# Simple Lotka-Volterra Slope Field in Phase Space

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.

• A possible reason is when you graph the slope field with only $x,y$ value, it does not contain the information of $t$. If you look at the arrows, their directions are correct except they all point to the positive $x$ values. That means in some sense it is treating $x$ as the time, so you cannot get a cycle as desired. – KittyL Jul 16 '15 at 9:47

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.