You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
p(n) = R * p(n - 1)
In which R < 1 predicts that the population will decline and eventually die out. Actually this is one place where the mathematics must be interpreted using common sense. The mathematics taken literally does not predict that the population will ever reach zero but rather that it will get smaller and smaller. Consider, for example, the following model
p(1) = 100
p(n) = 0.5 * p(n - 1)
With this model
p(10) = 0.1953125
p(20) = 0.00019073
p(30) = 0.00000019
p(40) = 0.000000000182
No matter how large n is p(n) will never reach zero. In practice, however, a real world population will die out completely because eventually these models predict there will be less than one whole individual. This general observation is sometimes obscured by the fact that populations are measured using many different units. Sometimes p(n) is the number of individuals, but sometimes it is the number of thousands of individuals, or the number of millions of individuals, and sometimes population is measured in terms of biomass rather than numbers. Whatever the units used, however, an exponential model with R < 1 eventually predicts less than one individual and, thus, it eventually predicts the disappearance of the species.
For this reason exponential models with R < 1 are not particularly useful. We don't see many species whose population growth can be described by such models because these species do die out.
There is, however, one situation in which we do see a variation on this theme. Consider a small inhospitable habitat, perhaps an island, that is located near a more hospitable habitat.
In this situation if the inhospitable habitat were isolated, the population might very well be described by an exponential model with R < 1 and it would eventually die out. However, when the inhospitable habitat is near a hospitable habitat then there is often some migration from the hospitable habitat into the inhospitable one. This situation can be modeled by a change equation of the form
p(n) = R * p(n - 1) + M
and is called an exponential model with immigration.
Because the inhospitable model is inhospitable R < 1 and if there were no immigration the population would die out. The constant M represents the immigration into the inhospitable habitat each year. For example, if we were measuring population in numbers of individuals and the population left by itself would decrease by 20% each year and 1000 individuals would immigrate each year we would have the change equation.
p(n) = 0.80 * p(n - 1) + 1000
With an initial population of 2000 this model predicts
p(1) = 2000
p(2) = 2600
p(3) = 3080
Twenty generations of this model are shown in the graph below.
IMPORTANT: Even though we often talk about population in terms of numbers of individuals we usually do allow fractions and decimals. There are several reasons for this besides mathematical convenience. For example, fractions and decimals make sense when we are measuring biomass or when the units are healthy individuals.
p(1), p(2), ... p(20)
p(n) = 0.90 * p(n - 1) + 1000
p(n) = 0.90 * p(n - 1) + 500
p(n) = 0.90 * p(n - 1) + 4000
p(n) = 0.90 * p(n - 1) + 1000
p(n) = 0.80 * p(n - 1) + 1000
p(n) = 0.70 * p(n - 1) + 1000
p(n) = 0.70 * p(n - 1) + 1000
p(n) = 0.80 * p(n - 1) + 1000