2024/12/19 by Jesús García de Dueñas, Dueñas, Jesús, Carmen Núñez +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #37B55 #37G35 #37N25 #Advanced Differential Equations and Dynamical Systems #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2412.14667
openalex publication_date 2024/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ b(t) and convex on x≤ b(t), where b is a C1 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 ρ0 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.