## Introduction

The neutral theory has become one of the pillars of macroecology (Watterson 1974; Caswell 1976; Hubbell 2001; reviews by Chave 2004; Alonso *et al.* 2006; Etienne & Alonso 2006; Hu *et al.* 2006). However, many ecologists doubt that the variety of life can be properly described by a theory based on the assumption that there are no ecological differences among species (according to the standard definition; Hubbell 2001, p. 7; Hu *et al.* 2006). Here, we introduce a radical change of perspective and start from the opposite assumption. We rigorously derive the species abundance distribution (SAD) to be expected when neglecting all ecological similarities among species, instead of neglecting their differences. We call our species ‘idiosyncratic’, in contraposition to the ‘equivalent’ species of the neutral theory. Strikingly, we find exactly the same SAD that is found in simple neutral models: the log-series. We could trace an imaginary line between the extremes of strict neutrality and strict idiosyncrasy and all models on this line would display a log-series, while moderate departures away from the line would lead us to the power law and the skewed log-normal. This suggests a general explanation for virtually all empirical SADs, and, indirectly, for the main types of species–area relationship (SAR).

These findings come after a series of observations in the literature indicating that multiple models, both neutral and non-neutral, lead to similar diversity patterns (McKane *et al.* 2000; Chave *et al.* 2002; McGill 2003a; Mouquet & Loreau 2003; Tilman 2004; Volkov *et al.* 2005; Pueyo 2006a; Nekola & Brown 2007; Zillio & Condit 2007). These patterns transcend particular models and can be best understood by using approaches that also transcend particular models.

The conventional approach to ecological theory is based on mechanistic modelling. The use of mechanistic models often forces us to choose either ignoring the complexity of nature or using so many parameters that hardly any reliability and generality can be expected. However, complexity is not intrinsically incompatible with reliability and generality. If species with diverse ecological features coexist, their singularities may cancel out in community-level measures and give rise to robust regularities. A promising alternative to the analysis of particular models is the study of the statistical properties of large ensembles of complex ecological models, with the aim of identifying such regularities. This is also in the spirit of the log-normal hypothesis, but this hypothesis relies on the precise assumptions of the central limit theorem, and there is no clear justification why these should apply to SADs (Williamson & Gaston 2005). Here we give new results specifically for SADs, using the maximum entropy formalism (MaxEnt) and other related tools, which are well established in statistical physics.

The use of MaxEnt in ecology has a venerable but little known history. Shortly after Jaynes (1957) introduced this method to statistical physics, MacArthur (1960) used a mathematically identical procedure and obtained the ‘broken stick’. However, this is not a realistic SAD. The right solution could not be possibly obtained without the key findings that Jaynes (1968) later added to MaxEnt theory (see ‘The prior distribution’ section). Thereafter, there have been a few isolated attempts to apply MaxEnt to species diversity (Alexeyev & Levich 1997; Levich 2000; Pueyo 2006a; Shipley *et al.* 2006; see also McGill 2006) and related areas (e.g. Luriè & Wagensberg 1983; Wagensberg *et al.* 1991; Hernández *et al.* 2006; Hijmans & Graham 2006; Phillips *et al.* 2006; Pearson *et al.* 2007) but, as far as we know, the way we use it to predict the SAD is entirely new. We compare it with earlier approaches in Appendix A.

Figure 1 places the idiosyncratic theory in the context of other previous views of community assemblage. The word ‘niche’ is used in a broader sense than usual, including not only resources but also, e.g. environmental conditions, consumers, infectious diseases and mutualists.

Neutral models assume that all species have the same niche, so neutrality corresponds to ‘simple niche apportionment rules’ and ‘high niche overlap’ (Fig. 1). Some parts of Hubbell's (2001) book seem to imply a wider definition of neutrality, but all mathematical results are based on models without niche differentiation (this also applies to the recent extensions of the theory allowing for species-dependent vital rates; Solé*et al.* 2004; Etienne *et al.* 2007; see also Pueyo 2006a, p. 395). The SADs in these models are mainly shaped by a particular mechanism: demographic noise. In principle, a high niche overlap is needed for this mechanism to dominate.

Engen & Lande (1996a) gave some useful tools to predict SADs in more complex models. For example, the inset in the lower left end of Fig. 1 has been obtained with their method, assuming the classical logistic equation plus a moderate environmental noise, but no demographic noise. The absence of demographic noise means that there is no niche overlap and that this model is not neutral. Indeed, the predicted SAD is completely different from that of neutral models. However, we used the same parameter values for all species (*r*, *K* and environmental noise variance *ɛ*^{2}), thus introducing a strong symmetry among them. As each species has a different niche, this symmetry does not imply a common resource use or shared interactions of any kind, unlike the main symmetries of neutral models. Therefore, it is a qualitatively different, more abstract type of symmetry, which we call ‘non-neutral symmetry’. The inset in Fig. 1 is one of the simplest examples, but we could design a non-neutral symmetric model for any conceivable SAD. The set of niche apportionment models in Tokeshi (1990; e.g. dominance pre-emption or dominance decay), in which a fix and simple rule is sequentially applied to each of the species in the community, are also non-neutral symmetric models.

Idiosyncrasy is defined by the non-existence of symmetries, either of the neutral or the non-neutral type. Each species is ‘idiosyncratic’ because it is fundamentally different from any other species. Engen & Lande (1996b) gave an important step to idiosyncrasy by extending their equations to sets of species with heterogeneous parameter values, which were assigned at random. However, this method does not necessarily give a fully idiosyncratic SAD. For example, if we used a logistic with *K* following a Gaussian distribution of parameters *μ*_{K} and *σ*_{K}, and applied a similar criterion to *r* and *ɛ*, we would still be assuming particular values for {*μ*_{K}, *σ*_{K}, *μ*_{r }, *σ*_{r }, *μ*, *σ*}, and also ignoring possible deviations from the logistic equation, so we would have a residual of non-neutral symmetry. In this paper, we derive the SAD that results from randomness in a more fundamental sense, free of any such residual.

The SAD gives the probability that an unspecified species will have some given abundance *n*. It has two components:

- 1The probability for a species chosen at random to display some given ecological features.
- 2The probability that a species with some given ecological features has abundance
*n*.

By assuming that all species are ecologically equivalent, the neutral theory assumes minimum variability in the first component and maximum in the second. The idiosyncratic theory assumes maximum variability in the first, and either small or large variability in the second. The net result is maximum variability in species abundances in both theories, for completely different reasons.