# Consider the predator / prey model x' = 7 x - x^2 - x y, y' = -5 y + x y. Find all critical...

## Question:

Consider the predator / prey model {eq}\displaystyle x' = 7 x - x^2 - x y,\ y' = -5 y + x y {/eq}.

Find all critical points in order of increasing x-coordinate.

## Find Critical Points:

The critical points of the function {eq}f(x,y) {/eq} are the solutions to the system of equations of the form {eq}\nabla f(x,y) = 0 {/eq}. That is, the solutions to the system of equations of the partial derivatives set equal to {eq}0 {/eq}.

We are given the partial derivatives of the function. To find the critical points, we must set each partial derivative equation equal to {eq}0 {/eq} and solve.

\begin{align*} x' &= 0 \\ 0 &= 7x - x^2 - xy \\ 0 &= x(7 - x - y) \\ x &= 0 \\ y &= 7 - x \\ y' &= 0 \\ 0 &= -5y + xy \\ \text{Let } x &= 0 \\ 0 &= -5y + (0)y \\ 0 &= -5y \\ y &= 0 \\ \text{Let } y &= 7 - x \\ 0 &= -5(7-x) + x(7-x) \\ 0 &= -35 + 12x - x^2 \\ 0 &= x^2 - 12x + 35 \\ 0 &= (x - 7)(x - 5) \\ x &= 7 \\ x &= 5 \end{align*}

The critical points of the function are {eq}(0,0), (5,2), \text{ and } (7,0) {/eq}. 