Mathematics & Science
Applications of Differential Equations
A Theoretical Introduction
You should have thoroughly absorbed by now the fact that differential equations are used to create mathematical models of any real world system in which rates of change are involved. One very obvious situation in which this sort of modeling would apply is in the study of how populations shrink and grow. These may be populations of people, animals, bacteria, or the number of HIV infected cells in a patient's bloodstream.
The purpose of this laboratory is to study two of the simplest models that have been created for the analysis of population dynamics—the Malthusian, or exponential model, and the Verhulst, or logistic model.
The Malthusian Model
One of the first researchers into population dynamics was Thomas Malthus, (1766-1834). Malthus was an Anglican minister as well as a professor of history at Cambridge University in England. His idea, which seems fairly obvious over 150 years later, was that the rate at which a population grows is directly proportional to its current size. This was a ground-breaking concept when he first published it!
To justify Malthus' idea we might argue, quite simplistically, as follows. Populations grow because people have babies. The more people there are, the more babies they'll have. This means that the number of babies being born is a constant multiple of the number of people present in the population. So, the rate of population growth is directly proportional to the size of the population.
You might want to point out that there's also a death rate that we haven't considered. However, this rate too will simply be a certain percentage of the population's size at any given time. As a consequence, we might think of combining the positive birth rate with the negative death rate, and we'd still end up with Malthus' conclusion.
Let's take the Malthus idea and formulate it in the language of differential equations. To repeat:
the rate at which a population grows is directly proportional to its current size
To create the model we'll need to introduce some variables. We'll choose them so that:
Clearly, then, the rate of change of the population, or the growth rate, would be represented by the quantity dp/dt, or p'.
Introducing these quantities, Malthus' idea becomes:
dp/dt= k p
This differential equation is quite easy to solve, but by itself it will yield a whole family of solutions. In order to nail down a specific solution we need an initial condition. This is quite easy to come up with. A known value of the population at the beginning of the "experiment" would be fine. So say that the population when t=to is po. Then our initial condition is:
Putting this together with the original differential equation gives us an initial value problem for the Malthus model:
dp/dt= k p
So what is it's solution? Even by hand the work is easy. The differential equation can be solved by separation of variables, and the initial condition can be substituted into this result to eliminate one of the unknown constants. The other unknown constant can only be eliminated if we can get hold of one other piece of population data for some later time. We'll see how to do all of this on the computer in a few minutes.
The Verhulst Model
Pierre-Francois Verhulst (1804-1849) was a Belgian mathematician who generalized the Malthusian model by allowing for the fact that populations encounter internal competition as they grow within a closed environment, and this competition has a tendency to retard the rate of growth. His idea says that while the population will continue to grow as time goes on, the rate at which it does this growing gets smaller. This is a slightly more realistic approach than that of Malthus, whose idea actually predicts that populations will grow exponentially, and without bound—a prospect that defies physical limitations. (Malthus was aware that this kind of growth couldn't continue—he wasn't stupid. The whole thing really scared him as he looked forward to global famine.)
So how could we account for the competition component within the model? Again, we'll argue things through fairly simplistically:
Competition happens when one member of the population encounters any other member of the population, and competes with it for resources such as food, land, water, and I suppose in today's economy you might even say jobs. So how many encounters of this kind could possibly occur? Well focus on a single individual for a moment, call her Bertha. If we use the same variable names that we did when we analyzed the Malthus model, then from Bertha's point of view there are p-1 other individuals in the population with whom she must compete. Now we expand our analysis beyond just Bertha. Every member of the population could look at the rest of the population just like Bertha does, i.e. they all see p-1 competitors. So, summarizing:
all p members of the population have p-1 competitors
So how many possible competitive encounters are we looking at? Well wouldn't it be just the product of the two quantities we just mentioned? In other words:
possible number of encounters = p * (p-1)
Hold on a second though, there's something wrong! We've actually counted every encounter twice. Focusing on Bertha again: say she encounters Eddy. We count this as the "Bertha-Eddy" encounter. Now look at things from Eddy's point of view: when he encounters Bertha we count this as the "Eddy-Bertha" encounter. But aren't they both just the same encounter? See, we've accidently counted the encounter twice! So, scratch our last equation—it should really be divided by two to give the correct number of possible encounters, and it should read as:
possible number of encounters = p(p-1)/2
OK, we've managed to come up with a number for the possible number of competitive encounters within the population, but now we have to absorb this idea into the growth rate model that we're building. Realistically only a small proportion on the encounters that we've mentioned ever takes place. (After all, have you ever encountered every other person living in the United States?) Even out of the encounters that do occur, not all of them result in competition for resources. Our conclusion then is that the actual competition component of our population growth model is just some small multiple of p(p-1)/2, say, c p(p-1)/2.
Putting this new competition component together with the old Malthus model is actually quite easy. You might say:
rate of growth = birth rate - competition rate
or, combining our new competition value together with the old Malthus equation:
dp/dt= k p - c p(p-1)/2.
This is actually more complicated than it needs to be. We could expand the product here to get:
dp/dt= k p - c p2/2 + c p/2.
Now the first and last terms are both multiples of p. Combining them gives us:
dp/dt= (k+c/2)p - c p2/2.
Or, to tidy it up even further:
dp/dt= (k+c/2)p - (c/2)p2.
Now since all of the constants in this expression are currently unknown, it would pay us to rename the combinations with simpler names. Let's use a=(k+c/2) and b=(c/2). Then the differential equation becomes:
dp/dt= ap - bp2.
And that's it! Put this together with the same initial condition we used for the Malthus model, and we have the Verhulst initial value problem:
dp/dt= ap - bp2.
This equation already contains two unknown constants, and a third will come from the integration required to solve the differential equation. Because of this, we'll need two other data points in addition to the original initial condition in order to have enough equations to solve for all of the unknown constants.
Once again we have a problem that is solvable by separation of variables, just like with Malthus. This time, however, the integration involved is extremely messy, so you'll be glad to hear that it will be the computer that does the work for us.
Let's move on to explore the problem with Mathematica.
ODE Laboratories: A Sabbatical Project by Christopher A. Barker
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