Predator-Prey Simulator

What is the Lotka-Volterra predator-prey model?

The Lotka-Volterra model is a pair of equations for a prey population x and a predator population y that makes the two rise and fall in a cycle that repeats for ever: dx/dt = αx − βxy and dy/dt = δxy − γy. Prey multiply at rate α when left alone and are eaten at a rate βxy that grows with how often the two meet. Predators die out at rate γ with nothing to eat and grow at a rate δxy by eating. This simulator solves both equations and shows the answer twice: as the two populations against time in the graph, and as a loop of predators against prey on the stage above it.

The cycle is easy to follow in words. Plenty of prey feed the predators, so the predators multiply. More predators eat the prey faster than they breed, so the prey decline. With less to eat the predators starve and decline in turn, and with fewer predators the prey recover. Then it starts again. Nothing in the equations mentions a cycle or a period; both come out of the solution.

The populations here are in thousands and time is in years, and the defaults are the estimates from Bob Carpenter’s 2018 fit of these equations to the numbers of snowshoe hare and Canada lynx pelts the Hudson’s Bay Company collected between 1900 and 1920: α = 0.55, β = 0.028, γ = 0.80 and δ = 0.024, with α and γ per year and β and δ per thousand animals per year, starting from his 33.956 and 5.933 thousand rounded to 34 and 5.9. The equations do not care about the units. Scale both populations by the same factor and divide β and δ by it and the curves are the same, so a textbook problem written in single animals can be typed straight in.

Where the populations balance

Set both rates to zero and the one point where nothing changes drops out: x* = γ/δ prey and y* = α/β predators. The prey stop changing when predators number exactly α/β, which is the horizontal dashed line on the stage, and the predators stop changing when prey number exactly γ/δ, the vertical one. Each is drawn in the colour of the population it holds still, so every time the loop crosses the blue line the prey are at a peak or a trough, and every time it crosses the orange line the predators are.

Look at which parameters set which. The prey level depends only on the predators’ death rate and gain, and the predator level only on how fast the prey breed and are caught. Make the prey breed faster and the prey do not become more common on average: the predators do. Volterra also showed that over any complete cycle each population averages exactly its equilibrium value, however large the swings, which is why each dashed line on the graph sits where its curve has equal area above and below it over a cycle, not halfway between the highs and lows.

Worked example: the hare and lynx defaults

Take the defaults as the page opens: α = 0.55, β = 0.028, γ = 0.80 and δ = 0.024, starting from 34 thousand prey and 5.9 thousand predators.

1. The equilibrium. x* = 0.80 / 0.024 = 33.33 thousand prey and y* = 0.55 / 0.028 = 19.64 thousand predators, the Equilibrium readout.
2. The small-swing period. 2π/√(αγ) = 2π/√(0.55 × 0.80) = 9.472 years. That is the period of a cycle too small to see.
3. How big this cycle is. Writing X = x/x* and Y = y/y*, the start is at X = 34/33.33 = 1.020 and Y = 5.9/19.64 = 0.3004, and the quantity the model conserves is H = γ(X − 1 − ln X) + α(Y − 1 − ln Y) = 0.80 × 0.000197 + 0.55 × 0.5031 = 0.2769 per year. It stays at that value for as long as the simulation runs, which the stage prints in its corner.
4. The highs and lows. Prey are at their extremes when Y = 1, so solving 0.80(X − 1 − ln X) = 0.2769 gives X = 2.077 and X = 0.3809: prey swing between 12.70 and 69.23 thousand. Predators are at theirs when X = 1, and 0.55(Y − 1 − ln Y) = 0.2769 gives Y = 2.364 and Y = 0.3002: predators swing between 5.898 and 46.43 thousand. Starting almost exactly at x* put the start almost at the predators’ low.
5. The period. Integrating the time taken once round the loop gives 10.15 years, 7.2 percent longer than the small-swing figure, and the Cycle period readout shows it. Within the cycle the predators peak 1.64 years after the prey, about a sixth of the period rather than the quarter a tiny cycle would give.

Averaged over those 10.15 years, the prey come to exactly 33.33 thousand and the predators to 19.64 thousand, the equilibrium values, even though the prey spend nearly three fifths of the cycle below 33.33 thousand and make up for it with a peak that rises further above it.

Reading the phase plane

Every starting point gives a closed loop around the equilibrium, travelled anticlockwise: prey rise along the bottom, predators rise up the right-hand side, prey fall along the top and predators fall down the left. Drag the start point on the stage and a different loop appears. There is no preferred cycle and no natural amplitude; the size of the cycle is set entirely by where it starts, which is what ecologists mean by calling it neutrally stable.

