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.
3. Lotka-Volterra Dynamics and Population Waves¶
What this covers¶
This chapter finishes the two-species Lotka–Volterra competition story that the previous lecture set up, then asks two questions that go beyond it: what happens with more than two competing species, and what can non-transitive ("rock-paper-scissors") competition do for the maintenance of diversity, in a well-mixed population and in a spatially structured one. It closes with a different kind of question — not who wins a competition, but how fast a population's edge moves outward when local growth is coupled to spatial spreading. It assumes the two-species Lotka–Volterra competition equations and the practice of classifying fixed points by stability and eigen-directions, both introduced in the previous lecture, together with the idea of a nullcline.
Two-species competition: the four outcomes¶
The model carried over from last time is
Competition means each species' presence lowers the other's growth, i.e. \(\beta_{12},\beta_{21}>0\): \(\beta_{12}\) is how much a unit of species 2 reduces species 1's growth, and \(\beta_{21}\) the reverse. Depending on the relative sizes of \(K_1\), \(K_2\), \(K_1/\beta_{12}\) and \(K_2/\beta_{21}\), there are exactly four possible long-run outcomes: species 1 always wins, species 2 always wins, stable coexistence, or bistability (the only case in which the outcome depends on where you start, provided you start with some of each species — obviously a species that starts at zero density never appears).
The same four outcomes show up when analysing frequency-dependent selection between two strategies within a single asexual population, in the sense used in Martin Nowak's Evolutionary Dynamics. That is not a coincidence of notation: it is a real structural link between an ecological question (different species competing) and an evolutionary one (allele or genotype frequencies changing inside one species), even though the two settings model different biological processes.
Rule of thumb for which outcome you get: coexistence needs the competition to be weak — small \(\beta\)'s — relative to the ratio of carrying capacities. If \(K_1=K_2\), this reduces to simply asking whether each \(\beta\) is above or below 1, i.e. whether a member of the other species harms you more or less than a member of your own species would.
Reading a phase portrait from its nullclines¶
The nullcline \(\dot N_1=0\) is the line \(N_1+\beta_{12}N_2=K_1\): it meets the \(N_1\)-axis at \(K_1\) and the \(N_2\)-axis at \(K_1/\beta_{12}\). The nullcline \(\dot N_2=0\) is \(N_2+\beta_{21}N_1=K_2\), meeting the axes at \(K_2\) and \(K_2/\beta_{21}\) respectively. The whole point of drawing them is a simple rule: on the \(\dot N_1=0\) line, \(N_1\) is momentarily not changing, so the local flow can only be vertical (only \(N_2\) moves); on the \(\dot N_2=0\) line the flow can only be horizontal. Along the axes themselves the system reduces to ordinary single-species logistic growth, so the flow there is easy: \(N_1\) grows toward \(K_1\) along the \(N_1\)-axis, \(N_2\) toward \(K_2\) along the \(N_2\)-axis. Stitching these directions together across the plane — which is exactly what the class did, pointing an arm to vote on "up or down" and then "left or right" at successive points — is enough to read off the whole phase portrait without solving anything explicitly.
Take the case where the two nullclines do not cross inside the positive quadrant because \(K_1>K_2/\beta_{21}\) and \(K_1/\beta_{12}>K_2\) (equivalently \(\beta_{21}>K_2/K_1\) and \(\beta_{12}<K_1/K_2\): species 1 harms species 2 a lot, and species 2 does not harm species 1 enough to matter). There are exactly three fixed points: the origin, \((K_1,0)\), and \((0,K_2)\).
- The origin is an unstable node in both directions, with eigenvectors along the two axes. That is just the statement \(r_1,r_2>0\): each species can grow from near-zero density on its own.
- \((K_1,0)\) is stable. One of its eigen-directions is along the \(N_1\)-axis (drop \(N_2=0\) and it is pure logistic growth of species 1); the other is tilted into the interior, and it is the direction along which trajectories starting with a real mixture of both species curve in.
- \((0,K_2)\) is unstable transverse to the axis: species 2 alone sits at its own carrying capacity, but any amount of species 1 present grows and eventually displaces it.
