Population Growth and Decay using Differential Equations

Population Growth and Decay using Differential Equations

Four years ago, a few classmates and I undertook an investigatory project which involved culturing microorganisms with agar as a growth medium. Since this was four years ago, I’m not entirely sure what the experiment was about, but I do remember having to count each colony by eye (boring and tedious). From our observations, we then had to compare the growth of the bacteria in each agar plate to measure the effects of our independent variable. Rejoice, Ken from four years ago! There’s actually another way to measure the growth of microbial colonies than tediously counting by eye! For this, we will be using differential equations, which I learned from my lovely CS130 class (fun and intuitive)! :D

I. What do derivatives mean?

Well, the derivative of any function at a certain point is the function’s slope at that same point. This means that given some function f(t), which represents the position of something at a given point in time, then we can say that the derivative of f(t), which is f’(t) can represent the rate at which the position changes at any point t,. This function f’(t) can be interpreted as the velocity of the object.

For example, given a function:

Then the derivative must be:

From this, we can say that for every unit time, the velocity of the object is equal to 2m/s.

II. How does this relate to population?

First of all, let’s mull over the properties of the quantity known as ‘population’. For starters, we know that given the ideal situation and enough time, the population will grow. The rate at which the population grows also factors in the current population. Hence, as the population grows, the rate at which the population grows also grows! We can write that as an equation like so:

in this equation, y represents the current population, y’ represents the rate at which the population grows, and k is the proportionality constant. This is known as the exponential growth model. Note that both y and y’ are both functions of time (t). We can further refine the equation above to relate the functions of y to time (t).

In this new equation, we have a new variable C, which is a constant of integration.

Let’s try this new equation out with a sample problem! :D

Suppose that there is a living population of 200 in paradise, with infinite resources and space. In a week, the population grew to 800. How big will the population be in 2 months?

To begin, we have to recognize that initially, there are 200 living organisms in the population. It is also given that after a week, there will be 800 living organisms. From this, we can conclude that

From here, we need to solve for the constant of integration. For this, we look at the case y(0), where y = 200 and t = 0

Now that we have C, we can now solve for k. For this, we can use the case y(1), where y = 800 and t = 1

Now that we have k, we can complete our equation

Lastly, we solve for the population at 8 weeks by plugging in t = 8

Thus, the population at 8 weeks is a whopping 12,910,075 (though it was actually 12,910,075.07, but you can’t have a fraction of a living organism).

That’s great! We solved it! However, the model used above isn’t really applicable to an agar plate because agar plates actually limit the population’s growth due to the size of the petri dish. Hence, I’ll introduce another model for you to use.

First of all, how does the population relate to the growth of the population in a petri dish? Well, it’s similar to the exponential growth model at the beginning since the walls of the petri dish don’t really limit how much the bacteria can grow when the population starts out small. However, as the population of the bacteria continues to grow, the walls of the petri dish grow closer, and the rate of growth would also slow down. We could say that the population has an upper limit due to the size of the petri dish, and we can call this value the carry capacity C. We can change the exponential model to represent this interaction between the carry capacity and the rate of growth by adding in another factor that gets smaller as the population approaches carry capacity.

From this model, we can see that the rate of growth of the population decreases as the population size grows closer to the carry capacity. This is called the logistic growth model. Similar to the exponential growth model, we can move this equation around to relate the functions of y to time (t).

From here, we integrate both sides to get the following equation (the integration is left as an exercise for the reader, haha!)

Where L is the constant of integration. Well, the next step is to use this with the previous sample problem! :D

First, we have to employ some sort of carry capacity C. For microbial measurements, this value can come with a unit of meters squared (m2), where C is the total area of the petri dish, and y is a function in terms of time that gives the area covered by the colonies of bacteria. Of course, that’s not the only way to represent the population of the bacteria quantitatively. For simplicity’s sake in this problem, however, let’s use the previous values, and set our carry capacity to be 200,000.

With that out of the way, let’s solve for our constant of integration, L, by plugging in y = 200 and t = 0

Now that we have our L, we can solve for k by plugging in y = 800 and t = 1

With solving k, we now have our full equation, and can finally plug in t = 8

From here, since the population y is always positive and less than carry capacity C, we can turn the 2nd term into

Continuing with our solution,

As you can see, the difference of the carrying capacity is wildy apparent, with us being left with only 133,066 (actually 133,066.4649, but again, you can’t have a fraction of a living organism) living organisms compared to the previous 12,910,075

For the case of my investigatory project, I would have to measure the initial population size, observe it for a few days, measure the population size again, and compare the proportionality constants of the different agar plates. From there, we could infer the size of the populations of the different agar plates at any point in time.

III. Conclusion

Of course, this isn’t the only application of differential equations. It is actually one of the things that can be used to model a good number of physical phenomenon happening in the real world, such as heating and cooling of objects, the dynamics of predator and prey, kinematics, and all sorts of interesting things! With all the cool things differential equations are, they aren’t perfect for certain models. This is due to the failure to model the problem correctly or some unforeseen consequences of the model that was used. I can’t deny, however, that differential equations are indeed useful and one of the more fun lessons in CS130! :)

by Manolo Hernandez and Ken Tabagan