Problem 30 - Calculus for Biology and Medicine

Question

Spruce budworms are a major pest that defoliates balsam fir. They are preyed upon by birds. A model for the per capita predation rate is given by $$ f(N)=\frac{a N}{k^{2}+N^{2}} $$ where \(N\) denotes the density of spruce budworms and \(a\) and \(k\) are positive constants. Find \(f^{\prime}(N)\), and determine where the predation rate is increasing and where it is decreasing.

Step-by-Step Solution

Answer

The predation rate is increasing for \(-k < N < k\) and decreasing for \(N > k\).

Step 1: Differentiate the Function

Given the function \(f(N)=\frac{a N}{k^{2}+N^{2}}\), we need to find the derivative \(f'(N)\). Use the quotient rule: if \(y=\frac{u}{v}\), then \(y' = \frac{u'v - uv'}{v^2}\). Set \(u = aN\) and \(v = k^2 + N^2\). Find \(u' = a\) and \(v' = 2N\). Now, apply the quotient rule:\[f'(N) = \frac{(a)(k^2 + N^2) - (aN)(2N)}{(k^2 + N^2)^2}\].

Step 2: Simplify the Derivative

Simplify the expression obtained in Step 1:\[f'(N) = \frac{a(k^2 + N^2) - 2aN^2}{(k^2 + N^2)^2}\].Combine like terms:\[f'(N) = \frac{ak^2 + aN^2 - 2aN^2}{(k^2 + N^2)^2}\].Simplify further:\[f'(N) = \frac{ak^2 - aN^2}{(k^2 + N^2)^2}\].Factor out \(a\) if needed to simplify:\[f'(N) = \frac{a(k^2 - N^2)}{(k^2 + N^2)^2}\].

Step 3: Determine Where the Function is Increasing or Decreasing

The function \(f(N)\) is increasing where \(f'(N) > 0\) and decreasing where \(f'(N) < 0\). So, solve:\[a(k^2 - N^2) > 0\].Since \(a > 0\), the inequality simplifies to:\[k^2 - N^2 > 0\].This implies:\[k^2 > N^2\].Taking square roots, we find:\[-k < N < k\].Therefore, the function is increasing for \(N < k\) and decreasing for \(N > k\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative provides information about how the function's output changes with respect to changes in its input. Essentially, it's a measure of the function's rate of change or its slope at any given point. This is particularly useful in biology, where you may want to understand how certain factors change over time or under different conditions.

In the context of the predation rate model for spruce budworms, differentiating the function helps us analyze how the predation rate changes as population density increases. The function provided, \(f(N) = \frac{aN}{k^2 + N^2}\), and its derivative, \(f'(N)\), are crucial in assessing whether the rate of predation increases or decreases as the population changes.

Quotient Rule

The quotient rule is a technique used in differentiation, specifically for functions that are expressed as a ratio of two separate functions. When you have a function \(y = \frac{u}{v}\), where both \(u\) and \(v\) are functions of a particular variable (like \(N\)), you can find its derivative using the formula: \[y' = \frac{u'v - uv'}{v^2}\].

In the exercise about spruce budworm predation, we applied the quotient rule to differentiate \(f(N) = \frac{aN}{k^2 + N^2}\). Here, \(u = aN\) and \(v = k^2 + N^2\). By finding the derivatives of \(u\) and \(v\) separately, we can use this quotient rule to establish \(f'(N)\), thus providing insights into the function’s behavior regarding the population density.

Predation Rate Model

Biological systems are often modeled using mathematical functions to predict certain behaviors under various conditions. The predation rate model for spruce budworms, \(f(N) = \frac{aN}{k^2 + N^2}\), is a great example of this.

This model helps predicts how many budworms are preyed upon per capita as the population density \(N\) changes. The constants \(a\) and \(k\) allow for adjustments based on specific ecological settings or species characteristics. By analyzing the function, biologists can infer the levels of predation and potential outcomes on the budworm population without experimental or field interventions. Understanding the derivative and consequent behavior of this model can lead to more informed decisions regarding pest management and ecological predictions.

Increasing and Decreasing Functions

Understanding where a function is increasing or decreasing is crucial for determining the behavior of a system. For the predation rate model, the derivative \(f'(N) = \frac{a(k^2 - N^2)}{(k^2 + N^2)^2}\) can be used to identify these regions.

The function is increasing where \(f'(N) > 0\) and decreasing where \(f'(N) < 0\). For \(f'(N)\), this translates to solving \(k^2 - N^2 > 0\). Thus, taking square roots gives us the range \(-k < N < k\). Therefore, as the density \(N\) is less than \(k\), the predation rate increases, indicating that the birds are likely taking advantage of increased prey availability. However, when \(N\) exceeds \(k\), the predation rate decreases, possibly due to saturation or decreased capture efficiency. This understanding helps in making precise observations and predictions about the population dynamics of spruce budworms.