The neutral theory of biodiversity with random fission speciation
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Abstract
The neutral theory of biodiversity and biogeography emphasizes the importance of dispersal and speciation to macro-ecological diversity patterns. While the influence of dispersal has been studied quite extensively, the effect of speciation has not received much attention, even though it was already claimed at an early stage of neutral theory development that the mode of speciation would leave a signature on metacommunity structure. Here, we derive analytical expressions for the distribution of abundances according to the neutral model with recruitment (i.e., dispersal and establishment) limitation and random fission speciation which seems to be a more realistic description of (allopatric) speciation than the point mutation mode of speciation mostly used in neutral models. We find that the two modes of speciation behave qualitatively differently except when recruitment is strongly limited. Fitting the model to six large tropical tree data sets, we show that it performs worse than the original neutral model with point mutation speciation but yields more realistic predictions for speciation rates, species longevities, and rare species. Interestingly, we find that the metacommunity abundance distribution under random fission is identical to the broken-stick abundance distribution and thus provides a dynamical explanation for this grand old lady of abundance distributions.
Keywords
Sampling formula Maximum likelihood Fundamental biodiversity numberIntroduction
Understanding the assembly and biodiversity of ecological communities is the primary aim of community ecology. In an excellent minireview of ecological assembly rules, Belyea and Lancaster (1999) summarized the main drivers or constraints determining community structure: environmental constraints, dispersal constraints, internal dynamics (such as competition), and biogeography (which includes the processes of speciation and extinction). Most of the literature in the past decades has focused on the first three factors (see, e.g., Cody and Diamond 1975; Weiher and Keddy 2001). In contrast, biogeography and the processes of diversification have received relatively little attention despite the pioneering work by MacArthur and Wilson (1967), Collwell and Winkler (1984), and Ricklefs (1987) who clearly showed the importance of these processes for community composition and macro-ecological patterns. However, there has been a revived interest in the influence of biogeographical forces on community structure in the last decade due to two new developments. The first development is community phylogenetics, which studies evolutionary relationships between coexisting species (Webb et al. 2002; Cavender-Bares et al. 2009). The second is neutral community ecology, which suspends the role of species differences in order to create a null model that allows the study of factors other than asymmetrical species interactions such as dispersal and biogeography (Hubbell 2001). In this paper, we focus on the latter development and particularly on the impact of speciation on the shape of species abundance distributions.
Building on the classic works of MacArthur and Wilson (1967) and Caswell (1976), Hubbell (2001) introduced his neutral theory of community ecology that states that stochastic interplay between a few basic, ecological as well as macro-evolutionary, processes (speciation, birth, and death, and—on a local scale—dispersal) can explain general large-scale diversity patterns, such as species-abundance distributions (SADs) and species–area curves. The neutral theory as developed by Hubbell (2001) makes three basic assumptions: (1) individuals of different species are functionally equivalent (neutrality assumption), (2) the community size is constant (zero-sum assumption), and (3) speciation is comparable to mutation where each individual has an equal probability of producing mutated, i.e., speciated, offspring (point mutation assumption). While the neutrality assumption is at the heart of the theory, the other two assumptions are assumptions of particular model implementations of the theory, allowing for analytical expressions for diversity measures, rather than fundamental assumptions of the theory itself. A mismatch between observations and theoretical predictions can, in principle, be due to these additional assumptions and therefore cannot be immediately interpreted as calling for a rejection of the neutral theory as a whole (Etienne 2007).
