Automata network predator-prey model with pursuit and evasion

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Abstract
An automata network predator-prey model with pursuit and evasion is studied. The local rule consists of two subrules. The first, applied synchronously, models predation, birth, and death processes. The second, applied sequentially, describes predator pursuit to catch evading preys. The model contains six parameters: the birth and death rates of preys and predators and two parameters characterizing the motion of preys and predators. The model has three fixed points. The first is trivial; it corresponds to a stationary state with no living individuals. The second characterizes a state with no predators. The third describes a state with nonzero densities of preys and predators. Moreover, the model may exhibit oscillatory behavior of the local prey and predator densities as functions of time through a Hopf bifurcation. In this particular case spatial coherence is lost. Spatial correlations decay with a finite correlation length ξ. Although local densities, measured over a range of the order of ξ, oscillate, collective variables are stationary.