Question: Lotka-Volterra system

[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

 

[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.

 

[HallsofIvy]

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