A useful check the class ran on this case: does the outcome change if the growth rates \(r_1,r_2\) change? No — a class vote confirmed it. Which species wins depends only on the \(\beta\)'s and \(K\)'s; the \(r\)'s only change the shape and speed of the approach, not the destination. The example given for why this matters is Strogatz's sheep-and-rabbits illustration of two competing species: rabbits divide faster than sheep, so starting from a mix of both, the trajectory can rise up toward what looks like a rabbit victory, before curving back and ending with the sheep displacing the rabbits. This is competitive exclusion: two species competing for close to the same resource generically end with only one of them surviving, however dramatic the transient looks along the way.
A digression: does stochasticity change this?¶
A student asked whether stochastic extinction could break the deterministic picture — in particular, whether a species being driven toward very low numbers near the unstable fixed point could go extinct by chance before the deterministic dynamics ever "resolve" the competition. Several points came out of the discussion:
- The Lotka–Volterra description here is continuous and deterministic: it allows fractional individuals and has no built-in randomness at all.
- In the case being analysed, the trajectories keep moving toward larger \(N_1\) once the process has really got going, so stochastic loss of the losing species is most likely to matter right at the start, at low density — not partway through, near the unstable fixed point.
- Whether a stochastic simulation of this same equation can even produce extinction depends entirely on how you split the net growth term \(rN(1-N/K)\) into an explicit birth rate and death rate before feeding it to something like a Gillespie algorithm. If you implement the whole expression as a birth rate and add no separate death process, extinction is impossible by construction — there is nothing that can reduce the population. A more faithful implementation writes the net rate as \(B-D\) for some birth rate \(B\) and death rate \(D\); at the level of the ordinary differential equation the two choices are identical, but they are not identical once you build a Fokker–Planck approximation or an actual stochastic simulation, because the rate of stochastic extinction grows with the sizes of \(B\) and \(D\) individually — bigger birth and death rates mean bigger fluctuations even at a fixed net growth rate.
Many competing species¶
Non-dimensionalising by carrying capacities — \(x_i=N_i/K_i\) — converts the two-species equations into a general \(N\)-species form
where the normalisation is chosen precisely so that each species alone follows simple logistic growth with carrying capacity 1. Everything about the ecology is now carried by the off-diagonal entries of the matrix \(\alpha_{ij}\). Being fluent in moving between a model in its raw parameters and this non-dimensionalised version is a skill in its own right — the kind of thing that showed up repeatedly on the second exam, in questions about how a parameter changes when the strength of some expression process changes.
Restrict to the purely competitive case, \(\alpha_{ij}\ge0\) for all \(i,j\) (some entries may be zero — species need not all interact — but none can help another). Three facts about this system were stated as known results to look up rather than derived in the lecture:
- The unit cube is invariant. If every \(x_i(0)\in[0,1]\), then \(x_i(t)\in[0,1]\) for all later \(t\). Staying non-negative is obvious; staying below 1 (i.e. below your own carrying capacity) is not, since nothing in the setup literally forbids starting or ending up above it — it is a genuine fact about this class of equations, not a triviality.
- The long-run dynamics live on an \((N-1)\)-dimensional surface. Specifying every possible trajectory still needs all \(N\) coordinates, since the starting point can be anywhere in \(N\)-dimensional space — but the steady-state behaviour for given parameters (fixed points, limit cycles, chaotic attractors) is confined to one dimension fewer than the number of species. A student pushed back on this in the \(N=2\) case, since with no oscillation at all the steady states there are just points, which trivially sit on a lower-dimensional set regardless; the claim is meant to bite once oscillatory or chaotic long-run behaviour is possible, and that is where the next two facts come in. (Whether there is some general conserved-quantity argument behind the claim was raised and left open.)
- Consequently, a limit cycle needs two dimensions to close up without crossing itself, so getting one in this framework needs \(N\ge3\) competing species; chaos needs three dimensions to fold without crossing, so it needs \(N\ge4\). Both bounds are known to be tight: limit cycles occur already at \(N=3\) and chaos already at \(N=4\). A further result says that for \(N\ge5\) essentially any dynamical behaviour is possible — including, apparently, a four-dimensional torus attractor, left as something to look into independently.
\(N\ge4\) making chaos possible does not mean every four-species competitive system is chaotic — only that some are. What makes this worth dwelling on is that pairwise Lotka–Volterra competition is about the simplest model one could write down for species interacting, and it is already rich enough to produce genuinely complicated dynamics.
