Qualitative Analysis of Systems

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Textbook Reference

Quick Reference

Nullclines for $\dot{x} = f(x,y)$, $\dot{y} = g(x,y)$:

• $x$-nullcline: $f(x,y) = 0$ (where $x$ stops changing; $\dot{x} = 0$)
• $y$-nullcline: $g(x,y) = 0$ (where $y$ stops changing; $\dot{y} = 0$)

Equilibria lie at intersections of $x$-nullclines and $y$-nullclines.

Between nullclines, determine the sign of $\dot{x}$ and $\dot{y}$ in each region to sketch phase-plane arrows.

Lotka-Volterra cycles: trajectories near the nontrivial equilibrium $(c/d, a/b)$ are closed curves (periodic orbits). The predator peak lags the prey peak by one-quarter cycle.

Motivation

For a two-variable system, the phase line generalizes to the phase plane: each point $(x, y)$ in the plane represents a state of the system, and the solution $t \mapsto (x(t), y(t))$ is a trajectory through that plane. Instead of arrows on a number line, you draw arrows at points in the plane, each arrow pointing in the direction $(\dot{x}, \dot{y})$.

Nullclines divide the plane into regions where $x$ is increasing or decreasing and $y$ is increasing or decreasing. Combining these signs gives the qualitative direction of motion in each region without solving the equations. This technique reveals whether the system oscillates, spirals inward, or spirals outward -- the major behaviors of two-variable systems.

Key Concept: Nullclines and Phase-Plane Arrows

For the Lotka-Volterra system $\dot{x} = ax - bxy$, $\dot{y} = -cy + dxy$:

$x$-nullclines: $x(a - by) = 0 \Rightarrow x = 0$ (vertical axis) or $y = a/b$ (horizontal line).

$y$-nullclines: $y(-c + dx) = 0 \Rightarrow y = 0$ (horizontal axis) or $x = c/d$ (vertical line).

In the biologically relevant region $x > 0$, $y > 0$, the nullclines $y = a/b$ and $x = c/d$ divide the first quadrant into four sectors:

Actually, let me state this correctly:

• $y < a/b$: $a - by > 0$, so $\dot{x} > 0$ (prey increasing)
• $y > a/b$: $a - by < 0$, so $\dot{x} < 0$ (prey decreasing)
• $x < c/d$: $-c + dx < 0$, so $\dot{y} < 0$ (predators decreasing)
• $x > c/d$: $-c + dx > 0$, so $\dot{y} > 0$ (predators increasing)

In each sector, both $\dot{x}$ and $\dot{y}$ have definite signs, giving the direction of the trajectory arrow. The pattern produces a counterclockwise circulation around the nontrivial equilibrium.

Worked Example

For $\dot{x} = 2x - xy$ and $\dot{y} = -y + xy$ (so $a=2, b=1, c=1, d=1$), sketch the nullclines and describe the phase-plane flow in each sector.

$x$-nullclines: $x = 0$ and $y = 2$. $y$-nullclines: $y = 0$ and $x = 1$.

Nontrivial equilibrium: $(1, 2)$.

Sectors of the first quadrant (excluding axes and nullclines):

• $0 < x < 1$, $0 < y < 2$: $\dot{x} = x(2-y) > 0$, $\dot{y} = y(-1+x) < 0$. Arrows point right and down.
• $x > 1$, $0 < y < 2$: $\dot{x} > 0$, $\dot{y} > 0$. Arrows point right and up.
• $x > 1$, $y > 2$: $\dot{x} < 0$, $\dot{y} > 0$. Arrows point left and up.
• $0 < x < 1$, $y > 2$: $\dot{x} < 0$, $\dot{y} < 0$. Arrows point left and down.

Following the arrows counterclockwise: a trajectory starting in sector 1 (low prey, few predators) moves right and down into sector 2 (more prey, more predators), then left and up into sector 3 (prey declining, predators high), then left and down back toward sector 4. The cycle repeats -- a closed orbit around $(1, 2)$.

Leveled Practice

Problem 1. For $\dot{x} = x(4 - 2y)$ and $\dot{y} = y(-3 + x)$, find the nullclines and the nontrivial equilibrium.

Mastery Checklist

• Find the $x$-nullclines and $y$-nullclines for a two-variable system
• Determine the sign of $\dot{x}$ and $\dot{y}$ in each sector between nullclines
• Sketch the direction of phase-plane arrows in each sector and predict whether solutions oscillate