Logistic population models are one of the most important tools in mathematical biology because they describe how populations grow when resources are limited. Unlike simple exponential growth models, logistic models capture the realistic behavior of populations slowing down as they approach a maximum sustainable size, known as the carrying capacity. From ecology and epidemiology to economics and engineering, these models provide a powerful framework for understanding real-world systems.
In this blog post, you will learn how to develop a mathematical model for logistic population growth step by step. We will begin by exploring the key ideas behind logistic growth and how they differ from exponential models. Then, we will build the governing differential equation and interpret the meaning of each parameter in the model.
Modeling the Problem
Suppose a population only changes based on birth and death rates.
Let B(t) = The number of births per unit population per unit of time t.
Let D(t) = the number of deaths per unit population per unit of time t.
This means the following:
This means the change in population is the following:
which means that
The error should approach zero as delta t approaches 0, which yields the following differential equation:
Now, suppose that the birthrate is a linear decreasing function of the population size, so B = B_0 – B_1*P where B_0 and B_1 are constants and greater than zero. Also suppose the death rate is constant, then the following holds:
Therefore, this results in the following equation where a = B_0 – D_0 and b = B_1
It is useful to rewrite as the following where k=b and M = a/b. Here M is the limiting population.
Solving for a Solution
The equation is separable, so it is separated as follows:
Now, we just integrate both sides. Also, note: for the left hand side, one must use partial fractions to integrate. You should get the following result:
Solving for P, you should get the following the result:
where B is equal to the following when t = 0,
Plugging that value for B, we obtain the final result for the population:
Solving a Logistic Population Problem
Suppose in 1885 the population of a certain country a population of 50 million with a growing rate of 750k per year. Suppose the population in 1940 is 100 million with a growing rate of 1 million per year.
Given the following equation:
we can solve the following two equations simultaneously:
Doing so will give the values M = 200 and k = 0.0001
Now, let 1940 be t=0. Since M is the limiting population, we have that M = 200 million and P_0 = 100 million
If we want to know what the population will be in the year 2000, we can let t=60, and plug the values into the equation we solved in the previous section. This becomes
Therefore, the population is approximately 153.7 million people in the year 2000.
Final Thoughts
In this blog post, you learned how to model logistic populations, and you learned how to solve problems using the derived equation.