Critical Transitions for Asymptotically Concave or d-Concave Nonautonomous Differential Equations with Applications in Ecology

Abstract

The occurrence of tracking or tipping situations for a transition equation \(x'=f(t,x,\Gamma (t,x))\) with asymptotic limits \(x'=f(t,x,\Gamma _\pm (t,x))\) is analyzed. The approaching condition is just \(\lim _{t\rightarrow \pm \infty }(\Gamma (t,x)-\Gamma _\pm (t,x))=0\) uniformly on compact real sets, and so there is no restriction to the dependence on time of the asymptotic equations. The hypotheses assume concavity in x either of the maps \(x\mapsto f(t,x,\Gamma _\pm (t,x))\) or of their derivatives with respect to the state variable (d-concavity), but not of \(x\mapsto f(t,x,\Gamma (t,x))\) nor of its derivative. The analysis provides a powerful tool to analyze the occurrence of critical transitions for one-parametric families \(x'=f(t,x,\Gamma ^c(t,x))\). The new approach significatively widens the field of application of the results, since the evolution law of the transition equation can be essentially different from those of the limit equations. Among these applications, some scalar population dynamics models subject to nontrivial predation and migration patterns are analyzed, both theoretically and numerically. Some key points in the proofs are: to understand the transition equation as part of an orbit in its hull which approaches the -limit and -limit sets; to observe that these sets concentrate all the ergodic measures; and to prove that in order to describe the dynamical possibilities of the equation it is sufficient that the concavity or d-concavity conditions hold for a complete measure subset of the equations of the hull.