Population Models, Limit Cycles, and Chaos

Two species sharing an environment obey a pair of coupled nonlinear rate equations with several critical points. The phase plane and the linearization and Liapunov tools answer whether the species coexist, which one survives, and whether the populations settle or cycle, with no closed-form solution. The same methods reach phenomena with no linear analogue: the self-sustaining oscillation of a limit cycle and the chaotic wandering of the Lorenz attractor.

Competing species

Two species that consume a common food supply, without preying on each other, each grow logistically in isolation and each depress the other's growth. Writing $x$ and $y$ for the two populations gives the competition system¹

with all six constants positive: ${ϵ}_i$ are the growth rates, ${\sigma }_i$ measure each species' self-limitation, and ${\alpha }_i$ measure the interference from the competitor. Only nonnegative $x,y$ are meaningful, and because the coordinate axes are themselves trajectories, a state that starts in the first quadrant stays there.

Setting both right-hand sides to zero gives up to four critical points: the origin $(0,0)$, the two single-species states $({ϵ}_1/{\sigma }_1,0)$ and $(0,{ϵ}_2/{\sigma }_2)$, and — when the nullclines cross in the interior — a coexistence point with both populations positive. Because each right-hand side is a quadratic polynomial, the system is locally linear at every critical point, and the Jacobian classifies each one.

Coexistence or exclusion

The interior critical point $(X,Y)$, when it exists, determines the outcome, and its character reduces to a single comparison. Evaluating the Jacobian there and using ${ϵ}_1-{\sigma }_{1}X-{\alpha }_{1}Y=0$ and its counterpart, the eigenvalues come from

The product of the eigenvalues is $({\sigma }_1{\sigma }_2-{\alpha }_1{\alpha }_2)XY$, so its sign determines the type.²

• Weak competition, ${\sigma }_1{\sigma }_2>{\alpha }_1{\alpha }_2$. The eigenvalues are real and negative: the coexistence point is an asymptotically stable node, the three boundary points are unstable, and every trajectory in the first quadrant converges to coexistence. The two species share the environment.
• Strong competition, ${\sigma }_1{\sigma }_2<{\alpha }_1{\alpha }_2$. The eigenvalues are real with opposite signs: the coexistence point is a saddle. The two single-species nodes are asymptotically stable, and the two trajectories entering the saddle form a separatrix dividing the quadrant into two basins. One species drives the other to extinction; which one wins depends on the starting populations.

The biological reading is direct. The $\sigma$'s measure how much each population inhibits itself; the $\alpha$'s measure how much it inhibits its rival. When self-limitation dominates cross-limitation, competition is too weak to exclude, and coexistence is stable. When cross-limitation dominates, competition is strong and the two cannot share — the principle of competitive exclusion.

Predator and prey

Now let one species eat the other. The prey $x$ grows in the predator's absence; the predator $y$ starves in the prey's absence; encounters, proportional to the product $xy$, feed the predator and cost the prey. These assumptions give the Lotka–Volterra equations:⁴

with $a,c,\alpha ,\gamma >0$. There are two critical points: the origin, and the coexistence point $(c/\gamma ,a/\alpha )$. Linearizing at the origin gives eigenvalues $a>0$ and $-c<0$, a saddle — a small population of either species does not simply die out. At the coexistence point the Jacobian is

pure imaginary: a center for the linear system. This is the delicate case the linearization theorem leaves undecided, so we go back to the full system. Dividing the two equations eliminates $t$,

which separates and integrates to the conserved relation

For each $C$ this is a closed curve around $(c/\gamma ,a/\alpha )$, so the coexistence point is a center for the nonlinear system too: predator and prey populations cycle indefinitely.⁵

Near the coexistence point the linear approximation gives sinusoidal oscillations of period $2\pi /\sqrt{ac}$, independent of amplitude, with the predator lagging the prey by a quarter cycle. Averaged over a full cycle the populations equal the equilibrium values $c/\gamma$ and $a/\alpha$.⁷ The mechanism is a feed back loop: abundant prey feed a predator boom, heavy predation crashes the prey, the starved predators decline, and the released prey recover. Cyclic predator–prey data — the lynx and snowshoe hare records of the Hudson's Bay Company, with their nine-to-ten-year period — show exactly this out-of-phase oscillation.

Because a center is structurally fragile, the Lotka–Volterra cycle must be read with caution. The smallest change to the model, such as a self-limiting prey term, can turn the neutral center into a spiral, so real populations rarely trace the perfect closed loops the idealized equations predict.

