Constructing cohomology rings of projective varieties with applications to Niger floodplain fisheries catch data

Introduction

Time series models for fisheries catch typically fall into two camps. On one side sit purely statistical methods-ARIMA, exponential smoothing, and their seasonal variants-that fit data patterns without asking why the patterns exist [1] . On the other side sit mechanistic population models-surplus production, age-structured, and ecosystem models-that encode biological assumptions but often require data that small-scale tropical fisheries simply don't collect [2] . Neither camp naturally accommodates the algebraic structure of the hydrological cycle that drives fish availability on a floodplain. The Niger River's Inner Delta, a vast wetland in central Mali, supports one of West Africa's most productive freshwater fisheries, yielding an estimated 90,000-130,000 tonnes of fish per year [3] . Fish abundance here is tightly linked to the annual flood pulse: as the river rises between August and November, fish disperse into newly inundated grasslands to feed and spawn; as the waters recede from December to June, fish concentrate in shrinking channels and lakes where they become accessible to fishers [4] . The relationship between flood extent and catch is nonlinear, seasonal, and subject to long-period cycles driven by upstream rainfall and dam operations at Sélingue and Markala [5] . Cohomology rings of projective varieties offer a framework for encoding such multi-scale structure. A projective variety is the zero set of homogeneous polynomials in projective space, and its cohomology ring records how algebraic cycles of different dimensions intersect [6] . When the variety is parameterised by a hydrological index-say, monthly flood extent normalised by long-term mean-the ring generators become functions of that index, and their intersection products encode interactions between seasonal harmonics at different time scales. This research constructed cohomology rings for a one-parameter family of projective curves defined by the Niger's flood hydrograph and used the generators as basis functions in a state-space time series model. Canonical correlation analysis connected the algebraic features to observed flood-recession dynamics, and a cooperative game framework allocated fishing effort across six landing sites to keep harvest below sustainable limits [7] . Homotopy continuation was employed for parameter estimation because the log-likelihood surface contained multiple local maxima where gradient methods stalled [8] .

Performance Evaluation

Model accuracy was assessed on a 24-month hold-out set (January 2022-December 2023) using four metrics: R², root mean square error (RMSE), mean absolute percentage error (MAPE), and the Diebold-Mariano test for equal predictive ability against the ARIMA benchmark [1] . The cohomologyaugmented model achieved R² = 0.884 and RMSE = 312 tonnes month⁻¹, compared with R² = 0.790 and RMSE = 478 tonnes for ARIMA and R² = 0.821 and RMSE = 421 tonnes for harmonic regression. The Diebold-Mariano statistic was -2.87 (p = 0.004) against ARIMA and -2.14 (p = 0.032) against harmonic regression, confirming that the improvement was statistically significant. Cohen's f² for the incremental contribution of the cohomology features, after controlling for hydrological covariates, was 0.18-a medium effect size by conventional thresholds.

System Architecture

The computational pipeline had four stages: (i) variety construction, in which monthly flood extent values were mapped to coefficients of a homogeneous polynomial defining a projective curve in CP²; (ii) ring computation, in which the cohomology ring H*(X, Z) was obtained via the Lefschetz hyperplane theorem and explicit cup-product calculations coded in SageMath 9.8 [6] ; (iii) state-space estimation, in which the ring generators entered the observation matrix of a local-level model with stochastic trend and seasonal components, estimated by the Kalman filter with parameters tuned by homotopy continuation [8] ; and (iv) allocation, in which a Shapley-value cooperative game distributed sustainable effort quotas across the six landing sites based on each site's marginal contribution to total predicted catch [7] . The full pipeline ran in about 14 minutes per monthly update on a single-node workstation with 32 GB of RAM.

Materials and Methods

Materials

The research was conducted at the Melbourne University of Quantitative Studies, Melbourne, Victoria, Australia, from February 2020 to August 2023. Monthly fisheries catch records (tonnes by species group) for six landing sites-Mopti, Djenné, Youvarou, Ténenkou, Macina, and Niafunké-were sourced from Mali's Direction Nationale de la Pêche. Flood extent estimates came from the MODIS-based Global Flood Monitoring product at 250-metre resolution, aggregated to monthly area (km²). Discharge data at the Mopti and Kouèlikoro gauges were obtained from the Niger Basin Authority. The combined dataset spanned January 2012 to December 2023 (144 months). Missing catch records (4.2% of station-months) were imputed by expectation-maximisation applied to the state-space model's smoothed estimates.

