Population dynamics on random networks: simulations and analytical models

Abstract

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.

Failure-recovery model with competition between failures in complex networks: a dynamical approach

“…For the parameters used in Fig. 3, the system reaches a steady state, however, we will show that for a specific region of parameters the system exhibits an oscillatory behavior, similarly to the ones found in some epidemic models [19][20][21] and in a model of neural networks [12]. In our model, these oscillations are a consequence of a competition between inactive internal nodes and inactive external nodes, with the aim of transforming the living nodes to their own state, as we will explain below.…”

“…Notice that in the case γ I < γ E , we cannot use the Lyapunov function given by Eq. (20), since for this case the parameter a in Eq. (18) is negative.…”

Cluster approximations for infection dynamics on random networks

“…Black lines are sample averages of the densities as obtained from 10 3 realizations of a RRG-4 with N = 10 6 nodes. For the SIRS model the agreement of the solutions of the PA deterministic equations with the averaged dynamics on RRGs is sensitive to the rate of immunity waning γ, viz it deteriorates with decreasing γ [34]. The upper panels in Fig.…”

“…As mentioned in the previous section, in Ref. [34] we have compared the data of the SIRS stochastic process obtained from Monte Carlo simulations on RRGs with the solutions of the standard PA equations and have shown that the PA describes correctly the global behavior of the model in the limit where rate of immunity waning γ ≫ 1 but it fails to capture the dynamics for γ ≪ 1. The question is then whether a cluster approximation of the next order can explain the suppression of global oscillations predicted by the PA for γ ≪ 1 and, in particular, whether it can describe stationary states correctly.…”

Maximum-entropy moment-closure for stochastic systems on networks

“…Therein lies a problem of non-uniqueness: depending on the local configuration of the network, there may be several such factorisations to choose from and no logical motivation for making one choice over another. For a discussion of this problem in relation to an approximation based on triples, see [14].…”