Simple reaction-diffusion population model on scale-free networks

Abstract

We study a simple reaction-diffusion population model (proposed by A. Windus and H. J. Jensen, J. Phys. A: Math. Theor. 40, 2287 (2007)) on scale-free networks. In the case of fully random diffusion, the network topology does not affect the critical death rate, whereas the heterogenous connectivity makes the steady population density and the critical population density small. In the case of modified diffusion, the critical death rate and the steady population density are higher, at the meanwhile, the critical population density is lower, which is good for survival of species. The results are obtained with a mean-field framework and confirmed by computer simulations.Recently, Windus and Jensen [1, 2] introduced a model for population on lattices with diffusion and birth/death according to 2A → 3A and A → φ for an individual A . They found that the model displays a phase transition from an active to an absorbing state which is continuous in 1 + 1 dimensions and of first-order in higher dimensions [1]. They also investigated the importance of fluctuations and that of the population density, particularly with respect to Allee effects in regular lattices [2]. It was found that there exists a critical population density below which the probability of extinction is greatly increased, and the probability of survival for small populations can be increased by a reduction in the size of the habitat [2].In the study of complex networks [3], an important issue is to investigate the effect of their complex topological features on dynamical processes [4], such as the spread of infectious diseases [5] and the reaction-diffusion (RD) process [6]. For most real networks, the connectivity distribution has powerlaw tails P (k) ∼ k −γ , namely, a characteristic value for the degrees is absent, hence the scale-free (SF) property. In this Brief Report, we shall study the simple RD population model [1] on SF networks.In an arbitrary finite network which consists of nodes i = 1, . . . , N and links connecting them. Each node is either occupied by a single individual (1) or empty (0). We randomly choose a node. If it is occupied, the individual dies with probability p d , leaving the node empty. If the individual does not die, a nearest neighbor-node is randomly chosen. If the neighboring node is empty, the particle moves there; otherwise, the individual reproduces with probability p b producing a new individual on another randomly selected neighboring node, conditional on that node being empty. A time step is defined as the number of network nodes N . In homogeneous networks (such as regular lattices and Erdös-Rényi (ER) random networks [7]), the mean-field (MF) equation for the density of active nodes ρ(t) is given by [2] For At p d = p dc , the stationary density jumps from 1/2 to 0, resulting in a first-order phase transition. In order to study analytically this process on SF networks in which the degree distribution has the form P (k) ∼ k −γ and nodes show large degree fluctuations, we are forced to consider the partial densities ρ ...

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