# Thread: logistic model for population

1. ## logistic model for population

Hello. This is a sample question for a test I have coming up on Monday.
It being the weekend, my prof doesn't have office hours at this point.

Consider the logistic model of population dynamics xn+1 = xn + Rxn(1 - xn/K),
where R is the growth rate, K is the carrying capacity, and xn is the population size at time n. Given fixed K, find the values of the growth rate R for which the population becomes extinct.

I would normally try to offer what I think I should do, but I am really clueless, so even a simple suggestion could put me in the right direction. Thanks for any help.

2. would R be between -1 and 0?

if its less than -1, youre getting into negative values
and if you are are 0, it doesnt change
if you are above 0, then it is increasing

by having R between [-1,0]
then you are approaching zero population

is this the right thinking?
if so, is there a better way to prove it rather than using trial examples?

3. Originally Posted by chrisc
Hello. This is a sample question for a test I have coming up on Monday.
It being the weekend, my prof doesn't have office hours at this point.

Consider the logistic model of population dynamics xn+1 = xn + Rxn(1 - xn/K),
where R is the growth rate, K is the carrying capacity, and xn is the population size at time n. Given fixed K, find the values of the growth rate R for which the population becomes extinct.

I would normally try to offer what I think I should do, but I am really clueless, so even a simple suggestion could put me in the right direction. Thanks for any help.
When $x_n$ is small the term $x_n/K$ is negligable and the difference equation becomes:

$x_{n+1}=(1+R)x_n$

Now if $-1 the population is shrinking, but as we have a continuous model of population this never becomes zero, so we have no extinction (in reality the population will drop below $1$ (or $2$ ) and that would be extinction).

If $R\le -1$ the population will go extinct in one step.

CB

4. Originally Posted by CaptainBlack
If $R\le 0$ the population will go extinct in one step.
Do you mean if

$R\le -1$

because like you said, if its in the interval of [-1, 0] is will approach zero but never get there (and if it is 0, it will never change)

Yes

CB