The evasion of tipping: pattern formation near a Turing-fold bifurcation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912454267633664 |
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| author | Staal, Dock Doelman, Arjen |
| author_facet | Staal, Dock Doelman, Arjen |
| contents | Model studies indicate that many climate subsystems, especially ecosystems, may be vulnerable to 'tipping': a 'catastrophic process' in which a system, driven by gradually changing external factors, abruptly transitions (or 'collapses') from a preferred state to a less desirable one. In ecosystems, the emergence of spatial patterns has traditionally been interpreted as a possible 'early warning signal' for tipping. More recently, however, pattern formation has been proposed to serve a fundamentally different role: as a mechanism through which an (eco)system may 'evade tipping' by forming stable patterns that persist beyond the tipping point.
Mathematically, tipping is typically associated with a saddle-node bifurcation, while pattern formation is normally driven by a Turing bifurcation. Therefore, we study the co-dimension 2 Turing-fold bifurcation and investigate the question: 'When can patterns initiated by the Turing bifurcation enable a system to evade tipping?'
We develop our approach for a class of phase-field models and subsequently apply it to $n$-component reaction-diffusion systems -- a class of PDEs often used in ecosystem modeling. We demonstrate that a two-component system of modulation equations governs pattern formation near a Turing-fold bifurcation, and that tipping will be evaded when a critical parameter, $β$, is positive. We derive explicit expressions for $β$, allowing one to determine whether a given system may evade tipping. Moreover, we show numerically that this system exhibits rich behavior, ranging from stable, stationary, spatially quasi-periodic patterns to irregular, spatio-temporal, chaos-like dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The evasion of tipping: pattern formation near a Turing-fold bifurcation Staal, Dock Doelman, Arjen Dynamical Systems 37G10 Model studies indicate that many climate subsystems, especially ecosystems, may be vulnerable to 'tipping': a 'catastrophic process' in which a system, driven by gradually changing external factors, abruptly transitions (or 'collapses') from a preferred state to a less desirable one. In ecosystems, the emergence of spatial patterns has traditionally been interpreted as a possible 'early warning signal' for tipping. More recently, however, pattern formation has been proposed to serve a fundamentally different role: as a mechanism through which an (eco)system may 'evade tipping' by forming stable patterns that persist beyond the tipping point. Mathematically, tipping is typically associated with a saddle-node bifurcation, while pattern formation is normally driven by a Turing bifurcation. Therefore, we study the co-dimension 2 Turing-fold bifurcation and investigate the question: 'When can patterns initiated by the Turing bifurcation enable a system to evade tipping?' We develop our approach for a class of phase-field models and subsequently apply it to $n$-component reaction-diffusion systems -- a class of PDEs often used in ecosystem modeling. We demonstrate that a two-component system of modulation equations governs pattern formation near a Turing-fold bifurcation, and that tipping will be evaded when a critical parameter, $β$, is positive. We derive explicit expressions for $β$, allowing one to determine whether a given system may evade tipping. Moreover, we show numerically that this system exhibits rich behavior, ranging from stable, stationary, spatially quasi-periodic patterns to irregular, spatio-temporal, chaos-like dynamics. |
| title | The evasion of tipping: pattern formation near a Turing-fold bifurcation |
| topic | Dynamical Systems 37G10 |
| url | https://arxiv.org/abs/2506.22251 |