General Properties of a System of $S$ Species Competing Pairwise

We consider a system ofN individuals consisting of S species that interact pairwise: xm + xl → 2xm with arbitrary probabilities plm. With no spatial structure, the master equation yields a simple set of rate equations in a mean field approximation, the focus of this note. Generalizing recent findings of cyclically competing threeand four-species models, we cast these equations in an appealingly simple form. As a result, many general properties of such systems are readily discovered, e.g., the major difference between even and odd S cases. Further, we find the criteria for the existence of (subspaces of) fixed points and collective variables which evolve trivially (exponentially or invariant). These apparently distinct aspects can be traced to the null space associated with the interaction matrix, plm. Related to the leftand rightzero-eigenvectors, these appear to be “dual” facets of the dynamics. We also remark on how the standard Lotka-Volterra equations (which include birth/death terms) can be regarded as a special limit of a pairwise interacting system. Introduction. Population dynamics is a venerable subject, dating back two centuries to Malthus, Verhulst, Lotka, Volterra, and many others[1, 2, 3]. Nonetheless, new and interesting phenomena are continually being discovered. For example, many studies of cyclic competition between 3 species (with no spatial structure, e.g., a well-mixed system) attracted considerable recent attention [4]. In fact, in 2009, Science Daily popularized this topic[5] by branding it “Survival of the Weakest.” We extended this investigation to a system with 4 species[6], which displayed no such counter-intuitive behavior. Instead, we found an intuitively understandable principle which underpins all systems with cyclically competing species, namely, “The prey of the prey of the weakest is least likely to survive.” In the case of cyclically competing 3 species, the prey of one’s prey is also one’s predator. Thus, its demise is indeed “good news” for the weakest and leads to the eye-catching headline. In this short note, we considered a wider range of systems of S species interacting pairwise, with arbitrary rates. Focusing only on a mean field description (i.e., rate equations), we find remarkable general properties, such as fixed points, invariant manifolds, collective variables with simple time dependence, as well as a (possibly new) form of “duality.” We begin by specifying the individual based stochastic model, from which our MF approximation is derived. This note is devoted only to the properties of solutions to the MF equations, however. Individual based model. Consider a system with N individuals, each being a member of one of S species. Let us denote the species by xm, with m = 1, ..., S, and the number General Properties of a System of S Species Competing Pairwise 2 of individuals of each by Nm. We allow only pairwise interactions, i.e., xm + xl pm −→ 2xm . (1) where pm are arbitrary probabilities for a “predator” xm to consume a “prey” xl. Note that, if we wish to model bi-directional interactions, then these p’s represent the net consumption of the dominant species. Thus, there are at most S (S − 1) /2 such positive quantities. When two individuals encounter, the role of each is well defined: one is the predator, the other is the prey. To emphasize, we illustrate with p1 = 0.7, and both x1 + x5 and x5 + x1 becomes 2x1 with probability 0.7 (and unchanged with probability 0.3). Defining an update of our system as randomly choosing a pair and letting them interact, we see that the Nm’s change by ±1 or 0 with N = ∑ mNm remaining a constant for all times. Depending on the pm’s, there are at least S absorbing states. The appropriate quantity for describing this stochastic evolution is, of course, P ({Nm} ; τ), the probability to find the system with the set {Nm} after updating it τ times from some given initial P ({Nm} ; 0). The change in one step, P ({Nm} ; τ + 1)− P ({Nm} ; τ), is given by 2/N (N − 1) times