Predator-Prey Models

Definition

Predator-prey models (Lotka-Volterra equations) are a specific system of autonomous differential equations modeling the interaction between two species where one consumes the other: $\frac{dx}{dt} = ax - bxy, \quad \frac{dy}{dt} = -cy + dxy$

• How to read: “The derivative of x with respect to time is equal to the constant a times x minus the constant b times x times y; and the derivative of y with respect to time is equal to the negative constant c times y plus the constant d times x times y.”
• Meaning: Lotka-Volterra system—prey $x$ grows logistically but is eaten; predators $y$ decline without prey but grow by hunting. where $x$ is the prey, $y$ is the predator, and $a, b, c, d > 0$.

Why It Matters

No population exists in a vacuum. If you don’t understand the “Phase Lag” between predators and prey, you will over-harvest at exactly the moment the system is most fragile. These models reveal that “Stability” is actually a series of cycles. Ignoring the coupling between variables is the fastest way to trigger a “Sudden Extinction” event in biology or business.

Core Concepts

• Interdependence: The prey’s growth is limited by predation ($-bxy$), while the predator’s growth is driven by prey consumption ($dxy$).
• Periodic Trajectories: In the phase plane, solutions typically form closed loops around an interior equilibrium point, representing sustained oscillations.
• Phase Lag: Predator populations typically peak after prey populations, creating a cyclic lag in the dynamics.
• Equilibrium Points: The trivial equilibrium $(0,0)$ and the non-trivial coexistence equilibrium $(c/d, a/b)$.
  • How to read: “The equilibrium points are at the origin zero, zero and the point given by the ratio of c to d and the ratio of a to b.”
  • Meaning: Extinction state and balanced coexistence where prey and predator populations are steady.

Connected Concepts

• Mathematical Models: Building Functions
• Quadratic Optimization
• Financial Models: Compound Interest