CSDC Seminar: Rate-induced tipping in non-autonomous reaction-diffusion systems: An invariant manifold framework and shifting habitats

The mathematical modelling of tipping points - large and sudden changes in the state of a system that arise in response to small and slow changes in the external inputs – has mainly focused on ordinary differential equation (ODE) models. In this talk, I will begin with a brief overview of tipping points in ODEs. I will then consider reaction-diffusion equations (RDEs) with time-dependent (nonautonomous) and space-dependent (heterogenous) reaction terms that decay in space (asymptotically homogeneous). Such models are likely to exhibit new and interesting tipping mechanisms, but their analysis is more challenging and requires new techniques.

As an illustrative example, we analyse a conceptual model of a habitat patch in one spatial dimension, that features an Allee effect in population growth and is geographically shrinking or shifting due to human activity and climate change. We identify two classes of tipping points to extinction: bifurcation-induced tipping (B-tipping) when the shrinking habitat falls below some critical length, and rate-induced tipping (R-tipping) when the shifting habitat exceeds some critical speed.

To facilitate the analysis of tipping points in RDEs, such as the moving habitat model, we propose a new mathematical framework. This framework is underpinned by a special compactification of the moving-frame coordinate in conjunction with Lin’s method for constructing heteroclinic orbits along intersections of stable and unstable invariant manifolds of saddles. This allows us to (i) obtain multiple coexisting pulse and front solutions for the RDE by computing heteroclinic orbits connecting saddles from negative and positive infinity, (ii) detect tipping points as bifurcations of such heteroclinic orbits, and (iii) obtain two-parameter tipping diagrams by numerical continuation of such bifurcations.