The textbook predator-prey cycle is fragile. Spread over a landscape, the populations still boom and bust locally, but the totals stay nearly steady.
The previous essay's agents wrote a field that spread across space; here the animals themselves spread. The Lotka-Volterra equations treat foxes and rabbits as two numbers in a well-stirred pot, and they cycle. Run on a landscape of patches, each patch still booms and crashes but the total holds steady; among individual animals on a lattice, predators also become harder to sustain.
With prey \(x\) and predators \(y\), prey grow at rate \(a\), predators eat them at rate \(b\), turn a fraction \(e\) of the catch into offspring, and die at rate \(m\):
These equations conserve \(V = e b\,x - m\ln x + b\,y - a\ln y\), so every trajectory is a closed loop and a nudge moves the system to another loop for good. Give prey a carrying capacity \(K\) and the loops become inward spirals. Let predators become satiated too, with the Holling type II response \(bx/(1 + b h x)\), and these Rosenzweig-MacArthur equations have either a stable fixed point or one limit cycle that every start reaches.
Figure 1. The well-mixed model, integrated with RK4. Drag the ringed dot, or tap the plot, to change the starting populations. Dashed lines are the nullclines; their crossing is the fixed point. The strip below plots both populations along the trajectory (prey green, predators red). With satiation on and \(K\) above about 1.14, every start converges to one cycle, whose prey range the readout shows.
The fixed point loses stability at \(K = 2x^* + 1/(bh) = 1.14\), a Hopf bifurcation. Past it the trough of the cycle falls about tenfold for each 0.5 of \(K\) above 2 (Figure 2). This is Rosenzweig's paradox of enrichment (1971): feeding the prey destabilises the system, and in a real population a trough of \(3\times10^{-5}\), reached at \(K = 3\), is extinction.
Figure 2. The Rosenzweig-MacArthur model of Figure 1 at 121 values of \(K\). For each, the run continues from the previous \(K\)'s end state, discards 1,500 time units and records the highest and lowest prey density over the next 500 (green band, log scale). The flat line is the fixed point \(x^*\), solid where it is stable and dashed where it is not; the amber line is the Hopf point. Click, or focus the plot and use the arrow keys, to set \(K\) here and in Figure 1. The tick marked "Fig. 3" is the landscape's default \(K\).
Huffaker's (1958) mites on oranges behaved this way: on a small, open array the predators ate every prey mite and starved. The two species went through three oscillations together only on a larger array broken up by barriers, so different patches sat at different points of the cycle.
Put the Rosenzweig-MacArthur model on a grid. Each cell holds prey \(H\) and predators \(P\), runs the local equations, and exchanges animals with its four neighbours by diffusion:
Below the Hopf point the grid goes flat. (It cannot make Turing spots: the predators' growth rate does not depend on their own density, so nothing self-enhancing is there for diffusion to amplify.) Above it, noise organises into waves of prey chased by predators that curl into spirals, as in the excitable media of the Belousov-Zhabotinsky reaction.
Figure 3. Rosenzweig-MacArthur on a 160×160 periodic grid (green: prey, red: predators, both on fixed colour scales). Drag across the field to draw a barrier that blocks diffusion. "Invasion" fills the grid with prey at \(K\) and releases predators at the centre. The strip plots prey at the cell under the crosshair (thin line) against the grid mean (thick line); the dashed band is the prey range of the well-mixed cycle at the same \(K\). Parameters: \(a{=}1\), \(b{=}3.2\), \(h{=}0.5\), \(e{=}0.6\), \(m{=}0.35\), \(D_H{=}0.02\), \(D_P{=}0.4\).
In Figure 3's strip one cell runs through most of the well-mixed boom and bust while the grid mean barely moves. How big must the landscape be for that?
Figure 4. The equations and parameters of Figure 3 at \(K = 2\) on periodic grids 8 to 192 cells across, each started from the same kind of noise, run for 300 time units and then measured for 300. Right: how much the prey density swings over the measuring window (its standard deviation), for the grid mean (filled) and for one cell (hollow); the dashed line has slope \(-1\), the fall expected if the grid were a patchwork of independent patches of fixed size. Left: the grid-mean trajectory for the size picked on the right, over the cycle of a single cell on its own (dashed). Three starts per size run when the figure comes into view and "Another seed" adds one more; lines join the means.
