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fix(ch14): correct Lotka-Volterra phase portrait coordinates

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Worker Agent
2026-06-04 18:49:10 -05:00
parent 42d59b1a1a
commit d79d93374b
1 changed files with 19 additions and 19 deletions
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38
chapters/ch14_nonlinear_systems.tex
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@@ -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