisbet, R.M. ; Gurney, William ; van den Bosch, F. / Simple models of cannibalism : a critique and a new approach. In: Mathematical Biosciences. 1986 ; Vol. 78, No. 1. pp. 21-46.
@article{bdf09f71adcb4d2e81c6a2c3f39bbb87,
title = "Simple models of cannibalism: a critique and a new approach",
abstract = "We show how to incorporate a functional response in recent models of Gurtin, Levine, and others for egg cannibalism. Starting from a relatively complicated model with vulnerability spread over an age interval of finite duration ϵ, we arrive at a much simpler model by passing to the limit ϵ ↓ 0. It turns out that survivorship through the vulnerable stage is implicitly determined by the solution of a scalar equation. Subsequently we study the existence and stability of steady states, and we find (analytically in a simple case, numerically in more general situations) curves in a two-dimensional parameter space where a nontrivial steady state loses its stability and a periodic solution arises through a Hopf bifurcation.",
keywords = "cannibalism, egg cannibalism, Hopf bifurcation, scalar equation",
author = "O. Diekmann and R.M. Nisbet and William Gurney and {van den Bosch}, F.",
year = "1986",
month = "3",
doi = "10.1016/0025-5564(86)90029-5",
language = "English",
volume = "78",
pages = "21--46",
journal = "Mathematical Biosciences",
issn = "0025-5564",
number = "1",

}

Simple models of cannibalism : a critique and a new approach. / Diekmann, O.; Nisbet, R.M.; Gurney, William; van den Bosch, F.

In: Mathematical Biosciences, Vol. 78, No. 1, 03.1986, p. 21-46.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Simple models of cannibalism

T2 - Mathematical Biosciences

AU - Diekmann, O.

AU - Nisbet, R.M.

AU - Gurney, William

AU - van den Bosch, F.

PY - 1986/3

Y1 - 1986/3

N2 - We show how to incorporate a functional response in recent models of Gurtin, Levine, and others for egg cannibalism. Starting from a relatively complicated model with vulnerability spread over an age interval of finite duration ϵ, we arrive at a much simpler model by passing to the limit ϵ ↓ 0. It turns out that survivorship through the vulnerable stage is implicitly determined by the solution of a scalar equation. Subsequently we study the existence and stability of steady states, and we find (analytically in a simple case, numerically in more general situations) curves in a two-dimensional parameter space where a nontrivial steady state loses its stability and a periodic solution arises through a Hopf bifurcation.

AB - We show how to incorporate a functional response in recent models of Gurtin, Levine, and others for egg cannibalism. Starting from a relatively complicated model with vulnerability spread over an age interval of finite duration ϵ, we arrive at a much simpler model by passing to the limit ϵ ↓ 0. It turns out that survivorship through the vulnerable stage is implicitly determined by the solution of a scalar equation. Subsequently we study the existence and stability of steady states, and we find (analytically in a simple case, numerically in more general situations) curves in a two-dimensional parameter space where a nontrivial steady state loses its stability and a periodic solution arises through a Hopf bifurcation.

KW - cannibalism

KW - egg cannibalism

KW - Hopf bifurcation

KW - scalar equation

U2 - 10.1016/0025-5564(86)90029-5

DO - 10.1016/0025-5564(86)90029-5

M3 - Article

VL - 78

SP - 21

EP - 46

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

IS - 1

ER -