Problem 18 - Calculus for Biology and Medicine

Question

Plant Growth The rate of growth of a plant depends on the amount of light available to it, which depends on whether it is growing in shade or full sun. Let the light that the plant receives be \(x, x\) could, for example, be measured in lumens (lumen is a unit for total light absorption). Growth rate also depends on how many herbivorous insects graze on the plant. Denote the number of nearby herbivorous insects by \(y(y\) could, for example, represent the number of herbivorous insects per square meter of habitat). A simple model is that the rate of growth, \(r\) is a linear function of \(x\) and \(y\); that is: $$ r(x, y)=a x+b y $$ for some constants \(a\) and \(b\). (a) Would you expect \(a>0\) or \(a<0 ?\) What sign do you expect \(b\) will have? (b) Consider the following data: A plant that receives an average of 3000 lumens of light, and that has 10 herbivorous insects per square meter, grows at a rate of \(20 \mathrm{~cm} / \mathrm{yr}\). A different plant, receiving an average of 5000 lumens of light, has 40 herbivorous insects per square meter, and grows at a rate of \(10 \mathrm{~cm} / \mathrm{yr}\). Use these data to fit \(a\) and \(b\). (c) How accurate is this model likely to be? In other words name some other factors besides light and number of herbivorous insects that could affect \(r\) and therefore should be in the model.

Step-by-Step Solution

Answer

(a) Expect \( a > 0 \), \( b < 0 \). (b) \( a = 0.01 \), \( b = -1 \). (c) Factors include soil fertility, water, temperature, and plant health.

Step 1: Analyze the signs of a and b

For the given function \( r(x, y) = ax + by \), we would expect \( a > 0 \) because a higher light intensity (\( x \)) should enhance plant growth, resulting in a positive contribution to growth rate. Conversely, \( b \) is expected to be negative \( b < 0 \) because an increase in herbivorous insects (\( y \)) would typically decrease plant growth due to grazing.

Step 2: Model the equations with given data

Using the data provided, we set up two equations based on the growth formula:1. For the first plant: \[ 20 = 3000a + 10b \]2. For the second plant: \[ 10 = 5000a + 40b \]

Step 3: Solve the system of equations

Use the system of equations from Step 2 to solve for \( a \) and \( b \). From Equation 1: \( 20 = 3000a + 10b \) \( \Rightarrow b = 2 - 300a \).Substitute \( b = 2 - 300a \) in Equation 2:\[ 10 = 5000a + 40(2 - 300a) \]\[ 10 = 5000a + 80 - 12000a \]\[ -70 = -7000a \]\( a = 0.01 \).Substitute \( a = 0.01 \) back into \( b = 2 - 300a \) to find \( b \):\[ b = 2 - 300(0.01) = -1 \].

Step 4: Evaluate the model's accuracy

The model may have limitations as it only considers light and herbivorous insects. Additional factors such as soil fertility, water availability, temperature, plant age, and diseases can also impact growth but are not included in the model. Therefore, while it provides a basic understanding, it might not predict growth rate accurately in all situations.

Key Concepts

Linear Functions

Understanding linear functions is crucial to modeling plant growth in terms of light intensity and herbivorous insects. A linear function of the form \( r(x, y) = ax + by \) describes a relationship where the growth rate \( r \) is dependent on variables \( x \) (light intensity) and \( y \) (herbivorous insect presence). In this equation, the coefficients \( a \) and \( b \) determine how each variable affects the growth rate. Because a linear function assumes a constant rate of change, it simplifies the relationship, making it easier to predict outcomes from changes in \( x \) and \( y \). However, in reality, plant growth might not always change at a constant rate due to the complexity of ecological interactions.

Light Intensity

Light intensity is a vital factor for plant growth as it directly influences photosynthesis—the process by which plants create food. Measured in lumens, it represents the amount of light energy available. More light typically leads to increased photosynthetic activity, thus enhancing growth. In our model, this is represented by the positive coefficient \( a \) in the linear function \( r(x,y) = ax + by \).

This positive relationship reflects that as light intensity increases, so does the growth rate. Plants in full sun receive more lumens compared to those in shade, enabling them to grow faster and stronger. However, too much light, particularly if accompanied by high temperatures, can also lead to issues like heat stress, showing that while light intensity is important, it remains just one piece of the plant growth puzzle.

Herbivorous Insects

Herbivorous insects pose a challenge to plant growth since they consume parts of the plant, directly reducing biomass and indirectly affecting photosynthesis and other physiological processes. In our function \( r(x,y) = ax + by \), the coefficient \( b \) is negative, meaning that an increase in the number of insects negatively impacts growth.

Insect herbivory can vary widely in its effects depending on the insect species, their population density, and the plant species. Some insects may cause significant damage, while others might have a minimal impact. This negative relationship underscores the importance of managing herbivory, whether through natural predators, chemical controls, or other methods, to minimize its impact on plant growth.

System of Equations

When trying to determine the coefficients \( a \) and \( b \) in our linear function, we use a system of equations derived from real data. Given plant growth rates and their relationships with light and insects, we form two linear equations:

• \( 20 = 3000a + 10b \)
• \( 10 = 5000a + 40b \)

By solving these simultaneously, we find the values of \( a = 0.01 \) and \( b = -1 \).

This technique showcases how real-world data can be used to tailor mathematical models to specific scenarios, enhancing the model’s predictive power. Nevertheless, it's important to recognize that this model is limited to the conditions and assumptions it was built upon and might not be applicable universally.