Unit 6: Mathematical Models

1. Agricultural Systems and Mathematical Models

1.1 Concept of a System

An agricultural system is a complex arrangement of components (soil, crops, livestock, pests, weather, machinery, and management practices) interacting within boundaries to produce agricultural products (food, fiber, fuel).

Because agricultural systems depend on biological processes and environmental variables, they are inherently variable and complex. To manage them, we use models.

1.2 Definition of a Mathematical Model

A mathematical model is a description of a system using mathematical concepts and language. It translates a real-world problem into equations, inequalities, and logical relationships to simulate behavior, predict outcomes, or optimize processes.

1.3 Role of Models in Agriculture

1. Prediction: Forecasting crop yields based on rainfall and fertilizer use.
2. Optimization: Determining the precise amount of feed for livestock to maximize weight gain while minimizing cost.
3. Control: Designing irrigation schedules to maintain soil moisture without wasting water.
4. Understanding: Analyzing the spread of plant diseases (epidemiology).

2. Classification of Mathematical Models

Models are categorized based on their structure, behavior, and the nature of the variables used.

2.1 Static vs. Dynamic Models

• Static Models:
  • Represent a system at a single point in time.
  • Time is not a variable.
  • Example: Calculating the optimal fertilizer mix for a field based on current soil test results.
  • Equation form: .
• Dynamic Models:
  • Represent how a system changes over time.
  • Time () is an essential variable.
  • Example: Modeling the growth of a maize plant over a season or the population dynamics of a pest.
  • Equation form involves derivatives or difference equations: .

2.2 Deterministic vs. Stochastic Models

• Deterministic Models:
  • Assume certainty. The same set of inputs will always produce the exact same output.
  • No random variables are involved.
  • Example: Calculating the area of a field or the theoretical flow rate of water through a pipe.
• Stochastic (Probabilistic) Models:
  • Incorporate randomness and uncertainty.
  • Outputs are probability distributions or ranges, not single fixed numbers.
  • Example: Predicting rainfall for the next month or simulating crop failure risks due to random weather events.

2.3 Empirical vs. Mechanistic Models

• Empirical (Statistical) Models:
  • Based on observation and experimental data without necessarily understanding the underlying biological principles.
  • "Black box" approach: describes what happens, not why.
  • Example: A regression line relating the amount of nitrogen applied to yield quantity.
• Mechanistic (Explanatory) Models:
  • Based on theoretical understanding of the underlying physical, chemical, or biological processes.
  • Describes why things happen.
  • Example: Modeling photosynthesis rates based on leaf area index, solar radiation, and CO2 concentration physics.

2.4 Continuous vs. Discrete Models

• Continuous: Variables change smoothly over time (e.g., plant height). Uses calculus (differential equations).
• Discrete: Variables change in steps (e.g., number of cows in a herd, generation of insects). Uses difference equations.

3. Fitting Models to Experimental Data

Curve fitting is the process of constructing a mathematical function that has the best fit to a series of data points. The most common method used is the Method of Least Squares.

Objective: Minimize the sum of the squares of the vertical differences (residuals) between the observed data points and the fitted curve.

3.1 Fitting a Linear Model

The linear model is the simplest form of regression, representing a straight-line relationship.

Equation:

• = Dependent variable (e.g., Yield)
• = Independent variable (e.g., Fertilizer)
• = Slope (Rate of change)
• = y-intercept (Value of when )

Method of Least Squares for Linear Fit:
Given data points , we solve for and using the "Normal Equations":

Calculation Steps:

1. Create a table with columns: , , , .
2. Sum the columns to get , , , .
3. Solve the system of equations for and , or use the direct formulas:

3.2 Fitting a Quadratic Model

Used when the data shows a parabolic trend (e.g., crop yield increases with fertilizer up to a point, then decreases due to toxicity).

Equation:

Method:
To find coefficients , , and , we set up a system of three linear equations based on the least squares criterion:

Note: This requires calculating higher powers of (). The system is usually solved using matrices (Gaussian elimination or Cramer’s rule).

3.3 Fitting an Exponential Model

Used for growth or decay processes (e.g., bacterial growth, population increase, radioactive decay).

Equation:

• = Initial value
• = Growth rate constant

Method (Linearization):
Non-linear models are difficult to fit directly using standard least squares. We transform the equation into a linear form using logarithms.

1. Take the natural logarithm () of both sides:
   

   
   
   
   

2. Map to Linear Equation ():

   • Let (The transformed dependent variable)
   • Let
   • Let (The intercept)
   • Let (The slope)

3. Procedure:

   • Convert all observed values to .
   • Perform standard linear regression on and to find and .
   • Convert back to original parameters:

4. Summary Table of Model Fitting

5. Review Questions

1. Differentiate between a deterministic and a stochastic model using agricultural examples.
2. Why is the method of least squares preferred for fitting curves to experimental data?
3. Given the data points for crop height over weeks, outline the steps to fit an exponential growth model.
4. Explain why a quadratic model might be more appropriate than a linear model for analyzing fertilizer-yield relationships.