Derivation and Analysis of a Discrete Predator-Prey Model

Area of Science:

• Ecology
• Mathematical Biology
• Population Dynamics

Background:

• Predator-prey models are fundamental to understanding population dynamics.
• Existing models often use continuous differential equations, which may not capture discrete population changes.
• A discrete model derived from first principles is needed to better reflect natural populations.

Purpose of the Study:

• To derive a novel discrete predator-prey model from fundamental ecological assumptions.
• To analyze the stability of the prey-only equilibrium and the conditions for coexistence.
• To investigate the dynamics of the discrete model, including bifurcations and oscillations.

Main Methods:

• Derivation of a discrete predator-prey model using an economic technique for continuous-discrete compounding.
• Extension of phase plane analysis with a novel 'next iterate root-curve'.
• Application of Lyapunov functions and bifurcation analysis (Neimark-Sacker bifurcation).

Main Results:

• The prey-only equilibrium is globally asymptotically stable below a specific predator consumption-energy rate threshold.
• Above this threshold, a stable coexistence equilibrium emerges, with solutions oscillating around it.
• The study identifies conditions for local stability and destabilization via a supercritical Neimark-Sacker bifurcation, bounding oscillation amplitudes.

Conclusions:

• The derived discrete predator-prey model offers a more realistic representation of population interactions.
• The model accurately predicts transitions from stable prey populations to oscillating coexistence.
• Bifurcation analysis reveals complex dynamics, including the emergence and stability of coexistence equilibria.