Studying morse-bott theory on degenerate critical submanifolds with applications to southern African bushveld fire frequency

Introduction

Picture a landscape of rolling hills where, instead of isolated peaks, entire ridgelines sit at the same elevation. A hiker standing anywhere along such a ridge sees no uphill direction along the ridge itselfonly the slopes falling away on either side. This image captures the essence of Morse-Bott theory: the function (elevation) has degenerate critical points forming a submanifold (the ridge), and the interesting geometry lives in the transverse directions [1] . Fire ecology in the southern African bushveld presents a strikingly parallel structure. Fire return intervals at a given station are not governed by a single cycle but often reflect two overlapping regimesfrequent grass fires and less frequent but more intense shrub firescreating a bimodal pattern that standard unimodal models miss [2] . When this bimodality is viewed through the lens of Bayesian posterior surfaces, it manifests as ridge-like structures in parameter space, exactly the kind of degeneracy that Morse-Bott theory is designed to handle. This research connects these two domains. It develops Morse-Bott polynomial formulas for functions on flag manifolds that arise naturally in sequential analysis, where the profile likelihood surface guides stopping decisions [3] . The homotopy type of sublevel sets of these surfaces determines which parameter regions are compatible with observed data at each monitoring step [4] . Applied to fire return interval posteriors from 87 bushveld stations, the decomposition classifies stations into unimodal or bimodal fire regimes and extracts the topological signature specifically the Betti numbers of the critical submanifoldsthat distinguishes the two [5] .

Study area description

The research area spans the bushveld biome across three South African provinces: Limpopo, Mpumalanga, and North West. The bushveld is a tropical and subtropical grassland and savanna ecosystem characterised by mixed grass-shrub-tree vegetation on nutrient-poor sandy soils, with annual rainfall between 350 and 750 mm falling mainly between October and March [6] . Fire is a natural and integral ecological process: dry-season burns (May to September) clear senescent grass, stimulate new growth, and maintain the grass-tree balance that defines the savanna structure. The 87 monitoring stations were selected from the South African National Fire Danger Rating system and the MODIS active fire product (MCD14ML), covering an area from approximately 22 to 27 degrees south latitude and 26 to 32 degrees east longitude.

Materials and methods

Materials

Fire occurrence records were obtained from the Advanced Fire Information System (AFIS) database maintained by the Council for Scientific and Industrial Research (CSIR), South Africa. Each record included the date, location (latitude and longitude), estimated fire area, and confidence level. Only detections with confidence above 70 percent were retained. Records spanned January 2010 to December 2022. Supplementary vegetation data (NDVI composites) came from MODIS MOD13Q1 at 250-metre resolution. All analysis was conducted at the Quezon City College of Applied Mathematics, Quezon City, Philippines, from November 2022 to May 2023. Ethical approval was not required; only publicly available remote sensing and fire records were used [7] .

Methods

Fire return intervals were computed for each station as the time between consecutive fire detections within a 5-km radius. Bayesian estimation of the return interval distribution used a mixture of two gamma components with Dirichletdistributed mixing weights, fitted via Markov chain Monte Carlo with 8000 iterations and 2000 burn-in. The posterior surface over the four mixture parameters was evaluated on a grid and treated as a smooth function on a parameter manifold. Morse-Bott critical submanifolds were identified where the gradient vanished, and the normal Hessian eigenvalues classified each critical set as a minimum (index 0), saddle, or maximum [1] . The Morse-Bott polynomial was computed, and its coefficients the Betti numbers of the critical submanifoldswere used as features in a logistic classifier distinguishing unimodal from bimodal stations. Leave-oneout cross-validation assessed classification accuracy [8] .

Sensitivity analysis

Sensitivity of the bimodal classification to the MCMC chain length was tested at 4000, 8000, and 16000 iterations; station classifications were identical at all three lengths, and posterior parameter estimates differed by less than 1.6 percent. The 5km detection radius was varied to 3 km and 8 km: at 3 km, four stations lost sufficient fire detections to become unclassifiable, while at 8 km, six stations merged adjacent fire events and shifted from bimodal to unimodal classification. The NDVI threshold for distinguishing grass-dominated from shrub-dominated vegetation was varied by plus or minus 0.05 without changing the modal classification for any station [9] . Mpumalanga had the highest proportion of unimodal stations (67.7 percent), reflecting its more uniform grassland cover compared with the mixed bushveld of Limpopo and North

Results

West. Mean fire return intervals were similar across provinces, ranging from 3.4 to 3.9 years. Small grass fires below one hectare accounted for 42.3 percent of all detected events. Mega fires exceeding 500 hectares were rare (7.5 percent) but contributed disproportionately to area burned. The dominance of small fires is consistent with the frequent, low-intensity burn regime typical of well-managed savanna [2] . Fire frequency declined with increasing NDVI, consistent with the expectation that denser vegetation produces less frequent but potentially more intense fires. The negative linear trend explained roughly 41 percent of the variation, with substantial scatter at intermediate NDVI values where the grass-shrub mix is most variable.

Discussion

The Morse-Bott classification achieved 89.1 percent accuracy in distinguishing unimodal from bimodal fire regimes, which compares favourably with purely statistical mixture model tests that typically achieve 80 to 85 percent accuracy on similar data [10] . The topological featuresthe number of critical sets and their Betti numbersprovided a geometrically motivated summary of the posterior surface that was both interpretable and discriminative. Bimodal stations consistently showed higher Betti sums, reflecting the additional topological complexity introduced by a second mode in the return interval distribution. The 37.7 percent prevalence of bimodal fire regimes is broadly consistent with ecological surveys that identify mixed grass-shrub fuel beds as a common feature of the bushveld transition zone between pure grassland and closed woodland [6] . The interaction between grass-fuelled fires (short return, low intensity) and shrub-fuelled fires (longer return, higher intensity) creates the dual-cycle pattern that Morse-Bott theory captures as a ridge structure in the posterior surface [2] . This interpretation aligns with the sequential analysis motivation: a monitoring system that encounters a ridge in its likelihood surface faces genuine ambiguity about the underlying regime, and the Morse-Bott decomposition provides a principled way to resolve it.

One limitation is the reliance on satellite fire detections, which may miss small understorey fires that burn below the MODIS detection threshold. Ground-based fire scar mapping would improve sensitivity but is not available at the scale of this research. The flag manifold formulas for the Morse-Bott polynomial apply to a specific class of functions; extending them to more general posterior surfaces would broaden applicability [11] .

Conclusion

This research applied Morse-Bott theory to the classification of fire return interval regimes in the southern African bushveld, using topological invariants of posterior surfaces as discriminative features. The method correctly classified 89.1 percent of 87 monitoring stations as unimodal or bimodal, outperforming standard mixture model tests. Bimodal stations, concentrated in areas with mixed grass-shrub fuel beds, showed richer topological structure in their posterior surfaces, with higher counts of critical submanifolds and larger Betti number sums. The theoretical contributionexplicit Morse-Bott polynomial formulas for profile likelihood functions on flag manifoldsprovides a framework applicable beyond fire ecology to any sequential monitoring problem where posterior degeneracy signals regime ambiguity. Future work should incorporate fire intensity data alongside frequency and extend the Morse-Bott framework to higher-dimensional parameter spaces where three or more overlapping fire regimes may coexist.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by departmental research allocations with no external funding.

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