Critical Transitions in D-Concave Nonautonomous Scalar Ordinary Differential Equations Appearing in Population Dynamics

Abstract

A function with finite asymptotic limits gives rise to a transition equation between a "past system" and a "future system". This question is analyzed in the case of nonautonomous coercive nonlinear scalar ordinary differential equations with concave derivative with respect to the state variable. The fundamental hypothesis is the existence of three hyperbolic solutions for the limit systems, in which case the upper and lower ones are attractive. All the global dynamical possibilities are described in terms of the internal dynamics of the pullback attractor: cases of tracking of the two hyperbolic attractive solutions or lack of it (tipping) may arise. This analysis, made in the language of processes and also in terms of the skewproduct formulation of the problem, includes cases of rate-induced critical transitions, as well as cases of phase-induced and size-induced tipping. The conclusions are applied in models of mathematical biology and population dynamics. Rate-induced tracking phenomena causing extinction of a native species or invasion of a non-native one are described, as well as population models affected by a Holling type III functional response to predation where tipping due to the changes in the size of the transition may occur. In all these cases, the appearance of a critical transition can be understood as a consequence of the strength of Allee effect.