1. ## Competition system

$\displaystyle \frac{dx}{dt}=a_1x-b_1x^2-c_1xy=x(a_1-b_1x-c_1y)$

$\displaystyle \frac{dy}{dt}=a_2y-b_2y^2-c_2xy=y(a_2-b_2y-c_2x)$

$\displaystyle a_1, a_2, b_1, b_2, c_1, c_2$ are positive constants.

Consider the nonzero critical point $\displaystyle (x_E, y_E)$, which is the solution of the system $\displaystyle a_1-b_1x-c_1y=0$ and $\displaystyle a_2-b_2x-c_2y=0$.

The stability of $\displaystyle (x_E, y_E)$ depends on whether $\displaystyle c_1c_2<b_1b_2$ or $\displaystyle c_1c_2>b_1b_2$. Could someone please direct me to a proof of this or guide me through it if it is not too computationally tedious?

2. ## Re: Competition system

Originally Posted by alexmahone
$\displaystyle \frac{dx}{dt}=a_1x-b_1x^2-c_1xy=x(a_1-b_1x-c_1y)$

$\displaystyle \frac{dy}{dt}=a_2y-b_2y^2-c_2xy=y(a_2-b_2y-c_2x)$

$\displaystyle a_1, a_2, b_1, b_2, c_1, c_2$ are positive constants.

Consider the nonzero critical point $\displaystyle (x_E, y_E)$, which is the solution of the system $\displaystyle a_1-b_1x-c_1y=0$ and $\displaystyle a_2-b_2x-c_2y=0$.

The stability of $\displaystyle (x_E, y_E)$ depends on whether $\displaystyle c_1c_2<b_1b_2$ or $\displaystyle c_1c_2>b_1b_2$. Could someone please direct me to a proof of this or guide me through it if it is not too computationally tedious?
Consider a series expansion about the critical point.

CB

3. ## Re: Competition system

Originally Posted by CaptainBlack
Consider a series expansion about the critical point.

CB
But there is no series in my question!

4. ## Re: Competition system

Originally Posted by alexmahone
But there is no series in my question!
Let $\displaystyle (x_0,y_0)$ be the critical point. Put $\displaystyle (x,y)=(x_0+\varepsilon, y_0+\delta)$. Now consider the solution to the system as a power series in $\displaystyle \varepsilon$ and $\displaystyle \delta$ (and truncate after the first non-zero (lowest order) non-constant terms)

Then if: $\displaystyle \partial_t\varepsilon \ne 0$ and $\displaystyle \partial_t\delta \ne 0$ you have a stable critical point iff:

$\displaystyle \partial_t\varepsilon < 0$

and

$\displaystyle \partial_t\delta < 0$

CB