Richards Growth Model and Multi-Metaheuristic Optimization for Precision Irrigation and Fertilization Scheduling: An Extension of the Becker-Zohdi Framework

Authors

https://doi.org/10.48313/bic.vi.77

Abstract

Agricultural input costs are rising, and weather patterns are becoming more variable. These trends create a need for better methods to allocate water and fertiliser in farming. This paper extends the crop model developed by Becker and Zohdi [1]. We replace the logistic growth equation with the Richards model. The Richards model has a shape parameter that allows asymmetric growth. We also compare five metaheuristic algorithms. These are Genetic Algorithm (GA), Particle Swarm Optimisation (PSO), Differential Evolution (DE), Grey Wolf Optimiser (GWO), and a Hybrid GA-PSO. The study uses corn production data from Iowa. We test the methods on 21 stochastic weather scenarios. The Richards model captures the vegetative and reproductive phases of corn more accurately than the logistic model. Among the algorithms, GWO gives the highest average revenue. It achieves $903 per acre. This is higher than GA ($876/acre), DE ($884/acre), PSO ($891/acre), and Hybrid GA-PSO ($898/acre). This represents a 7% improvement over the original GA-logistic result of $842/acre. Statistical tests confirm the superiority of GWO with 95% confidence. The main findings are that asymmetric growth modelling improves optimisation results, especially under stress conditions. GWO provides a good balance between exploration and exploitation. This work provides a benchmark for metaheuristic performance in precision agriculture. It also shows that the Richards model is a better choice than the logistic model for this type of problem. 

Keywords:

Precision agriculture, Richards growth model, Metaheuristic algorithms, Grey wolf optimiser, Irrigation scheduling, Crop optimisation

