Lecture notes. Written from the material of MIT 8.591J 2014, licensed CC BY-NC-SA 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.
23. Predator-Prey Cycles and Their Fragility¶
What this covers¶
This chapter asks why the standard textbook model of a predator and its prey — the Lotka–Volterra equations — is, in a mathematical biologist's own words, "a bad model but profoundly important," and works through what that means in practice: which of its assumptions are doing the work, why its oscillations are a fragile borderline case rather than a robust prediction, and what happens once that fragility is repaired in two different directions. It then follows two places where a real system's departure from the simple model turned out to be the interesting result rather than an embarrassment: a laboratory rotifer–algae system whose cycles had the wrong period and the wrong phase, and a demographic-noise effect that manufactures sustained oscillations out of a model that, deterministically, has none. It assumes the reader can linearize a nonlinear system at a fixed point and read stability off the Jacobian's eigenvalues, has met the chemical master equation as an individual-based alternative to a differential equation, and knows the Poincaré–Bendixson theorem by name.
The Lotka–Volterra model and its four assumptions¶
The history, as told in lecture: in the mid-1920s the marine biologist Humberto D'Ancona had spent about a decade (roughly 1912–1923) recording the composition of fish sold at markets across Italy. He noticed that the fraction of selachians — sharks and shark-like, broadly predatory fish — rose sharply during World War I and fell again afterward, and wanted a mechanistic explanation rather than an appeal to "the general zeitgeist." He happened to be engaged to Luisa Volterra, daughter of the mathematician Vito Volterra, and asked his future father-in-law about it. Volterra wrote down a model and analyzed D'Ancona's data with it. Independently, Alfred Lotka had studied essentially the same pair of equations some fifteen years earlier, first in the context of autocatalytic chemical reactions and only later for population dynamics — hence the two names attached to one model.
Writing \(x\) for the prey and \(y\) for the predator, the model is
with \(a,b,c,d>0\). The lecture's own way of getting at what this equation actually says is to name the assumptions that had to go in to write it down, rather than treat it as given:
- The interaction term is \(xy\). This is a mass-action assumption on the encounter rate between predator and prey, and it is worth keeping separate from "the population is well-mixed." A population can be spatially resolved — tracked as densities that vary with position, as in a reaction–diffusion or Turing-pattern model — and still have its local interaction rate written as \(x(\mathbf{r})y(\mathbf{r})\) at each point. Conversely, being well-mixed does not by itself force the interaction term to be a product of the two densities. What is really being assumed here is that predator and prey encounter each other, and something happens, at a rate proportional to the product of their numbers or densities — nothing about saturation, and nothing about whether space is being tracked explicitly.
- The prey grows exponentially in the predator's absence (\(\dot x = ax\) when \(y=0\)). The rate \(a\) is a net rate — births minus deaths — not a birth rate alone, so it is not quite right to say "the prey doesn't die without the predator"; only that its net growth rate is assumed positive.
- The predator dies exponentially in the prey's absence (\(\dot y = -cy\) when \(x=0\)), at rate \(c\).
- Predation is one predator eating one prey at a time. There is no group-hunting effect built in — nothing here would capture how a pack of wolves brings down a buffalo. This is bundled into the same \(xy\) term as assumption 1, but it is a separate claim: assumption 1 is about how often predator and prey meet, this one is about what a single meeting does.
Fixed points, and why "neutral" is a trap¶
The model has two fixed points. The origin is one, since \(x\) appears in every term of the top equation and \(y\) in every term of the bottom one. Along the \(y\)-axis (\(x=0\)) the dynamics reduce to \(\dot y = -cy\), decaying to the origin — stable in that direction, since the predator dies out without prey. Along the \(x\)-axis (\(y=0\)) the dynamics reduce to \(\dot x = ax\), growing without bound — unstable in that direction, since the prey grows freely without a predator. So the origin is a saddle, with eigenvectors along the two axes.
