Ocean Trophic Web Dynamics: Four Levels, One Recycled Nutrient Pool

Beyond two species

The Lotka-Volterra equations, written down independently by Alfred Lotka and Vito Volterra in the 1920s, describe the oldest story in ecology: a prey population grows exponentially until a predator population eats into it, the predators then starve once prey gets scarce, and the cycle repeats forever with a fixed period and amplitude set by the initial conditions. That two-species picture is elegant, but a real ocean has more than two players stacked on top of each other.

This simulation extends the idea to four trophic levels: phytoplankton at the base convert dissolved nutrients and sunlight into biomass, zooplankton graze on phytoplankton, small and mid-size fish eat zooplankton, and apex predators — think tuna or sharks — eat the fish. Each level is coupled to the one above and below it by a mass-action predation term, exactly like classical Lotka-Volterra, just chained four times instead of twice.

The equations, level by level

Each population's growth rate is the sum of what it gains from eating the level below and what it loses to being eaten by the level above, plus its own natural death rate. In shorthand, for phytoplankton P, zooplankton Z, fish F and apex predators A, with nutrient pool N:

dN/dt = -uptake(N,P) + recycling(P,Z,F,A)
dP/dt =  uptake(N,P)  - graze(P,Z)      - deathP*P
dZ/dt =  eff1*graze(P,Z)  - predate(Z,F) - deathZ*Z
dF/dt =  eff2*predate(Z,F) - predate(F,A) - deathF*F
dA/dt =  eff3*predate(F,A) - deathA*A

The efficiency terms eff1, eff2 and eff3 are always well below 1 — typically around 10% in real marine food webs — because most of the energy consumed at each level is lost to metabolism and movement rather than converted into new biomass. This is the ecological reason apex predator populations are always tiny compared to the phytoplankton at the base: energy is lost by roughly an order of magnitude at every trophic transfer, a pattern known as the ten percent rule and visualized as the ecological pyramid.

Closing the loop: nutrient recycling

A pure predator-prey chain is an open system — biomass that dies just vanishes from the model, and eventually the whole web starves. Real oceans avoid this because death and excretion at every level return dissolved nutrients — nitrogen, phosphorus, iron — back into the water column, where phytoplankton can take them up again. The recycling term in the nutrient equation above sums a fraction of the natural mortality and metabolic loss from every level and feeds it back to N, closing the loop and letting the whole system reach a sustained oscillation or a stable equilibrium instead of collapsing.

Top-down vs bottom-up control

A four-level web lets you see two very different ways an ecosystem can be regulated. In bottom-up control, nutrient supply sets the ceiling: more N grows more phytoplankton, which ripples upward through every level. In top-down control, apex predators suppress fish, which releases zooplankton from predation, which then over-grazes phytoplankton — a cascading pattern called a trophic cascade. Removing apex predators (as overfishing does in real oceans) can flip a system from top-down to bottom-up regulation, and the shift shows up in this model as a change in which level's population swings the widest.

Frequently asked questions

Try it live

Everything above runs in your browser — open Ocean Trophic Web Dynamics and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.