Up to 16 cells across, diffusion pulled every start into step and the mean swung as far as a lone cell; at 24 and 32, two starts of three. From 48 cells up the grid breaks into waves, and the swing of the mean falls nearly fourfold by 192 cells, close to the \(1/L\) of a patchwork of independent patches on a grid \(L\) cells across, while one cell keeps swinging. On a big enough landscape the boom and bust is entirely local.
The spread of phases also keeps prey alive: an emptied patch is refilled from neighbours in other phases, while in the well-mixed model the trough comes everywhere at once. Released at one end of a strip of prey, predators advance as a front and leave regular travelling waves that, further back, break into what Sherratt, Lewis and Fowler (1995) called chaos in the wake of invasion.
Figure 5. The equations of Figure 3 on a strip of 800 cells with closed ends, prey at \(K\) everywhere and predators released in the five leftmost cells. Position runs left to right, time downward (one row every 2.5 time units, 1,000 time units in all); brightness is prey density on a fixed scale, dark where predators have eaten them out. The readout fits the front's position (the rightmost cell where prey have dropped below \(K/2\)) against time over the second half of the run and compares it with the linear spreading speed of predators into prey at \(K\).
The front moves at the speed a thin edge of predators grows into untouched prey: 0.970 cells per time unit against 0.972 predicted at \(K = 2\), within 0.003 from \(K = 1.3\) to 3.
A cell that has run down to a thousandth of a rabbit has no rabbits. In the stochastic lattice Lotka-Volterra model each site is empty or holds one prey or one predator. Pick a site at random. A prey fills a random empty neighbour with offspring with probability \(\sigma\). A predator dies with probability \(\mu\); otherwise, if a random neighbour holds a prey, it eats it and puts an offspring there with probability \(\lambda\). One sweep of the grid is one unit of time.
Prey can only breed into empty sites, so the mean-field version of these rules has a stable fixed point. The lattice never settles, though the swing in its totals tends to zero as it grows (Mobilia, Georgiev and Täuber 2007).
Figure 6. The grid is the stochastic lattice model on a 100×100 periodic grid with \(\sigma = 0.5\), \(\mu = 0.1\) (green prey, red predators, dark empty). The chart plots stationary densities against predation rate. Each dot is a separate 64×64 lattice, run when the figure scrolls into view for 1500 sweeps and averaged over the last 300. Dashed curves are the mean-field fixed point of the same rules. The hollow ring is the live lattice, averaged over its last 100 sweeps. Near the threshold the lattice relaxes slowly, so those points are noisy.
Below a critical predation rate \(\lambda_c\) predators die out for good. Mean field puts the threshold where a lone predator among prey just replaces itself, \(\lambda(1-\mu) = \mu\), or 0.111 here. The lattice needs nearly twice that, partly because a predator's offspring land beside it and compete for the same prey. Near the edge a lattice can take longer to die than Figure 6's runs.
Figure 7. Five 256×256 lattices with the rules and \(\sigma, \mu\) of Figure 6, at predation rates from 0.205 to 0.23, each started from the same random mix (30% prey, 15% predators) and run side by side for 5,000 sweeps. Predator density against time on log-log axes; a cross marks extinction. The square shows the lattice at \(\lambda = 0.214\) (amber curve). The dashed line has slope \(-0.451\), the decay exponent of two-dimensional directed percolation, drawn at half the amber curve's value at \(t = 300\). The readout fits each curve's slope over the last decade. "Another seed" runs all five again and plots the mean over runs.
The transition is in the directed-percolation class, generic for a single absorbing state (Antal and Droz 2001; Mobilia, Georgiev and Täuber 2007). At the threshold the density decays as \(t^{-\delta}\), \(\delta = 0.451\) in two dimensions, and curvature on log-log axes locates it (Hinrichsen 2000). At \(\lambda = 0.214\) the slope over the last decade was \(-0.44\) to \(-0.49\) in five of six runs (\(-0.65\) in the sixth), so \(\lambda_c \approx 0.214\). One decade after a plateau is too short to measure \(\delta\) itself.
Rock-paper-scissors species show the same contest: well mixed, a finite population loses two of the three, while on a lattice all three persist as spirals as long as individuals move slowly enough for a spiral to fit inside the habitat (Reichenbach, Mobilia and Frey 2007).
This landscape makes waves but no Turing spots, because it has no self-enhancing activator. The next essay starts from a reaction that has one, and compares it with the other explanation biology offers for how an embryo becomes patterned.