Question: Lotka-Volterra system

1. Oct 4, 2006

mathboy20

Hi

I need help intepreting the following.

Given Lotka-Volterra model system

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

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

Show that the system is integratable, which supposedly implies that there exist a $$C^1$$-function $$F:U \rightarrow \mathbb{R}$$ where $$U \subseteq K$$ is open, and close in K.

According to my professor "close" implies that for every point in K, there exist a sequence of socalled "limitpoints", who's elements belongs to K. Also as a consequence of "close" $$\nabla F \neq 0$$ for all $$x \in U$$, 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 K?

Sincerley Yours
Mathboy20

Last edited: Oct 4, 2006
2. Oct 19, 2006

lapin

Volterra

To begin with you need to find a function of two variables (x_1, x_2) which is a constant of motion. i.e. its time derivative is zero. Then you worry where it is defined.

3. Oct 19, 2006

HallsofIvy

Staff Emeritus
You might also want to look up the definition of "integrable" since that is what you are asked to show!