logistic equation: biologists stocking a lake with 400 fish

[jsaxman]

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7200. The number of fish doubled in the first year.

a) Assuming that the size of the fish population satisfies the logistic equation:
dP/dt=kP(1-P/K)

determine the constant k, and then solve the equation to find an expression for the size of the population after t years.

b)How long will it take for the population to increase to 3600?

 

[Deleted member 4993]

Re: logistic equation

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 7200. The number of fish doubled in the first year.

a) Assuming that the size of the fish population satisfies the logistic equation:

dP/dt=kP(1-P/K) <<< You have "k" and "K" - are those same or different?

determine the constant k, and then solve the equation to find an expression for the size of the population after t years.

b)How long will it take for the population to increase to 3600?

Please show us your work and exactly where you are stuck - so that we know where to begin to help you.

 

[jsaxman]

Re: logistic equation

ok, big K is the carrying capacity little k is constant i believe so far i got:
dP/(P(1-P/K))=kdt
integral of (1/P)dP+(1/K) * integral of (1/(1-(P/K)))dP=kt+c this is where I'm stuck....

-jsaxman

 

[Deleted member 4993]

Re: logistic equation

ok, big K is the carrying capacity little k is constant i believe so far i got:
dP/(P(1-P/K))=kdt
integral of (1/P)dP+(1/K) * integral of (1/(1-(P/K)))dP=kt+c this is where I'm stuck....

-jsaxman

Use

\(\displaystyle \int \frac{dx}{x} \, = \, ln(x) \, + \, C\)

 

[jsaxman]

Re: logistic equation

Use

\(\displaystyle \int \frac{dx}{x} \, = \, ln(x) \, + \, C\)[/quote]

How would i incorporate that equation to what i have??
jsaxman

 

[Deleted member 4993]

Re: logistic equation

Use

\(\displaystyle \int \frac{dx}{x} \, = \, ln(x) \, + \, C\)

How would i incorporate that equation to what i have??
jsaxman[/quote]

You have:

integral of (1/P)dP

and

(1/K) * integral of (1/(1-(P/K)))dP

You don't see the similarities between these and dx/x ???

 

[jsaxman]

im just a bit confused where to go from there; how would i go about setting it up to solve what it is asking?
jsaxman

 

[Deleted member 4993]

im just a bit confused where to go from there; how would i go about setting it up to solve what it is asking?
jsaxman

First tell us what did you get (out of those integrals) using my hint.

look at:

http://calclab.math.tamu.edu/~belmonte/ ... c9/L91.pdf

for some worked out examples.

 

[jsaxman]

i even attempted the problem on matlab and i get
y=dsolve('Dy=K*y*(7200-y)',' y(0)=400', 't');

i get [(7200)/(1+17exp(-7200*k)]-800

when i solve for k i get k=.000105; when i enter that value for k in the homework online says its wrong
please helpp
jsaxman

 

[Deleted member 4993]

i even attempted the problem on matlab and i get
y=dsolve('Dy=K*y*(7200-y)',' y(0)=400', 't');

i get [(7200)/(1+17exp(-7200*k)]-800 <<< Something wrong - where did 't' go

when i solve for k i get k=.000105; when i enter that value for k in the homework online says its wrong
please helpp
jsaxman

 

[jsaxman]

well i preceded to the next step accidentally:
i get (7200)/(1+17exp(-k*t))

SOLVED!!!!!!!!!!