Rate Equations

This site presents an example of the use of "Rate Equations" to model growth in biological systems.

A rate equation is a kind of equation which you may not have seen before. Most equations you have been dealing with are probably the kind which describe the relationship between quantities. For example you might have an equation which related the cost of a number of apples, C, to the cost of an individual apple. If each apple cost 10 cents then the equation for the cost of N appleswould be given by the equation

In this equation, the terms on both sides of the equation are quantities. If you know how many apples are being bought, then the equation enables you to calculate the total cost. A rate equation is an equation which describes, not a relationship between 2 quantities, but rather how a quantity changes over time. Using a rate equation one can calculate the change in a quantity over time. What this means will become clear by considering an example.

A Simple Example of a Rate Equation --- Rabbits

The growth of an idealized population of rabbits is a simple biological example which can be modeled by a rate equation. To see how to develop such a model let's first consider what parameters are needed to describe how the rabbit population increases. We know that it takes a pair of rabbits to reproduce, but what is not determined is how many babies each pair will have in a litter. So the first parameter we need is the number of babies per litter, which we will call the procreation coefficient, p. If each pair of rabbits had, for example, 4 babies a year, then p = 4. A second parameter which we need is the number of rabbits we have to begin with. Let's use the symbol n to represent the number of rabbits. We will call the year when we start our calculation the year 0. To represent the number of rabbits which we have in the year 0 we will use the symbol n(0). This stands for n, the number of rabbits in the year 0.

Now let's formulate an equation which would describe the change in the number of rabbits in year 0 in terms of the number of rabbits we start with, n(0). The number of pairs of rabbits will be n(0)/2, half the total number of rabbits. Each pair of rabbits will have p babies, so that means that the number of babies is p * n(0)/2. This last quantity, which is the change in the number of rabbits per year is such an important quantity that we will give it a special symbol. We will use the  '  symbol, called "prime", to denote change per unit time. Thus if I write the expression, n'(0), this means "the change in the number of rabbits in year 0". So, in the case we just described we have the following equation for n'(0)

This is our first example of a rate equation.

Since we have an equation for the change in the number of rabbits in the year 0, we can use it to write an equation for the number of rabbits which we will have in the year 1. In the year 1 we will have the number of rabbits we started out with, n(0), plus the increase in the number of rabbits in the year 0, n'(0). If we write the number of rabbits in the year 1 as n(1), we have

which becomes, using Eq.(1.) for n'(0) ,

Factoring out the n(0), we get

So to summarize what we have got so far, we have Eq.1, the rate equation, which gives us an expression for the change in the number of rabbits in the year 0. And we have Eq.2 which enables us to calculate the number of rabbits in the year 1 in terms of the number of rabbits which we had in the year 0. We will call this second equation the iterating equation. Iterating means that the equation enables us to use the number of rabbits in year 0, and the rate equation, to calculate the number of rabbits in the next year, thus we can iterate the number of rabbits from one year to the next.

To illustrate how these equation work lets take a specific example. To do this we have to choose values for the 2 parameters of our model, n(0), and p. Let's take 4 rabbits to start with, n(0) = 4, and say that each pair of rabbits has 2 babies, p = 2. If we plug these specific choices for the parameters into our rate equation above we get

So the change in the rabbit population in year 0 is an increase of 4 rabbits, given the specific assumptions we have made concerning p and n(0).

To calculate the number of rabbits which we will have in the year 1 we use Eq. 2,

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