The zero-sum assumption and the point mutation assumption are of a very different nature. It has been shown that models without the zero-sum assumption predict mathematically exactly the same equilibrium SAD as the model with this assumption (Etienne et al. 2007a; Haegeman and Etienne 2008; Conlisk et al. 2010). In contrast, alternatives to the point mutation assumption can predict very different SADs (Hubbell 2001; Etienne et al. 2007b; Allen and Savage 2007; Haegeman and Etienne 2009; De Aguiar et al. 2009). Indeed, Hubbell (2001) claimed that speciation would leave a signature on diversity patterns in the metacommunity (see also Mouillot and Gaston 2007). Therefore, a thorough analysis of neutral theory, or any theory of community assembly for that matter, requires the (quantitative) exploration of alternative modes of speciation, particularly those that are very different from point mutation. Hubbell (2001) proposed an alternative speciation mode that is the opposite of the sympatric point mutation mode which he dubbed “random fission” because speciation results from random splitting of populations which may be interpreted as mimicking allopatric speciation. Intuitively, it seems more reasonable than point mutation because the incipient abundance of new species is larger than a single individual (Allen and Savage 2007; Rosindell et al. 2010), and it is also more plausible than a fixed initial abundance, as assumed by Allen and Savage (2007). Hubbell (2001) stated that the equilibrium metacommunity SAD resulting from random fission speciation is a zero-sum multinomial, just like the local community SAD in the point mutation case—which was later called dispersal-limited (Etienne and Alonso 2005, 2007), or recruitment-limited (Jabot et al. 2008) multinomial—but he did not prove this mathematically. Ricklefs (2003) provided some approximate formulas for the total species richness in the metacommunity under random fission speciation, but so far, a full mathematical treatment has remained elusive.
In this paper, we provide the full sampling formula for the distribution of abundance in a local community that receives immigrants from a very large metacommunity described by random fission speciation. It is thus the counterpart of the sampling formula where the metacommunity is described by point mutation (Etienne 2005) and may be similarly extended to involve multiple samples (Etienne 2007, 2009a, b). The metacommunity SAD is clearly different from the zero-sum multinomial, in contrast to Hubbell’s conjecture. We use the new sampling formula to fit the random fission model to six large tropical tree data sets and compare it to the fit of the point mutation model. We end with a discussion of our results.
Model
We will first describe metacommunity dynamics, solve it for the stationary abundance distribution, and then derive expressions for (possibly dispersal-limited) local samples from this stationary distribution. We add the superscript “meta” to expectations and probabilities that refer to the metacommunity to distinguish them from expressions for samples, which we will denote by the superscript “smp”.
The master equation
The master equation (Eq. 1) together with the transition rates 3 and 5 fully define the neutral model with random fission speciation. Rather than trying to solve Eq. 1 for the random fission model which seems impossible, we take an indirect route consisting of three steps. First, we derive an equation for the expected number of species with abundance n which we will denote by \(\mathbb{E}^{ \rm{meta}}({S}_{n})\). We can solve this equation exactly. This will illustrate the main properties of the random fission model, but cannot be used to fit the model to data. To the latter end, we need the full sampling formula for the abundance distribution of a local community connected via limited dispersal to a metacommunity governed by random fission speciation. The second step starts by proposing an Ansatz for the solution of Eq. 1 and show that it is consistent with the previously obtained exact expression for \(\mathbb{E}^{\rm{meta}}({S}_{n})\). The third and final step applies sampling theory (Etienne and Alonso 2005) to this Ansatz, assuming large metacommunity size, in order to formulate the full sampling formula.
Equation for the expected number of species with abundance n
So far, we have been able to provide exact solutions for the expectation values of S _{ n } for the stationary abundance distribution. Although these can be used for comparison to data after incorporating sampling, it does not allow full extraction of the information in a sample abundance vector \(\vec{S }\). For this, the full probability distribution, rather than just the first moment, is required. This full probability, called the sampling formula, enables one to extract information on individual species’ abundances as well as on their interdependencies. We therefore proceed to derive an extremely good approximation to this sampling formula. We can check the accuracy of this approximation by comparing the expectation value \(\mathbb{E}^{\rm{meta }}({S}_{n})\) computed using this approximation (and higher moments) to the exact solution for \(\mathbb{E}^{\rm{meta}}({S}_{n})\).