The dimension-counting argument rests on one fact about continuous, deterministic systems: trajectories cannot cross themselves (solutions are unique). A limit cycle is a closed loop that never repeats a point except by returning to its start, and a line cannot do that without crossing itself, so it needs a second dimension to curl into. Chaos additionally needs trajectories to stretch and fold without ever touching, which a flat plane cannot support either — hence a third dimension. This connects to, but should not be over-read from, the Poincaré–Bendixson theorem: an unstable fixed point surrounded by a trapping region containing no other fixed point does guarantee a limit cycle inside that region. The converse is not a theorem — a stable fixed point does not, in general, rule out a limit cycle elsewhere. In the specific two-species predator–prey model discussed earlier in the course, it happens that oscillations disappear exactly when the interior fixed point becomes stable, but that is proved for that model by a separate divergence-type argument, not read off from Poincaré–Bendixson. A related but distinct question — raised by a student — is what separates chaos from a trajectory that merely spirals in toward a limit cycle without ever crossing itself: that distinction is usually made with a Lyapunov exponent, which measures whether a small blob of nearby initial conditions in phase space contracts (as it does approaching a stable limit cycle) or stretches and folds while staying bounded (chaos); this was flagged as belonging properly to a course in nonlinear dynamics rather than developed here.
Non-transitive interactions: rock-paper-scissors¶
Three strategies or species \(A,B,C\) interact non-transitively if \(A\) beats \(B\), \(B\) beats \(C\), and \(C\) beats \(A\) in pairwise competition — "beats" meaning that whichever pair is put together, one drives the other extinct. This is exactly rock-paper-scissors, and it can be encoded in the Lotka–Volterra framework by an appropriate choice of the interaction coefficients.
Non-transitive competition has been proposed as a mechanism that can stabilise the coexistence of several species or strategies through the kind of cyclic — even chaotic — dynamics discussed above. How much of the diversity actually observed in nature this explains is an open question; what is not in doubt is that it is a real, demonstrated phenomenon, and that spatial structure seems to matter a great deal to whether it actually maintains diversity. Two studies were assigned as readings and discussed at length.
Side-blotched lizards: three male mating strategies¶
Sinervo and Lively's 1996 study of side-blotched lizards in the mountains of Merced County, California, identified three genetically determined, heritable male morphs, distinguished by throat colour:
- orange-throat — highly aggressive, defends a very large territory with many females, and fights off rival males;
- blue-throat — less aggressive, with a smaller, more defensible territory;
- yellow-striped "sneaker" — mimics the female's colouring and appearance, holds no territory at all, and instead sneaks into other males' territories to mate.
Pairwise, orange beats blue (a bigger territory simply passes on more genes). Sneakers beat orange: an orange male's territory is too large to defend effectively against sneaking. Blue beats sneakers: a smaller, well-defended blue territory keeps sneakers out. That closes the cycle orange → blue → sneaker → orange. Tracking morph frequencies (and the number of females held per territory) over roughly seven years, from about 1990 to 1996, showed the frequencies cycling around in strategy-frequency space — a rock-paper-scissors dynamic actually observed unfolding in a wild, spatially structured population.
Bacterial colicin warfare¶
Kerr and colleagues (Nature, 2002) studied three strains of E. coli differing only by a mutation or plasmid — a single species playing rock-paper-scissors through genotype rather than through separate species:
- C, colicinogenic — carries a plasmid encoding colicin, a toxin that punches holes in the membranes of other bacteria. Colicin is released only when the producing cell itself lyses (bursts open), so producing it is unambiguously costly to the individual and can only be favoured by a kin- or group-selection argument: helping clone-mates, who share the plasmid and its immunity gene, outcompete rivals, at the cost of the producer's own life.
- R, resistant — resistant to colicin, at some smaller fitness cost.
- S, sensitive — ordinary bacteria, no cost, but no defence either.
Pairwise: R beats C (resistance is cheaper than suicide), S beats R (resistance carries a cost that plain sensitivity avoids), and C beats S (colicin kills sensitive cells) — closing the cycle C → S → R → C. It is worth noting how carefully the original paper phrases this: it says the fitnesses are such that this ordering holds, rather than claiming it always will for any isolate you might pick up — a real reminder to read a paper's hedges rather than over-generalise its claim.