What labels each loop is the conserved quantity H, zero at the equilibrium and larger on every loop further out. It is what lets the page give an exact period and exact highs and lows without trusting the animation, and the faint loop under the trajectory is drawn from it, so any gap between the two would be integration error. The period grows with H, as Jörg Waldvogel proved in 1986, so a large cycle is always slower than 2π/√(αγ), the way a pendulum swinging wide is slower than its small-angle formula; the simple pendulum simulator shows the same relationship for a swing.

Take the start far from the equilibrium and the loop hugs the axes. The prey then crash to a small fraction of their peak and spend most of the cycle there, the predators starving slowly, before a short explosive recovery. Without predators the prey would grow exponentially at rate α, doubling every ln 2/α years, 1.26 years at the default α, the same arithmetic the cell doubling time calculator does for a culture.

What a carrying capacity changes

Switch on Limit prey growth and the prey’s own growth becomes logistic, αx(1 − x/K), so prey compete for food even with no predators about. The closed loops disappear. As long as K is above γ/δ, every start now settles to a single steady state, at the same x* = γ/δ prey and fewer predators, y* = (α/β)(1 − γ/(δK)).

With the defaults and K = 100 thousand the populations settle at 33.33 thousand prey and 13.10 thousand predators, in swings that halve about every 7.56 years and repeat every 11.77 years, the readouts under the stage. Lower K towards γ/δ and the spiral tightens into a steady approach with no swings at all; take K to or below γ/δ and there is no longer enough prey to sustain any predators, which die out while the prey settle at K.

That is the most important thing the simulator can show about the classic model. Its cycles are a knife-edge: the tiniest bit of competition among the prey turns them into a spiral, and most other small changes to the equations break them too. A model whose cycles persist in nature needs something that holds their size in place, which the classic equations do not have.

Volterra’s fishing puzzle

Volterra came to these equations through a question from the biologist Umberto D’Ancona. The catch landed at the Adriatic port of Fiume held a far larger share of sharks, skates and rays while fishing was cut back during the First World War: 11.9 percent of the catch in 1914 and 36.4 percent in 1918, in the figures Martin Braun reproduces in his textbook. Less fishing should have helped every species, so why did it help the predators more?

Because fishing both kinds of fish at the same rate e is the same as lowering the prey’s growth rate to α − e and raising the predators’ death rate to γ + e. The averages move to (γ + e)/δ prey and (α − e)/β predators, so fishing harder means more prey and fewer predators, and fishing less the reverse. Try it here: from the defaults, set α to 0.45 and γ to 0.90, the effect of fishing at 0.1 a year, and the equilibrium moves from 33.33 and 19.64 to 37.5 thousand prey and 16.07 thousand predators.

Common mistakes

• Swapping which parameters set which equilibrium. The prey level is γ/δ, the predators’ numbers, and the predator level is α/β, the prey’s. Feeding the prey better raises the predators.
• Using 2π/√(αγ) as the period of a real cycle. It is the small-swing limit, and every larger cycle is slower. The defaults’ cycle takes 10.15 years, not 9.47.
• Expecting the predator peak a quarter of a cycle after the prey peak. True only for a very small cycle. On a large one the predators peak sooner after the prey and take longer to recover from the trough.
• Treating β and δ as the same number. β is the prey lost per predator and δ the predators gained per prey, and catching a prey does not make a predator. In the defaults β is 0.028 and δ is 0.024.
• Reading the size of the cycle as a property of the species. In this model it is set by the starting numbers, and any disturbance moves the populations to a different cycle for good.
• Believing a trough of a fraction of an animal. The equations treat populations as smooth numbers, so they recover from any low, however small. When a cycle takes either population below one animal the page says so, because a real population would be extinct.

What this model leaves out

Predators that fill up. Here each predator eats in proportion to the prey available, however many there are. Real predators can only catch and eat so fast, which Holling described in 1959 as a saturating functional response, and the curve is the same shape as the Michaelis-Menten rate in the enzyme kinetics simulator. Add that and a carrying capacity, the model Rosenzweig and MacArthur analysed in 1963, and the swings either die away or settle into a cycle of one size, rather than keeping whatever size they started at.

Chance, and whole animals. Births, deaths and hunts happen one at a time and at random, which matters most when numbers are small and a run of bad luck ends a population. The radioactive decay simulator shows how chance scatters a small count around the smooth curve an equation predicts.

Everything else in the ecosystem. Age structure, the time a predator takes to mature, weather, disease, space and every other species. The real hare cycle, about ten years long, comes from the interaction of predation with the hares’ food supply, as Krebs and colleagues showed with large-scale experiments in the Yukon, reported in 2001.

What the data actually counted. The Hudson’s Bay Company records are pelts collected, not animals counted, and Charles Gordon Hewitt, who analysed them in 1921, noted that lynx are easier to trap when they are hungry, which is when hares are scarce. A good fit of these equations to those numbers is a description of the data, not a census.