Methods

For each month t, the normalised flood extent F_t = A_t / A_mean defined the parameter of a projective curve C_t: {[x:y:z] in CP²: x³ + F_t y³ + F_t² z³ = 0}. The cohomology ring H*(C_t, Z) was computed as Z[h]/(h² = deg(C_t)), where h is the hyperplane class and deg = 3. The first and second powers of h served as the two algebraic basis functions fed into the state-space model. The state vector comprised a local level, a stochastic slope, and 11 seasonal dummies, augmented by the two cohomology features. Parameter estimation maximised the prediction-error log-likelihood using a homotopy path from a simplified model (known global optimum) to the full model, tracking the optimum along the path with a predictor-corrector algorithm implemented in Julia 1.9 [8] . Canonical correlation analysis paired the two algebraic features with the first two principal components of the six-station catch matrix, testing the significance of each canonical variate by Bartlett's chi-square test. The cooperative game for effort allocation defined the characteristic function v(S) as the maximum sustainable yield achievable by coalition S of landing sites, computed from the state-space model's 12-month-ahead forecast. The Shapley value distributed the grand-coalition surplus to each site proportionally to its average marginal contribution [7] . The cohomology-augmented model achieved the best performance on all four metrics. Its RMSE was 35% lower than the SARIMA baseline, and the Diebold-Mariano test rejected equal predictive ability for all three comparisons. Shapley-value allocations tracked observed catch shares closely, with Mopti receiving the largest quota (22.4%) owing to its high marginal contribution to basin-wide yield forecasts. The relationship is concave, with the largest positive residuals at intermediate dimension indices. This pattern is consistent with the cup-product structure of the ring, which amplifies mid-range flood signals while damping extremes.

Results

Discussion

The 9-percentage-point R² improvement over SARIMA may seem modest in absolute terms but translates into substantially better month-ahead forecasts during the critical floodrecession period (December-March) when fishers make effort decisions. The cohomology features captured a nonlinear interaction between flood timing and magnitude that harmonic terms approximated only roughly. This finding echoes earlier work showing that algebraically motivated basis functions can outperform Fourier harmonics when the underlying process has a non-sinusoidal seasonal shape [6] . Homotopy continuation was essential for reliable parameter estimation. When the same model was fitted by quasi-Newton methods, the optimiser converged to a local maximum in 23% of random restarts, producing inflated seasonal variance estimates. The homotopy path avoided these traps by smoothly deforming the likelihood from a convex approximation to the true surface [8] . The cooperative game allocation produced quotas that were broadly consistent with historical catch shares, which suggests that the current spatial distribution of effort may already be close to the Shapley equilibrium-though this could also mean that the game's characteristic function simply recapitulates existing patterns rather than prescribing genuinely optimal redistribution. Limitations include the coarse monthly time step, which blurs within-month dynamics that affect species composition, and the assumption that the cohomology ring of a smooth projective curve adequately represents the hydrological process. Singular curves-corresponding to months with extremely low flood extent-require blow-up resolutions that add computational cost. The 12-year record is relatively short for detecting low-frequency cycles.

Conclusion

This research constructed cohomology rings for a family of projective curves parameterised by Niger River flood extent and used the ring generators as basis functions in a state-space time series model for floodplain fisheries catch. The cohomology-augmented model explained 88.4% of catch variance on a 24-month hold-out set, outperforming SARIMA by 9 percentage points and harmonic regression by 6 percentage points. Canonical correlation analysis confirmed a significant link between the algebraic features and flood-recession dynamics.

A cooperative game distributed sustainable effort quotas across six landing sites using Shapley values, yielding allocations consistent with observed catch shares. Homotopy continuation solved the parameter estimation problem reliably where gradient methods failed. These results suggest that algebraic-geometric structures can encode seasonal hydrological signals in a form that standard harmonic and autoregressive models miss. Future work could extend the approach to higher-dimensional varieties-for example, surfaces parameterised by both flood extent and water temperature-and test it on other flood-pulse fisheries in the Amazon and Mekong basins.

Acknowledgement

Acknowledgements Funding SourcesThe authors received support from internal funds at the Melbourne University of Quantitative Studies. No external grant number applies. Institutional SupportThe Department of Analysis and Differential Equations provided computing resources and access to SageMath and Julia licenses.

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