I'm trying to plot the slope field in phase space of a simple (all constants set equal to $1$) Lotka-Volterra system described by the following differential equations:
$$\frac{dw}{dt} = w-wr$$ $$\frac{dr}{dt} = -r+wr$$
Where r represents the population of predators and w represents the population of prey.
In order to plot the slope field in phase space, I believe I need $\frac{dw}{dr}$, so I get the following:
$$\frac{dw}{dr} = \frac{\frac{dw}{dt}}{\frac{dr}{dt}} = \frac{w-wr}{-r+wr} = -\frac{w}{r} \frac{1-r}{1-w}$$
Which I've plotted using Grapher to get the following slope field and solution where $y(1)=2$: Grapher Lotka-Volterra Plot
This can be compared to the following plot from Wikipedia: Wikipedia Lotka-Volterra Plot
(Unfortunately I don't yet have enough reputation to post images directly.)
My understanding is that I should be getting getting a slope field that gives orbitals such as those shown in the plot from Wikipedia, but in my slope field the directionality abruptly changes along the line $y=1$.
Please let me know if you see what I'm doing incorrectly.
Finding $\frac{dw}{dr}$ is useful when you want to find a (usually implicit) relation that defines the trajectories. But for plotting slope fields, you should just input the equations as stated i.e. $$ \Delta \begin{bmatrix} y\\x \end{bmatrix} = \begin{bmatrix} y-yx\\ -x+yx \end{bmatrix} $$ assuming $y=w, x=r$. In your input you said that $x'=1$, which forces trajectories to always be increasing in the $x$ direction, which means you could never have an orbit.