MHB How Do Steady State Equations Predict Animal Population Dynamics?

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Anyone able to calculate the steady states?
 
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Do you not know what a "steady state" IS? A steady state is a constant solution to the differential equation. And if it is constant, $\frac{du*}{dt}= 0$
So $\frac{du*}{dt}= 0= u*(1- u*)(1+ u*)- Eu*$.

Solve that third degree equation for u*.
 
Country Boy said:
Do you not know what a "steady state" IS? A steady state is a constant solution to the differential equation. And if it is constant, $\frac{du*}{dt}= 0$
So $\frac{du*}{dt}= 0= u*(1- u*)(1+ u*)- Eu*$.

Solve that third degree equation for u*.
Cheers, it was the wording of the question I struggled with, because it said as a function of E. So this is what I did:

\[ Eu*=u*(1-u*)(1+u*) \]
\[ E=(1-u*)(1+u*) \]
\[ ∴u*=1-E, u*=E-1 \]

I think that u*=0 would be a solution, but I'm not sure with the wording of "function of E" what it wants me to do, cheers
 
Then you should have said, in your first post, that your difficulty was with solving the cubic equation, not with the "steady state" of the differential equation! I am concerned that, in a problem involving differential equations, your difficulty is with solving a quadratic equation! In fact, your difficulty is with solving an equation of the form "$x^2= a$".

Yes, Eu*= u*(1- u*)(1+ u*). u*= 0 is a solution. If u* is not 0 we can divide by u* to get E= (1- u*)(1+ u*).

But that does NOT give u*= 1- E or u*= 1+ E as solutions! You would have seen that if you had checked: If u*= 1- E then 1- u*= 1-1+ E= E and 1+ E= 2- E. $(1- u*)(1+ u*)= E(2- E)= 2E- E^2$.

From $E= (1- u*)(1+ u*)= 1- u*^2$ we get $u*^2= 1- E$ so that $u*= \pm\sqrt{1- E}$.
 
Country Boy said:
Then you should have said, in your first post, that your difficulty was with solving the cubic equation, not with the "steady state" of the differential equation! I am concerned that, in a problem involving differential equations, your difficulty is with solving a quadratic equation! In fact, your difficulty is with solving an equation of the form "$x^2= a$".

Yes, Eu*= u*(1- u*)(1+ u*). u*= 0 is a solution. If u* is not 0 we can divide by u* to get E= (1- u*)(1+ u*).

But that does NOT give u*= 1- E or u*= 1+ E as solutions! You would have seen that if you had checked: If u*= 1- E then 1- u*= 1-1+ E= E and 1+ E= 2- E. $(1- u*)(1+ u*)= E(2- E)= 2E- E^2$.

From $E= (1- u*)(1+ u*)= 1- u*^2$ we get $u*^2= 1- E$ so that $u*= \pm\sqrt{1- E}$.

Meaning therefore we have 3 steady state solutions?
\[ u*=0 \]
\[ u*=sqrt(1-E) \]
\[ u*=-sqrt(1-E) \]

Thanks for your help by the way
 
Yes.
 
Country Boy said:
Yes.
Thanks, do you know much about how you would interpret relevant steady states? So the steady state equation was about an animal population. If a steady state is 0 is this indicating that there's no population? and the negative square root steady state is indicating a decline in the population?
 
mt91 said:
Thanks, do you know much about how you would interpret relevant steady states? So the steady state equation was about an animal population. If a steady state is 0 is this indicating that there's no population?
The population is u(t). If for any t, u(t)= 0, there is no population so, of course, it cannot decrease and is not going to increase so stays 0. 0 is always a "steady state" for a population. (Unless you allow "spontaneous generation"!)

and the negative square root steady state is indicating a decline in the population?
No, the whole point of "steady state" is that it is steady, it is NOT changing, neither decreasing nor increasing. A negative value here is a mathematically a solution to the equation but is not a realistic solution- the number of animals cannot be negative.
 

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