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We propose a framework to study tipping points in reaction-diffusionequations (RDEs) in one spatial dimension, where the reaction term decays inspace (asymptotically homogeneous) and varies linearly with time(nonautonomous) due to an external input. A compactification of themoving-frame coordinate together with Lin's method to construct heteroclinicorbits along intersections of stable and unstable invariant manifolds allows usto (i) obtain multiple coexisting pulse and front solutions for the RDE bycomputing heteroclinic orbits connecting equilibria at negative and positiveinfinity in the compactified moving-frame ordinary differential equation, (ii)detect tipping points as dangerous bifurcations of such heteroclinic orbits,and (iii) obtain tipping diagrams by numerical continuation of suchbifurcations. We apply our framework to an illustrative model of a habitatpatch that features an Allee effect in population growth and is geographicallyshrinking or shifting due to human activity or climate change. Thus, weidentify two classes of tipping points to extinction: bifurcation-inducedtipping (B-tipping) when the shrinking habitat falls below some critical lengthand rate-induced tipping (R-tipping) when the shifting habitat exceeds somecritical speed. We explore two-parameter R-tipping diagrams to understand howthe critical speed depends on the size of the habitat patch and the dispersalrate of the population, uncover parameter regions where the shifting populationsurvives, and relate these regions to the invasion speed in an infinitehomogeneous habitat. Furthermore, we contrast the tipping instabilities withgradual transitions to extinction found for logistic population growth withoutthe Allee effect.

Rate-Induced Tipping in Heterogeneous Reaction-Diffusion Systems: An Invariant Manifold Framework and Geographically Shifting Ecosystems