EQ-LOTKA-VOLTERRA · Population Dynamics
Lotka–Volterra Predator-Prey Equations
alpha*x - beta*x*y = 0
Variables
variable
alpha
prey intrinsic growth rate
- Object
- prey_population
- Property
- PopulationGrowthRate
- Context
- well_mixed
variable
beta
predation rate coefficient
- Object
- predator_population
- Property
- PredationRate
- Context
- well_mixed
variable
x
prey population (using mole as generic count unit until biology-specific dim added)
- Object
- prey_population
- Property
- PopulationCount
- Context
- well_mixed
variable
y
predator population
- Object
- predator_population
- Property
- PopulationCount
- Context
- well_mixed
Axioms
algebraic bounded_growth classical conservative_count deterministic dynamical galilean_invariant nonlinear
Assumptions
- Well-mixed, no spatial structure
- Prey grow exponentially in absence of predators (unbounded food)
- Predator mortality is linear in predator count
- Predation rate is bilinear in prey × predator (mass-action)
- No carrying capacity (logistic effects ignored)
Derivation
- Lotka, J. Phys. Chem. 14 (1910), 271 (autocatalytic reactions)
- Volterra, Nature 118 (1926), 558 (fish populations in the Adriatic)
- Mass-action law applied to prey-predator encounters
- Has a conserved quantity V = δx − γ ln x + βy − α ln y (first integral)
References
- Murray, Mathematical Biology I, 3rd ed., §3.1
- Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., §6.5
- Hirsch, Smale & Devaney, Differential Equations, Dynamical Systems, §12