Problem 60 - Calculus for Biology and Medicine

Question

Suppose that the population size at time \(t\) is $$ N(t)=N_{0} e^{r t}, \quad t \geq 0 $$ where \(N_{0}\) is a positive constant and \(r\) is a real number. (a) What is the population size at time \(0 ?\) (b) Show that $$ \frac{d N}{d t}=r N $$

Step-by-Step Solution

Answer

(a) The population size at time 0 is \( N_0 \). (b) It is shown that \( \frac{dN}{dt} = rN \).

Step 1: Understanding the Function at Time Zero

We begin with the population function given by \[ N(t) = N_0 e^{rt} \]To find the population size at time \( t=0 \), we substitute \( t = 0 \) into the equation. This gives us:\[ N(0) = N_0 e^{r imes 0} = N_0 e^0 \]Since \( e^0 = 1 \), the calculation simplifies to:\[ N(0) = N_0 \]Therefore, the population size at time \( t=0 \) is \( N_0 \).

Step 2: Showing the Derivative Formula

The task is to show that the derivative of the population size \( N(t) \) with respect to time \( t \) is equal to \( rN \). Start by differentiating the population function \( N(t) = N_0 e^{rt} \) with respect to \( t \). Using the chain rule, the derivative is:\[ \frac{dN}{dt} = \frac{d}{dt}(N_0 e^{rt}) = N_0 \cdot \frac{d}{dt}(e^{rt}) \]The derivative of \( e^{rt} \) with respect to \( t \) is \( re^{rt} \) due to the chain rule. Substituting this back gives:\[ \frac{dN}{dt} = N_0 \cdot re^{rt} = rN_0 e^{rt} \]Since \( N(t) = N_0 e^{rt} \), we have\[ \frac{dN}{dt} = rN \]Thus, the equation \( \frac{dN}{dt} = rN \) is verified.

Key Concepts

Population Dynamics

Population dynamics examines how populations of organisms, such as bacteria, animals, or humans, change over time. It's a dynamic field that helps us understand how population sizes grow, shrink, or remain stable based on a variety of factors.

In this context, the given equation \( N(t) = N_0 e^{rt} \) models exponential growth. Here, \( N_0 \) represents the initial population size at time \( t = 0 \), meaning the starting number of individuals in the population. The constant \( r \) is the growth rate, determining how fast the population increases over time. If \( r > 0 \), the population grows, while if \( r < 0 \), the population decreases.

Exponential growth occurs in environments where resources are abundant, allowing the population to double at a constant rate. This type of model is useful for understanding potential growth patterns and forecasting future population sizes, providing insights into ecological and evolutionary changes.

Differential Equation

A differential equation is a mathematical equation involving functions and their derivatives, describing how a rate of change of a quantity is related to the quantity itself.

In the context of the population model, the differential equation \( \frac{dN}{dt} = rN \) describes the rate of change of the population size \( N(t) \) over time \( t \). Here, \( \frac{dN}{dt} \) represents how quickly the population is changing at any given time.

This specific equation is a first-order linear differential equation that describes exponential growth or decay. It's solvable and gives insights into how the population changes with respect to time.

• If \( r > 0 \), the solution describes exponential growth.
• If \( r < 0 \), the solution describes exponential decay.

This fundamental concept is critical in many biological, ecological, and engineering applications where growth patterns are analyzed.

Chain Rule Differentiation

Chain rule differentiation is a crucial concept in calculus, used to differentiate composite functions. It helps in finding the derivative of a function that is applied to another function.

When we analyze the population equation \( N(t) = N_0 e^{rt} \), the goal is to find \( \frac{dN}{dt} \). Notice the exponential component \( e^{rt} \). To differentiate it, we apply the chain rule.

The exponential function \( e^{rt} \) undergoes differentiation in two steps:

• First, we differentiate the outer function \( e^{x} \) to get \( e^{rt} \).
• Then, we differentiate the inner function \( rt \) with respect to \( t \), resulting in \( r \).

Combining these results, the derivative \( \frac{d}{dt}(e^{rt}) = re^{rt} \).

Therefore, using the chain rule allows us to find that \( \frac{dN}{dt} = rN \), confirming the rate of change equation for the given exponential function.