Limit cycles

The predator–prey orbits are closed curves, but each sits in a nested family, and a perturbation slides the state onto a neighboring loop. A limit cycle is isolated instead: a closed trajectory that nearby trajectories spiral toward.

A limit cycle is a self-sustaining oscillation with a fixed amplitude set by the system itself, not by the initial conditions — unlike the center, whose amplitude is whatever the starting point dictates. The cleanest example is

In polar coordinates this decouples into

The radial equation has $r^{\prime }>0$ for $r<1$ and $r^{\prime }<0$ for $r>1$: the unit circle $r=1$ is an asymptotically stable limit cycle, and every trajectory except the origin spirals onto it.⁸

Existence and nonexistence of closed orbits

Three theorems bracket the question of when closed trajectories can occur, all for $x^{\prime }=F(x,y)$, $y^{\prime }=G(x,y)$ with continuous first partials.⁹

The first two are tests for non-existence: a region with no critical point, or with only a saddle, or across which the divergence keeps its sign, holds no closed orbit. The Poincaré–Bendixson theorem is the positive result — build an annular trapping region with inflow on both boundaries and no equilibrium inside, and a limit cycle must live there. This is a phenomenon peculiar to the plane; it has no analogue in three dimensions, where chaos first becomes possible.

The van der Pol oscillator

The archetype of a system with a limit cycle is the van der Pol equation, governing a triode oscillator's current $u$:¹⁰

Its damping coefficient $-\mu (1-u^2)$ is the mechanism. For large $u$ it is positive and drains energy, shrinking large oscillations; for small $u$ it is negative and pumps energy, growing small ones. Between the two lies a single oscillation of intermediate size that all others approach — an asymptotically stable limit cycle. Written as a system with $x=u$, $y=u^{\prime }$,

The origin is the only critical point: an unstable spiral for $0<\mu <2$, an unstable node for $\mu \ge 2$, so every trajectory is pushed away from it, while far out the trajectories turn inward — the Poincaré–Bendixson picture. As $\mu$ grows the limit cycle deforms from a near-circle into a sharply cornered relaxation oscillation, holding amplitude near $2$ while its period lengthens. When a parameter crossing turns a stable spiral into an unstable one and spawns a surrounding limit cycle, the transition is a Hopf bifurcation.

Chaos: the Lorenz equations

Everything so far is two-dimensional, and in the plane the trajectories' inability to cross forces order: a trapped trajectory must approach an equilibrium or a closed orbit, and nothing wilder is possible. In three dimensions that constraint lifts, and a deterministic system can behave unpredictably. The canonical example is the Lorenz equations, a drastic reduction of atmospheric convection:¹¹

with $\sigma =10$ and $b=8/3$ standard, and $r$ (proportional to the imposed temperature difference) the control parameter. Here $x$ measures the intensity of convective motion and $y,z$ the temperature variations. The system has one critical point at the origin for $r\le 1$; as $r$ passes $1$ the origin loses stability and two new critical points $P_1,P_2$ appear at $(\pm \sqrt{b(r-1)},\pm \sqrt{b(r-1)},r-1)$, representing steady convection rolls turning one way or the other.

The eigenvalues at $P_1$ and $P_2$ stay in the stable half-plane until a second threshold. For $\sigma =10$, $b=8/3$, that threshold is $r_2=470/19\approx 24.74$: beyond it, $P_1$ and $P_2$ each acquire a pair of complex eigenvalues with positive real part, and all three critical points are unstable.¹² Although all three critical points are unstable, every solution stays bounded, drawn toward a set of zero volume with an intricate layered structure: a strange attractor.

The defining feature is sensitive dependence on initial conditions. Two solutions starting a hundredth of a unit apart, at $(5,5,5)$ and $(5.01,5,5)$, track together until about $t=10$, then diverge and bear no resemblance thereafter. The Lorenz equations are deterministic — the future is fixed by the present — yet arbitrarily small uncertainty in the present grows until long-range prediction is impossible.¹³ This was the observation that led Lorenz to conclude that detailed long-range weather forecasts cannot be made, and it gave the field its name.

The projection above only hints at the structure; the true trajectory lives in three dimensions and never crosses itself, the apparent crossings being artifacts of the flattened view. In the plane the qualitative methods force a trapped trajectory toward an equilibrium or a closed orbit; in one more dimension the same equations produce behavior that is deterministic and bounded yet permanently unpredictable.