Any kind of organism in theory could take over the earth just by reproducing. For instance, just assume that we start with a single pair of male and female rabbits. If these rabbits and their descendants for seven years reproduce at a massive speed like bunnies, without any deaths, there would be enough rabbits to populate an entire state of Rhode Island. This actually pales in comparison if we use E. coli instead, if we start with just one bacterium, we would later, in just 36 hours have enough bacteria to cover the earth with a 1-foot layer.

As you may have notices, that there is no 1-foot layer of bacteria covering the earth, at least not from where I can see it, nor have bunnies taken over Rhode Island. Then, why don’t we see these populations grow as they theoretically could. Specific resources are needed by rabbits, E. coli and other living organisms, such as suitable environments and nutrients, in order to reproduce and mainly survive. A population can only reach a size that matches the availability of resources in its environment, meaning these resources are not unlimited.

A variety of mathematical methods are used by population ecologists, to model the population dynamics, or know how the populations change in composition and size overtime. Some of these models include ceilings that are determined by limited resources and others represent growth without environmental constraints. To predict future changes and to accurately describe the changes that are occurring a population, mathematical models of populations can be used.

There are two kinds of growth models – logistic and exponential.

Exponential growth – is when the per capita rate of increase (r), regardless of the size of the population takes the same positive values.

Logistic growth – is when the per capita rate of increase (r), as the population increases toward a maximum limit, decreases.

Let’s learn about exponential growth and logistic growth in detail.

Exponential Growth

An excellent example for exponential growth is the bacteria grown in the lab. In proportion to the size of the population, the population’s exponential growth rate increases over time, in exponential growth. How does this work? Let’s take a look. For many bacterial species the, the bacteria reproduce by binary fission, that is splitting in half, the time this takes is abut an hour. Let’s start by placing 1000 bacteria in a flask with unlimited supply of nutrients, to see how this is exponential growth.

After an hour, each bacterium will divideand yield an increase of 1000 bacteria, that is 2000 bacteria. After a couple hours, each of the 2000 bacteria will now divide and produce 2000 more bacteria, that is 4000 bacteria, and after 3 hours, each of the 4000 bacteria will divide and produce 4000 more bacteria, that in total makes 8000 bacteria.

The key concept of exponential growth is that the number of organisms added in each generation or population growth rate, gets larger as the population increases. And as we’ve seen, the results can be dramatic after only 1 day or 24 cycles of division, the bacterial population will have grown from 1000 to over 16 billion. When N, the population size, is plotted over time, a growth curve that is j-shaped is made.

Now, what is the way to model the exponential growth of a population? As we already know, we get exponential; growth when the per capita of increase rate r for our population is constant and positive. While any constant, positive r can lead to exponential growth, we will more often than not see exponential growth being represented with a r or rmax.

The maximum per capita rate of increase for a particular species under ideal conditions is rmax, and it varies from species to species. For instance, humans cannot reproduce as fast as bacteria, and thus, bacteria would have a higher maximum per capita rate of increase. The biotic potential, is the maximum population growth rate for a species, it is express in the following equation,

dNdT= rmaxN

Logistic Growth

Since it depends on an infinite number of resources, which tend to not exists in the real world, exponential growth is not very sustainable. It is possible for exponential growth for happen for a while, especially when there are few individuals and a lot of resources. But the resources start to get used up when the number of individuals gets large enough, showing the growth rate. Eventually the growth will level off or plateau, making an S-shaped curve. The maximum population size a particular environment can support is represented by the population size at which it plateaus. This is called K or, the carrying capacity.

By modifying our equation for exponential growth, we can mathematically model logistic growth, using per capita growth rate, r that depends on population size N and how close it is to carrying capacity K. When it is very small, we can assume that the population has a base growth rate of rmax, we express it in the following equation,

dNdT= rmax(K-N)KN