Lost in translation? – Caveat to the application of the voter model in ecology and evolutionary biology

Abstract

The voter model is a paradigmatic model of competition between alternative states within groups. Its properties have been intensively studied in statistical physics. Due to its generality, the model lends itself to various applications in ecology and evolutionary biology. I briefly review these opportunities, but call attention to a frequently occurring misinterpretation: it is often assumed that the agents in the model represent individual organisms. I argue that this assumption only holds under very specific conditions, and thus the meaning of the agents is often ‘lost in translation’ between physics and biology. Instead of an individual-based view, I propose that an alternative, site-based approach is more plausible. I suggest that the biological applicability of the model could further be broadened by considering the transitional states of the agents (sites) explicitly and letting the network evolve according to the agents’ states.

Introduction

The voter model is an agent-based stochastic model. It can be considered as a paradigmatic model of competition between alternative states within groups. The group's structure is typically represented by a network. The agents have discrete states (red/blue in Figure 1). Each updating step starts by selecting a focal agent (F) randomly and uniformly from all the agents, and then choosing a neighbor (N) also randomly and uniformly from those agents that are linked to F. In the classical voter model, a Death-Birth (DB) rule is applied ¹ : the original state of F (blue) is annihilated, and N's state (red) is adopted. In the reverse voter model, ² also known as ‘invasion process’, ³ a Birth-Death (BD) rule is used: F transfers its state to N. Repeating the updating step (DB or BD) in a finite, connected network leads to a consensus, in which all the agents are in the same state. Typical questions concern the consensus time, and the states’ probabilities of winning from various initial conditions.^1–5

Figure 1.

Example for an update in the voter model. The state of each agent is shown in red/blue. F is the focal agent, N is its randomly selected neighbour. Under the Death-Birth (DB) rule, F adopts the state of N, while under the Birth-Death (BD) rule, it transfers its state to N.

After the introduction of the basic voter model, ⁶ many model versions were developed, especially in statistical physics (see Refs.^1,3,4 for reviews). Most of these have assumed that the competition is neutral (i.e. red and blue are equally strong), and the edges are typically undirected and uniform; but some model versions have permitted the updating rule to depend on the state of F and/or N.^1,3,7 It would be an important step towards real-life applications to let the network structure change, that is, to permit the appearance and disappearance of vertices and edges, especially according to the local states. The study of voter models on such evolving networks is in a pioneering stage yet. ⁸

Most applications of the voter model have been suggested in social sciences.^2,5 The use of the model in this field is fairly straightforward: the vertices represent individual people, and the links denote communication channels. The state of a vertex (red/blue) indicates an opinion, for example, a political view. This can be transferred from one agent to another through communication. Alternative opinions can thus spread within the network and compete.

Competition between alternative states is a central topic in biology as well. Nevertheless, the biological applications of this powerful model are only sparse and are mainly restricted to mathematical biology without any real-life examples (see Ref. ⁹ as an exception). Most of the biological applications have stemmed from the Moran process, a voter-like mean field model introduced in 1958 for the study of genetic drift in populations. ¹⁰ Later, many other model versions were introduced, including extensions to networks with various topologies^4,11,12 and the incorporation of both BD and DB rules.^1,4,12 In general, the voter model can be viewed as a general model of competition, including competition between species in ecosystems, or between genotypes within a species during evolution. Despite this potential, the model has not been widely used in biology (see examples in ecology^9,13–16 and evolutionary biology^1,4,11,12). Its broader application is an exciting matter of future research. To facilitate this, let me consider what the network's elements can represent in biological populations and communities.

Some challenges in the biological interpretation

The state of a vertex (red/blue in Figure 1) generally corresponds to a species in an ecological setting or a genotype in an evolutionary setting. It is noteworthy that these states are inherited. This is a fundamental difference from the usual social science setting, in which the state can be learned, that is, changed during individual life. A person can adopt the opinion of his neighbour; meanwhile, he can preserve his position in the network, for example, can keep his acquaintances. Conversely, an individual plant or animal cannot change its species or genotype. For a change, it must die and be replaced by another individual. This jeopardizes the biological realism of the voter model in fixed networks for the following reasons. A) It is difficult to imagine that an individual would be outcompeted by another individual in such a way that the winner could get exactly the same links that the loser had (e.g. all of its social relations). Note that the winner can belong to another genotype or species. Furthermore, it is a newborn, young individual. So, its position in the network is likely to be different from that of the former individual. B) It is not easy to find a plausible mechanism to ensure that whenever an individual dies, another individual would be born immediately. Such a linkage between birth and death can be imagined under the BD rule, assuming specifically that the birth of an individual is the direct cause of the death of another; but this does not work in the case of the DB rule. Although problem A is specific to networks, problem B emerges more generally, in mean field models as well. Altogether, the assumption that the vertices represent individuals does shrink the applicability of the model considerably.

