Generalized predator-prey oscillations in ecological and economic equilibrium

Area of Science:

• Ecology
• Mathematical Biology
• Theoretical Ecology

Background:

• The classical Volterra predator-prey model assumes unlimited resources and simple interactions.
• Existing models often lack the complexity to capture real-world ecological dynamics, such as resource limitation and stochasticity.

Purpose of the Study:

• To generalize the standard predator-prey model beyond the Volterra linear-log form.
• To explore the mathematical structures and dynamic behaviors of generalized ecological models.
• To investigate the impact of environmental limitations and stochasticity on ecological stability.

Main Methods:

• Generalization of the Volterra model to include non-linear interactions and more than two species.
• Conversion of ecological models to a variational Hamiltonian form.
• Analysis of small vibrations around equilibrium points.
• Inclusion of limited space, inorganic matter, diminishing returns, and increasing returns.
• Introduction of exogenous stochastic elements (e.g., weather shocks).

Main Results:

• Deduced conservative oscillations and conversion to a variational Hamiltonian form for generalized predator-prey systems.
• Demonstrated that small vibrations around equilibrium in multi-species Hamiltonian systems are of undamped sinusoidal type.
• Showed that limited resources and space disrupt autonomous periodicity, rendering classical statistical mechanics inapplicable.
• Found that diminishing returns lead to damped motions sustained by stochastic shocks.
• Revealed that increasing returns near equilibrium induce self-exciting oscillations towards a stable limit cycle, with stochastic forcing enabling a predictable long-run ergodic state.

Conclusions:

• Generalized predator-prey models can exhibit complex dynamics, including Hamiltonian structures and non-autonomous oscillations.
• Environmental realism (resource limitation, stochasticity) is crucial for understanding ecological stability and predictability.
• The study provides a framework for analyzing more realistic ecological systems using advanced mathematical techniques.