The second, interior fixed point comes from solving \(a-by=0\) and \(-c+dx=0\) simultaneously:
Linearizing about \((x^*,y^*)\) gives the Jacobian
whose eigenvalues are \(\lambda = \pm i\sqrt{ac}\) — purely imaginary. Near the fixed point this looks like a center: trajectories trace closed loops rather than spiraling in or out, with an approximate period \(2\pi/\sqrt{ac}\). It is worth pausing on that period: it depends only on \(a\) and \(c\), the two intrinsic growth/death rates, and not at all on \(b\) or \(d\), the two interaction rates — which is not obvious just from staring at the equations.
But purely imaginary eigenvalues are a borderline case, and the lecture was explicit that this matters: a linear analysis that returns a center at a nonlinear system's fixed point does not by itself prove that the full nonlinear system actually has neutrally stable closed orbits away from the fixed point. The nonlinear terms that the linearization threw away could just as easily turn the picture into a stable spiral (orbits decaying to the fixed point) or a genuine limit cycle (orbits converging onto one particular closed curve). For the Lotka–Volterra system specifically it happens to be true that the orbits are neutrally stable — there is an exact quantity conserved along trajectories, with a different value on each orbit, which is what pins every orbit in place rather than letting nearby orbits drift into one another. That conservation law is left as an exercise rather than derived in lecture. The orbits run counterclockwise around \((x^*,y^*)\), there are infinitely many of them nested inside one another, and — a fact stated as something the reader should be able to show, not proved in lecture — the time average of \(x(t)\) and of \(y(t)\) over one circuit of any of these orbits comes out to exactly \(x^*\) and \(y^*\). That is what makes \(x^*,y^*\) worth reasoning about even though the system, generically, never actually sits at that point.
The paradox of hunting the predator, and D'Ancona's fish¶
Suppose a wildlife manager wants to help a struggling prey population by shooting the predator — raising the predator's death rate, \(c\). In this model that changes \(x^* = c/d\) upward (more prey, as intended) but leaves \(y^* = a/b\) completely unchanged: the time-averaged predator population is untouched by how hard you hunt it. And the effect on the prey is not permanent — stop hunting and \(c\) returns to its original value, and the prey population comes back down. A parallel case raised in class: to reduce a predator population at equilibrium (framed, for fun, as thinning out zombies) the model says you want to decrease \(a\) — the predator's supply of new individuals via \(y^*=a/b\) — not increase the death rate \(c\), which does nothing to \(y^*\) at all. That inversion of the naive intuition was offered with an explicit caution: it is worth checking whether a model's assumptions are any good before turning it into public policy.
That fixed-point algebra is also Volterra's own explanation of D'Ancona's data. To first order, the war's effect was not biological but economic: fewer fishermen went out. Less fishing removes fewer prey fish, raising their net growth rate — \(a\) goes up. Less fishing also removes fewer predator fish, lowering their net death rate — \(c\) goes down. By the formulas above, \(y^*=a/b\) then rises (more predatory fish) while \(x^*=c/d\) falls (fewer prey fish) — exactly the shift D'Ancona saw in the market data during the war. The class also raised, fairly, that this is not a controlled experiment: an equally plausible story is that fishermen simply avoided the deeper waters where predatory fish live, and confidence in the mechanistic story would really need many independent "wars" to average over.
Small changes, different endings¶
Because the neutral orbits sit on a nonhyperbolic borderline, almost any modification to the model's assumptions can tip the outcome toward decay or toward a genuine, self-correcting cycle. Two modifications from lecture:
Logistic prey growth. Replace \(ax\) with \(ax(1-x/K)\), leaving everything else the same:
This looks like a modest, purely local repair of assumption 2 above (real prey populations saturate, they don't grow to infinity), and for large \(K\) the dynamics near the fixed point are barely changed. But its effect on the long-run picture is not modest at all: the neutral orbits are destroyed, and every trajectory now spirals into \((x^*,y^*)\) — damped oscillations rather than sustained ones. The decay timescale grows as \(K\) grows, and in the limit \(K\to\infty\) that timescale diverges, recovering the original neutrally stable orbits as a degenerate limiting case.
