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June 24th, 2014, 06:31 PM  #1 
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  Does the stability of a trivial fixed point change?
Course: SelfStudy Textbook: A Course in Mathematical Biology Question: At what value of a does the stability of the trivial fixed point change? I have this equation $f(x) = axe^{x}$, where $a = e^r$. To find the fixed points, $f(b) = b$, then we have $$\begin{align} abe^{b} &= b \\ abe^{b}  b &= 0 \\ b\left[ae^{b}  1 \right] &= 0 \end{align}$$ Thus, our fixed points are $b_1=0$ (trivial) or $b_2=\ln(a)$ (nontrivial). To determine the stability of the fixed points, we have $$f'(x) = ae^{x}[1x]$$ so that $$\begin{align} f'(b_1) &= f'(0) = a = e^r\\ f'(b_2) &= f'(\ln(a)) = 1  \ln(a) = 1 r \end{align}$$ Since $f'(b_1) = e^r$, $f'(b_1)>1$, thus the trivial fixed point, $b_1$, is always unstable. So the absolute value of $f'(b_1)$ is never less than one. So the stability of the trivial fixed point never changes, right? 
June 25th, 2014, 03:08 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,155 Thanks: 731 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
I'm not 100% sure if you're right, but if you are, it's worth pointing out that your conclusions are based on the assumption that r is positive. Is this the case?

June 25th, 2014, 04:48 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,657 Thanks: 2635 Math Focus: Mainly analysis and algebra 
Yes, I think so. \begin{align*} f^\prime(x) \ge 1 \implies \frac{f(x+h)  f(x)}{h} &\ge 1 &\text{for small $h$} \\ f(x+h)  f(x) &\ge h & h \ge 0 \\ F(x+h)  f(x) &\le h & h \le 0 \\ \end{align*} so a small perturbation in $x$ grows with iterations and $f(x)$ is unstable. I think your result holds for any $r$, but I could be wrong. Last edited by v8archie; June 25th, 2014 at 04:55 AM. 
June 25th, 2014, 10:26 AM  #4 
Member Joined: Jun 2012 From: San Antonio, TX Posts: 84 Thanks: 3 Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems  

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change, fixed, point, stability, trivial 
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