Two environments were compared:
- Well-mixed (test tube). The three strains did not coexist. The sensitive strain died out first — killed fastest once colicin accumulated — leaving a straight two-strain contest between C and R, which R wins, so only the resistant strain survives long-term. (A caution attached to this: there is no reason these particular strain dynamics should be well described by a smooth Lotka–Volterra model at all — if colicin accumulates enough, sensitive cells can be wiped out en masse in a way a continuous model does not naturally capture.)
- Spatially structured (agar plate). Strains were arranged in a grid of patches and propagated for about a week by replica-plating with velvet stamps onto fresh plates each day. All three strains coexisted throughout, in a spatial pattern consistent with each strain chasing the one it beats around the plate — sensitive into resistant territory, resistant into colicinogenic territory, colicinogenic into sensitive territory — though the boundary between S and R specifically could not be seen directly, since the two grow to similar colony density; only the colicinogenic strain was visually distinct. Distinguishing S from R in practice meant scraping cells off and re-plating them to test directly for colicin sensitivity, rather than reading a visible boundary.
The conclusion drawn in lecture: for these three particular strains, non-transitive competition alone is not enough to maintain three-way coexistence in a well-mixed environment, but local dispersal is. That does not show non-transitive competition can never support coexistence without spatial structure — reasonable rock-paper-scissors-type equations exist (of the kind discussed in Nowak's book) that spiral inward to a stable coexistence fixed point in a perfectly well-mixed setting. It only shows it does not happen for this system, under these conditions.
A practical aside: despite roughly 500 citations, this system has attracted very few follow-up experimental papers, in part because it has never become a "domesticated" laboratory model — no fluorescent markers, no clean cloning plasmid. The resistant strain, for instance, is obtained simply by exposing sensitive cells to colicin-laden supernatant and seeing which cells grow, and the underlying mutation (often in a surface receptor colicin uses to enter the cell) can differ between isolates. The lecturer's own group has worked with these strains and found them, in his words, "a little bit messy" — a case where the field could use cleaner, fluorescently labelled strains of the same three-way system. A pointer was also given to computational work by Erwin Frey varying the mobility of individual agents in a spatial rock-paper-scissors model: there is a critical mobility below which spatial structure supports coexistence, and above which the system behaves like the well-mixed case and loses diversity.
Population waves¶
Spatial structure matters broadly in ecology and evolution, and a common, tractable way to model the movement of organisms over length and time scales much larger than any individual step is to treat it as an effective diffusion process. This is justified whenever the distribution of individual step sizes is not long-tailed, so that the central limit theorem applies to the accumulated displacement; step-size distributions that are long-tailed — as has been argued for the spread of disease via modern air travel, where most people stay local but a small, non-decaying fraction fly across the world — behave qualitatively differently and diffusion is the wrong picture for them (Oskar Hallatschek's work on epidemic spread with long-tailed dispersal was given as a pointer for that regime).
The model — due originally to Fisher in the 1920s, who introduced it to describe the spatial spread of a beneficial allele rather than a population, another instance of the same mathematics describing an evolutionary and an ecological process — adds a diffusive term to logistic growth:
where \(n(x,t)\) is now a population density, \(r\) the per-capita growth rate at low density, \(K\) the local carrying capacity, and \(D\) the diffusion coefficient. Starting from a single individual at a point, the population divides up locally to saturation at \(K\), and a front then spreads outward whose shape stays fixed over time in a co-moving frame: \(n(x,t)=f(x-vt)\) for a fixed profile \(f\) and velocity \(v\). That stationarity of shape is what earns it the name "wave."
The wave speed, by dimensional analysis¶
\(r\) has units of one over time, \(D\) has units of length squared over time, and \(v\) must have units of length over time. The only combination that works is \(v\propto\sqrt{rD}\), and Fisher's exact result fixes the constant:
(A question raised in class about whether the derivatives themselves carry hidden units: they do not — \(\partial/\partial t\) and \(\partial^2/\partial x^2\) are pure operators, and all the length and time dimensions in the equation are already accounted for by \(D\).) Both ingredients behave as expected: faster individual growth and greater mobility each speed up the wave, and the velocity is a genuinely population-level property — it is not simply "growth" or simply "motion," but a result of the two coupled together. To leading order, in this deterministic treatment, the velocity does not depend on the carrying capacity \(K\) at all.
