Predator-Prey Model Calculator

Simulate population dynamics using the Lotka-Volterra predator-prey model

Initial Populations

Prey Parameters

Natural growth rate without predation

Rate at which predators consume prey

Predator Parameters

Efficiency of converting prey to predator offspring

Natural death rate without prey

Time units for simulation (days, months, years, etc.)

The Lotka-Volterra Model

The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first-order nonlinear differential equations that describe the dynamics of biological systems in which two species interact: one as a predator and the other as prey. Developed independently by Alfred J. Lotka (1925) and Vito Volterra (1926), these equations form the foundation of modern ecological modeling and demonstrate how predator and prey populations oscillate in characteristic cycles.

Model Assumptions

  • Prey population grows exponentially in the absence of predators
  • Predator population depends entirely on prey for food
  • Predation rate is proportional to the encounter rate between predator and prey
  • The environment provides unlimited resources for prey (no carrying capacity)
  • No immigration or emigration occurs
  • Predators have unlimited appetite and never get satiated
  • Both populations are uniformly distributed in space

Characteristic Dynamics

Phase 1: Abundant Prey

When prey are abundant, predators have plentiful food. The predator population grows, increasing predation pressure on the prey population.

Phase 2: Prey Decline

High predation causes the prey population to decline. As prey become scarce, predators struggle to find food.

Phase 3: Predator Decline

With limited prey, the predator population declines due to starvation. Predation pressure on prey decreases.

Phase 4: Prey Recovery

With fewer predators, the prey population recovers and grows. The cycle begins again as predators eventually respond to the increased prey availability.

Equilibrium and Stability

The Lotka-Volterra model has two equilibrium points where both populations remain constant:

Trivial Equilibrium

N = 0, P = 0

Both populations extinct. This equilibrium is stable but ecologically trivial—if both populations disappear, they remain extinct.

Coexistence Equilibrium

N* = γ/δβ, P* = α/β

Both populations coexist at non-zero levels. This equilibrium is neutrally stable—populations oscillate around it in closed loops (conservative oscillations).

Important Property: Neutral Stability

The coexistence equilibrium is neutrally stable, meaning the system neither converges to nor diverges from equilibrium. Instead, populations oscillate indefinitely in closed cycles. The amplitude and period of oscillations depend on initial conditions—starting further from equilibrium produces larger oscillations.

Classic Real-World Examples

Canadian Lynx and Snowshoe Hare

Historic fur trapping data from the Hudson's Bay Company (1845-1935) revealed cyclic oscillations in lynx and hare populations with an approximate 10-year period.

  • • Hare populations peak first
  • • Lynx populations peak 1-2 years later
  • • Both populations then decline
  • • Classic example of predator-prey cycles

Isle Royale Wolves and Moose

Long-term study (1958-present) on isolated island ecosystem showing predator-prey dynamics influenced by additional factors.

  • • Wolf population tracks moose availability
  • • Genetic factors affect wolf survival
  • • Climate influences moose food supply
  • • Demonstrates model limitations

Didinium and Paramecium

Laboratory experiments with protists showing simplified predator-prey interactions in controlled environments.

  • • Rapid population oscillations
  • • Often leads to extinction in simple setups
  • • Refuges stabilize the system
  • • Validates basic model predictions

Plankton Dynamics

Marine phytoplankton (prey) and zooplankton (predator) exhibit oscillations consistent with Lotka-Volterra predictions.

  • • Seasonal bloom patterns
  • • Consumer-resource cycles
  • • Foundation of marine food webs
  • • Critical for fisheries management

Model Limitations and Extensions

Limitations of the Basic Model

  • No carrying capacity for prey (unrealistic unlimited growth)
  • Predators have no alternative food sources
  • No age structure or life history complexity
  • Assumes constant parameters (no environmental variation)
  • No spatial structure or migration
  • Ignores other species interactions (competition, mutualism)
  • Neutral stability rarely observed in nature

Common Extensions and Improvements

Rosenzweig-MacArthur Model

Adds carrying capacity for prey and type II functional response for predators

Multiple Species

Extends to food webs with multiple predators, prey, and trophic levels

Spatial Models

Incorporates space, movement, and habitat heterogeneity

Stochastic Models

Adds random environmental variation and demographic stochasticity

References

The Lotka-Volterra model and predator-prey dynamics are foundational in ecology and mathematical biology:

Related Calculators

Population Growth Rate CalculatorCarrying Capacity CalculatorSpecies Density CalculatorGrowth Rate CalculatorBiomass Estimator

Note: This calculator implements the classic Lotka-Volterra predator-prey model, which provides a simplified representation of species interactions. Real ecosystems involve additional complexity including environmental stochasticity, spatial structure, multiple species, and adaptive behaviors. Use this model as an educational tool to understand fundamental predator-prey dynamics and as a starting point for more sophisticated ecological modeling.

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