While AquaCrop is widely used, GK's impression is that a clearer focus, in the documentation at least, on differential equation, and delay d.e., etc., etc. modelling would improve matters a lot.
Aspects of older software include "simplifications", piecewise linearisation to get to approximations based on the functions in C's math.h, Fortran, etc.
There is a lot of training material for AquaCrop.
In spite of its name, people do use it for modelling dry-land farming.
I am not aware of comparative studies of the various simulators
Background: https://en.wikipedia.org/wiki/Theoretical_production_ecology
FSPM refers to Functional-Structural Plant Models
Comparison of the python version of AquaCrop and of Python Crop Simution Environment (PCSE) both, free open-source might be interesting or useful.
From google
AquaCrop, the Food and Agriculture Organization (FAO) crop water productivity model, has been applied and validated for major broadacre crops in Australia, including studies relevant to the Mediterranean-type climate of the Western Australian (WA) wheatbelt.
https://www.fao.org/aquacrop/en
https://openknowledge.fao.org/server/api/core/bitstreams/723a4043-dbd1-4ce6-989a-d419797a9aa7/content
Key applications and context for AquaCrop in Western Australia include:
Crop Model Calibration: Studies have validated AquaCrop for canola in Australian conditions, including at Wagga Wagga (NSW), where it accurately predicted canopy cover, biomass accumulation, and grain yield. This validation is critical for applying the model in similar dryland environments like WA.
Ketema Tilahun Zeleke, David Luckett, Raymond Cowley
Calibration and Testing of the FAO AquaCrop Model for Canola
Agronomy Journal 2011
https://doi.org/10.2134/agronj2011.0150
Mediterranean Climate Adaptation: The WA wheatbelt, characterized by 300–700 mm of annual rainfall and winter/spring cropping (May–October), fits the water-limited conditions that AquaCrop is designed to model.
Wheat Yield Modeling: AquaCrop has been used to simulate the impact of water-limited conditions on wheat, with studies indicating high accuracy in predicting yield under varying rainfall scenarios.
Dry Sowing Strategy Analysis: The model is useful for analyzing farming strategies, such as dry sowing, which has become a common, yet risky, practice in the WA wheatbelt to cope with climate variability.
AquaCrop assists in analyzing how climate variability, such as shifting rainfall patterns in Western Australia, affects water-limited crop yields, helping in the evaluation of management practices like earlier sowing.
end of this bit from google
Other references
Steduto, P., T.C. Hsiao, D. Raes, and E. Fereres. 2009.
AquaCrop— the FAO crop model to simulate yield response to water: I. Concepts and underlying principles.
Agron. J. 101: 426–437
Raes, D., P. Steduto, P., T. C. Hsiao and E. Fereres. 2009.
AquaCrop—he FAO crop model to simulate yield response to water: II. Sotware. Agron. J. 101:438–447.
Nicholas Dercas Nicolas R. Dalezios Stamatis I. Stamatiadis Nicholaos Tserlikakis
AquaCrop Simulation of Winter Wheat under Different N Management Practices
March 2022Hydrology 9(4):56
DOI: 10.3390/hydrology9040056
Hadisseh Rahimikhoob, Teymour Sohrabi, Mojtaba Delshad
Simulating crop response to Nitrogen-deficiency stress using the critical Nitrogen concentration concept and the AquaCrop semi-quantitative approach
Scientia Horticulturae
Volume 285, 27 July 2021,
https://doi.org/10.1016/j.scienta.2021.110194
SpatialAquaCrop, an R Package for Raster-Based Implementation of the AquaCrop Model
https://pmc.ncbi.nlm.nih.gov/articles/PMC9654151/
AquaCrop-OS matlab
https://pure.manchester.ac.uk/ws/files/48339277/AquaCropOS_Foster.pdf
ResearchGate
Evaluating the water productivity by Aquacrop model of wheat under irrigation systems and algae
July 2020Open Agriculture 5(1):262-270
DOI: 10.1515/opag-2020-0029
Hani Abdel-Ghani Mansour Maybelle Saad Gaballah Osama Nofal
AquaCropPlotter: A Shiny app for visualizing and analyzing AquaCrop simulation results
PLoS ONE
Sanguankiattichai, Nattapong; Setti, Andrea; Alvar-Beltrán, Jorge; Salman, Maher; Raes, Dirk; Soldan, Riccardo
Vol. 20 Issue 12, p. e0337705, 2025.
More from google
In AquaCrop, the canopy growth (measured as green canopy cover, CC) is modeled using a piecewise approach that mimics the logistic differential equation. While the paper by Steduto et al. (2009) is the primary biological reference, its mathematical structure is effectively a "switching" variant of the classical logistic growth model.
1. The Differential Equation Variants
AquaCrop splits canopy development into two distinct phases to account for the biological reality that plant growth rates differ during early expansion versus late-stage maturation.
Phase 1: Exponential Expansion (the "Start-up")For \(CC\le CC_{x}/2\) (where \(CC_{x}\) is maximum canopy cover), growth follows a first-order kinetic equation where the rate is proportional to existing cover:
\(\frac{dCC}{dt}=CGC
\cdot CC\)CGC is the Canopy Growth Coefficient, representing the relative growth rate.
Phase 2: Exponential Decay of Growth (the "Slow-down")
For \(CC>CC_{x}/2\), the model switches to a variant that describes the "gap-filling" process as the canopy approaches its carrying capacity (\(CC_{x}\)):
\(\frac{dCC}{dt}=CGC\cdot (CC_{x}-CC)\)
This is an exponential decay function of the remaining potential growth space.
2. Applied Mathematician's Perspective
From a mathematical standpoint, this is a discontinuous-rate logistic variant. Instead of the standard Verhulst form
\\(\frac{dN}{dt}=rN(1-\frac{N}{K})\),
AquaCrop uses a symmetric switch at the inflection point (\(0.5\cdot CC_{x}\)).
The Switch: By integrating these two pieces, the model ensures a smooth sigmoidal curve while keeping the parameters (\(CGC\)) physically intuitive for agronomists.
Stress Modulation: In applied terms, these ODEs are non-autonomous. Environmental stresses (water, temperature) act as multiplicative "shifters" (e.g., \(K_{s,exp}\)) that reduce the effective growth coefficient in real-time.
3. Formal Documentation For a formal account written for those focused on the algorithmic and mathematical details, refer to the AquaCrop Reference Manual (Raes et al., 2011). Chapter 3 specifically details the transition between the exponential and decay functions and the mathematical handling of the "stress" multipliers.
Older FAO programs exist.
AquaCrop is considered the successor to CROPWAT for more advanced, dynamic, and accurate simulation of crop water productivity and yield response, especially under water-limited conditions. While CROPWAT (notably version 8.0) is still used for basic irrigation scheduling, it is a calculation-based tool rather than a dynamic growth model.