The goal of Population Biology is to mathematically model the way in which animal or plant populations change with time. In the simplest models we use four variables:

N = total population (total number of organisms making up the population)

t = time               b = number of births              d = number of deaths.

In this model any change in our population will depend on the relative values of "b" and "d". Consider your own family. If you only have one sibling then your parents are simply replacing themselves (2 parents, 2 children) and over time (once your parents pass away) the size of your family will basically stay unchanged. If, on the other hand, a couple chooses not to have children, then over time the size of their family will decrease to zero (once they die); while a couple that has three or more children will see their family increase over time (despite their own deaths). We refer to the difference between the birth and death rate as "r", the intrinsic population growth rate, where r = (b – d). So if "r" is positive the population increases, if "r" is negative it decreases, and if "r"=0 it stays the same.

Overall, in order to determine the change in the population with time, we use the equation

dN/dt = rN (equation 1) (NOTE: "d" is basically shorthand for "the change in")

In other words, the amount of change in the total number of organisms in the population (dN) during a given time period (dt = the amount of change in time) will be equal to the intrinsic population growth rate (r) times the original number of organisms (N). Since "r" can be either positive, negative or zero, the population can increase, decrease or remain the same.

GRAPH 1.

If we were to plot the change in population size versus time using equation 1 (assuming a positive "r") we would end up with a graph like the one above, in which the curve quickly becomes exponential and heads toward an infinite population size. Now this obviously cannot happen in the real world because any population of organisms consumes resources (food, water, etc.) which limit the population growth.

Ecologists use the variable "K" (the carrying capacity of the environment) to mark the maximum population size for a species based on the resources available to that species in a particular environment. Note that "K" will vary for any given species depending on the environment and even seasonally within a given environment. Now our equation becomes

 DN/dt = rN((K – N)/K)            (equation 2)

So that when the number of organisms in our population equals the Carrying Capacity, K – N = 0, and we have no change in the size of the population. In actuality, as N gets closer to K the growth rate will slow although there is usually some expected overshooting of K (note that in the graph below, our population curve passes above the "K" line, the drops below, then passes above until it finally settles at "K").

 GRAPH 2

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 An Aside – "r" and "K" strategy among organisms.

Ecologists have used the two variables from our population model ("r" and "K") to define end members in the spectrum of animal behavior. As you might expect, "r"-selected species rely on a high intrinsic population growth rate in order to succeed. These are "weedy" species which disperse quickly to find newly opened environments, have very many offspring (but small – only a very small portion of the parents reproductive resources are devoted to any single juvenile), and are generally poor competitors. "K"-selected species are just the opposite. The are weak dispersers, devote a large amount of their reproductive resources to only one or a few offspring, and are strong competitors. Note in the figure below (Survivorship curves) that Type III species are "r"-strategists which have large numbers of offspring, but with a very low survival rate. Type I species would represent "K"-strategy, in which most individuals survive into "old" age.

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While the value of "r" clearly depends on how close the population size "N" is to the carrying capacity "K", "r" is actually dependent on a number of variables which, for simplicity, do not appear in equation 2 . These "hidden" independent variables can be lumped together into the category of stochastic events.

These are events which deal with probabilities. For instance, when you watch the weather forcast on your local news, you often hear that the chance of rain is 70%. What you are being told is that meteorological records show that in the past, on 7 out of 10 days with weather conditions similar to current conditions, it rained.  So while you have a 70% chance of rain, you also have a 30% chance of no rain.

The most important stochastic category for our population’s "r" values will be Environmental Stochastic Events. Basically, weather trends. Major storms, droughts, 100 year floods, are all stochastic events, as are summers of bright sunny days with gentle rains every night. In other words good years and bad years for organisms are stochastic events. You can also have Demographic Stochastic Events (for instance a year in which only male offspring are born) but these are more important to small populations and will be discussed later. How are these stochastic events incorporated into Graph 2?  Well, they are not.  Each point along the curve represents an average "r" value, but if we wanted to be more accurate, each point would be represented by a bell curve like the one below.

In the curve the average value of "r" is at the peak of the curve, which means that it has the highest probability of occurring; but it does not mean that it has to occur. There are also probabilities that a very low or a very high "r" value can occur during any given time interval.

What does this mean for our population? Consider the randomly generated curves below.

 All these curves are based on our population growth equation (curve #2), but "r" was allowed to vary stochastically. The result is some populations which growth much more rapidly than expected (curve #3), and many which grow less rapidly, or even decline (having suffered from a series of "bad years", such as curve #1).

Finally, we can add one more variable to our equation – "M". "M" simply represents the Minimum Viable Population. If the population declines below this number extinction is unavoidable.

DN/dt = rN((K-N)/K)((N-M)/N).     (equation 3)