Saturating predation. Replace the linear \(xy\) predation term with one that saturates in \(x\) (a Michaelis–Menten-type functional response, so a predator's rate of eating doesn't grow without limit as prey become abundant). With this change the interior fixed point becomes unstable, while trajectories far from it are still driven inward — with no food at all, neither population can sustain itself, so a large enough region of the \((x,y)\) plane has trajectories entering but none leaving. The Poincaré–Bendixson theorem then applies: a bounded trapping region containing no fixed point other than a single unstable one must contain at least one periodic orbit. That guarantees a genuine limit cycle — a single closed orbit with its own characteristic amplitude and period, attracting trajectories from both inside and outside.
A more general statement, named but not derived in lecture: writing a predator–prey pair through its per-capita growth rates, \(\dot x = x f(x,y)\) and \(\dot y = y g(x,y)\), Kolmogorov gave a set of four conditions on the partial derivatives of \(f\) and \(g\) with respect to \(x\) and \(y\) — holding at least in certain limits — under which a stable limit cycle is guaranteed. The lecture pointed to the assigned reading for the precise statement rather than stating it.
One question from the room was left open rather than answered: could the Poincaré–Bendixson argument above be evaded by semi-stable limit cycles — orbits attracting from one side and repelling from the other — nested so that trajectories are still trapped without any single orbit being fully stable? The lecturer agreed the picture could apparently be drawn, noted that semi-stable orbits are not robust to any amount of noise anyway (a real, noisy system only ever "sees" the fully-stable cycles), but did not resolve on the spot whether this specific construction is actually consistent with the usual statement of the theorem, and said he would check.
Rapid evolution and long, out-of-phase cycles: the Yoshida experiments¶
The lecture turned next to a laboratory system where the simple model's failure was the interesting result: Yoshida et al., "Rapid Evolution Drives Ecological Dynamics in a Predator-Prey System," Nature (2003), studying a rotifer feeding on algae in a chemostat.
An earlier paper from the same group had varied the chemostat's dilution rate and watched the system cross a Hopf bifurcation: at low dilution, a stable coexistence fixed point; at intermediate dilution, sustained oscillations; at high dilution, collapse of the predator–prey pair. That matched a simple model reasonably well, with the dilution rate essentially setting \(c\) (the rotifer barely dies on its own, so its death rate is set by being washed out of the chemostat) and \(a\) set by the algal growth rate at that dilution minus the dilution rate itself.
Two features of the oscillations that this simple picture did not explain:
- Much longer period than predicted — five to ten times longer than expected from the measured rates, which mattered practically, since it meant runs had to last around six months to see even a couple of cycles.
- The wrong phase. Any two-variable differential-equation predator–prey model generically predicts the predator peaking a quarter-cycle (90°) after the prey — that is simply what going around a closed orbit in the \((x,y)\) plane near a spiral or center looks like. Experimentally the lag was often closer to 180°: predator and prey nearly anti-correlated rather than a quarter-cycle apart.
Follow-up modeling found that letting the prey population itself be heterogeneous — more than one algal type present at once, differing in traits — reproduced both anomalies. The lecture was careful about the word "evolution" here: what is meant is not new mutants arising and spreading over the course of the experiment, which is what an experimental microbial-evolution researcher might expect the word to imply, but a shift in the relative frequencies of prey variants that may already be present — which is evolution by the textbook definition (a change in allele frequency over time) even without new mutation on the timescale of a run. Ecologist and evolutionary biologist frame the same mechanism differently — "ecological dynamics between fixed types" versus "evolution of a variable population" — and which label is reached for is, in the lecturer's words, largely a matter of taste.
Direct confirmation came from a controlled version of the experiment: starting from an isogenic (clonal) prey population rather than a natural mixture made both anomalies disappear — the period returned to what the measured rates predicted, and the phase lag returned to the generic 90°. That is strong evidence that heterogeneity within the prey population, rather than some overlooked physiological detail, was the cause.
A later paper from the same research program (an Ecology Letters paper from a few years before this lecture) made the underlying trade-off concrete. Two algal types were tracked alongside the rotifer: one divided rapidly but existed as free single cells; the other divided more slowly but grew in clumps that were harder for the rotifer to eat. All three sub-populations — fast singles, slow clumps, and rotifer — were followed simultaneously and shown to oscillate together, giving a direct picture of the growth-versus-defense trade-off that the earlier modeling had only inferred.
