Problem 17 - Calculus for Biology and Medicine

Question

\(N(t)\) is the size of the population at time \(t\). The population is modeled using a differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a constant. (a) According to the differential equation what is the growth rate of the population? (b) Assume that \(r>0\). Explain why the population size at time \(t=1\) will be larger than the population size at time \(t=0\). (c) If \(r>0\), will the growth rate at time \(t=1\) be larger or smaller than the growth rate at time \(t=0\) ? (d) Answer (c) again but comparing the per capita growth rates at times \(t=0\) and \(t=1\).

Step-by-Step Solution

Answer

(a) The growth rate is \( rN \). (b) Population will be larger at \( t=1 \) because \( r>0 \). (c) The growth rate at \( t=1 \) is larger. (d) The per capita growth rates are equal at \( t=0 \) and \( t=1 \).

Step 1: Understanding the Differential Equation

The differential equation given is \( \frac{dN}{dt} = rN \). This indicates that the rate of change of the population \( N \) with respect to time \( t \) is proportional to the current population size \( N \), with \( r \) being the constant of proportionality.

Step 2: Deriving the Growth Rate

In the equation \( \frac{dN}{dt} = rN \), \( rN \) is the expression for the growth rate of the population. Thus, the growth rate of the population is \( rN \).

Step 3: Evaluating Population Size for \( r>0 \)

With \( r>0 \), the differential equation indicates that the population is growing. As time increases from \( t=0 \) to \( t=1 \), the population size \( N(t) \) increases because the rate of change \( rN \) is positive, hence \( N(1) > N(0) \).

Step 4: Comparing Growth Rates at Times \( t=0 \) and \( t=1 \)

To compare growth rates, note that the growth rate is \( rN(t) \). Since the population grows, \( N(1) > N(0) \), this implies \( rN(1) > rN(0) \). Thus, the growth rate at \( t=1 \) is larger than at \( t=0 \).

Step 5: Per Capita Growth Rate Comparison

The per capita growth rate is the growth rate divided by the population size, \( \frac{rN}{N} = r \). Since \( r \) is constant, the per capita growth rate remains the same at both \( t=0 \) and \( t=1 \), indicating equal per capita growth rates at these times.

Key Concepts

Differential Equation

Differential equations are mathematical tools that help us model changing systems over time. In our exercise, the differential equation is \( \frac{d N}{d t} = r N \). This equation tells us how the population size \( N \) changes concerning time \( t \).

The left side of the equation, \( \frac{d N}{d t} \), represents the rate at which the population size is changing, also known as the "derivative" of \( N \) with respect to time. The right side, \( r N \), shows us that this rate depends on two things – a constant \( r \) and the current population size \( N \).

In simpler terms, the equation tells us that the population change isn't just random; it grows at a rate proportional to its current size. If the number of rabbits in a field is growing, according to our equation, it isn’t just the constant factor \( r \) working alone. It's \( r \) times however many rabbits are already there. The greater the population, the more it can grow!

Growth Rate

The term "growth rate" in this context describes how quickly the population is increasing at any given time. According to the equation we have, \( \frac{d N}{d t} = r N \), the growth rate is given by the expression \( rN \).

Let's break it down:

• \( r \): This is a constant, sometimes referred to as the "growth factor." It's a constant rate at which each individual in the population contributes to the total population growth.
• \( N \): This represents the current size of the population.

When multiplied together, \( rN \) gives us how rapidly the whole population is growing at that point in time.

With \( r > 0 \), more means faster. As the population size \( N \) increases, \( rN \), the growth rate, does too. That's why by the time we get to \( t=1 \), the growth rate is larger than it was at \( t=0 \) because the population itself has grown in size.

Per Capita Growth Rate

Per capita growth rate refers to the average growth rate per individual in the population. To find it, we divide the growth rate \( rN \) by the population size \( N \), which simplifies back down to \( r \).

Let's walk through it:

• The growth rate formula: \( rN \).
• Divide by population \( N \): \( \frac{rN}{N} = r \).

What's interesting here is that, unlike the overall growth rate, which increases as the population size grows, the per capita growth rate remains constant because \( r \) is constant.

This constancy means each individual in the population contributes equally to the growth of the population, regardless of the population's current size.

Therefore, even though more individuals mean a higher total growth rate, each individual's contribution to that growth stays the same from \( t=0 \) to \( t=1 \). This is key in understanding how population dynamics work over time based on differential growth models.