Growth, Decay, and the Logistic Equation
We have seen that the solution for dy/dx = y is an exponential function. On this page we explore this a bit more.
1. Exponential growth and decay
The applet shows the slope field for dy/dx = ky. Move the k slider and see what happens to the slope field and to the solution graph. What happens when k is big? Close to 0? Negative? You can work out through separation of variables that the general solution to this differential equation is
This represents growth when k is positive and decay when k is negative. This is a very common differential equation in modeling different kinds of problems, including population growth, interest accumulation, and radioactive decay. In these types of problems the independent variable is usually t (for time) instead of x.
In words, the differential equation says "the rate of change of y with respect to x is proportional to y," with k as the constant of proportionality. For positive k, the rate of change gets bigger with bigger y. Also note that P_{0} is the population for x = 0, usually called the initial population.
2. Logistic growth
Select the second example from the drop down menu, showing dy/dx = ky(1y/L). Move the k slider to see how this effects the solution curve. Also move the L slider (but keep L > 1) and notice what happens. One of the problems with exponential growth models is that real populations don't grow to infinity. The differential equation in this example, called the logistic equation, adds a limit to the growth. Here, k still determines how fast a population grows, but L provides an upper limit on the population. The solution can be found through separation of variables and is
where P_{0} is the initial population.
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Other 'Differential Equations' topics
Acknowledgements
Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.
