If we want to calculate the change in population over a period of time, we take the number of births and we subtract the number of deaths. And we can calculate the birth rate or how many births are registered per some unit, say number of births per thousand individuals and number of deaths. as number of deaths per thousand individuals and come up with a rate to multiply by our population size so say we've got 10,000 individuals and we have a birth rate of 0.003 or three births per thousand individuals and a death rate of say 0.001 or one death per thousand individuals we can figure out if over a given period of time how that's going to affect our population size.
So we can change our estimate of how a population will grow to the birth rate times the population size minus the death rate times our population size is going to give us our change in the size of the population in a period of time. but we can simplify this seeing as we've got n in both of these terms and these are rates if we can just figure out the birth rate and subtract the death rate that'll give us an overall rate r so zero population growth zpg occurs when the birth rate and the death rate are equal so if we get three births per thousand and three deaths per thousand individuals That means our R is going to be 0. So we can revise our previous estimate of the population size change. And we can say the change in the size of our population N over a certain period of time is going to be this figure R, which remember is the birth rate minus the death rate, our base population size N.
We can also calculate our instantaneous growth rate, which is what is happening in a very small calculus differential. period of time. So under idealized conditions we have unlimited food, unlimited space, unlimited resources. We would expect population, if the number of births is greater than number of deaths, to increase exponentially.
Which means the more individuals we have the faster we're going to add individuals. and we can come up with an equation for the exponential growth rate. That's going to give us a characteristic J-shaped curve because it's more quickly.
And if this rate, this intrinsic rate of growth, is very high, we can see this curve goes up a lot more quickly if it's half as much the rate then the shape of the J it lags out and increases more slowly and sometimes we do see this J-shaped exponential growth curve in natural environments It can't go on forever because we live in, well we live in an open system, we do not have infinite resources. But if we look at a population after it's been reintroduced from a very low number, or once some sort of... aspect of the death rate, for example hunting, is ceased as occurred in the Kruger National Park in South Africa, the elephant population began to show an exponential rate of growth. So once the hunting of elephants was banned, the rate of increase in elephant populations, since there weren't that many deaths, but the same number of births, led to an exponential increase in the size of the elephant population. And this led to policies that had to be put in place to control elephant populations because they were there were going to be so many elephants that they were becoming damaging to the park.
There were too many elephants. And believe me, if you have too many elephants around, they're going to wreck up the place. So exponential growth cannot be sustained indefinitely. It usually can't be sustained for very long because organisms live in an environment where resources are limited.
So we can come up with a different type of model considering that only so many individuals, a population can only reach a certain size. before resources are depleted. We call this maximum population size the carrying capacity and we refer to that carrying capacity as K. That's going to vary based on how many what the situation is with resources such as food and space and water and things like that. So we can come up with a different growth model called a logistic population growth model different from the exponential growth model because it takes into account the idea that there's only enough resources for a certain number of individuals.
So the mathematical expression is similar to our logistic growth model only we introduce these factors to take into account that there is a carrying capacity there is an upper bound to what the population is. So our rate of growth times our population is going to be the carrying capacity minus the number of individuals over that carrying capacity. So as long as this number is, the number of individuals is less than k, then we're going to still have growth in the population. So if we have a carrying capacity of 10,000, we've got 9,000 individuals. 10,000 minus 9,000 leaves us with 1,000, that's a positive number, over K, which means we're still going to have a growing population with a positive intrinsic rate of growth.
However, if our carrying capacity is 10,000 and our number in the population is 11,000, this is going to be a negative number, which is going to decrease. principle anyway is going to decrease the number of individual joules in the population. So the population will decrease in the face of limited resources.
So when n is small compared to k this term k minus n over k is close to one. So say we've got a carrying capacity of 10 000 and n is 1000 and you've got 9000 which is k minus n over 10,000 that's very close to 1. If n were only say 10, imagine 10,000 minus 10 is 990 over 10,000 that's going to be very close to 1 and the rate of growth is going to be close to the maximum but when n is large compared to k This term is going to be close to zero, which means the increase in the per capita rate is going to be small. So imagine that we've got a carrying capacity of 10,000 and our number of individuals is 9,999, which means this term is going to be 1 over 10,000, which is close to zero, which means our per capita rate of increase will therefore be very small. And when n equals k, the population cannot grow anymore, at least in principle.
So let's take a... a hypothetical population with a carrying capacity of 1500 individuals if we've got 25 in the population and the maximum rate of increase is 1 if we look at this term k minus n so 1500 minus 25 over k it's going to be 0.98, our per capita rate of increase is 0.98 which when we plug that into our Model means we're going to add 25 individuals to the population. Okay, so a very fast rate of growth.
If we have 100 individuals, again, our maximum rate of increase is 1.0. This term is going to get a little bit lower because we have more individuals to start with. We're getting further away from one. We're going to add a lot of individuals, but not the exact number that we started with.
So here we added as many individuals as we started off with. Here we add fewer individuals than we started off with. If it's 250, again this number, this term k minus n over k is going down, which means we're adding fewer individuals as a proportion of our population size, to the point where if we get to 1000 individuals, this k minus n over k term is only one third.
which means our per capita rate of increase is only one-third which means we can only add a third of the number of individuals relative to the initial population size and when we graph that out, that rate of growth over time what we're going to see is that instead of having a J-shaped curve we have a sigmoid or S-shaped curve So new individuals are added to the population most rapidly during the intermediate population size. We'll see what that means in a second in the graph. And the population growth rate is going to decrease as the number in the population approaches the carrying capacity.
So here we see this sigmoid shaped curve reflecting logistic growth. So if our carrying capacity is 1500, the closer we get to that carrying capacity, the rate of growth is going to keep decreasing and decreasing. And in principle there is an asymptote here, and if you remember from your mathematics class, an asymptote means it's never going to actually get to 0% rate of growth, but it'll get very very close to it. So the fastest rate of population growth is right in the middle of this sigmoid So when you have 750 individuals in the population, that's when the rate of increase is the fastest. It begins to slow down after you get past half of your carrying capacity.
So we can actually measure growth in populations of certain organisms in the laboratory, perform experiments with things like paramecia. Paramecia is a ciliate. Paramecia are ciliates like Tetrahymena.
We can grow them in a constant environment and keep feeding them, remove predators and competitors, and see what happens to the rate of growth. If we don't keep adding food, the resources are going to be limited, and we're going to end up having a nice sigmoid growth curve. Sometimes we can overshoot our carrying capacity for a limited period of time for this experiment with Daphne.
I remember Daphne from our experiments in the lab and what's going to happen is the number of growths may overshoot K but then population is going to decline for a little while until it starts to normalize and get closer to the carrying capacity. So some populations can overshoot. their carrying capacity, settling down to a relatively stable density.
Some populations are going to fluctuate widely, and we don't necessarily know what their carrying capacity is because resources may be fluctuating as well. Another thing that can occur is something called an Allee effect. Which, when there's only a few number of individuals, it makes it difficult for survival and reproduction to occur. Because you don't have a big enough clump to make...
the growth of the population occur.