How can you determine isoclines using slope fields?

[ssb]

I can find many websites that show a slope field, an answer, and the isoclines but for the life of me I cannot figure out the relationship between slope fields and isoclines!

I need a nudge in the right direction please!

 

[AiRAVATA]

When you have a system of ode's, your solution will be a parametrized curve in the plane (space), i.e. $\{x(t),y(t)\}$. If you derivate such curve, you obtain a vector tangent to such curve given by $T=\{\dot{x}(t),\dot{y}(t)\}$, where the dot denotes derivations with respect to time. From your calc & geometry classes, you should remember that the slope of the tangent vector is given by (using the chain rule):

$$m=\frac{d y/dt}{dx/dt}=\frac{dy}{dx}$$.

And there you go. If you have a given system

$$\begin{array}{l} \dot{x}(t)=f(x,y,t) \\ \dot{y}(t)=g(x,y,t)\end{array}$$

then the isoclines will be the curves where the slope

$$m=\frac{g(x,y,t)}{f(x,y,t)}$$

remains constant. Of particular importance are the nullclines ($m=0$ and $m=\infty$). (why?)