Understanding the One-Dimensional Map Used by Ecologists

[sapiental]

Homework Statement

1) Determine the qualitative properties of the One-Dimensional maps:

x_n+1 = f(x_n) = x_n * e^(r * (1 - x_n))

This function has been used by ecologists to study a population that is limited at high densities by the effect of epidemics. Although it is more complicated than the map we have been using, its advantage is that the population remains positive no matter what positive value is taken for the initial population.

2) There are no restrictions on the maximum value of r, but if r becomes sufficiently large, x eventually becomes effectively zero. What is the behavior of the time series of this function for r = 1.5, 2, 2.7. Does f(x) have a maximum?

Homework Equations

x_n+1 = f(x_n) = x_n * e^(r * (1 - x_n))

The Attempt at a Solution

1) I don't really understand the question.. Are there any websites about this equation that goes into more detail about each part of the equation? Thanks

2) If you increase the parameter r to a higher value, the time series will increase as well. I wouldn't say that f(x) has a maximum, rather a equilibrium value that it reaches.

 

[Mindscrape]

Oh, I was really confused looking at the equation at first. I take it that the equation looks like this:

$$x_{n+1}=f(x_n)=x_n e^{r(1-x_n)}$$

1. Is there a question? Maybe it wants you to take the limits as r approaches infinity and zero, and as x_n approaches infinity and zero.

2. This is wrong, if r is sufficiently large x_n is not zero. You have to look at two different cases. If x_n is greater than 1 then you will have a decreasing function, which I think represents that your population has crossed some kind of epidemic threshold and is dying; if x_n equals 1 then your population is at some constant; and if x_n is less than 1 then your population isn't sick enough to die out and continues to grow. r doesn't really mean anything other than how fast your population will grow or die.

Having a max or min will depend on what I said above. I'll let you think about the implications of what I said and how that translates to maxes and mins.

As far as the equation, it looks like a logistics equation. Typically it can be derived through differential equations, this one is particularly easy because it is a first order, linear, constant coefficient diff eq. You can look it up if you really want, but you would have to take a detour to learn how to solve first order DEs (not hard, but ultimately will take time that is probably not worth one problem).

 

[tim_lou]

first order DE? I don't think this is a differential equation at all... rather, it is a recursion equation...unless if there is some way that you can transform this equation into a differential equation.