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

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 ⩾ κ ( t ) and convex on x ⩽ κ ( t ) , where κ is a C 1 map, is considered. The different long-term dynamics that may appear are analysed 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 applied to describe a population dynamics model for which a sufficiently large increase in the peak of a temporary rise in predation leads to extinction, and justified by showing that concave–convex models fit some laboratory experimental data correctly, by means of a technical and very detailed spline-based model construction.