How i can solve limit this ODE question .

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Jumanah Al omran
Jumanah Al omran on 1 May 2022
Commented: Sam Chak on 8 May 2022
The population P(x) of a species satisfies the logistic differential equation.the initial population P(0)= 3000 and t is the in years. How I can find the lim x -> infinity P(t) ? this is the equation
  4 Comments
Sam Chak
Sam Chak on 7 May 2022
Based on your description, P(x) is the population as a function of x (the years...).
And dx/dt?
By the way, I plotted this to help you visualize the rate dx/dt. You see, x grows initially but the growth starts declining after x > 500K. It is still growing but getting slower and slower until dx/dt comes to a halt when it reaches 0.
x = linspace(0, 1e6, 100001);
dx = x/100 - x.^2/1e8;
plot(x, dx)

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Answers (3)

Bjorn Gustavsson
Bjorn Gustavsson on 7 May 2022
This is a Riccati equation. Convert it to a standard linear second-order differential equation and solve it that way, check what transformation to use from: Riccati-equation
Solve the ODE by separation of variables.
At what level will the source and loss-terms on the RHS be equal?
HTH

Torsten
Torsten on 7 May 2022
I think you mean
lim (t->oo) P(t)
instead of
lim (x->oo) P(t)
don't you ?
Seems the limit is not 0, but 1e6:

Sam Chak
Sam Chak on 8 May 2022
Looks like the population saturates at 1 million.
t = linspace(0, 1500);
P = (3000000*exp(t/100))./(997 + 3*exp(t/100));
plot(t, P)
  2 Comments
Sam Chak
Sam Chak on 8 May 2022
Thanks @Bjorn Gustavsson. I forgot to factor the quadratic function.

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