Abstract
The standard predator-prey model is generalized beyond the Volterra linear-log form. Conservative oscillations are deduced and also conversion to a variational Hamiltonian form. Generalization to more than two species is also castable into Hamiltonian form, with small vibrations around equilibrium being of undamped sinusoidal type by virtue of associated characteristic exponents all being pure imaginaries. However, introduction into ecological equilibrium of a recognition of limited space and inorganic matter destroys the autonomous periodicity of the motions and makes inapplicable the elegant formalisms of classical statistical mechanics. Introduction of simple diminishing returns leads to damped motions that are kept cyclically alive by shocks of the weather and other exogenous stochastic elements. Introduction of increasing returns solely in an interval near equilibrium leads to autonomous self-exciting oscillations near a stable limit cycle; under stochastic forcing functions, a long-run ergodic state becomes predictable.
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