Effects of local population structure in a reaction-diffusion model of a contact process on metapopulation networks

We investigate the effects of local population structure in reaction-diffusion processes representing a contact process (CP) on metapopulations represented as complex networks. Considering a model in which the nodes of a large scale network represent local populations defined in terms of a homogeneous graph, we show by means of extensive numerical simulations that the critical properties of the reaction-diffusion system are independent of the local population structure, even when this one is given by a ordered linear chain. This independence is confirmed by the perfect matching between numerical critical exponents and the results from a heterogeneous mean-field theory suited, in principle, to describe situations of local homogeneous mixing. The analysis of several variations of the reaction-diffusion process allows us to conclude the independence from population structure of the critical properties of CP-like models on metapopulations, and thus of the universality of the reaction-diffusion description of this kind of models.

Figure 1

Reaction-diffusion process in heterogeneous metapopulations represented with a networked inner structure. Populations are depicted by large blue vertices, while the vertices inside each population are represented by circles. Particles in connected populations are exchanged at a rate $\mu$, create offspring in their nearest empty neighbors inside their respective populations at a rate $\lambda$, and become empty at a unitary rate.

Figure 2

Moment ratios for the reaction-diffusion processes on metapopulations connected by UCM networks with minimum degree $k_0=6$ and cutoff $k_c=N^{1/2}$. In the main plot, the migration rate is $\mu =0.10$, degree exponent is $\gamma =2.75$, and the local population structure of the populations are RRNs with size $L=250$ and $z=2$. In the inset, the parameters $\mu =0.50$, $\gamma =2.25$ and regular chains of size $L=250$ and $z=2$ are used.

Figure 3

Finite size scaling of the critical density for distinct structured local populations. Results for (a) RRN and (b) linear chain are shown. The migration rate is $\mu =0.1$ and the remaining fixed parameters are as in Table . The curves $\overline{\rho }$ vs $N$ are represented by open symbols while curves $\rho$ vs $gN$, $g=\langle k^2\rangle /{\langle k\rangle }^2$, are represented by the filled ones. Inset in (b) shows the critical density against $L$ for $N=256000$ and $\gamma =2.75$ for RRN (circles) and linear chains (squares). Filled symbols were shifted to improve visibility. Dashed lines represent the power law $X^{-1/2}$.

Figure 4

Finite size scaling of the critical characteristic time for distinct structured local populations. Results for (a) RRN and (b) linear chain are shown. The fixed parameters are as in Fig. 3. The curves $\tau$ vs $N$ are represented by open symbols while curves $\tau$ vs $N/g$, $g=\langle k^2\rangle /{\langle k\rangle }^2$ are represented by the filled ones. Filled symbols were shifted to improve visibility. Dashed lines represent the power law $X^{1/2}$.

Figure 5

Finite size scaling of the critical density $\overline{\rho }$ and characteristic time $\tau$ in the variants of the RD process. In both variants, the structured local populations are represented by chains with $z=2$ nearest neighbors. The migration rate is $\mu =0.1$ and the degree exponent is $\gamma =2.75$. Power laws ${\left(Ng\right)}^{-1/2}$ and ${(N/g)}^{1/2}$ are shown for comparison.