### Abstract

We study a predator–prey model with different characteristic time scales for the prey and predator populations, assuming that the predator dynamics is much slower than the prey one. Geometrical Singular Perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig–MacArthur predator–prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on Blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.

Original language | English |
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Pages (from-to) | 1-22 |

Number of pages | 22 |

Journal | Journal of Mathematical Biology |

DOIs | |

Publication status | E-pub ahead of print - 20 Feb 2019 |

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### Keywords

- Blow-up
- Canard solution
- Invariant manifolds
- Predator–prey dynamics
- Singular perturbations

### Cite this

*Journal of Mathematical Biology*, 1-22. https://doi.org/10.1007/s00285-019-01337-4

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*Journal of Mathematical Biology*, pp. 1-22. https://doi.org/10.1007/s00285-019-01337-4

**Analysis of a predator–prey model with specific time scales : a geometrical approach proving the occurrence of canard solutions.** / Poggiale, Jean Christophe; Aldebert, Clément; Girardot, Benjamin; Kooi, Bob W.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - Analysis of a predator–prey model with specific time scales

T2 - a geometrical approach proving the occurrence of canard solutions

AU - Poggiale, Jean Christophe

AU - Aldebert, Clément

AU - Girardot, Benjamin

AU - Kooi, Bob W.

PY - 2019/2/20

Y1 - 2019/2/20

N2 - We study a predator–prey model with different characteristic time scales for the prey and predator populations, assuming that the predator dynamics is much slower than the prey one. Geometrical Singular Perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig–MacArthur predator–prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on Blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.

AB - We study a predator–prey model with different characteristic time scales for the prey and predator populations, assuming that the predator dynamics is much slower than the prey one. Geometrical Singular Perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig–MacArthur predator–prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on Blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.

KW - Blow-up

KW - Canard solution

KW - Invariant manifolds

KW - Predator–prey dynamics

KW - Singular perturbations

UR - http://www.scopus.com/inward/record.url?scp=85061920540&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061920540&partnerID=8YFLogxK

U2 - 10.1007/s00285-019-01337-4

DO - 10.1007/s00285-019-01337-4

M3 - Article

SP - 1

EP - 22

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

ER -