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Crop Production Forecasts and Groundwater Trends Based on the Predator–Prey Model

Assaad Mrad

Author

Assaad Mrad

Title

Crop Production Forecasts and Groundwater Trends Based on the Predator–Prey Model

Description

Proposal of a dynamical systems framework capable of explaining past trends in groundwater-based irrigation and providing forecasts of food production.

Crop Production Forecasts and Groundwater Trends Based on the Predator–Prey Model

December 10, 2020 — Assaad Mrad, Guest Author

Ever since Thomas Robert Malthus’s book An Essay on the Principle of Population, scientists have sought to determine the limit to the growth of human population due to finite resources. One such resource is groundwater. About 40% of global food production ultimately depends on irrigation and, increasingly, the source of irrigation water is groundwater aquifers. Groundwater irrigation allows farmers to increase crop yields, maintain them in dry spells and overcome temporal mismatches between growing seasons and seasonal rain. In many parts of the world, groundwater withdrawal (or pumping from wells) exceeds recharge, leading to groundwater depletion. In these regions, the “lifespan” of groundwater aquifers is limited, putting a bound on the amount of irrigation per year and the sustainability of groundwater-based agriculture. The goal of this study was to propose a dynamical systems framework capable of explaining past trends in groundwater-based irrigation and providing forecasts of food production.

Why the High Plains Aquifer?

Coauthors and I chose the US High Plains (HP) aquifer (also called the Ogallala aquifer) as a testbed for two reasons:

First, the HP aquifer extends from the Texas panhandle to South Dakota and underlies eight states, therefore representing differing levels of interaction among groundwater withdrawal, recharge, climate and yearly crop yield improvements.

The aquifer underlies hot and dry (south) to cold and wet (north) climates.

Conditions in Nebraska (representing the north) are amenable for high recharge rates while Texas (south) and Kansas (center) possess recharge rates of about four and eight times higher than Nebraska, respectively.

Second, there is a lot of data on crop production and groundwater withdrawal and use thanks to the U.S. Geological Survey (USGS) and the U.S. Department of Agriculture (USDA).

The HP aquifer underlies parts of eight US states and is the largest aquifer in the US, with an area of almost 176,000 square miles.

Using the Wolfram Language, we can get the HP aquifer’s outer polygon from the USGS and transform it with proper geoprojection:

In[]:=

ogallala=Module[{url="https://water.usgs.gov/GIS/dsdl/ds543.zip",proj,completePolygons},proj=Import[url,"Projection"]/.("ReferenceModel"_)("ReferenceModel""NAD831986");completePolygons=Query["Geometry",1][First[Import[url,"Data"]]]/.Line[pnts_]Line[GeoPosition@GeoGridPosition[pnts,proj]];completePolygons/.FilledCurve[{{Line[GeoPosition[x_]]},__}]Polygon[GeoPosition[x]]];ogallala//GeoArea

Out[]=

175867.

We can then obtain the US states overlapping with the HP aquifer and geovisualize it:

In[]:=

Module{outer,usPoly,usStatesHP},outer=ogallala/.Polygon[GeoPosition[x_]]Polygon[x];usPoly=EntityValue

all US states with District of Columbia

ADMINISTRATIVE DIVISIONS

,"Polygon","EntityAssociation"/.Polygon[GeoPosition[x_]]Polygon[x];usStatesHP=Keys@Select[RegionDisjoint[#,outer]&/@usPoly,!#&];GeoGraphics{EdgeForm[Black],Tooltip[#["Polygon"],#["Name"]]&/@usStatesHP,FaceForm[Blue],EdgeForm[Blue],ogallala},GeoProjection"Mercator",GeoRange

United States

COUNTRY

,PlotLabel"High Plains Aquifer (blue) and overlapping US states"

Out[]=

Texas High Plains Agriculture

In this blog post, I will focus on the case of Texas High Plains agriculture. We found it most interesting because it offered a surprise during analysis. For an analysis of Kansas and Nebraska, please refer to the published paper.

Here is a map showing the HP aquifer and the different USDA agricultural districts of Texas:

In[]:=

ShowGeoListPlotValues@

Texas Agricultural Statistical Districts

,PlotLegendsKeys

Texas Agricultural Statistical Districts

,PlotStyleRandomColorLength

Texas Agricultural Statistical Districts

,PlotMarkersAutomatic,GeoGraphics[{EdgeForm[Black],ogallala},GeoProjection"Mercator"],PlotLabel"TX agricultural districts and High Plains Aquifer"

Out[]=

	Northern High Plains
	Southern High Plains
	Northern Low Plains
	Southern Low Plains
	Cross Timbers
	Blacklands
	North East Texas
	South East Texas

	Trans-Pecos
	Edwards Plateau
	South Central
	Coastal Bend
	Upper Coast
	South Texas
	Lower Valley

In the two districts that overlie the aquifer (the northern and southern High Plains), the four most irrigated crops are corn, cotton, sorghum and winter wheat. Plotting the rate of groundwater extraction and crop production, we get:

Looking at the data, we notice three prominent features:

◼

Groundwater withdrawal and crop production tend to rise, peak and then decline.

