Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1703.09266

Canard Phenomenon in a modified Slow-Fast Leslie-Gower and Holling type scheme model

B. Ambrosio·M. A. Aziz-Alaoui·R. Yafia
View original source ↗Read source document ↗Export metadata ↓

Abstract

Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie-Gower type for which we consider that prey reproduces mush faster than predators. This naturally leads to introduce a small parameter $\epsilon$ which gives rise to a slow-fast system. This system has a special folded singularity which has not been analyzed in the classical work of Krupa-Szmolyan. We use the blow-up technique to visualize the behavior near this fold point $P$. Outside of this region the dynamics are given by classical singular perturbation theory. This allows to quantify geometrically the attractive limit-cycle with an error of $O(\epsilon$) and shows that it exhibits the \textit{canard} phenomenon while crossing $P$.

Explore related subjects

math.DS

Keep this discovery

BibTeXRIS

B. Ambrosio, M. A. Aziz-Alaoui, R. Yafia. 2017-03-27. Canard Phenomenon in a modified Slow-Fast Leslie-Gower and Holling type scheme model. https://arxiv.org/abs/1703.09266

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