One subtlety raised in class: if growth is switched off entirely (\(r\to0\)) at some instant, the population does not stop spreading — pure diffusion still spreads it. This does not contradict \(v=2\sqrt{rD}\to0\), because that formula describes the velocity of a genuine travelling wave with a fixed, self-similar shape; switch off growth and the profile itself starts changing shape, so a formula that assumes a fixed shape no longer applies.
Pulled waves and pushed waves¶
Compare several different growth curves — the per-capita growth rate as a function of density — that agree everywhere except very close to \(n=0\) and very close to \(n=K\). The wave speed depends only on \(r\equiv g(0)\), the growth rate exactly at zero density, regardless of how the curve behaves at higher density. The reason is that the very leading edge of the wave is always at vanishingly low density, growing essentially exponentially at rate \(r\) with a characteristic decay length
and it is this leading edge that "pulls" the rest of the front along at \(v=2\sqrt{rD}\). Waves of this kind are called pulled waves (Fisher waves): the entire wave's velocity and front-decay length are set by the dynamics at the front, not by the bulk behind it.
Now suppose growth has a strong Allee effect: the per-capita growth rate is negative — net death — at low density, becoming positive only once density passes some threshold. A naive argument would say such a population cannot expand at all, since the growth rate right at \(n=0\) is negative and the leading edge should die back rather than grow. Expansion is nonetheless possible: the front itself is dying, but diffusion out of the healthy, growing bulk behind it pushes the population forward. This is a pushed wave — qualitatively different from a pulled wave because it is driven by the dense bulk rather than the sparse front.
The distinction has a genetic consequence, mentioned but not derived here (and pointed to further in an assigned Physics Today reading on loss of heterozygosity during range expansions): pulled waves have a smaller effective population size during expansion than pushed waves, because in a pulled wave the population that matters for the front is the sparse one right at the edge, while in a pushed wave it is the dense bulk.
Sources¶
All content is from the transcript of one lecture in MIT 8.591J Systems Biology (Fall 2014),
recordings/3eizij6qncy.md (course computational-biology/mit-ocw/8591j-2014); administrative
boilerplate at the start was cut. This is a transcript-only recording — whatever was written on the
board (nullcline diagrams, the wave-profile sketch, the dimensional-analysis "cards") is not in the
source and is not shown here; the two figures above are reconstructions built from the verbal
description of what was drawn, not the board itself.
- Two-species competition recap, nullclines, worked "species 1 wins" case, and the stochastic extinction digression: 00:00–24:44. This continues a two-species Lotka–Volterra model set up in the previous lecture, which is referred to here but was not supplied as input.
- General \(N\)-species competition, the invariant cube, the two dimension theorems, and the Poincaré–Bendixson digression: 24:44–37:08.
- Rock-paper-scissors framing and the two case studies: 37:08–1:01:28.
- Population waves, dimensional analysis, and pulled vs. pushed waves: 1:01:28–end (1:20:37).
Referred to in the lecture but not supplied as input, and not reproduced here beyond what was said aloud about them:
- Strogatz's textbook chapter on competing species (the sheep-and-rabbits example), ~17:48.
- Martin Nowak, Evolutionary Dynamics — the frequency-dependent-selection comparison, ~02:22 and ~54:42.
- Sinervo, B. and Lively, C. M. (1996), "The Rock-Paper-Scissors Game and the Evolution of Alternative Male Strategies," Nature — assigned reading, discussed ~41:42–47:35.
- Kerr, B. et al. (2002), "Local Dispersal Promotes Biodiversity in a Real-Life Game of Rock-Paper-Scissors," Nature — assigned reading, discussed ~47:35–1:01:28.
- Erwin Frey's computational study of rock-paper-scissors dynamics versus agent mobility, ~57:09.
- Oskar Hallatschek's work on epidemic spread with long-tailed dispersal kernels, ~1:03:39.
- An assigned Physics Today article on loss of heterozygosity in pulled versus pushed range expansions, ~1:19:32.
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