Problem 16 - Calculus for Biology and Medicine

Question

16\. Population Growth Assume that \(N(t)\) denotes the size of a population at time \(t\), and that in some conditions \(N(t)\) satisfies the differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a constant. (a) Find the per capita growth rate. (b) Assume that \(r<0\) and that \(N(0)=20\). Is the population size at time 1 greater than 20 or less than \(20 ?\) Explain your answer.

Step-by-Step Solution

Answer

(a) The per capita growth rate is \( r \). (b) The population size at time 1 is less than 20, as \( r < 0 \) makes \( e^r < 1 \).

Step 1: Understanding Per Capita Growth Rate

The per capita growth rate is given by the derivative of the population size, divided by the population size itself: \( \frac{dN/dt}{N} \). We know from the differential equation that \( \frac{dN}{dt} = rN \). Thus, the per capita growth rate is \( \frac{rN}{N} = r \).

Step 2: Solving for the Population Size after One Time Unit

Given the differential equation \( \frac{dN}{dt} = rN \), we want to find \( N(t) \). This is a separable differential equation, and its solution is given by \( N(t) = N(0) e^{rt} \).

Step 3: Determine the Population Size at Time t = 1 for r < 0

Substitute the initial condition and time into the population size function: \( N(1) = 20 e^{r \cdot 1} = 20 e^r \). Since \( r < 0 \), \( e^{r} \) is less than 1, resulting in \( N(1) < 20 \). Thus, the population size at time 1 is less than 20.

Key Concepts

Differential Equation

A differential equation is a mathematical equation that involves an unknown function and its derivatives. In the context of population growth, it models how the population changes over time. The differential equation given in the problem is \( \frac{dN}{dt} = rN \). This particular form is called a first-order linear differential equation. It relates the rate of change of the population size, denoted as \( \frac{dN}{dt} \), to the population size \( N \) at any time \( t \).The constant \( r \) in this equation is crucial. It represents the growth rate of the population and can be positive, zero, or negative. This equation suggests that the rate at which the population changes is proportional to the current population size itself, a characteristic of exponential growth. To solve this equation, we need to separate variables, integrate both sides, and apply the initial conditions to find a specific population function, which gives us insights into the dynamics of population growth and decline.

Per Capita Growth Rate

The concept of per capita growth rate is central in understanding how each individual in a population contributes to the overall growth. Essentially, it is the rate of growth per individual in the population. Mathematically, the per capita growth rate is derived by dividing the rate of change of the population, \( \frac{dN}{dt} \), by the population size, \( N \). In our scenario, given the differential equation \( \frac{dN}{dt} = rN \), the per capita growth rate simplifies to \( \frac{rN}{N} = r \). The unit of per capita growth rate is "per time unit," such as "per year."

• If \( r \) is positive, it indicates that each individual is contributing positively, leading to population growth.
• If \( r \) is negative, each individual represents a decline, leading to a shrinking population.

Understanding this measurement is helpful in making predictions about how the population might evolve over time given a specific growth rate.

Exponential Growth

Exponential growth is a specific type of growth pattern where the size of the population at any time can be described by an exponential function. If we consider our differential equation \( \frac{dN}{dt} = rN \), solving this gives us the function \( N(t) = N(0) e^{rt} \), where \( N(0) \) is the initial population size and \( e \) is the base of the natural logarithm.This equation reveals that population growth follows an exponential trend:

• If \( r > 0 \), the population grows exponentially, doubling in size at regular time intervals, often referred to as "doubling time."
• When \( r = 0 \), the population remains constant over time.
• If \( r < 0 \), the population declines, which can be seen as exponential decay or shrinking.

Analyzing exponential growth helps us understand phenomena such as how quickly a population might reach a certain size or how changes in growth rates impact long-term population dynamics.