References

  1. [1] Becker, C. J., & Zohdi, T. I. (2026). Optimizing irrigation and fertilization strategies for crop growth: A comparative study of genetic algorithm and model predictive control under weather uncertainty. Archives of computational methods in engineering, 1–31. https://doi.org/10.1007/s11831-026-10640-5
  2. [2] Economic Research Service. (2024). Farm income and wealth statistics. https://www.ers.usda.gov/data-products/farm-income-and-wealth-statistics
  3. [3] Zohdi, T. I. (2024). A machine-learning enabled digital-twin framework for next generation precision agriculture and forestry. Computer methods in applied mechanics and engineering, 431, 117250. https://doi.org/10.1016/j.cma.2024.117250
  4. [4] Mengi, E., Becker, C. J., Sedky, M., Yu, S. Y., & Zohdi, T. I. (2024). A digital-twin and rapid optimization framework for optical design of indoor farming systems. Computational mechanics, 74(1), 31–43. https://doi.org/10.1007/s00466-023-02421-9
  5. [5] Goodrich, P., Betancourt, O., Arias, A. C., & Zohdi, T. (2023). Placement and drone flight path mapping of agricultural soil sensors using machine learning. Computers and electronics in agriculture, 205, 107591. https://doi.org/10.1016/j.compag.2022.107591
  6. [6] Betancourt, J. O., Li, I., Mengi, E., Corrales, L., & Zohdi, T. I. (2024). A computational framework for precise aerial agricultural spray delivery processes: JO Betancourt et al. Archives of computational methods in engineering, 31(7), 4149–4162. https://doi.org/10.1007/s11831-024-10106-6
  7. [7] Verhulst, P. F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondence mathematique et physique, 10, 113–129. https://cir.nii.ac.jp/crid/1570009749935841536
  8. [8] Singh, A. (2012). An overview of the optimization modelling applications. Journal of hydrology, 466–467, 167–182. https://doi.org/10.1016/j.jhydrol.2012.08.004
  9. [9] Epperson, J. E., Hook, J. E., & Mustafa, Y. R. (1993). Dynamic programming for improving irrigation scheduling strategies of maize. Agricultural systems, 42(1-2), 85–101. https://doi.org/10.1016/0308-521X(93)90070-I
  10. [10] Wang, L., Xu, Y., & Xu, J. (2020). Realization of wireless charging in intelligent greenhouse with orthogonal coil system uniform magnetic field. Computers and electronics in agriculture, 175, 105524. https://doi.org/10.1016/j.compag.2020.105524
  11. [11] Goldberg, D. E. (1989). Genetic algorithms in search, optimization, and machine learning. Addison-Wesley Publishing Company. https://www.amazon.com/Genetic-Algorithms-Optimization-Machine-Learning/dp/0201157675
  12. [12] Jones, J. W., Hoogenboom, G., Porter, C. H., Boote, K. J., Batchelor, W. D., Hunt, L. A., ... & Ritchie, J. T. (2003). The DSSAT cropping system model. European journal of agronomy, 18(3-4), 235–265. https://doi.org/10.1016/S1161-0301(02)00107-7
  13. [13] Holzworth, D. P., Huth, N. I., deVoil, P. G., Zurcher, E. J., Herrmann, N. I., McLean, G., ... & Keating, B. A. (2014). APSIM–evolution towards a new generation of agricultural systems simulation. Environmental modelling & software, 62, 327–350. https://doi.org/10.1016/j.envsoft.2014.07.009
  14. [14] Van Diepen, C. V., Wolf, J. V., Van Keulen, H., & Rappoldt, C. W. O. F. O. S. T. (1989). WOFOST: A simulation model of crop production. Soil use and management, 5(1), 16–24. https://doi.org/10.1111/j.1475-2743.1989.tb00755.x
  15. [15] Gebbers, R., & Adamchuk, V. I. (2010). Precision agriculture and food security. Science, 327(5967), 828-831. https://doi.org/10.1126/science.1183899
  16. [16] Richards, F. J. (1959). A flexible growth function for empirical use. Journal of experimental botany, 10(2), 290–301. https://doi.org/10.1093/jxb/10.2.290
  17. [17] Yin, X., Goudriaan, J. A. N., Lantinga, E. A., Vos, J. A. N., & Spiertz, H. J. (2003). A flexible sigmoid function of determinate growth. Annals of botany, 91(3), 361–371. https://doi.org/10.1093/aob/mcg029
  18. [18] Zeide, B. (1993). Analysis of growth equations. Forest science, 39(3), 594–616. https://doi.org/10.1093/forestscience/39.3.594
  19. [19] Shi, J. X., Goldschmidt, E. E., Goren, R., & Porat, R. (2007). Molecular, biochemical and anatomical factors governing ethanol fermentation metabolism and accumulation of off-flavors in mandarins and grapefruit. Postharvest biology and technology, 46(3), 242–251. https://doi.org/10.1016/j.postharvbio.2007.05.009
  20. [20] Vikram, P., Swamy, B. M., Dixit, S., Ahmed, H., Cruz, M. S., Singh, A. K., ... & Kumar, A. (2012). Bulk segregant analysis:“An effective approach for mapping consistent-effect drought grain yield QTLs in rice.” Field crops research, 134, 185–192. https://doi.org/10.1016/j.fcr.2012.05.012
  21. [21] Holland, J. H. (1975). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press. https://mitpress.mit.edu/9780262581110/adaptation-in-natural-and-artificial-systems/
  22. [22] Wardlaw, R., & Bhaktikul, K. (2004). Application of genetic algorithms for irrigation water scheduling. Irrigation and drainage: The journal of the international commission on irrigation and drainage, 53(4), 397–414. https://doi.org/10.1002/ird.121
  23. [23] Yu, Z., & Fu, W. (2025). Optimization of nitrogen fertilization strategies for drip irrigation of cotton in large fields by DSSAT combined with a genetic algorithm. Applied sciences, 15(7), 3580. https://doi.org/10.3390/app15073580
  24. [24] Salcedo, R., Zhu, H., Ozkan, E., Falchieri, D., Zhang, Z., & Wei, Z. (2021). Reducing ground and airborne drift losses in young apple orchards with PWM-controlled spray systems. Computers and electronics in agriculture, 189, 106389. https://doi.org/10.1016/j.compag.2021.106389
  25. [25] Eberhart, R., & Kennedy, J. (1995). Particle swarm optimization. Proceedings of the IEEE international conference on neural networks (Vol. 4, pp. 1942–1948). IEEE. https://doi.org/10.1109/ICNN.1995.488968
