§ B-12 Ecosystem Dynamics: Lotka–Volterra Predator–Prey Oscillations & Trophic Energy Transfer (Grade 11 · Ecology: Lotka–Volterra Model)
Predator–Prey Differential Equations, Phase-Space Attractor, Population Waves & Lindeman’s 10% Pyramid. “Coupled interactions between prey x(t) and predators y(t) generate sustained periodic oscillations in which predator population peaks systematically lag behind prey peaks by a quarter cycle.”
Theory Summary & Scientific Mechanism
The Lotka–Volterra predator–prey model (1925–1926) governs the coupled population dynamics of prey x(t) and predators y(t) via the nonlinear system (dx)/(dt) = α x - β x y and (dy)/(dt) = δ x y - γ y, where α is prey intrinsic growth rate, β is predation rate, γ is predator mortality, and δ is trophic conversion efficiency.
Setting (dx)/(dt) = 0 and (dy)/(dt) = 0 yields the non-trivial interior equilibrium point (Volterra’s principle): **x^* = (γ)/(δ) and y^* = (α)/(β)**. Consequently, time-averaged prey abundance is controlled by predator parameters (γ, δ), while average predator abundance is controlled by prey parameters (α, β).
The conserved quantity V(x,y) = δ x - γln x + β y - αln y = const confines trajectories to closed orbits around (x^*, y^*) with period T ≈ (2π)/(√(αγ)), in which predator peaks lag behind prey peaks by a quarter cycle (π/2). Across trophic levels, Lindeman’s 10% ecological efficiency rule dictates that only sim 10% of net production is transferred to the next trophic tier.
Everyday Analogy
Picture snowshoe hares and lynx in the boreal forest: a boom in hares provides abundant food so lynx numbers surge a quarter-cycle later. Overpredation then drives hare numbers down, causing a subsequent lynx decline that lets hares recover in an endless ecological wave around (x^*, y^*)!
Interactive Question & Derivation
The Lotka–Volterra predator–prey system is governed by (dx)/(dt) = x(α - β y) for prey and (dy)/(dt) = y(δ x - γ) for predators. What are the coordinates of the non-trivial equilibrium point (x^*, y^*) where (dx)/(dt) = 0 and (dy)/(dt) = 0, and how is the predator oscillation wave y(t) phase-shifted relative to the prey wave x(t)?
- A: Equilibrium depends only on initial population sizes, and both populations peak simultaneously in phase
- B: The equilibrium point is determined solely by interaction parameters: x^* = γ/δ and y^* = α/β, and predator peaks systematically lag behind prey peaks by approximately one-quarter period (π/2)
- C: Predator biomass is always 10 times larger than prey biomass by Lindeman’s rule
- undefined: Increasing prey birth rate α decreases the equilibrium predator density y^*
Study Hint: Equilibrium occurs at x^* = γ/δ and y^* = α/β, with predator population peaks lagging behind prey peaks by one-quarter period.
Proof & Explanation: Setting (dx)/(dt) = 0 and (dy)/(dt) = 0 for x > 0, y > 0 yields α - β y^* = 0 ⇒ y^* = α/β and δ x^* - γ = 0 ⇒ x^* = γ/δ. Remarkably, the average prey density x^* = γ/δ is governed by predator mortality and conversion efficiency, whereas average predator density y^* = α/β is governed by prey growth and predation rate (Volterra’s principle). In the (x, y) phase plane, trajectories form closed orbits circulating counterclockwise around (x^*, y^*): abundant prey x fuels predator growth y with a quarter-cycle lag (≈ π/2), which then depletes prey, causing predator decline and allowing prey recovery.
Key Equations & 3D Simulation Parameters
V(x,y) = δ x - γln x + β y - αln y = const, T_cycle ≈ (2π)/(√(αγ))
- Prey intrinsic growth rate α: 0.5 … 2 yr⁻¹ (default: 1.1 yr⁻¹)
- Predation encounter rate coefficient β: 0.02 … 0.08 (default: 0.04)
- Predator natural mortality rate γ: 0.4 … 1.6 yr⁻¹ (default: 0.9 yr⁻¹)
- Trophic biomass conversion efficiency δ: 0.01 … 0.05 (default: 0.02)