Concave-convex nonautonomous scalar ordinary differential equations: from bifurcation theory to critical transitions

TL;DR

This paper develops a theory for scalar nonautonomous differential equations with concave and convex regions, analyzing bifurcations and critical transitions, and applies it to population dynamics and experimental data.

Contribution

It introduces a new theoretical framework for concave-convex scalar equations, linking bifurcation analysis to critical transitions in nonautonomous systems.

Findings

Identifies conditions for critical transitions in concave-convex models.

Demonstrates the theory's applicability to laboratory data.

Shows how parameter changes can lead to extinction in population models.

Abstract

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 $x'=f(t,x)+\lambda$. The results are used to establish conditions on a concave-convex map $h$ and a nonnegative map $k$ ensuring the existence of a value $\rho_0$ giving rise to the unique critical transition for the parametric family of equations $x'=h(t,x)-\rho\,k(t,x)$, 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.