From d79d93374b1959fa34b30d7c3db9f490c66317bd Mon Sep 17 00:00:00 2001
From: Worker Agent <worker@diff-eq-handbook.local>
Date: Thu, 4 Jun 2026 18:49:10 -0500
Subject: [PATCH] fix(ch14): correct Lotka-Volterra phase portrait coordinates

---
 chapters/ch14_nonlinear_systems.tex | 38 ++++++++++++++---------------
 1 file changed, 19 insertions(+), 19 deletions(-)

diff --git a/chapters/ch14_nonlinear_systems.tex b/chapters/ch14_nonlinear_systems.tex
index 60ff2d4..028c044 100644
--- a/chapters/ch14_nonlinear_systems.tex
+++ b/chapters/ch14_nonlinear_systems.tex
@@ -777,42 +777,42 @@ The resulting flow is counterclockwise around the equilibrium.
   \draw[->, thick] (0,-0.3) -- (0,5) node[above] {$y$ (predator)};
 
   % Nullclines
-  \draw[dashed, thick, blue!60] (1,0) -- (1,4.5) node[above, font=\scriptsize, blue!60] {$x=1/3$};
-  \draw[dashed, thick, red!60] (0,1.875) -- (4.8,1.875) node[right, font=\scriptsize, red!60] {$y=1.5$};
+  \draw[dashed, thick, blue!60] (0.333,0) -- (0.333,4.5) node[above, font=\scriptsize, blue!60] {$x=1/3$};
+  \draw[dashed, thick, red!60] (0,1.5) -- (4.8,1.5) node[right, font=\scriptsize, red!60] {$y=1.5$};
 
   % Closed orbits (ellipses centered at (1/3, 1.5))
   % Inner orbit
-  \draw[thick, purple!80] (1,1.875) ellipse (0.6 and 0.6);
+  \draw[thick, purple!80] (0.333,1.5) ellipse (0.6 and 0.6);
   % Middle orbit
-  \draw[thick, purple!80] (1,1.875) ellipse (1.2 and 1.2);
+  \draw[thick, purple!80] (0.333,1.5) ellipse (1.2 and 1.2);
   % Outer orbit
-  \draw[thick, purple!80] (1,1.875) ellipse (1.9 and 1.8);
+  \draw[thick, purple!80] (0.333,1.5) ellipse (1.9 and 1.8);
 
   % Arrows on orbits (counterclockwise)
   % Inner orbit arrows
-  \draw[thick, purple!80, ->] (1.6,1.875) -- (1.65,2.05);
-  \draw[thick, purple!80, ->] (1,2.475) -- (0.83,2.45);
-  \draw[thick, purple!80, ->] (0.4,1.875) -- (0.4,1.65);
-  \draw[thick, purple!80, ->] (1,1.275) -- (1.17,1.295);
+  \draw[thick, purple!80, ->] (0.933,1.5) -- (0.983,1.68);
+  \draw[thick, purple!80, ->] (0.333,2.1) -- (0.163,2.08);
+  \draw[thick, purple!80, ->] (-0.267,1.5) -- (-0.267,1.28);
+  \draw[thick, purple!80, ->] (0.333,0.9) -- (0.503,0.92);
 
   % Middle orbit arrows
-  \draw[thick, purple!80, ->] (2.2,1.875) -- (2.28,2.12);
-  \draw[thick, purple!80, ->] (1,3.075) -- (0.76,3.05);
-  \draw[thick, purple!80, ->] (-0.2,1.875) -- (-0.2,1.55);
-  \draw[thick, purple!80, ->] (1,0.675) -- (1.24,0.70);
+  \draw[thick, purple!80, ->] (1.533,1.5) -- (1.613,1.75);
+  \draw[thick, purple!80, ->] (0.333,2.7) -- (0.093,2.68);
+  \draw[thick, purple!80, ->] (-0.867,1.5) -- (-0.867,1.18);
+  \draw[thick, purple!80, ->] (0.333,0.3) -- (0.573,0.32);
 
   % Equilibrium points
   \filldraw[black] (0,0) circle (2.5pt) node[below left=1pt] {$(0,0)$};
-  \filldraw[black] (1,1.875) circle (2.5pt) node[below right=1pt] {$(\gamma/\delta,\,\alpha/\beta)$};
+  \filldraw[black] (0.333,1.5) circle (2.5pt) node[below right=1pt] {$(\gamma/\delta,\,\alpha/\beta)$};
 
   % Region labels
-  \node[font=\scriptsize, align=center, blue!60] at (2.5,2.8) {Region I\\($x' > 0,\; y' > 0$)};
-  \node[font=\scriptsize, align=center, blue!60] at (0.4,0.9) {Region II\\($x' > 0,\; y' < 0$)};
-  \node[font=\scriptsize, align=center, blue!60] at (-0.5,2.5) {Region III\\($x' < 0,\; y' < 0$)};
-  \node[font=\scriptsize, align=center, blue!60] at (2.5,3.5) {Region IV\\($x' < 0,\; y' > 0$)};
+  \node[font=\scriptsize, align=center, blue!60] at (2.5,2.5) {Region I\\($x' > 0,\; y' > 0$)};
+  \node[font=\scriptsize, align=center, blue!60] at (2.5,0.5) {Region II\\($x' > 0,\; y' < 0$)};
+  \node[font=\scriptsize, align=center, blue!60] at (-0.7,2.5) {Region III\\($x' < 0,\; y' < 0$)};
+  \node[font=\scriptsize, align=center, blue!60] at (-0.7,0.5) {Region IV\\($x' < 0,\; y' > 0$)};
 
   % Legend
-  \node[font=\footnotesize, align=left, anchor=west] at (2.8,4.2) {
+  \node[font=\footnotesize, align=left, anchor=west] at (1.2,4.2) {
     \textcolor{blue!60}{\rule{6pt}{1pt}} $x$-nullcline \\
     \textcolor{red!60}{\rule{6pt}{1pt}} $y$-nullcline \\
     \textcolor{purple!80}{\rule{6pt}{1pt}} Closed orbits