Problem 44 - Calculus for Biology and Medicine

Question

Let \(N(t)\) denote the population size at time \(t\), and assume that \(N(t)\) is twice differentiable and satisfies the differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a real number. Differentiate the differential equation with respect to \(t\), and state whether \(N(t)\) is concave up or down.

Step-by-Step Solution

Answer

\( N(t) \) is concave up since \( \frac{d^2N}{dt^2} = r^2 N > 0 \) for positive \( N(t) \).

Step 1: Differentiate the given differential equation

We are given the differential equation \( \frac{dN}{dt} = rN \). To differentiate it with respect to \( t \), we apply the chain rule on the left-hand side, yielding \( \frac{d^2N}{dt^2} = r \frac{dN}{dt} \).

Step 2: Substitute the first derivative

From the given differential equation, we know \( \frac{dN}{dt} = rN \). Substitute this into the second derivative: \( \frac{d^2N}{dt^2} = r(rN) = r^2 N \).

Step 3: Determine concavity

The sign of \( \frac{d^2N}{dt^2} = r^2 N \) indicates the concavity of \( N(t) \). Since \( r^2 \) is always positive (as \( r^2 \geq 0 \) for any real number), \( N(t) \) is concave up for any positive \( N(t) \).

Key Concepts

Population Dynamics

Population dynamics is a fascinating area of study that explores how and why the population size of species changes over time. In mathematical terms, it often involves differential equations, which help model the rate of change of a population. In our exercise, the equation \( \frac{d N}{d t}=r N \) is central to understanding the dynamics of population growth.

This differential equation is known as the exponential growth model, where \( r \) is the growth rate. If \( r \) is positive, the population grows exponentially, while a negative \( r \) implies a decrease. This model assumes that the rate of population change is proportional to its size.

Such equations help predict future population sizes under various conditions. However, real-world factors can complicate these predictions, requiring more complex models for accurate forecasts.

Second Derivative

The second derivative of a function gives us valuable insights into how the rate of change of the function itself is changing. In the context of our population model, differentiating \( \frac{d N}{d t}=r N \) with respect to \( t \) reveals the second derivative, \( \frac{d^2 N}{d t^2} \).

By applying the chain rule, we find \( \frac{d^2 N}{d t^2} = r^2 N \). Here, \( r^2 \) represents a constant multiplier of the population size \( N \), signifying how the acceleration of the population size changes with respect to time.

Having a positive second derivative signals an increase in the growth rate, indicating an accelerating growth or decelerating decline depending on the sign of \( r \). However, since \( r^2 \) is always positive, this tells us specifically about the concavity of the function.

Concavity

Concavity is all about the direction the function is curving. It's a geometric property that tells us whether the graph of a function is curving upwards or downwards at any particular point.

For our function \( N(t) \), the second derivative \( \frac{d^2N}{dt^2} = r^2 N \) helps determine its concavity. Because the term \( r^2 \) is always non-negative, \( N(t) \) remains concave up for all positive values of \( N(t) \).

When a function is concave up, it means the slope is increasing, and the graph takes on a "smile" shape. This indicates that the population growth is accelerating, providing insights into how sustainable and rapid the growth might be in future scenarios. A positive second derivative is crucial in these interpretations, highlighting ever-increasing growth rates, a significant aspect of understanding population dynamics.