Coexistence and critical behavior in a lattice model of competing species

Abstract

In the present paper we study a lattice model of two species competing for the same resources. Monte Carlo simulations for d = 1,2, and 3 show that when resources are easily available both species coexist. However, when the supply of resources is on an intermediate level, the species with slower metabolism becomes extinct. On the other hand, when resources are scarce it is the species with faster metabolism that becomes extinct. The range of coexistence of the two species increases with dimension. We suggest that our model might describe some aspects of the competition between normal and tumor cells. With such an interpretation, examples of tumor remission, recurrence, and different morphologies are presented. In the d = 1 and d = 2 models, we analyze the nature of phase transitions: they are either discontinuous or belong to the directed-percolation universality class, and in some cases they have an active subcritical phase. In the d = 2 case, one of the transitions seems to be characterized by critical exponents that differ from directed-percolation ones, but this transition could be also weakly discontinuous. In the d = 3 version, Monte Carlo simulations are in a good agreement with the solution of the mean-field approximation. This approximation predicts that oscillatory behavior occurs in the present model but only for d 2. For d 2, a steady state depends on the initial configuration in some cases.

Critical behavior of a tumor growth model: Directed percolation with a mean-field flavor

“…Some preliminary results on this model were already reported [13]. Closely related models but without any preference in nutrient supply (∆ = 0) were also studied [12].…”

“…The general strategy to develop such an approach is to write a master equation and then solve it using some decoupling procedure [12,20,21]. In the first step one sums the states of all but one site.…”

“…In some cases the critical points in these models were shown to belong to the directed percolation universality class [9][10][11][12]. However, an implementation of biologically relevant factors, such as, for example, inter-or intraspecies competition, food supply, aging, birth, or death, results in rather complex dynamics.…”

“…Absorbing states appear also in some ecological or tumor growth models, where they correspond to the extinction of one or more species or types of cells. In some cases the critical points in these models were shown to be-long to the Directed Percolation universality class [9][10][11][12]. However, an implementation of biologically relevant factors, such as, for example, inter-or intra-species competition, food supply, aging, birth or death, results in rather complex dynamics.…”

Tumor proliferation and diffusion on percolation clusters

“…However, they did not compare the differences in the homogeneity of the tissues surrounding a tumor. Wendykier et al [54], Lipowski et al [55], Ferreira et al [56], and Moglia et al [57] used percolation to examine the critical behavior of avascular tumor growth, in which the survival and reproduction of tumor cells depended on the supplied nutrients. However, these studies did not systematically investigate the influence of the inhomogeneity of the surrounding tissue on tumor proliferation and diffusion.…”