Lotka-Volterra Equations
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 4.6: “Predator-Prey Systems” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/4-6-predator-prey-systems |
| Textbook used in class | Stewart, Calculus, Section 9.6: “Predator-Prey Systems” (Examples 1-2) |
Quick Reference
Lotka-Volterra system (prey $x$, predator $y$, all constants positive): $$\frac{dx}{dt} = ax - bxy, \qquad \frac{dy}{dt} = -cy + dxy.$$
- $ax$: prey grows exponentially without predators
- $bxy$: prey is eaten (rate proportional to encounters)
- $-cy$: predators die without prey
- $dxy$: predators grow from eating prey
Equilibria: $(x,y) = (0,0)$ and $(x,y) = (c/d,\; a/b)$.
Slope of trajectories in the phase plane: $$\frac{dy}{dx} = \frac{-cy + dxy}{ax - bxy} = \frac{y(-c + dx)}{x(a - by)}.$$
Motivation
A single population with unlimited resources grows exponentially. A single population with limited resources grows logistically. When two species interact -- one eating the other -- neither model applies, and the two populations cycle together over time. The Lotka-Volterra equations are the simplest mathematical model of this predator-prey cycle.
Their structure is intuitive: prey increase naturally ($ax$) but decrease through encounters with predators ($bxy$); predators die naturally ($-cy$) but increase through the same encounters ($dxy$). The encounter rate $xy$ appears in both equations, coupling them.
Key Concept: Understanding the System
Each term in the equations has a biological meaning:
| Term | Equation | Meaning |
|---|---|---|
| $ax$ | prey | prey reproduce (proportional to prey count) |
| $-bxy$ | prey | prey killed by predators (proportional to encounters) |
| $-cy$ | predator | predators die without food |
| $dxy$ | predator | predators reproduce from eating prey |
When predators are absent ($y = 0$): prey grows exponentially ($\dot{x} = ax$). When prey are absent ($x = 0$): predators decay exponentially ($\dot{y} = -cy$).
The nontrivial equilibrium $(c/d, a/b)$ is found by setting both equations to zero: $$ax - bxy = x(a - by) = 0 \implies y = a/b \text{ (for } x > 0).$$ $$-cy + dxy = y(-c + dx) = 0 \implies x = c/d \text{ (for } y > 0).$$
Worked Example
Suppose rabbits ($x$, thousands) and foxes ($y$, hundreds) satisfy $\dot{x} = 0.08x - 0.001xy$ and $\dot{y} = -0.02y + 0.00002xy$. Find the nontrivial equilibrium.
Identify: $a = 0.08$, $b = 0.001$, $c = 0.02$, $d = 0.00002$.
Nontrivial equilibrium: $$x^* = \frac{c}{d} = \frac{0.02}{0.00002} = 1000 \text{ (thousands of rabbits)}, \qquad y^* = \frac{a}{b} = \frac{0.08}{0.001} = 80 \text{ (hundreds of foxes)}.$$
At the equilibrium, both populations remain constant. Near it, they cycle: rabbit and fox populations oscillate together, with peaks and valleys out of phase.
Reading the slope: at a point like $(x, y) = (2000, 80)$ (double the equilibrium rabbits, equilibrium foxes): $$\frac{dy}{dx} = \frac{80(-0.02 + 0.00002\cdot 2000)}{2000(0.08 - 0.001\cdot 80)} = \frac{80(0.02)}{2000(0)} = \text{undefined (tangent is vertical).}$$
The vertical tangent at $x = c/d$ means the predator population is at an extremum when prey equals the equilibrium value -- the cycles’ peaks and valleys of $y$ occur when $x = c/d$.
the Lotka-Volterra model predicts that if predators increase, prey immediately decrease. Both populations change continuously, with a phase lag. When prey are abundant, predators grow -- but by the time predators peak, prey have already started declining. Then predators decline (food becomes scarce), and prey recover. The cycle continues. This lag is built into the differential equations: the predator’s peak follows the prey’s peak, and the prey’s recovery follows the predator’s decline. You cannot read the populations at one moment and conclude what happens next without solving the full system.
Leveled Practice
Problem 1. For the system $\dot{x} = 3x - xy$ and $\dot{y} = -y + xy$, find both equilibria.
Show answer
Set $\dot{x} = 0$: $x(3 - y) = 0 \Rightarrow x = 0$ or $y = 3$.
Set $\dot{y} = 0$: $y(-1 + x) = 0 \Rightarrow y = 0$ or $x = 1$.
Equilibria: $(0, 0)$ and $(x, y) = (1, 3)$.