How i can solve limit this ODE question .

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

[Sam Chak]

What is x(t)?

[Jumanah Al omran]

 the years

[Jan]

@Jumanah Al omran: You are funny. The unit does not matter. For ->Inf you get the same result for years or seconds, and even for bananas. :-)

You can determine the equation for x(t) easily. Then it is easy to set x to Inf are determine the result.

A deeper look on the equation for dx/dt helps also. x is growing, -x^2 is shrinking even faster. What dows a curve with a rapidly shrinking derivative do for large values?

[Sam Chak]

Hey @Jumanah Al omran

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)

Answer 1

[Bjorn Gustavsson]

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

Answer 2

[Torsten]

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:

https://www.wolframalpha.com/input?i=y%27%28t%29%3Dy%28t%29%2F100-y%28t%29%5E2%2F1e8%2Cy%280%29%3D3000

Answer 3

[Sam Chak]

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)

[Bjorn Gustavsson]

Yes. This is obvoius if we rewrite the ODE as:

dxdt = x/100.*(1-x/1e6);

First factor will be positive for all positive x, second factor goes to zero as x->1e6.

[Sam Chak]

Thanks @Bjorn Gustavsson. I forgot to factor the quadratic function.