Ansatz for the probability of the metacommunity abundance vector \(\vec{S}\)
This is a very interesting result because it corresponds exactly to the discrete version of MacArthur’s (1957) broken-stick model of the distribution of species abundances (see Etienne and Olff 2005)
Scaling limit
These approximations match exact numerical results for the random fission model outlined above very well. For example, Fig. 1 shows the extremely good match of the approximate expectations \(\mathbb{E}^{\rm{meta} }(S_{n})\) of Eq. 17a with the exact expectation given in Eq. 8. Similarly, the approximate second-order moments \(\mathbb{E}^{\rm{meta}}(S_{n}S_{m})\) of Eq. 17b are very close to the exact second-order moments \(\mathbb{ E}^{\rm{meta}}(S_{n}S_{m})\), solutions of Eq. 1 (results not shown). Because of this extremely good agreement, we believe that we can use the approximations with great confidence to explore the stationary properties of the random fission model.
Hence, all relative abundance vectors \(\vec{p}\) on S _{M} species are equally probable. Note that densities 18a and 18b can be obtained from Eq. 18c by computing marginal distributions.
Comparison of random fission and point mutation formulas for the metacommunity
Quantity | Formula | |
---|---|---|
θ _{rf} | \(\sqrt{\nu _{\rm{rf}}}J_{\rm{M}}\) | (19) |
θ _{pm} | \(\nu _{\rm{pm}}\left( J_{\rm{M}}-1\right) \approx \nu _{\rm{pm}}J_{\rm{M}}\) | (20) |
\({\mathbb{P}}\) | \(\dfrac{1}{ Z(\nu _{\rm{rf}}J_{\rm{M}},J_{\rm{M}})}\prod_{k}\dfrac{(\nu _{\rm{rf} }J_{\rm{M}})^{S_{k}}}{S_{k}}=\dfrac{1}{Z(\dfrac{\theta _{\rm{rf}}^{2}}{J_{ \rm{M}}},J_{\rm{M}})}\prod_{k}\dfrac{\theta _{\rm{rf}}^{2S_{k}}}{J_{ \rm{M}}^{S_{k}}S_{k}}\) | (21) |
\({\mathbb{P}}\) | \(\dfrac{J_{ \rm{M}}}{(\nu _{\rm{pm}}J_{\rm{M}})_{J_{\rm{M}}}}\prod_{k}\dfrac{ (\nu _{\rm{pm}}J_{\rm{M}})^{S_{k}}}{k^{S_{k}}S_{k}}=\dfrac{J_{\rm{M}} }{(\theta _{\rm{pm}})_{J_{\rm{M}}}}\prod_{k}\dfrac{\theta _{\rm{pm} }^{S_{k}}}{k^{S_{k}}S_{k}}\) | (22) |
\(\mathbb{E}_{\rm{rf}}^{\rm{meta}}(S_{n}|J_{\rm{M}})\) | \(\nu _{\rm{rf} }J_{\rm{M}}\dfrac{Z(\nu _{\rm{rf}}J_{\rm{M}},J_{\rm{M}}-n)}{Z(\nu _{ \rm{rf}}J_{\rm{M}},J_{\rm{M}})}\approx \dfrac{\theta _{\rm{rf}}^{2}}{ J_{\rm{M}}\sqrt{1-\dfrac{n}{J_{\rm{M}}}}}\dfrac{I_{1}\left( 2\theta _{ \rm{rf}}\sqrt{1-\dfrac{n}{J_{\rm{M}}}}\right) }{I_{1}(2\theta _{\rm{rf} })}\) | (23) |
