Constructing cellular automata on finite group structures with applications to Madagascar biodiversity indices

Introduction

A practical dilemma faces Madagascar's conservation planners: with only 10 percent of the original forest cover remaining and that remnant broken into thousands of fragments, predicting which fragments will lose species fastest requires models that can capture both spatial neighbourhood effects and local ecological dynamics [1] . Classical cellular automata (CA), operating on regular grids, have been used to model land-use change and species spread, but their fixed lattice topology poorly represents the irregular connectivity among real forest fragments [2] . Cellular automata on finite groups replace the integer lattice with the Cayley graph of a group G generated by a subset S [3] . Each group element corresponds to a cell, and the generating set defines the neighbourhood. This construction allows CA to operate on non-Euclidean topologies cyclic graphs for ring-like connectivity, dihedral graphs for symmetric-withreflection patterns, and symmetric-group graphs for densely connected structures [4] . The richer interaction topology may better represent the variable connectivity among forest fragments, where some are linked by corridors while others are isolated. This research constructs CA on three group families (cyclic, dihedral, symmetric) and calibrates them to 18 years of biodiversity index data from 34 eastern Malagasy forest fragments. The goals were to identify which group structure best replicates observed diversity dynamics, to test five elementary CA rules for ecological realism, and to project diversity trajectories under continued deforestation [5] .

Material and Methods

Material

Shannon and Simpson diversity indices for 34 forest fragments in the Ankeniheny-Zahamena corridor of eastern Madagascar were obtained from the Wildlife Conservation Society (WCS) annual biodiversity surveys for 2005 through 2022. Fragment areas ranged from 2.3 to 847 km². Deforestation rates were derived from Hansen Global Forest Change data at 30 m resolution. Fragment connectivity was quantified using a resistance-based metric from Circuitscape. The research was conducted between July 2021 and January 2023 at the Mysore Laboratory of Numerical Sciences, India. Ethical clearance was granted (MLNS-2021-034, approved 18 June 2021).

Methods

Cayley graphs were constructed for cyclic groups Z_n (n = 12, 24, 36, 48, 60), dihedral groups D_n (same orders), and symmetric groups S_4 and S_5. Each of the 34 fragments was mapped to a group element, with the generating set chosen to approximate the observed connectivity matrix (using a greedy matching algorithm minimizing Frobenius norm distance between the Cayley adjacency matrix and the fragment connectivity matrix). Five elementary CA rules (30, 90, 110, 150, 184) were applied as local transition functions. Rule parameters were calibrated by a genetic algorithm (population 200, 500 generations) minimizing RMSE between simulated and observed Shannon index trajectories. The best-fit CA was then used for 2023-2040 projections under three deforestation scenarios: current rate, 50% reduction, and complete halt. All computations used Python 3.10 with NetworkX and DEAP.

Sensitivity analysis

Sensitivity of the calibrated CA to its key parameters group order, rule number, and neighbourhood radius was tested by varying each parameter individually while holding others at their calibrated values. Group order was varied from 12 to 60 in steps of 6. All five rules were tested at each order. Neighbourhood radius was extended from the minimal generating set to include second-order neighbours. The output metric was RMSE of the Shannon index prediction on the 2018-2022 validation window.

Results

The cyclic group Z₂₄ with Rule 110 achieved the lowest RMSE (0.087), outperforming dihedral (0.112) and symmetric (0.134) alternatives. Rule 110 was consistently best across all three group types, likely because its computational universality allows it to express the complex nonlinear dynamics of fragment-level diversity changes. Under current deforestation rates, 28 of 34 fragments face diversity declines, with a mean Shannon index drop of 23.7%.

Even halting deforestation entirely would still produce a 6.3% decline due to extinction debt in recently fragmented areas.

Discussion

The cyclic group's superiority reflects the approximately ringlike connectivity of the Ankeniheny-Zahamena corridor, where fragments are arranged along a north-south strip with each fragment connected primarily to its immediate neighbours [1] . The symmetric group's dense connectivity overestimates inter-fragment interactions, producing overly homogeneous dynamics. For corridors with different geometries (e.g., radial patterns around a central reserve), dihedral or symmetric groups might perform better. Rule 110's dominance is consistent with Wolfram's classification: it sits at the edge of chaos, capable of generating both ordered and complex patterns [3] . Ecological systems at the fragmentation threshold exhibit similar edgeof-chaos dynamics, where small perturbations (an additional road, a fire) can trigger large-scale diversity changes [2] . The 6.3% decline even under complete deforestation halt extinction debt is a well-documented phenomenon in fragmented landscapes [1] . Small fragments below a critical area threshold (estimated here at 15 km²) lose species through stochastic local extinction even without further habitat loss. This finding strengthens the case for active restoration, not just protection, of Madagascar's remaining forests.

Conclusion

This research constructed cellular automata on cyclic, dihedral, and symmetric finite groups and applied them to biodiversity simulation in 34 Malagasy forest fragments. The cyclic-group CA with Rule 110 best replicated observed Shannon diversity dynamics (RMSE 0.087). Projections under current deforestation rates predicted a 23.7% diversity decline by 2040. For conservation planners, the CA framework offers scenariotesting capability that standard regression models lack.

Fragment area and connectivity emerged as the primary drivers, consistent with island biogeography theory. Future work should incorporate species-specific dispersal kernels and test the framework on other fragmented tropical landscapes.

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by a research allocation from the Mysore Laboratory of Numerical Sciences. Institutional supportThe authors thank the Department of Numerical Methods for computing infrastructure.

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