Problem 47 - Calculus for Biology and Medicine

Question

Let \(N(t)\) denote the size of a population at time \(t\). Differentiate $$ f(N)=r\left(a N-N^{2}\right)\left(1-\frac{N}{K}\right) $$

Step-by-Step Solution

Answer

The derivative \( f'(N) = ra - 2rN - \frac{2raN}{K} + \frac{3rN^2}{K} \).

Step 1: Recognize the Structure of f(N)

Given the function \( f(N) = r \left( aN - N^2 \right) \left( 1 - \frac{N}{K} \right) \), notice it is the product of \( u(N) = r(aN - N^2) \) and \( v(N) = 1 - \frac{N}{K} \). We'll apply the product rule to differentiate.

Step 2: Derivative of u(N)

Determine the derivative of \( u(N) = r(aN - N^2) \). Differentiating gives:\[ u'(N) = r(a - 2N) \].

Step 3: Derivative of v(N)

Determine the derivative of \( v(N) = 1 - \frac{N}{K} \). Differentiating gives:\[ v'(N) = -\frac{1}{K} \].

Step 4: Apply the Product Rule

The product rule states that \((u(N)v(N))' = u'(N)v(N) + u(N)v'(N)\). Substitute the expressions found in Step 2 and Step 3 into the product rule:\[ f'(N) = r(a - 2N) \left(1 - \frac{N}{K}\right) + r(aN - N^2)\left(-\frac{1}{K}\right) \].

Step 5: Simplify the Expression

Simplify each term separately:\[ r(a-2N) \left(1 - \frac{N}{K}\right) = r(a-2N) - r(a-2N)\frac{N}{K} \]\[ -\frac{r(aN-N^2)}{K} = -\frac{raN}{K} + \frac{rN^2}{K} \]Combine the results:\[ f'(N) = ra - 2rN - ra\frac{N}{K} + 2r\frac{N^2}{K} - \frac{raN}{K} + \frac{rN^2}{K} \]Combine like terms to get:\[ f'(N) = ra - 2rN - \frac{2raN}{K} + \frac{3rN^2}{K} \].

Key Concepts

Differentiation

Differentiation is a powerful mathematical tool used to find the rate at which a function changes. It helps us understand how variables in a function impact each other, which is why it's important in population dynamics and other fields. In the context of our exercise, we are differentiating a function related to population size, \( N(t) \), with respect to time \( t \). This involves determining how fast the population size changes as time progresses. The function given, \( f(N) = r(a N - N^2)(1 - \frac{N}{K}) \), represents a complex interaction of factors affecting population. Differentiation gives us \( f'(N) \), or the derivative, which tells us how \( f(N) \) changes with \( N \). To achieve this, we apply specific rules like the product rule, making the process systematic and structured. Differentiating functions often involves dealing with powers of variables and constants, which emphasizes its importance in analyzing dynamic systems.

Product Rule

The product rule is used when differentiating a function that is the product of two other functions. Let's say we have a function \( f(x) = u(x) \cdot v(x) \). The product rule states that the derivative, \( f'(x) \), is given by:

• \( f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

This means we differentiate each function separately and then combine the results. For the population function \( f(N) = r(a N - N^2)(1 - \frac{N}{K}) \), we identify \( u(N) = r(aN - N^2) \) and \( v(N) = 1 - \frac{N}{K} \). Each part is differentiated individually to find \( u'(N) \) and \( v'(N) \), and then inserted back into the product rule formula. The beauty of the product rule is it saves us time and errors, providing a straightforward path to finding derivatives of products without having to expand the entire expression.

Derivative

The derivative of a function is essentially the tool that provides us with the rate of change of one variable with respect to another. In our exercise, we're finding \( f'(N) \) by applying differentiation techniques, specifically focusing on product rule application. After determining \( u'(N) = r(a - 2N) \) and \( v'(N) = -\frac{1}{K} \), we use these to find the overall derivative of the function. The derivative gives us practical insights such as:

• Identifying growth rate or decline in population dynamics.
• Understanding the effect of different parameters, like \( a \), \( K \), and \( r \), on population change over time.

By calculating \( f'(N) \), we glimpse into the dynamic behavior of the population, particularly how changes in \( N \) affect the population's trajectory. Derivatives play a crucial role in models that predict population stability, oscillation, or extinction.

Simplification

After finding the derivative, the next step is simplification. This process involves streamlining the expression to make it easier to understand and work with. In our case, simplifying \( f'(N) \) after applying the product rule results in a clean, minimal version of the derivative. Simplification can involve:

• Combining like terms: Gathering terms with the same factors.
• Reducing fractions: Making fractional terms as simple as possible.
• Reorganizing terms: Rearranging terms to make expressions more comprehensible.

The original derivative expression was \( f'(N) = ra - 2rN - \frac{2raN}{K} + \frac{3rN^2}{K} \). Through simplification, we aim to reveal the core dynamics without unnecessary complexity, aiding deeper understanding and making further analysis feasible. Simplification not only results in a more elegant expression but also highlights the critical coefficients determining the rate of change in the population model.