An alternative, site-based approach

As an alternative approach, vertices can be considered as sites. The options that spatial populations can be modelled in a site-based vs. individual-based manner have been distinguished for decades in theoretical ecology (e.g. Ref. ¹⁷ ). I suggest rejecting the individual-based interpretation of the voter model (except for the rare cases in which the conditions mentioned in ‘Some challenges in the biological interpretation’ section are satisfied), sticking to the site-based view. In the simplest case, each vertex can correspond to a microsite (a small habitat patch) that can host a single individual (e.g. Refs.^13–16). A typical example of this approach is a case study in a tropical rainforest and a corresponding lattice model with voter dynamics. ¹⁰ The lattice constant was 1 m, ² and each lattice site represented a potential habitat for a single tree individual. Alternatively, each site may host a subpopulation (see below).

The voter model has been applied more intensively in evolutionary biology than in ecology, but primarily with an individual-based view. The investigations have partly stemmed from the classical voter model (e.g. Refs.^{1,7,11,12,18,19}), and partly from the Wright–Fisher model, in which the vertices represent demes (i.e. sub-populations ²⁰ ).

The individual-based approach is certainly disadvantageous when we cannot assume that the network's structure would be fixed (e.g. social connections may change; see above). Conversely, the assumption of fixed structure is more natural in a site-based model, because the vertices (sites) are anchored in space. It is noteworthy that classification of competition processes in ecology is closely related to this site-based interpretation of the voter model: the BD/DB rule corresponds to replacement/displacement competition. ²¹

In the case of the individual-based approach, the spatial resolution is necessarily fine, as it corresponds to the individuals’ size. Conversely, the site-based approach permits the consideration of coarser spatial scales as well. The vertices can represent habitat patches that are larger than a single individual; and thus, can host multiple individuals (e.g. Ref. ²⁰ ). This interpretation opens new opportunities: to apply the voter model in landscape ecology and metacommunity ecology, that is, in two branches of ecological research in which a network representation of habitat patches is becoming increasingly important.^22–25 In general, the voter model is one of the powerful tools in spatial ecology.^13,26 Within the rich variety of existing spatial dynamic models, it is specific in the sense that every ‘color’ (species, genotype, etc.) is active in terms of spreading (Figure 1). Thus the consideration of vacant sites is not self-evident: it requires an extension of the classical voter model.

Another promising, potential extension is also related to the set of states of the vertices. The original voter model does not contain any transitional state, in which both species/genotypes could be present. Therefore, it is necessary to implement a time scale separation, to assume that the establishment of individuals in new sites is fast relative to the time needed for reaching new sites. It would be an interesting matter of future research to relax this assumption. The consideration of transitional states has already started in social sciences (e.g. in the constrained voter model ²⁷ ) and could also broaden the biological applicability of the model significantly.

Summary

The voter model is a good candidate for becoming a baseline model for the study of competition in biological systems. A lot of knowledge has been built up in physics and mathematics about the model's behaviour, which could be utilized further in biology. This information includes, for example, the expected duration of competitive exclusion (in physics, the exit time), the local species turnover in habitat patches at different network positions, and the effect of displacement versus replacement on the dynamics of spreading. A considerable advantage of the voter model is that it is relatively simple and is open to various extensions towards the specificities of real-life systems.

For the sake of realism, I suggest considering the vertices as sites, not as individuals. Thus the states of vertices represent the kinds of species/genotypes that occupy the sites. Using this approach, the spatial resolution can be extended from the individual to the landscape scale, which broadens the applicability of the model. This could be facilitated further by considering the transitional states of the vertices explicitly and letting the network evolve according to the local states.