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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mi>κ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and convex on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mi>κ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>ρ</mml:mi> <mml:mstyle scriptlevel="0"/> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , which is assumed to approach <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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.