[MOVED] Given this Lotka-Volterra model system, show that

SophieToft

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Oct 3, 2006
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I need help intepreting the following:

Given Lotka-Volterra model system:

. . .\(\displaystyle \begin{array}{cc} x'_1\, =\, (a\,-\,bx_2)x_1 \\ x'_2 \,=\, (cx_1\, -\,d) x_2\end{array}\)

Look at the system on the open 1.Quadrant Q; where a, b, c, and d are all positive constants.

Show that the system is integratable, which implies that there exist a \(\displaystyle C^1\)-function \(\displaystyle F:V \rightarrow \mathbb{R}\) where \(\displaystyle V \subseteq Q\) is open, and close in Q.
According to my professor, "close" implies that for every point in Q, there exist a sequence of socalled "limitpoints", who's elements belongs to Q. Also as a consequence of "close" \(\displaystyle \nabla F \neq 0\) for all \(\displaystyle x \in V\), and F is constant on all trajectories of the system.

What is my first step here? Do I prove that there exist a solution for the system only in Q?

Sincerley Yours
Sophie Toft
 
Hi

I have looked that page many times. I know howto find fixpoint. But what troubles is proving existence of \(\displaystyle C^1\)-function. Any idears? Cause I have run of the road with this one.

Sincerely Yours and God Bless

Sophie Toft
 
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