Problem 39 - Calculus for Biology and Medicine

Question

\mathrm{\\{} L \text { loyd } (1987) studied how the likelihood of a chick surviving to adulthood depends on the amount of resource that the chick's parents invested in it. He proposed the following model for the relationship between the amount of resource \(R\) invested in the chick and the likelihood \(p(R)\) that it survives to adulthood: $$ p(R)=\frac{R^{2}}{k^{2}+R^{2}}, \quad R>0 $$ In the model, \(k>0\) is a coefficient that varies between different species and different environments (a) Show that for all values of \(k, p(R)\) is an increasing function of \(R\) (b) Assume now that \(k=1\). Show that if \(R>1 / \sqrt{3}\), then the \(p(R)\) is concave down. Explain why this means that for \(R\) in this interval, there are diminishing returns from increasing the investment in the chick.

Step-by-Step Solution

Answer

(a) \( p(R) \) is increasing for all \( k \). (b) For \( k=1 \), \( p(R) \) is concave down if \( R > \frac{1}{\sqrt{3}} \), showing diminishing returns in that interval.

Step 1: Analyze the Nature of p(R)

We're given the function \( p(R) = \frac{R^2}{k^2 + R^2} \). We need to show that this is an increasing function of \( R \). To determine if \( p(R) \) is increasing, we should find the derivative \( p'(R) \) and analyze its sign.

Step 2: Find the Derivative of p(R)

To find \( p'(R) \), use the quotient rule: if \( g(x) = \frac{f(x)}{h(x)} \), then \( g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{[h(x)]^2} \). Here, \( f(R) = R^2 \) and \( h(R) = k^2 + R^2 \). Accordingly, we have \(abla f(R) = 2R \) and \( abla h(R) = 2R \). Substitute into the quotient rule:\[p'(R) = \frac{2R(k^2 + R^2) - R^2(2R)}{(k^2 + R^2)^2} = \frac{2Rk^2 + 2R^3 - 2R^3}{(k^2 + R^2)^2} = \frac{2Rk^2}{(k^2 + R^2)^2}\]

Step 3: Determine if p'(R) is Positive

From the expression for \( p'(R) \) in Step 2, we have \( p'(R) = \frac{2Rk^2}{(k^2 + R^2)^2} \). Since \( R > 0 \) and \( k > 0 \), all components of this derivative are positive, meaning \( p'(R) > 0 \) for all \( R > 0 \). This confirms that \( p(R) \) is an increasing function of \( R \).

Step 4: Verify Concavity with k=1

We set \( k = 1 \) and need to check when \( p(R) \) is concave down. A function is concave down when its second derivative \( p''(R) \) is negative. Let’s find \( p''(R) \) through differentiation of \( p'(R) \) - first simplifying with \( k=1 \):\[p'(R) = \frac{2R}{(1 + R^2)^2}\]

Step 5: Calculate the Second Derivative p''(R)

Using the product rule and chain rule for differentiation:\[p''(R) = abla \left( \frac{2R}{(1 + R^2)^2} \right) = \frac{(2)(1 + R^2)^2 - 4R^2(1 + R^2)}{(1 + R^2)^4}\]Simplifying, this becomes:\[p''(R) = \frac{2 - 2R^2}{(1 + R^2)^3}\]

Step 6: Determine Conditions for Concavity

For \( p''(R) < 0 \), the numerator \( 2 - 2R^2 < 0 \). Solving for \( R \), we find:\[2 < 2R^2 \Rightarrow 1 < R^2 \Rightarrow R > \frac{1}{\sqrt{3}}\]This indicates \( p(R) \) is concave down for \( R > \frac{1}{\sqrt{3}} \).

Step 7: Explain Diminishing Returns

With \( p(R) \) concave down for \( R > \frac{1}{\sqrt{3}} \), increased investment leads to smaller increments in \( p(R) \) (probability of survival). This is characteristic of diminishing returns, where each additional unit of resource yields less benefit.

Key Concepts

Understanding Derivatives in Biological Models

Derivatives are a fundamental concept in calculus used to determine the rate at which a function is changing. In biological models, derivatives can help describe how changes in one variable affect another. For the function \( p(R) = \frac{R^2}{k^2 + R^2} \), the resource \( R \) invested directly impacts the survival probability \( p(R) \) of a chick. By taking the derivative, noted as \( p'(R) \), one can ascertain whether increasing the resources will increase the likelihood of the chick’s survival.To find this derivative, we utilize the quotient rule: if \( g(x) = \frac{f(x)}{h(x)} \), then \( g'(x) = \frac{f'(x)h(x) - f(x)h'(x)}{[h(x)]^2} \). Applying this to our function gives us a result of \( p'(R) = \frac{2Rk^2}{(k^2 + R^2)^2} \).

• This expression indicates the sign of \( p'(R) \) depends on \( R \) and \( k \).
• Since both \( R > 0 \) and \( k > 0 \), the derivative is positive: \( p'(R) > 0 \).
• Thus, \( p(R) \) is an increasing function, meaning more resources lead to higher survival chances.

The Role of Concave Functions in Analytics

A function is termed concave down when the slope of the tangent decreases as \( R \) increases, meaning each extra unit of \( R \) contributes less to the increase in \( p(R) \). For biological functions like \( p(R) \), the second derivative, \( p''(R) \), helps in determining this concavity.Given \( k = 1 \), simplifying \( p'(R) \) becomes \( \frac{2R}{(1 + R^2)^2} \). To discover where the function is concave down, one calculates:\( p''(R) = \frac{2 - 2R^2}{(1 + R^2)^3} \) and finds when it becomes negative.

• This results in \( p''(R) < 0 \) when \( 1 < R^2 \).
• Essentially, \( R > \frac{1}{\sqrt{3}} \).
• Here, \( p(R) \) stops increasing rapidly, and the function is concave down, signaling a change in resource effectiveness.

Applying Diminishing Returns in Biological Contexts

Diminishing returns occur when additional resources do not proportionately increase a desired outcome—in our case, chick survival probability. When \( R > \frac{1}{\sqrt{3}} \), \( p(R) \) demonstrates diminishing returns.This phenomenon is typically observed in nature, where there is a threshold beyond which extra resources yield decreasing benefits. In mathematical terms, when the second derivative \( p''(R) \) is negative, increases in \( R \) lead to less gain in \( p(R) \).

• This is indicative of a resource allocation strategy, where optimal investment is vital for maximizing survival without waste.
• It teaches biologists and ecologists how organisms might behave optimally concerning energy or resource investments.

In conclusion, understanding diminishing returns helps in resource management and strategic planning within biological ecosystems.