  26. [26] Nagesh Kumar, D., & Janga Reddy, M. (2007). Multipurpose reservoir operation using particle swarm optimization. Journal of water resources planning and management, 133(3), 192–201. https://doi.org/10.1061/(ASCE)0733-9496(2007)133:3(192)
  27. [27] Garg, N. K., & Dadhich, S. M. (2014). Integrated non-linear model for optimal cropping pattern and irrigation scheduling under deficit irrigation. Agricultural water management, 140, 1–13. https://doi.org/10.1016/j.agwat.2014.03.008
  28. [28] Noory, H., Liaghat, A. M., Parsinejad, M., & Haddad, O. B. (2012). Optimizing irrigation water allocation and multicrop planning using discrete PSO algorithm. Journal of irrigation and drainage engineering, 138(5), 437-444. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000426
  29. [29] Storn, R., & Price, K. (1997). Differential evolution–A simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341–359. https://doi.org/10.1023/A:1008202821328
  30. [30] Das, S., & Suganthan, P. N. (2010). Differential evolution: A survey of the state-of-the-art. IEEE transactions on evolutionary computation, 15(1), 4–31. https://doi.org/10.1109/TEVC.2010.2059031
  31. [31] Das, S., Mullick, S. S., & Suganthan, P. N. (2016). Recent advances in differential evolution–An updated survey. Swarm and evolutionary computation, 27, 1–30. https://doi.org/10.1016/j.swevo.2016.01.004
  32. [32] Chen, L., Xie, H., Wang, G., Qian, X., Wang, W., Xu, Y., ... & Yang, J. (2021). Reducing environmental risk by improving crop management practices at high crop yield levels. Field crops research, 265, 108123. https://doi.org/10.1016/j.fcr.2021.108123
  33. [33] Zhao, J., Han, T., Wang, C., Jia, H., Worqlul, A. W., Norelli, N., ... & Chu, Q. (2020). Optimizing irrigation strategies to synchronously improve the yield and water productivity of winter wheat under interannual precipitation variability in the North China Plain. Agricultural water management, 240, 106298. https://doi.org/10.1016/j.agwat.2020.106298
  34. [34] Yang, L., Bi, P., Tang, H., Zhang, F., & Wang, Z. (2022). Improving vegetation segmentation with shadow effects based on double input networks using polarization images. Computers and electronics in agriculture, 199, 107123. https://doi.org/10.1016/j.compag.2022.107123
  35. [35] Mirjalili, S., Mirjalili, S. M., & Lewis, A. (2014). Grey wolf optimizer. Advances in engineering software, 69, 46–61. https://doi.org/10.1016/j.advengsoft.2013.12.007
  36. [36] Niu, W. J., Feng, Z. K., Liu, S., Chen, Y. B., Xu, Y. S., & Zhang, J. (2021). Multiple hydropower reservoirs operation by hyperbolic grey wolf optimizer based on elitism selection and adaptive mutation: Niu W. j. et al. Water resources management, 35(2), 573–591. https://doi.org/10.1007/s11269-020-02737-8
  37. [37] Wu, L., Tian, J., Liu, Y., Wang, Y., & Zhang, P. (2024). Multi-objective planting structure optimisation in an irrigation area using a grey wolf optimisation algorithm. Water, 16(16), 2297. https://doi.org/10.3390/w16162297
  38. [38] Migallon, H., Jimeno-Morenilla, A., Sánchez-Romero, J. L., & Belazi, A. (2020). Efficient parallel and fast convergence chaotic Jaya algorithms. Swarm and evolutionary computation, 56, 100698. https://doi.org/10.1016/j.swevo.2020.100698
  39. [39] Wang, X. A., Tang, J., & Whitty, M. (2021). DeepPhenology: Estimation of apple flower phenology distributions based on deep learning. Computers and electronics in agriculture, 185, 106123. https://doi.org/10.1016/j.compag.2021.106123
  40. [40] Tagkopoulos, I., Brown, S. F., Liu, X., Zhao, Q., Zohdi, T. I., Earles, J. M., ... & Youtsey, G. D. (2022). Special report: AI institute for next generation food systems (AIFS). Computers and electronics in agriculture, 196, 106819. https://doi.org/10.1016/j.compag.2022.106819
  41. [41] Isied, R. S., Mengi, E., & Zohdi, T. I. (2022). A digital-twin framework for genomic-based optimization of an agrophotovoltaic greenhouse system. Proceedings: Mathematical, physical and engineering sciences, 478(2267), 1–17. https://www.jstor.org/stable/27203349
  42. [42] Popova, Z., Eneva, S., & Pereira, L. S. (2006). Model validation, crop coefficients and yield response factors for maize irrigation scheduling based on long-term experiments. Biosystems engineering, 95(1), 139–149. https://doi.org/10.1016/j.biosystemseng.2006.05.013
  43. [43] Sarshar, A., Tranquilli, P., Pickering, B., McCall, A., Roy, C. J., & Sandu, A. (2017). A numerical investigation of matrix-free implicit time-stepping methods for large CFD simulations. Computers & fluids, 159, 53–63. https://doi.org/10.1016/j.compfluid.2017.09.014
  44. [44] Zinnanti, C., Schimmenti, E., Borsellino, V., Paolini, G., & Severini, S. (2019). Economic performance and risk of farming systems specialized in perennial crops: An analysis of Italian hazelnut production. Agricultural systems, 176, 102645. https://doi.org/10.1016/j.agsy.2019.102645
  45. [45] Abendroth, L. J., Elmore, R. W., Boyer, M. J., & Marlay, S. K. (2011). Corn growth and development. https://crops.extension.iastate.edu/cropnews/2011/04/corn-growth-classic-rewritten-and-available-order
  46. [46] Nielsen, R. L. (2020). Tassel emergence & pollen shed. Purdue University, Department of Agronomy, Corny News Network. https://www.agry.purdue.edu/ext/corn/news/timeless/Tassels.html

Published

2026-09-12

Issue

Vol. 3 No. 3 (2026): Biocompounds

Section

Articles

How to Cite

Pourqasem, M. ., & Montazeri, F. Z. . (2026). Richards Growth Model and Multi-Metaheuristic Optimization for Precision Irrigation and Fertilization Scheduling: An Extension of the Becker-Zohdi Framework. Biocompounds, 3(3), 175-190. https://doi.org/10.48313/bic.vi.77

Similar Articles

1-10 of 21

You may also start an advanced similarity search for this article.