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.
Some of the most important problems we face involve populations -- the human populations in our cities, our countries, and our planet; and the populations of the various species of plants and animals with whom we share our world.
The populations of different species are described by different kinds of models. Some species -- for example, temperate zone insects -- have very distinct generations. Typically, the life cycle for each generation begins in the spring when the eggs hatch and it ends in the fall when the survivors lay the eggs that will become the next generation the following spring.
Because the generations are distinct, this situation is usually described by a sequence of numbers -- p(1), p(2), ... , p(n), ... -- with p(n) denoting the population of the n-th generation. For example, if we were studying the population of a species whose initial population was 10,000 and that was growing at a rate of 2% per year, we would have
p(1) = 10,000 p(n) = p(n - 1) + 0.02 p(n - 1) = 1.02 p(n - 1)
Any model, like this one, in which the population increases or decreases by a fixed percentage each year is called an exponential model and is described by a change equation of the form
p(n) = R p(n - 1)
where R is a constant. In the example above R is 1.02.
If the population is increasing then R > 1 and if it is decreasing then R < 1.
p(1) = 1,000 p(n) = 1.05 * p(n - 1) p(1) = 1,000 p(n) = 1.01 * p(n - 1) p(1) = 1,000 p(n) = 0.95 * p(n - 1) p(1) = 1,000 p(n) = 0.99 * p(n - 1)Describe your results. answer
Exponential models are not very useful. If R < 1 then the population will "die out" -- that is, it will decrease until it is very close to zero. If R > 1 then the population will increase without any bound. This is unrealistic since any real habitat will have only a finite supply of food, water, and other resources. Therefore, any model that predicts the population will increase without any bound is unrealistic. Finally, if R = 1 then the population will remain constant. Such a model, however, is in a very delicate balance. If R changes by the slightest amount then the model will predict declining population or increasing population without bound.
p(1) = 1,000 p(n) = 1.05 * p(n - 1) p(1) = 1,000 p(n) = 1.01 * p(n - 1) p(1) = 1,000 p(n) = 0.95 * p(n - 1) p(1) = 1,000 p(n) = 0.99 * p(n - 1)Suppose that these models are proposed as possible models for a species in a habitat that can support 5,000 individuals. Suppose that a species becomes "endangered" if its population falls below 50. How long will it take for each model before one of the following happens.
Any model in which the generations are distinct and thus can be described by a sequence of numbers is called a discrete dynamical system. Discrete dynamical systems appear deceptively simple. We will see examples whose behavior is surprizingly complex.
P(1) = 1000
P(n+1) = 1.05 P(n)
P(1) = 100
P(n+1) = 1.10 P(n)
P(1) = 500
P(n+1) = 0.90 P(n)
P(1) = 1000
P(2) = 1050
P(1) = 500
P(n+1) = 450