How the Volterra Predator‑Prey Model Explains Shark Surges During War

1 Volterra Model

During World War I, the proportion of captured sharks increased dramatically. Although war reduced overall fishing, both prey (fish) and predator (sharks) populations were expected to rise, yet sharks rose disproportionately. Biologist D. Ancona consulted mathematician Vito Volterra, who used differential equations to explain the phenomenon.

Let the prey (fish) and predator (sharks) populations at time t be denoted by x(t) and y(t). Assuming abundant resources, prey grow exponentially with intrinsic rate a, so dx/dt = a x.

The presence of predators reduces the prey’s growth rate proportionally to predator abundance, giving dx/dt = a x − b x y, where b reflects the predation efficiency.

Predators cannot survive without prey; their natural death rate is d, so dy/dt = −d y.

Prey availability lowers predator mortality and promotes growth proportionally to prey abundance, yielding dy/dt = −d y + c x y, where c measures how effectively prey support predator reproduction.

Combining these yields the classic Volterra predator‑prey system:

dx/dt = a*x - b*x*y

dy/dt = -d*y + c*x*y

This simple model captures the interdependence and constraints between prey and predator, ignoring self‑limiting growth.

The numerical iteration formula can be illustrated as follows:

Introducing a fishing effort coefficient f reduces the prey’s intrinsic growth rate (a → a − f) and increases predator mortality (d → d + f). Under war conditions, the capture coefficient drops, leading to new equilibrium values that predict a higher shark proportion.

2 Volterra Model with Logistic Term

Although the original Volterra model explains some observations, many ecologists note that real predator‑prey systems tend toward a stable equilibrium rather than perpetual cycles. Adding a logistic self‑limiting term to the prey equation introduces carrying‑capacity effects, allowing the model to simulate such stable behavior.