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

1 Volterra Model

During World War I, the proportion of captured sharks rose sharply, puzzling biologists because overall fishing decreased. To explain this, biologist D. Ancona consulted mathematician V. Volterra, who applied differential equations to model the predator‑prey interaction between prey (edible fish) and predators (sharks).

Assuming abundant resources, the prey grows exponentially with intrinsic rate r, while the presence of predators reduces the prey’s growth proportionally to the predator population, introducing a predation coefficient α. The prey equation becomes dX/dt = rX – αXY .

Predators cannot survive without prey; their natural death rate is d, but the availability of prey reduces this death rate and promotes growth proportionally to prey abundance, with a conversion coefficient β. The predator equation becomes dY/dt = –dY + βXY .

Combining these yields the classic Volterra system, describing the interdependence of prey and predator populations without considering intraspecific competition.

The numerical iteration can be illustrated as follows:

Introducing a fishing effort coefficient h reduces the prey’s growth rate (r → r – h) and increases the predator’s death rate (d → d + h). Under wartime conditions, the reduced capture of fish and increased predator mortality lead to a higher equilibrium proportion of sharks, matching the observed data.

2 Volterra Model with Logistic Term

Although the basic Volterra model captures some dynamics, many ecological systems do not exhibit the perpetual oscillations predicted; instead they tend toward a stable equilibrium. Adding a logistic self‑limiting term to the prey equation accounts for intraspecific competition and yields more realistic behavior.

References

ThomsonRen GitHub https://github.com/ThomsonRen/mathmodels