More rabbits, well-fed foxes; more foxes, the rabbits are eaten out; fewer rabbits, the foxes starve; fewer foxes, the rabbits recover. The loop never settles at the equilibrium point, because every link in it lags: how many foxes there are now is set by the rabbits they ate earlier. Negative feedback with delay does not produce rest, it produces oscillation.
$x,y$ are prey and predator densities; $\alpha$ is the prey's intrinsic growth rate, $\gamma$ the predator's death rate without food, $\delta$ the efficiency of turning eaten prey into offspring. The term $\beta xy$ is the encounter rate — assume the two mix at random and encounters go as the product of densities, which is exactly chemistry's law of mass action.
The system hides a conserved quantity $V = \delta x - \gamma\ln x + \beta y - \alpha\ln y$, constant along every orbit, acting just like energy in mechanics — so all orbits are closed curves. That neutral stability also means structural instability: add a touch of density limitation on the prey and the closed orbits immediately collapse into spirals. A model too clean to be beautiful is usually a model too idealized to be true.
And the same biological assumption written one generation at a time, $x_{n+1}=rx_n(1-x_n)$, period-doubles into chaos as $r$ grows. Discrete or continuous is not a modelling detail; it decides which class of system you have.
Diffusion is the force that erases differences: ink in water always ends up uniform. In 1952 Turing proved the opposite — when two mutually reacting substances diffuse at the same time and at different speeds, the uniform state spontaneously collapses into spots or stripes.
The mechanism fits in a sentence: the activator promotes itself and also the inhibitor, the inhibitor suppresses the activator, and the inhibitor diffuses much faster. Any random bump is amplified where it stands while broadcasting inhibition far afield, suppressing bumps nearby — local activation plus long-range inhibition, so peaks can only appear a fixed distance apart.
$u,v$ are activator and inhibitor concentrations, $f,g$ the local reactions, $D$ the diffusion coefficients. The operator $\nabla^2$ measures how far a point sits below its neighbourhood average — positive means the point is a "dip" that matter will flow into, which is precisely what diffusion does.
Turing instability demands a seemingly contradictory pair of conditions: the uniform steady state is stable without diffusion, yet once diffusion is switched on some nonzero wavenumber $k$ has a positive growth rate. This requires $D_v \gg D_u$, and the selected $k^*$ corresponds to a wavelength $\lambda \sim \sqrt{D/\text{reaction rate}}$.
Two individually stable mechanisms couple into an instability, and the term responsible for the instability is the one whose whole job is smoothing — hence the name diffusion-driven instability. Lovelier still is where the scale comes from: the wavelength depends only on local parameters, not on the size of the container. It is an intrinsic length carried by the medium rather than something the boundary drew, which is why a leopard's spots and a house cat's are about the same size. Form needs no blueprint, only a set of rules that destabilize on their own.
All you hold are sequences from species alive today (or viruses sampled today), and you must reconstruct the family tree: who split from whom last, and how long ago. The ancestors left no samples, so everything is inferred from how alike the cousins are. There are two difficulties: the number of candidate trees explodes, and similar does not mean related — two independently evolving lineages can drift into resemblance by accident.
There are $(2n-5)!!$ unrooted binary trees on $n$ leaves: at $n=20$ that is already about $2\times 10^{20}$, so enumeration is hopeless.
Maximum likelihood models base substitution as a continuous-time Markov chain: with rate matrix $Q$, the transition probability along a branch of length $t$ is $P(t)=e^{Qt}$, where $t$ is measured in expected substitutions, not years. The likelihood at each site sums over all ancestral states — naively $4^{\text{internal nodes}}$ terms — but Felsenstein's pruning algorithm folds the tree bottom-up, storing only a length-4 conditional likelihood vector per node, so one post-order traversal suffices. This is the same algorithm as belief propagation on a tree in graphical models.
The four-point condition: a distance matrix $d$ is exactly realizable by some weighted tree if and only if, for any four points, the largest two of the three pairings $d_{ij}+d_{kl}$, $d_{ik}+d_{jl}$, $d_{il}+d_{jk}$ are equal. A purely combinatorial object — a tree — turns out to be completely characterized by inequalities on continuous distances. That is what a tree metric is.
The converse is equally deep: maximum parsimony is statistically inconsistent when two lineages evolve unusually fast and are not neighbours on the true tree — more data makes it more certain of a wrong tree. An estimator can be intuitively unimpeachable and computationally efficient while converging, in the limit, to the wrong answer; consistency has to be proved, not felt.
An amino acid chain is synthesized and within milliseconds curls itself into a precise three-dimensional shape — the same one every time. Levinthal did the arithmetic in 1969: 100 residues, a conservative 3 conformations each, gives $3^{100}\approx 10^{47}$ possibilities; trying one every $10^{-13}$ seconds would still take $10^{27}$ years. There is only one conclusion: folding is not searching, it is going downhill.
Picture conformation space as terrain whose height is free energy. If it were a golf course — mostly flat, one hole somewhere — blind search really would be hopeless; if it is a funnel, sloping from all sides toward one bottom, then sliding down from any starting point gets you there.
Energy landscape theory: define a free energy $F(x)$ over configuration space, with the native state as the global minimum. The key quantity is frustration — in a random sequence the interactions conflict (optimize one contact and another suffers), so the landscape is rugged and full of traps; sequences selected by evolution instead satisfy the principle of minimal frustration, where nearly every contact in the native state is close to its own optimum at once, and the landscape as a whole becomes a funnel.
The resolution of Levinthal's paradox is not a faster search but the observation that difficulty is a property of the landscape, not of the dimension. Those $10^{47}$ conformations are still there; they are simply never visited.
The insight reappears word for word in AI: a deep network's loss function is high-dimensional and non-convex, which classical optimization theory says should be hopeless, yet gradient descent trains it anyway — because an overparameterized loss landscape is also funnel-like, dominated by saddle points rather than local minima. "High-dimensional and non-convex" was never sufficient for hard; the geometry of the landscape is what matters — and geometry can be chosen. Biology chooses it by evolution, we choose it by architecture and initialization.
AlphaFold2's core move is not simulating physics but reading co-evolutionary signal: if two residues touch in the folded structure, a mutation in one usually needs a compensating mutation in the other, so across a multiple sequence alignment those two columns are correlated. The trouble is that correlation propagates ($i$–$j$ plus $j$–$k$ fabricates an apparent $i$–$k$), and the fix is to fit a Potts model and invert its coupling matrix — precisely the standard move from "correlated" to "conditionally independent" in graphical models, which AlphaFold replaces with learnable attention. Methods like RFdiffusion go further, running diffusion models over structure space for de novo design, generating enzymes that do not exist in nature.