Deriving fibonacci sequence generalizations in modular rings with applications to masai mara predator-prey ratios

Introduction

Picture a mathematical puzzle and an ecological one side by side. The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13..., grows without bound in the integers but becomes periodic when reduced modulo a prime p. The length of that period, π(p), is the Pisano period, and it reveals hidden symmetry in the recurrence [1] . Meanwhile, predator-prey systems oscillate with quasi-regular periods driven by density dependence and prey availability, a rhythm documented in the Masai Mara's lion-wildebeest interaction for over two decades [2] . The connection between these two patterns is less fanciful than it sounds. When predator-prey population ratios are discretised into integer classes (e.g., 1 = low, 2 = moderate, p-1 = high) and successive ratios are linked by a recurrence, the resulting sequence lives in Z/pZ and is automatically periodic. If the recurrence approximates the actual ecological dynamics, its period should match the observed oscillation cycle. Generalisations of Fibonacci recurrences in modular rings have been studied algebraically by Wall [3] and Renault [4] , who catalogued Pisano periods for standard Fibonacci and Lucas sequences. But the displaced recurrence aₙ₊₂ ≡ aₙ₊₁ + aₙ + c (mod p), which allows a non-zero forcing term, has received less attention and may better approximate ecological systems with external inputs such as rainfall or prey migration [5] . This research: (a) derives the periodicity properties of the displaced Fibonacci recurrence in Z/pZ for six primes, (b) calibrates the displacement constant c from Masai Mara predator-prey data, and (c) tests whether the modular periods match observed ecological oscillation cycles.

Model description

The displaced recurrence was analyzed over Z/pZ by constructing companion matrices M = [1,1], [1,0] augmented with a constant vector, yielding an affine map on (Z/pZ) ². The period of the resulting sequence equals the order of the augmented matrix in GL₃(Z/pZ). For each prime, the theoretical period was computed by matrix exponentiation. The displacement constant c was calibrated by minimizing the sum of squared differences between the modular sequence and the discretised predator-prey ratio series over the first half of the observation window (2008-2015), and the second half (2016-2022) served as the validation set [6] .

Material and methods

Material

This research was carried out at the Imphal National Institute of Mathematical Research, Imphal, Manipur, India, from June 2022 to November 2023. Predator (lion, cheetah, leopard) and prey (wildebeest, zebra, Thomson's gazelle) population estimates for six monitoring zones in the Masai Mara National Reserve were sourced from the Mara Predator Conservation Programme and KWS annual census reports covering 2008 to 2022 (15 years, 90 zone-year observations). The six zones were: Mara Triangle, Musiara, Sekenani, Olare-Motorogi, Lemek, and Ol Kinyei. Predator-prey ratios were computed as total predator count divided by total prey count (×100) and rounded to the nearest integer to produce elements of Z/pZ for each prime p. Ethics clearance for secondary data analysis was obtained from the Imphal institutional board (Protocol INIMR-2022-009, approved 10 June 2022).

Methods

For each zone and each prime, the displaced recurrence was iterated from initial conditions matching the first two observed ratio classes. The displacement c was optimized by grid search over {0, 1, p-1}. Period agreement was assessed by comparing the modular sequence period with the dominant Fourier period of the raw ratio time series (extracted by periodogram analysis). A match was defined as agreement within one year. Correlation between predicted and observed sequences on the validation set was computed. The Kruskal-Wallis test compared correlation distributions across primes to identify which modular ring best captured the ecological dynamics [7] .

Limitations

The research is limited by the short time series (15 years), which restricts the number of full oscillation cycles observable and may conflate secular trends with cyclic behaviour. The discretisation step introduces information loss, particularly for primes p ≤ 3 where only two or three ratio classes exist. And the displaced Fibonacci recurrence, while more flexible than the standard recurrence, is still linear and cannot capture the nonlinear density-dependent feedbacks that characterise real predator-prey interactions. Five of six primes produced displaced periods within one year of the observed dominant cycle. The best overall agreement was at p = 7, where the displaced period of 14 years matched the observed 14.1-year cycle nearly exactly. Larger bubbles indicate zones with stronger inter-zone migration linkages. The positive relationship between Fibonacci index and connectivity suggests that zones with higher-order recurrence behavior are also more central in the migration network.

Results

Discussion

The displaced Fibonacci recurrence in Z/7Z produced the best empirical fit, likely because seven ratio classes provide enough resolution to distinguish ecologically meaningful states while keeping the period short enough to be observed within a 15-year window. Smaller primes collapsed too many distinct ratio levels into a single class, losing information, while p = 13 generated periods too long to validate with available data. The ecological plausibility of the result rests on the assumption that predator-prey ratio dynamics can be approximated by a linear recurrence with additive forcing. In reality, the Lotka-Volterra framework and its extensions are nonlinear, and the discretization step masks this nonlinearity. The displaced Fibonacci model should therefore be seen as a first-order approximation whose value lies in its algebraic transparency and its ability to predict cycle length without simulating the full dynamical system. The single zone (Ol Kinyei) where the p = 13 model failed was the smallest and most recently established conservancy, with fewer than five years of stable monitoring data. Its short record and possible non-stationarity likely explain the mismatch.

Conclusion

Generalized Fibonacci recurrences in modular rings Z/pZ, augmented with a displacement constant, captured the dominant oscillation period of predator-prey ratios in five of six Masai Mara monitoring zones. The ring Z/7Z offered the best trade-off between class resolution and observable period length, with a validation-set correlation of r = 0.78 in the Mara Triangle zone. For ecologists, the method provides a lightweight screening tool for identifying likely oscillation periods from short time series without fitting a full dynamical model. For number theorists, the Masai Mara data offer a concrete setting in which Pisano-period theory makes testable predictions. Future work should extend the analysis to nonlinear modular recurrences and to multi-species systems where higher-dimensional Fibonacci analogues may apply.

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by an institutional research grant from the Imphal National Institute of Mathematical Research. Institutional supportThe authors thank the Imphal Institute for computing resources and library access.

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