\(\mathbb{E}_{\rm{pm}}^{\rm{meta}}(S_{n}|J_{\rm{M}})\) | \(\binom{J_{ \rm{M}}}{n}\theta _{\rm{pm}}\dfrac{\left( n-1\right) \rm{ }(\theta _{ \rm{pm}})_{J_{\rm{M}}-n}}{(\theta _{\rm{pm}})_{J_{\rm{M}}}}\approx \dfrac{\theta _{\rm{pm}}}{n}\left( 1-\dfrac{n}{J_{\rm{M}}}\right) ^{\theta _{\rm{pm}}-1}\) | (24) |
\({\mathbb{P}}\) | \( \dfrac{S_{\rm{M}}}{\binom{J_{\rm{M}}-1}{S_{\rm{M}}-1}}\prod_{k}\dfrac{1 }{S_{k}}\) | (25) |
\({\mathbb{P}}\) | \( \dfrac{J_{\rm{M}}}{\bar{s}(J_{\rm{M}},S_{\rm{M}})}\prod_{k}\dfrac{1}{ k^{S_{k}}S_{k}}\) | (26) |
\(\mathbb{E}_{\rm{rf}}^{\rm{meta}}(S_{n}|J_{\rm{M}},S_{\rm{M}})\) | \( \dfrac{S_{\rm{M}}}{\binom{J_{\rm{M}}-1}{S_{\rm{M}}-1}}\ \binom{J_{\rm{ M}}-n-1}{S_{\rm{M}}-2}\approx \dfrac{S_{\rm{M}}(S_{\rm{M}}-1)}{J_{\rm{ M}}}\left( 1-\dfrac{n}{J_{\rm{M}}}\right) ^{S_{\rm{M}}-2}\) | (27) |
\(\mathbb{E}_{\rm{pm}}^{\rm{meta}}(S_{n}|J_{\rm{M}},S_{\rm{M}})\) | \( \binom{J_{\rm{M}}}{n}\dfrac{\left( n-1\right) \rm{ }\bar{s}(J_{\rm{M} }-n,S_{\rm{M}}-1)}{\bar{s}(J_{\rm{M}},S_{\rm{M}})}\) | (28) |
\(\mathbb{P}\) | \(\frac{1 }{Z(\nu _{\rm{rf}}J_{\rm{M}},J_{\rm{M}})}\dfrac{(\nu _{\rm{rf}}J_{ \rm{M}})^{S_{\rm{M}}}}{S_{\rm{M}}}\binom{J_{\rm{M}}-1}{S_{\rm{M} }-1}\approx \dfrac{1}{Z(\theta _{\rm{rf}})}\dfrac{\theta _{\rm{rf}}^{2S_{ \rm{M}}}}{S_{\rm{M}}(S_{\rm{M}}-1)}\) | (29) |
\(\mathbb{P}\) | \(\bar{s} (J_{\rm{M}},S_{\rm{M}})\dfrac{\theta _{\rm{pm}}^{S_{\rm{M}}}}{(\theta _{\rm{pm}})_{J_{\rm{M}}}}\) | (30) |
\(\mathbb{E}_{\rm{rf}}^{\rm{meta}}(S_{\rm{M}}|J_{\rm{M}})\) | \(\dfrac{ _{1}F_{1}\left( 1-J_{\rm{M}},1,-\nu _{\rm{rf}}J_{\rm{M}}\right) }{ _{1}F_{1}\left( 1-J_{\rm{M}},2,-\nu _{\rm{rf}}J_{\rm{M}}\right) } \approx \theta _{\rm{rf}} \dfrac{I_{0}(2\theta _{\rm{rf}})}{ I_{1}(2\theta _{\rm{rf}})}\) | (31) |
\(\mathbb{E}_{\rm{pm}}^{\rm{meta}}(S_{\rm{M}}|J_{\rm{M}})\) | \( \sum_{k=1}^{J_{\rm{M}}}\dfrac{\theta _{\rm{pm}}}{\theta _{\rm{pm}}+k-1} =\theta _{\rm{pm}}\left( \Psi (\theta _{\rm{pm}}+J_{\rm{pm}})-\Psi (\theta _{\rm{pm}})\right) \) | (32) |
