Optimisation of tipping pathways in a spatially heterogeneous world

Abstract

In spatially extended systems, tipping does not necessarily lead to a uniform, abrupt transition typical of low-dimensional conceptual climate models. Instead, spatially structured forms of tipping can emerge, for example, through front propagation and pinning, coexistence between alternative states, and pattern formation. Tipping can therefore remain partial or spatially heterogeneous rather than affecting the entire system. Existing work on spatial tipping in conceptual models remains mostly descriptive, and a general framework to influence tipping dynamics through spatial interventions is still lacking. Here, we introduce a constrained optimisation framework that systematically identifies spatial interventions designed to maximise resilience or promote recovery of a desirable state, subject to dynamical and resource constraints. The framework is general and applicable to a broad class of spatially extended systems. We illustrate it using the one-dimensional Allen-Cahn equation with spatial heterogeneity, a minimal bistable model in which the key spatial tipping processes in gradient systems can be analysed explicitly. In this setting, optimisation acts by shifting local tipping thresholds and controlling front dynamics. Our results show that small-scale local, targeted modifications can determine large-scale system outcomes: optimal interventions can prevent tipping, induce recovery of the desirable state, or confine collapse to limited regions of the domain.