The growth of the earth’s population is one of the pressing issues of our time. Will the population continue to grow? Or will it perhaps level off at some point, and if so, when? In this section, we look at two ways in which we may use differential equations to help us address these questions.
We will now begin studying the earth’s population. To get started, in Table 7.6.1 are some data for the earth’s population in recent years that we will use in our investigations.
On the face of it, this seems pretty reasonable. When there is a relatively small number of people, there will be fewer births and deaths so the rate of change will be small. When there is a larger number of people, there will be more births and deaths so we expect a larger rate of change.
Our work in Activity 7.6.2 shows that that the exponential model is fairly accurate for years relatively close to 2000. However, if we go too far into the future, the model predicts increasingly large rates of change, which causes the population to grow arbitrarily large. This does not make much sense since it is unrealistic to expect that the earth would be able to support such a large population.
We see that is the ratio of the rate of change to the population; in other words, it is the contribution to the rate of change from a single person. We call this the per capita growth rate.
In the exponential model we introduced in Activity 7.6.2, the per capita growth rate is constant. This means that when the population is large, the per capita growth rate is the same as when the population is small. It is natural to think that the per capita growth rate should decrease when the population becomes large, since there will not be enough resources to support so many people. We expect it would be a more realistic model to assume that the per capita growth rate depends on the population .
In the previous activity, we computed the per capita growth rate in a single year by computing , the quotient of and (which we did for ). If we return to the data in Table 7.6.1 and compute the per capita growth rate over a range of years, we generate the data shown in Figure 7.6.2, which shows how the per capita growth rate is a function of the population, .
From the data, we see that the per capita growth rate appears to decrease as the population increases. In fact, the points seem to lie very close to a line, which is shown at two different scales in Figure 7.6.3.
Graphing the dependence of on the population , we see that this differential equation demonstrates a quadratic relationship between and , as shown in Figure 7.6.4.
The equation is an example of the logistic equation, and is the second model for population growth that we will consider. We expect that it will be more realistic, because the per capita growth rate is a decreasing function of the population.
Indeed, the graph in Figure 7.6.4 shows that there are two equilibrium solutions, , which is unstable, and , which is a stable equilibrium. The graph shows that any solution with will eventually stabilize around 12.5. Thus, our model predicts the world’s population will eventually stabilize around 12.5 billion.
A prediction for the long-term behavior of the population is a valuable conclusion to draw from our differential equation. We would, however, also like to answer some quantitative questions. For instance, how long will it take to reach a population of 10 billion? To answer this question, we need to find an explicit solution of the equation.
The equilibrium solutions here are and , which shows that . The equilibrium at is called the carrying capacity of the population for it represents the stable population that can be sustained by the environment.
The graph shows the population leveling off at 12.5 billion, as we expected, and that the population will be around 10 billion in the year 2050. These results, which we have found using a relatively simple mathematical model, agree fairly well with predictions made using a much more sophisticated model developed by the United Nations.
The logistic equation is good for modeling any situation in which limited growth is possible. For instance, it could model the spread of a flu virus through a population contained on a cruise ship, the rate at which a rumor spreads within a small town, or the behavior of an animal population on an island. Through our work in this section, we have completely solved the logistic equation, regardless of the values of the constants ,, and . Anytime we encounter a logistic equation, we can apply the formula we found in Equation (7.6.2).
Consider the model for the earth’s population that we created. At what value of is the rate of change greatest? How does that compare to the population in recent years?
According to the model we developed, what will the population be in the year 2100?
According to the model we developed, when will the population reach 9 billion?
Now consider the general solution to the general logistic initial value problem that we found, given by
If we assume that the rate of growth of a population is proportional to the population, we are led to a model in which the population grows without bound and at a rate that grows without bound.
By assuming that the per capita growth rate decreases as the population grows, we are led to the logistic model of population growth, which predicts that the population will eventually stabilize at the carrying capacity.
(a) On a print-out of the slope field, sketch three non-zero solution curves showing different types of behavior for the population . Give an initial condition that will produce each:
(c) Considering the shape of solutions for the population, give any intervals for which the following are true. If no such interval exists, enter none, and if there are multiple intervals, give them as a list. (Thus, if solutions are increasing when is between 1 and 3, enter (1,3) for that answer; if they are decreasing when is between 1 and 2 or between 3 and 4, enter (1,2),(3,4). Note that your answers may reflect the fact that is a population.)
(a) A logistic model is a good one to use for these data. Explain why this might be the case: logically, how large would the growth in VCR ownership be when they are first introduced? How large can the ownership ever be?
What limiting value does this point of inflection predict (note that if the logistic model is reasonable, this prediction should agree with the data for 1990 and 1991)?
3.Finding a logistic function for an infection model.
The total number of people infected with a virus often grows like a logistic curve. Suppose that 25 people originally have the virus, and that in the early stages of the virus (with time, , measured in weeks), the number of people infected is increasing exponentially with . It is estimated that, in the long run, approximately 6000 people become infected.
(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point?
For organisms which need a partner for reproduction but rely on a chance encounter for meeting a mate, the birth rate is proportional to the square of the population. Thus, the population of such a type of organism satisfies a differential equation of the form
(Your answers may involve a and b. Give your answers as an interval or list of intervals: thus, if dP/dt is less than zero for P between 1 and 3 and P greater than 4, enter (1,3),(4,infinity).)
(b) Use this graph to sketch the shape of solution curves with various initial values: use your answers in part (a), and where is increasing and decreasing to decide what the shape of the curves has to be. Based on your solution curves, why is called the threshold population?
The logistic equation may be used to model how a rumor spreads through a group of people. Suppose that is the fraction of people that have heard the rumor on day . The equation
Suppose that measures the number of bacteria living in a colony in a Petri dish, where is measured in thousands and is measured in days. One day, you measure that there are 6,000 bacteria and the per capita growth rate is 3. A few days later, you measure that there are 9,000 bacteria and the per capita growth rate is 2.
Assume that the per capita growth rate is a linear function of . Use the measurements to find this function and write a logistic equation to describe .
What is the carrying capacity for the bacteria?
At what population is the number of bacteria increasing most rapidly?
If there are initially 1,000 bacteria, how long will it take to reach 80% of the carrying capacity?
Suppose that a long time has passed and that the fish population is stable at the carrying capacity. At this time, humans begin harvesting 20% of the fish every year. Modify the differential equation by adding a term to incorporate the harvesting of fish.
What is the new carrying capacity?
What will the fish population be one year after the harvesting begins?
How long will it take for the population to be within 10% of the carrying capacity?