Steady State

ol98

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Anyone able to work out this steady state question?
 

Subhotosh Khan

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ol98

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What is the characteristic of "steady-state"?

Please show us what you have tried and exactly where you are stuck.​
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sorry, I wasn't aware of the rules. So it's the wording of "as a function of E" that I'm not entirely sure with. I know for a steady state you want to set du/dt to 0. So my working out is as follows:

0=u*(1-u*)(1+u*)-Eu*
Eu*=u*(1-u*)(1+u*)
E=(1-u*)(1+u*)
Giving u*=1-E or u*=E-1

Any feedback on this would be great, thanks
 

Subhotosh Khan

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sorry, I wasn't aware of the rules. So it's the wording of "as a function of E" that I'm not entirely sure with. I know for a steady state you want to set du/dt to 0. So my working out is as follows:

0=u*(1-u*)(1+u*)-Eu*
Eu*=u*(1-u*)(1+u*)
E=(1-u*)(1+u*)
Giving u*=1-E or u*=E-1 .............................................How are you getting these?!

Any feedback on this would be great, thanks
No!

You have:

E=(1-u*)(1+u*)

This is a quadratic equation in u* \(\displaystyle \ \ \to \ \ \) solve for u* accordingly.

However, if you "play" with the equation - you could solve it in another (more intuitive) way.
 
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ol98

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u*=0, u*= sqrt(1-E), u*=-sqrt(1-E) ?
 

Subhotosh Khan

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u*=0, u*= sqrt(1-E), u*=-sqrt(1-E) ?
Correct - however

Why does your question use the term "biologically relevant"?

What is the significance of that term?
 

ol98

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that was what I was going to go onto next. The differential is referring to an animal population. If the steady state is 0 is this indicating that there is no change in the system population, and the negative square root term is indicating a decline in the population?
 

yoscar04

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that was what I was going to go onto next. The differential is referring to an animal population. If the steady state is 0 is this indicating that there is no change in the system population, and the negative square root term is indicating a decline in the population?
No, do you remember what u denotes? can a population be negative?
 

ol98

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No, do you remember what u denotes? can a population be negative?
u is denoting an animal population. So the steady state that is -sqrt(1-E) is biologically irrelevant as a population cannot be negative. The steady state 0 indicates the population of the animal species is extinct, and the positive sqrt steady state indicates an increase population?
 

yoscar04

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u is denoting an animal population. So the steady state that is -sqrt(1-E) is biologically irrelevant as a population cannot be negative. The steady state 0 indicates the population of the animal species is extinct, and the positive sqrt steady state indicates an increase population?
It sounds good to me besides the last part. Drop the "increase", this is just a steady state solution (positive constant value which is biological relevant).
 

ol98

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It sounds good to me besides the last part. Drop the "increase", this is just a steady state solution (positive constant value which is biological relevant).
great, thanks a lot
 

ol98

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It sounds good to me besides the last part. Drop the "increase", this is just a steady state solution (positive constant value which is biological relevant).
to determine the stability, would you look at a phase line plot?
 

yoscar04

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to determine the stability, would you look at a phase line plot?
You need to write the equation for a perturbation solution based on the steady state that you just found: u(t)=uss+upert. At the end you should get something like dup/dt=constant\(\displaystyle \cdot\)up. The stability will be determined by the sign of the constant. What textbook are you using? I recommend you Lee Segel and Leah Edelstein-Keshet: A primer on mathematical models in Biology (2013). I used an older version of this book many years ago in my class.
 
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