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Steady States

mt91

New member
Jul 30, 2020
15
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Any help would be appreciated
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
464
what do you need help with? Do you know what "steady state" means? Do you know what stability is? An interesting, non-mathematical, phrase here is "biological relevance". What do you think that means? There is the parameter, a, that is undefined. You might want to consider the cases a<1, a= 1, and a> 1 separately.

Do you know what a "separable differential equation" is? This equation is "separable"- it can be written $\frac{du}{u(1- u)(a+ u)}= dt$. The left side can be integrated using "partial fractions".
 
Last edited:

mt91

New member
Jul 30, 2020
15
what do you need help with? Do you know what "steady state" means? Do you know what stability is? An interesting, non-mathematical, phrase here is "biological relevance". What do you think that means? There is the parameter, a, that is undefined. You might want to consider the cases a<1, a= 1, and a> 1 separately.

Do you know what a "separable differential equation" is? This equation is "separable"- it can be written $\frac{du}{u(1- u)(a+ u)}= dt$. The right side can be integrated using "partial fractions".
I got 3 steady states, as u=0, u=1 and u=-a?

and then by plotting u(1-u)(a+u), indicated from the graph that u=-a is stable, u=0 is unstable and u=1 is stable. Not sure what you mean by looking at the cases for a, so are you ok to explain that?
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
464
That was because I had misread the problem and was looking at u= -1, 0, and -a!

Yes, the equilibrium points are at u= -a, 0, and 1. We can write u'= (u+ a)u(1-u)= (-1)(u+ a)u(u- 1). if u< -a then all four of those are negative so the product, and so u', is positive. u is increasing up to u(-a). If -a< u< 0 then -1, u, and u-1 are negative but u+ a is positive so the product, and so u', is negative. u goes down from u(-a) to u(0). If 0< u< 1, both u+ a and u are positive while -1 and u are negative so the product, and so u', is positive. u goes up to u(1). Finally, for u> 1, all except -1 are negative so u' is negative. u goes down from u(1).

That is, u goes up to u(-a) then down after that so u(-a) is a local maximum. u goes down to u(0) then up so u(0) is a local minimum. u goes up to u(1) then down so u(1) is a local maximum.