Population dynamics on random networks: simulations and analytical models |
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Authors: | G Rozhnova and A Nunes |
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Institution: | (1) Rockefeller College of Public Affairs and Policy, University at Albany, State University of New York, Albany, USA; |
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Abstract: | We study the phase diagram of the standard pair approximation equations for two different models in population dynamics, the
susceptible-infective-recovered-susceptible model of infection spread and a predator-prey interaction model, on a network
of homogeneous degree k. These models have similar phase diagrams and represent two classes of systems for which noisy oscillations,
still largely unexplained, are observed in nature. We show that for a certain range of the parameter k both models exhibit
an oscillatory phase in a region of parameter space that corresponds to weak driving. This oscillatory phase, however, disappears
when k is large. For k = 3, 4, we compare the phase diagram of the standard pair approximation equations of both models with
the results of simulations on regular random graphs of the same degree. We show that for parameter values in the oscillatory
phase, and even for large system sizes, the simulations either die out or exhibit damped oscillations, depending on the initial
conditions. We discuss this failure of the standard pair approximation model to capture even the qualitative behavior of the
simulations on large regular random graphs and the relevance of the oscillatory phase in the pair approximation diagrams to
explain the cycling behavior found in real populations. |
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