where P(t) is the population at (discrete) time t and b and d are the positive birth and death rates, respectively. Now that we've used our biological knowledge to come up with this equation, let's use some math knowledge to learn more about it and what it can tell us about bacteria growth.
First, we can collect like terms, and make the expression simpler. This is like using the fact that x + 2x = 3x, since we have like terms of x and 2x. Since most people become physically ill when algebra is mentioned, I'll just write down the collected terms in our P(t) formula, and get
P(t+1) = (1 + b - d) P(t) .
First, we can collect like terms, and make the expression simpler. This is like using the fact that x + 2x = 3x, since we have like terms of x and 2x. Since most people become physically ill when algebra is mentioned, I'll just write down the collected terms in our P(t) formula, and get
This "new" form of our population equation can tell us something about what will happen to our population in the long-run (that is, as we let t get larger and larger). The above expression is an example of a geometric sequence. A sequence is just a list of numbers, following some pattern. In a geometric sequence, each number in the sequence is related to the previous one by multiplying it by a constant value. Using another mathematical trick, let's make a substitution, and let
P(t+1) = k P(t) .
In this form, we can really see what will happen to our population. If we start with some initial population size, such as P(0) = 100, then using our simplified equation above, we can find P(1), which is
Then
P(1) = k P(0) = 100k.
Then
P(2) = k P(1) = k (100k) = 100 k^2
and
P(3) = k P(2) = 100k^3
and so on. If we look at the pattern here, we see that there is a general equation for
P(t) = P(0) k^t
if the initial population is P(0) and k = 1+ b - d. If you're still with me at this point, you might be asking "so what does this get us, exactly?". Well, if we know what k is, it tells us a lot. The growth in the population depends completely on what k is, which depends on what b and d are. This makes sense: the birth and death rates determine what will happen to the little bacteria culture we have growing. Let's look at an example to see how.
Suppose that b = 0.2 and d = 0.1, meaning that k = 1 + 0.2 - 0.1 = 1.1 in the equation above. Then if we look at powers of k, we get (1.1)^2 = 1.21, (1.1)^3 = 1.331, (1.1)^4 = 1.4641, and so on. You can check for yourself that each power gets larger, and is larger than the previous one. That means as t increases, (1.1)^t is also increases, and in turn, P(t) is increasing. This makes some intuitive sense, since this is a result of the birth rate being higher than the death rate. Since at each time step there are more births and deaths, the population will increase indefinitely. This is a little bit unrealistic, since there are limits to growth (the amount of food the bacteria have available to them, having no more space in the petri dish, etc). This is due to the simplicity of the model, by the strong assumption that birth and death rates are constant in time. Still, the model does tell us something that makes sense and is realistic.
Let's look at the opposite case now. Suppose that b = 0.1 and d = 0.2 so that k = 1 + 0.1 - 0.2 = 0.9. If we look at powers of this k, we get (0.9)^2 = 0.81, (0.9)^3 = 0.729, (0.9)^4 = 0.6561 and so on. Here, as t increases, (0.9)^t is decreasing, and since it can never be negative, it will shrink smaller and smaller until it is almost zero. Biologically this means the bacteria will eventually go extinct if there more deaths than births at each time step. Again, this is exactly what you'd expect. The model, while pretty simple, makes biological sense, and so if our assumptions are not too strict, it would be a good model of bacteria growing in a laboratory.
So, there's a simple model of population growth. Of course, there more complicated models, like the logistic equation, which has rich mathematical properties as well. Personally, I think this area of study is really cool. Math is as useful as it is fun!
Suppose that b = 0.2 and d = 0.1, meaning that k = 1 + 0.2 - 0.1 = 1.1 in the equation above. Then if we look at powers of k, we get (1.1)^2 = 1.21, (1.1)^3 = 1.331, (1.1)^4 = 1.4641, and so on. You can check for yourself that each power gets larger, and is larger than the previous one. That means as t increases, (1.1)^t is also increases, and in turn, P(t) is increasing. This makes some intuitive sense, since this is a result of the birth rate being higher than the death rate. Since at each time step there are more births and deaths, the population will increase indefinitely. This is a little bit unrealistic, since there are limits to growth (the amount of food the bacteria have available to them, having no more space in the petri dish, etc). This is due to the simplicity of the model, by the strong assumption that birth and death rates are constant in time. Still, the model does tell us something that makes sense and is realistic.
Let's look at the opposite case now. Suppose that b = 0.1 and d = 0.2 so that k = 1 + 0.1 - 0.2 = 0.9. If we look at powers of this k, we get (0.9)^2 = 0.81, (0.9)^3 = 0.729, (0.9)^4 = 0.6561 and so on. Here, as t increases, (0.9)^t is decreasing, and since it can never be negative, it will shrink smaller and smaller until it is almost zero. Biologically this means the bacteria will eventually go extinct if there more deaths than births at each time step. Again, this is exactly what you'd expect. The model, while pretty simple, makes biological sense, and so if our assumptions are not too strict, it would be a good model of bacteria growing in a laboratory.
So, there's a simple model of population growth. Of course, there more complicated models, like the logistic equation, which has rich mathematical properties as well. Personally, I think this area of study is really cool. Math is as useful as it is fun!
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