Phase transitions in a lattice population model

Abstract

We introduce a model for a population on a lattice with diffusion and birth/death according to 2A −→ 3A and A −→ φ for a particle A. We find 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 in agreement with the mean field equation. For the 1 + 1 dimensional case, we examine the critical exponents and a scaling function for the survival probability and show that it belongs to the universality class of directed percolation. In higher dimensions, we look at the first-order phase transition by plotting a histogram of the population density and use the presence of phase coexistence to find an accurate value for the critical point in 2 + 1 dimensions.

Cluster geometry and survival probability in systems driven by reaction–diffusion dynamics

“…From our mean field analysis, we therefore have a phase diagram as shown in figure 2. We note that in our earlier paper [12], we examined the case k = 0 and so were restricted, at the mean field level, to the first-order regime only.…”

“…To find the values of p d c in the first-order regime we adopt an approach inspired by Lee and Kosterlitz [15] (see also [12]), due to the lack of power-law behaviour. For k < k * we examine Plots showing the critical behaviour of the model through powerlaw behaviour for the continuous phase transitions (k k * ) and a histogram of population density for the first-order transitions (k < k * ).…”

“…Recently, we examined a simple reaction-diffusion model with the reactions A → φ and 2A → 3A [12] (see also [6,13]), which exhibited a phase transition that is continuous, and DP, in 1 + 1 dimensions, and first-order in higher dimensions. Although, from the mean field, the first-order phase transition was expected in all dimensions, the larger fluctuations in the (1 + 1)-dimensional case are likely to destabilize the ordered phase [14], resulting in the observed DP transition.…”

Simple reaction-diffusion population model on scale-free networks

“…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].…”

“…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.…”