Rank abundance relations in evolutionary dynamics of random replicators

Abstract

We present a non-equilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multi-species replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and anti-symmetric cases. RARs are obtained analytically via a generating functional analysis, describing fixed-point states of the system in terms of a small set of order parameters, and in dependence on the symmetry or otherwise of interactions and on the productivity of the community. Our work is an extension of Tokita [Phys. Rev. Lett. 93 178102 (2004)], where the case of symmetric interactions was considered within an equilibrium setup. The species abundance distribution in our model come out as truncated normal distributions or transformations thereof and, in some case, are similar to left-skewed distributions observed in ecology. We also discuss the interaction structure of the resulting food-web of stable species at stationarity, cases of heterogeneous co-operation pressures as well as effects of finite system size and of higher-order interactions.

Analytical theory of species abundance distributions of a random community model

“…8, Lyapunov function f in the symmetric case or L p ðxÞ in the antisymmetric case is not known and we cannot use the methods of equilibrium statistical mechanics such as the replica method (Mezard et al 1987;Nishimori 2001)in the case of symmetric interactions. There are, however, some techniques of nonequilibrium statistical physics using the path integral or the generating functional to analyze the global behavior of a system with random interactions (Rieger 1989;Opper and Diederich 1992;Galla 2005Galla , 2006Yoshino et al 2007Yoshino et al , 2008. Such techniques have been recently applied to cross-disciplinary areas such as neural networks (Gardner et al 1987;Rieger et al 1988;Düring et al 1998;Katayama and Horiguchi 2001;Mimura et al 2004), and the minority game in economics and social science (Challet et al 2005;Coolen 2005).…”

“…where À1 c 1 is the level of symmetry the same as in Eq. 8 (Opper and Diederich 1992;Galla 2006;Yoshino et al 2007Yoshino et al , 2008. The scaling factor 1=N is introduced for the same reason in Eq.…”

“…Parameter u([0) is productivity which is same as in the case of symmetric interactions in Eq. 12 and is also termed ''cooperation pressure'' because diversity (the number of coexisting species) is in general a monotonically increasing function of u (Diederich and Opper 1989;Opper and Diederich 1992;Tokita 2004;Yoshino et al 2007Yoshino et al , 2008. We also here use f ðtÞ ð1=NÞ P N j¼1 P N k¼1 a jk x j x k for the average fitness.…”

“…Some important results obtained through the analyses mentioned above (Opper and Diederich 1992;Galla 2006;Yoshino et al 2008) are summarized below and in Table 1.…”

“…In the field of community ecology, one of the mostly observed quantities is the species abundance distribution (SAD) FðxÞ, which is the frequency of species having a population of x (May 1975). Recently, two analytical theories on SAD have been progressed: the neutral theory (Hubbell 2001) and the random community model (May 1972;Tokita 2004;Yoshino et al 2008). In this paper, we review such studies on the random community model.…”