t = 0
POPULATION OVER TIME
last 300 ticks
click grid to seed · drag sliders to reshape dynamics
Predator-prey, exponential growth, carrying capacity — logistic curves you can break. Same equation, two views.
In 1925, the Italian mathematician Vito Volterra sat down to explain a question his future son-in-law had brought him: why, in the years just after the First World War, was the Adriatic Sea suddenly full of predatory fish? The fisheries had been quiet during the war. The prey species had recovered. The predators followed. The two populations chased each other in cycles he could draw on the back of a napkin.
Around the same time, the American chemist Alfred Lotka derived the same equations from completely different reasoning — chemical reaction rates. Two independent derivations, the same coupled differential equations, both published before either could find the other. Predator-prey dynamics earned the name Lotka–Volterra.
Two years later, in 1927, the Scottish physicians William Kermack and Anderson McKendrick were trying to explain why a plague outbreak in Bombay would burn fiercely for months and then die out before infecting everyone. They split a population into three compartments — susceptible, infected, recovered — and wrote a set of coupled rate equations relating the flows between them. The shape of every epidemic that has ever been modeled since — influenza, polio, AIDS, Ebola, COVID-19, the Memphis Triple Disaster scenarios in OPA's case-study library — comes out of the SIR model they wrote that year.
Lotka–Volterra and Kermack–McKendrick share a deep structural property: the dynamics live in the coupling, not in the individual species or stages. The same mathematical grammar that describes lynx and hare populations on Hudson Bay also describes the diffusion of a respiratory illness through a city. And both extend naturally to trophic webs — grass feeds zooplankton feeds fish feeds eagles — through a stack of coupled equations that turn into the modern field of ecosystem ecology.
The trophic mode runs four species (grass, algae-like primary producer, zooplankton-like consumer, fish-like top predator) on the same grid using a simplified Rosenzweig–MacArthur formulation. Same mathematical lineage, more species. The pattern is the same; the dimensions just grow.
A student who learns Lotka–Volterra from a textbook sees clean sinusoidal oscillations — predator population traces a phase-shifted curve behind prey, forever. A student who runs the same equations as a spatial agent simulation sees something the textbook never shows: local pockets where the species briefly go extinct, refuge corners where they survive, traveling waves of infection or grazing pressure that move across the landscape, and emergent stable patches the equations alone would never predict.
SIR disease. A susceptible population, an infected outbreak that spreads through neighbor contact, recovered individuals that no longer transmit. Click anywhere on the grid to plant a new infection and watch it propagate. Slide the transmission rate up and the outbreak burns through the population; slide the recovery rate up and the outbreak dies before reaching critical mass. This is the equation the city public health department runs in their head every winter.
Predator-prey. Lotka–Volterra on a 2D grid. Left-click adds prey, right-click adds predators. Watch the spirals form in the population chart — prey booms, predators follow, prey crashes, predators crash. Then watch what the grid does that the chart can't show: regional extinctions, refuge corners, traveling waves of predation. This is what wildlife biologists actually see in long-term field data — not smooth oscillations but noisy, patchy, regional cycles.
Trophic web. Four species coupled top to bottom. Grass grows. Zooplankton-like grazers eat the producers. Fish-like predators eat the grazers. Cascade effects propagate through the whole web when you push any one species. This is the Yellowstone-wolves-reintroduction story, the why-the-cod-fishery-collapsed story, the what-happened-when-we-killed-the-keystone story.
Same equation, two views. The aside chart shows the continuous population curves the textbook predicts. The main grid shows the spatial agent simulation the textbook pretends doesn't exist. The chart is the ideal. The grid is the reality. Both are honest. The student needs both.
A browser-based spatial agent simulation on an 80×80 grid (6,400 cells) coupled to a time-series chart of total population per species. Three modes (SIR, predator-prey, trophic) share the same grid engine but interpret each cell's state differently and apply different transition rules. All parameters are exposed as sliders. Click anywhere on the grid to seed additional agents.
The dynamics are qualitatively faithful to the published equations — SIR compartments transition by neighbor-contact infection and per-tick recovery; Lotka–Volterra runs as a stochastic agent process whose population average reproduces the standard sinusoidal phase-shifted oscillations; the trophic mode uses simplified Rosenzweig–MacArthur coupling. The dynamics are not calibrated to any specific species, disease, or ecosystem. Tick units are dimensionless. Suite standing rule: teach the shape, not the digits.
Section 4.3.3 · OPA Life Sciences Suite. Filed under Building 3 (Agriculture & Animal Intelligence), College III (Agriculture & Animal Sciences).
Sibling labs in the Engineering Suite (Horseshoe Vortex 4.10.35, Concert Hall 4.10.36, Live Beam 4.10.37) and the Physics Suite (Ripple Tank 4.9.4c, Double-Slit 4.9.4d, Edge Cases 4.9.8, Standing Question 4.00.9) share the OPA grammar but use distinct palettes per suite. The Life Sciences Suite's moss-green palette is unique to this lab and its future siblings.
Filed under Opathorlokan University, Birmingham, Alabama. Built by Travis Jenkins (User Zero) with Claude. The lab exists so that a student can hold the same equation in two hands at once — the continuous curve and the spatial agent — and feel the moment they stop being the same answer.