Problem 4 - Biocalculus Calculus for the Life Sciences

Question

4\. Soil contamination A crop is planted in soil that is contaminated with a pollutant. The pollutant gradually leaches out of the soil but is also absorbed by the growing crop. A simple model of this process is $$\frac{d s}{d t}=-\alpha s-\beta s \quad \frac{d c}{d t}=\beta s$$ where \(s\) and \(c\) are the amounts of pollutant in the soil and crop (in mg), respectively, and \(\alpha\) and \(\beta\) are positive constants. $$\begin{array}{l}{\text { (a) Suppose that } s(0)=s_{0} \text { and } c(0)=0 . \text { What is the solu- }} \\ {\text { tion to the initial-value problem? }} \\ {\text { (b) In the long term (that is, as } t \rightarrow \infty ), \text { what is the amount }} \\ {\text { of pollutant in the crop? }}\end{array}$$

Step-by-Step Solution

Answer

(a) \( s(t) = s_0 e^{-(\alpha + \beta)t} \) and \( c(t) = \frac{\beta s_0}{\alpha + \beta} (1 - e^{-(\alpha + \beta)t}) \). (b) As \( t \to \infty \), pollutant in the crop is \( \frac{\beta s_0}{\alpha + \beta} \).

Step 1: Understand the Initial Conditions

The initial conditions are given as \( s(0) = s_0 \) and \( c(0) = 0 \). This means at time \( t = 0 \), the initial amount of pollutant in the soil is \( s_0 \) mg, and the crop starts with no pollutant.

Step 2: Solve the Differential Equation for Soil

The first differential equation is \( \frac{ds}{dt} = -\alpha s - \beta s = -( alpha + \beta)s \). This is a separable first-order linear differential equation. We can solve it by separation of variables.\[ \frac{1}{s} \frac{ds}{dt} = -( alpha + \beta) \] Integrating both sides with respect to \( t \), we have: \[ \int \frac{1}{s} ds = -( alpha + \beta) \int dt \] which gives: \[ \ln |s| = -( alpha + \beta) t + C_1 \] Exponentiating both sides to solve for \( s \): \[ s(t) = C_2 e^{-( alpha + \beta) t} \] Using the initial condition \( s(0) = s_0 \), we find \( C_2 = s_0 \). So, the solution is: \[ s(t) = s_0 e^{-( alpha + \beta)t} \]

Step 3: Solve the Differential Equation for Crop

The second differential equation is \( \frac{dc}{dt} = \beta s \). Substitute \( s(t) \) from the earlier step: \[ \frac{dc}{dt} = \beta s_0 e^{-( alpha + \beta)t} \] Integrate with respect to \( t \) to find \( c(t) \). \[ c(t) = \int \beta s_0 e^{-( alpha + \beta)t} dt \] Let \( u = -( alpha + \beta) t \), so \( du = -( alpha + \beta) dt \). Rewriting the integral, \[ c(t) = -\frac{\beta s_0}{\alpha + \beta} \int e^{u} du \] We get: \[ c(t) = -\frac{\beta s_0}{\alpha + \beta} e^{u} + C_3 \] Back substitute for \( u \): \[ c(t) = -\frac{\beta s_0}{\alpha + \beta} e^{-( alpha + \beta)t} + C_3 \] Using initial condition \( c(0) = 0 \), solve for \( C_3 \): \[ 0 = -\frac{\beta s_0}{\alpha + \beta} + C_3 \] So, \( C_3 = \frac{\beta s_0}{\alpha + \beta} \). Thus, \[ c(t) = \frac{\beta s_0}{\alpha + \beta} \left( 1 - e^{-( alpha + \beta)t} \right) \]

Step 4: Analyze the Long-Term Behavior

As \( t \to \infty \), the term \( e^{-( alpha + \beta)t} \to 0 \) because the exponential term decays to zero. Therefore, the pollutant amount in the crop in the long term is: \[ \lim_{t \to \infty} c(t) = \frac{\beta s_0}{\alpha + \beta} \]

Key Concepts

Soil Contamination

Soil contamination occurs when harmful substances, such as pollutants, enter the soil and adversely affect its quality. In this context, the pollutant is absorbed by the soil and the crops, impacting the surrounding ecosystem. Understanding how these pollutants behave can help us devise strategies to minimize their impact.

In the exercise provided, the equations describe how the pollutant leaches out of the soil and is absorbed by the crop. These equations form a differential equation system, where:

• \( \frac{d s}{d t} = -\alpha s - \beta s \) describes how the pollutant amount in the soil decreases over time due to leaching and absorption by the crops.
• \( \frac{d c}{d t} = \beta s \) shows how the concentration in the crop increases as it absorbs the pollutant from the soil.

Contaminated soil directly affects agricultural productivity and environmental health. Managing soil contamination often involves monitoring and modeling these processes to mitigate the pollutant spread.

Biological Modeling

Biological modeling uses mathematical equations to represent biological processes, making it easier to understand, predict, and manage complex systems. The use of differential equations to model the movement of pollutants from soil to crops is an example of biological modeling.

Here, the model uses parameters \( \alpha \) and \( \beta \), which are positive constants representing different aspects of the leaching and absorption processes.

• \( \alpha \) represents the rate at which the soil loses pollutants naturally through processes like leaching.
• \( \beta \) is the rate of absorption by crops, showing how quickly the plants pull the pollutant from the soil.

These variables help us predict how long pollutants will remain active in the environment and how much may end up in crops over time. Such models are vital in developing sustainable agricultural practices and protecting public health.

Initial Value Problem

An initial value problem involves solving a differential equation using a set of initial conditions to find a specific solution that fits these conditions over time. In our exercise, the initial conditions are \( s(0) = s_0 \) and \( c(0) = 0 \), meaning that at the start (time \( t = 0 \)), the soil holds a certain initial amount of pollutant \( s_0 \), while the crop has none.

The step-by-step solution involves solving two differential equations given the initial conditions:

• Solving \( \frac{ds}{dt} = -\alpha s - \beta s \) leads us to the solution \( s(t) = s_0 e^{-(\alpha + \beta)t} \), indicating an exponential decrease in pollutant concentration in the soil over time.
• For the crop, \( \frac{dc}{dt} = \beta s \) is solved resulting in \( c(t) = \frac{\beta s_0}{\alpha + \beta} (1 - e^{-(\alpha + \beta)t}) \), showing the pollutant level in the crop rises over time and approaches a constant value.

The goal of solving initial value problems is to determine how a system evolves from an initial state, offering insights into long-term outcomes of biological and environmental processes.