Concave-convex nonautonomous scalar ordinary differential equations: from bifurcation theory to critical transitions
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| Format: | Preprint |
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2024
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| author | Dueñas, Jesús Núñez, Carmen Obaya, Rafael |
| author_facet | Dueñas, Jesús Núñez, Carmen Obaya, Rafael |
| contents | A mathematical modeling process for phenomena with a single state variable that attempts to be realistic must be given by a scalar nonautonomous differential equation $x'=f(t,x)$ that is concave with respect to the state variable $x$ in some regions of its domain and convex in the complementary zones. This article takes the first step towards developing a theory to describe the corresponding dynamics: the case in which $f$ is concave on the region $x\ge b(t)$ and convex on $x\le b(t)$, where $b$ is a $C^1$ map, is considered. The different long-term dynamics that may appear are analyzed while describing the bifurcation diagram for $x'=f(t,x)+λ$. The results are used to establish conditions on a concave-convex map $h$ and a nonnegative map $k$ ensuring the existence of a value $ρ_0$ giving rise to the unique critical transition for the parametric family of equations $x'=h(t,x)-ρ\,k(t,x)$, which is assumed to approach $x'=h(t,x)$ as time decreases, but for which no conditions are assumed on the future dynamics. The developed theory is justified by showing that concave-convex models fit correctly some laboratory experimental data, and applied to describe a population dynamics model for which a large enough increase on the peak of a temporary higher predation causes extinction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14667 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concave-convex nonautonomous scalar ordinary differential equations: from bifurcation theory to critical transitions Dueñas, Jesús Núñez, Carmen Obaya, Rafael Dynamical Systems 37B55, 37G35, 37N25 A mathematical modeling process for phenomena with a single state variable that attempts to be realistic must be given by a scalar nonautonomous differential equation $x'=f(t,x)$ that is concave with respect to the state variable $x$ in some regions of its domain and convex in the complementary zones. This article takes the first step towards developing a theory to describe the corresponding dynamics: the case in which $f$ is concave on the region $x\ge b(t)$ and convex on $x\le b(t)$, where $b$ is a $C^1$ map, is considered. The different long-term dynamics that may appear are analyzed while describing the bifurcation diagram for $x'=f(t,x)+λ$. The results are used to establish conditions on a concave-convex map $h$ and a nonnegative map $k$ ensuring the existence of a value $ρ_0$ giving rise to the unique critical transition for the parametric family of equations $x'=h(t,x)-ρ\,k(t,x)$, which is assumed to approach $x'=h(t,x)$ as time decreases, but for which no conditions are assumed on the future dynamics. The developed theory is justified by showing that concave-convex models fit correctly some laboratory experimental data, and applied to describe a population dynamics model for which a large enough increase on the peak of a temporary higher predation causes extinction. |
| title | Concave-convex nonautonomous scalar ordinary differential equations: from bifurcation theory to critical transitions |
| topic | Dynamical Systems 37B55, 37G35, 37N25 |
| url | https://arxiv.org/abs/2412.14667 |