Demographic noise can manufacture oscillations¶
The closing topic, picked up in the following problem set, was guided by McKane and Newman, Physical Review Letters (2005). Start from the model with logistic prey growth from earlier in the chapter — deterministically, this has no sustained oscillations, only a spiral decaying to the fixed point. Ask what happens once predators and prey are treated as discrete individuals, born, dying and eating each other at random times, rather than as smooth continuous densities.
There are two different ways to introduce "noise" into this picture, with very different consequences:
- Add a noise term directly onto the differential equations — for example a term proportional to \(x\) tacked onto the \(\dot x\) equation and one proportional to \(y\) onto the \(\dot y\) equation. This produces only a noisy version of the same spiral: a jittery path in toward the fixed point.
- Start instead from an individual-based description — a master equation, in the same style used earlier in the course for chemical reaction networks, in which birth, death and predation events happen stochastically to discrete individuals. Solved this way, the same underlying deterministic skeleton (a damped spiral) produces sustained, surprisingly large-amplitude oscillations that do not decay away.
The mechanism: demographic noise excites the system at every frequency, but the linearized dynamics near the fixed point has one preferred frequency, set by the (now complex, decaying) eigenvalues of the spiral. That frequency is selectively, resonantly amplified — what would otherwise be a spiral collapsing onto the fixed point instead gets continually kicked back outward at its own natural frequency, producing an oscillation that, over any short stretch, looks much like a genuine limit cycle even though nothing in the deterministic equations calls for one.
Because the noise is demographic, its size relative to the mean population falls off like \(1/\sqrt{N}\) for a system of order \(N\) individuals, so the effect is relatively weaker at larger population sizes. But the lecture stressed that the absolute effect can still be surprisingly large even for populations of order a thousand individuals — which is what the following problem set's simulations were built to demonstrate. In passing, the lecture noted that the same demographic-noise mechanism has since been used to produce noise-induced pattern formation in space, without giving further detail.
Sources¶
Transcript-only lecture: recording wtesorg5h-a (8.591J, Systems Biology, MIT OpenCourseWare),
timestamps [00:00]–[1:20:15]. No slides, notes or exercises were supplied with this lecture, and
nothing was written on a captured board — the phase portraits, the D'Ancona time-series sketch, and
the Jacobian matrix reproduced above are reconstructed from the spoken description and the algebra,
not from the original drawings.
- History, the four assumptions, the equations, fixed points, linearization and eigenvalues, hunting-the-predator, the zombie aside, and the World War I explanation: [03:21]–[49:30].
- Logistic prey growth, saturating predation, the Poincaré–Bendixson argument, the class's semi-stable-limit-cycle question, and Kolmogorov's conditions: [50:38]–[1:00:18].
- The Yoshida rotifer–algae experiments: [1:00:18]–[1:15:52].
- Noise-induced oscillations (McKane and Newman): [1:15:52]–[1:20:15].
Referred to in the lecture but not contained in this recording, and not supplied alongside it:
- Mark Kot, Elements of Theoretical Ecology — apparent source of the "bad model but profoundly important" framing, and presumably of the historical D'Ancona/Volterra/Lotka account as told.
- Yoshida, Jones, Ellner, Fussmann and Hairston, "Rapid Evolution Drives Ecological Dynamics in a Predator-Prey System," Nature (2003) — discussed at length, but its figures and full model are not shown here.
- The same group's earlier paper crossing a Hopf bifurcation in a chemostat predator–prey system, referred to only informally in the recording, without a full citation.
- A later Ecology Letters paper from the same research program tracking two algal morphotypes alongside the rotifer — exact citation not given in the recording.
- Kolmogorov's four conditions for a stable limit cycle in a general predator–prey system — named, not stated, with the assigned reading pointed to for the precise statement.
- McKane and Newman, Physical Review Letters (2005), on noise-induced oscillations in predator–prey systems, including the simulation plots the lecturer displayed but which are not captured in a transcript.
- The following problem set built on the McKane–Newman model, referred to repeatedly but not supplied with this lecture.
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