◼

Trends and peaks in crop production lag behind those for groundwater withdrawal.

◼

There are two “boom–bust” cycles before and after 1987.

Using Predator–Prey Equations

Having been trained as an environmental scientist in my Ph.D. and as a mechanical engineer in undergrad, I realize that the first two prominent features arise in predator–prey dynamics. So, we will use modified predator–prey equations (or Lotka–Volterra equations) to model the nexus of groundwater withdrawal and crop production:

WC+r

HPA

(

)

WC-

(

)

... where W and C represent the volume of accessible groundwater and the rate of crop production, respectively. In the equations shown, r represents the recharge rate per unit surface area, a value determined from the literature, and

HPA

represents the surface area of the HP aquifer underlying Texas. Groundwater recharge occurs because of percolating rainfall and surface water expressions like streams and playas.

The first terms in equations 1 and 2 encode the interaction between prey (groundwater) and predator (crops). The more predators there are (larger C), the faster the “population” of prey will dwindle (decreasing W). When W falls, physically it means that the groundwater table is deeper. This means that farmers will have to spend more energy to extract it. This reduces the withdrawal (represented by

) and crop production.

This second term in equation 2 encodes the “competition” among predators for the same prey. In other words, there is diminishing value in having more crop production such that, at some point, increasing the density of crops planted will hurt the total weight of grain produced because they will compete with each other.

Fitting the Data to Models

We fit the data such that

represents groundwater withdrawal and C represents the rate of crop production. This is to find the values of parameters

. We apply the fit before and after the rebound in groundwater withdrawal and crop production (1987):

The model is able to capture the peaks in groundwater withdrawal and crop production as well as the lags between withdrawal and production. Peak non-renewable resource use is a known phenomenon exemplified by Hubbert’s 1956 prediction of a peak in US crude oil production. The projection from the second model fit says that groundwater-based crop production is expected to decline from an average of about five million tons per year today to two million per year in 2050.

It seems ad hoc that we performed two model fits to the data, so how can we trust the projection? There is a good reason for having fits before and after the rebound. Notice how after the rebound, crop production per unit groundwater withdrawal is higher than before 1987. This strongly points to an increase in efficiency. This is true:

data

logistic fit

Around the same time when the rebound happened, sprinkler adoption in the Texas High Plains increased significantly. The improvements in efficiency increased farmer economic returns, thus making tapping into deeper groundwater profitable, driving the rebound. In general, relatively fast adoption of technologies is hard to predict. Therefore, we can trust the model forecasts, given that no disruptive technologies are adopted for irrigation-based agriculture.

Why is there a lag between peak groundwater withdrawal and peak crop production? There is a multitude of reasons. Crop yields keep improving steadily as fertilizers are used more efficiently and conditions are optimized. A more nuanced reason is crop shifting. Over the past decades, farmers have reduced sorghum irrigation in favor of corn:

	Corn
	Sorghum
	Wheat
	Cotton

Since irrigated corn has a higher yield than irrigated sorghum, crop production by weight has improved per unit water volume applied. Luckily, the predator–prey framework encodes gradual yield improvements

(C/

WC=1/

, increasing with decreasing W) but not disruptive ones (e.g. technology).

Application to Field Sustainability

In conclusion, this is one of the first steps in a computational approach to the field of sustainability. For more info about the underlying codes, please refer to the corresponding post on Wolfram Community.

While previous forecasts on crop production simply fit data to logistic curves without explanation, this conceptual dynamical systems approach is based on mechanistic understanding of the tight link between groundwater use and crop production. The simplicity of the equations means that one can apply a similar approach to areas facing different challenges, such as groundwater quality in India or different dynamics, e.g. government incentive programs, like in Saudi Arabia. One would need to include feedback terms into equations 1 and 2 to capture these effects. This is how we hope this work will fertilize the fields of agriculture, water and sustainability.

Acknowledgments

I’d like to thank Mads Bahrami for his help with preparation of this post. Also, my special thanks to the Wolfram Blog team of Wolfram Research for their invitation.

Guest author Dr. Assaad Mrad earned his Ph.D. from the Nicholas School of the Environment at Duke University and will be a postdoctoral fellow at the Samueli School of Engineering at the University of California, Irvine starting in January 2021. This post is based on a recent open-access publication that utilizes the Wolfram Language in the Proceedings of the National Academy of Sciences.

Read the research on this topic and join the conversation on Wolfram Community.

Cite this as: Assaad Mrad, "Crop Production Forecasts and Groundwater Trends Based on the Predator–Prey Model" from the Notebook Archive (2020), https://notebookarchive.org/2020-12-4o9ckiq