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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md

@@ -88,4 +88,13 @@ For $\bf J$:
 - $\Delta = \pm\omega^2$
 Then:
 - $\theta$ is 0, $\Delta = \omega^2 >0$, spiral. Stable
-- $\theta = n \pi$, $\Delta = - \omega^2 <0$, saddle. Unstable
+- $\theta = n \pi$, $\Delta = - \omega^2 <0$, saddle. Unstable
+---
+# Competing Species Problems
+We have a Species X vs. Species Y.
+$$\dot x = x(\beta-\delta x -\gamma y) = \beta x - \delta x^2 - \gamma xy$$
+$$\dot y = y(b - d y - c x) = by - dy^2 -cxy$$
+>[!note] Coupling Terms
+>$\gamma x y$ and $c x y$ are coupling terms.
+>These equations are coupled because of these. Without them x and y would just be doing their own thing.
+