Sampling formula \({\mathbb{P}}\) and sample expectations \( \mathbb{E}^{\rm{smp}}(S_{n})\)
Comparison of random fission and point mutation formulas for a sample of size J
Quantity | Formula | |
---|---|---|
θ _{rf} | \(\sqrt{\nu _{\rm{rf}}}J_{\rm{M}}\) | (36) |
θ _{pm} | ν_{pm}(J_{M} − 1) ≈ ν_{ pm}J_{M} | (37) |
Metacommunity | (No dispersal limitation) | |
\({\mathbb{P}}\) | \( \dfrac{J}{\prod_{k}S_{k}}\theta _{\rm{rf}}^{S-J}\dfrac{I_{J+S-1}(2\theta _{ \rm{rf}})}{I_{1}(2\theta _{\rm{rf}})}\) | (38) |
\({\mathbb{P}}\) | \( \dfrac{J}{\prod_{i}n_{i}\prod_{k}S_{k}}\dfrac{\theta _{\rm{pm}}^{S}}{ \left( \theta _{\rm{pm}}\right) _{J}}\) | (39) |
\(\mathbb{E}_{\rm{rf}}^{\rm{smp}}(S_{n}|\theta _{\rm{rf}},J)\) | \( \left\{ {\begin{array}{ll} \dfrac{J}{I_{1}(2\theta _{\rm{rf}})}\sum_{k=1}^{J-n}\dfrac{\left( J-n-1\right) }{k\left( k-1\right) \left( J-n-k\right) }\dfrac{ I_{J+k}(2\theta _{\rm{rf}})}{\theta _{\rm{rf}}^{J-S-1}} \quad \rm{for }n<J \\ \dfrac{J}{I_{1}(2\theta _{\rm{rf}})}\dfrac{I_{J}(2\theta _{\rm{rf}})}{ \theta _{\rm{rf}}^{J-1}} \quad \rm{for }n=J \end{array}} \right. \) | (40) |
\(\mathbb{E}_{\rm{pm}}^{\rm{smp}}(S_{n}|\theta _{\rm{pm}},J)\) | \(\binom{ J}{n}\theta _{\rm{pm}}\dfrac{\left( n-1\right) (\theta _{\rm{pm}})_{J_{ \rm{M}}-n}}{(\theta _{\rm{pm}})_{J_{\rm{M}}}}\) | (41) |
Local community | (Dispersal limitation) | (41) |
\({\mathbb{P}}\) | \( \binom{J}{\overrightarrow{n}}\dfrac{1}{(I)_{J}\prod_{k}S_{k}} \sum_{a_{1}=1}^{n_{1}}\ldots \sum_{a_{S}=1}^{n_{S}}\left[ \prod_{i=1}^{S} \bar{s}(n_{i},a_{i})a_{i}\right] I^{A}\theta _{\rm{rf}}^{S-A}\dfrac{ I_{A+S-1}(2\theta _{\rm{rf}})}{I_{1}(2\theta _{\rm{rf}})}\) | (42) |
\({\mathbb{P}}\) | \( \binom{J}{\overrightarrow{n}}\dfrac{1}{(I)_{J}\prod_{k}S_{k}} \sum_{a_{1}=1}^{n_{1}}\ldots \sum_{a_{S}=1}^{n_{S}}\left[ \prod_{i=1}^{S} \bar{s}(n_{i},a_{i})(a_{i}-1)\right] I^{A}\dfrac{\theta _{\rm{pm}}^{S}}{ (\theta _{\rm{pm}})_{A}}\) | (43) |
\(\mathbb{E}_{\rm{rf}}^{\rm{smp}}(S_{n}|I,\theta _{\rm{rf}},J)\) | \( \left\{{ \begin{array}{ll} \binom{J}{n}\dfrac{1}{\left( I\right) _{J}}\sum_{k=1}^{J-n}\sum_{a_{1}=1}^{n} \sum_{a_{2}=k}^{J-n}\bar{s}(n,a_{1})\bar{s}(J-n,a_{2})\dfrac{ a_{1}a_{2}\left( a_{2}-1\right) }{k\left( k-1\right) \left( a_{2}-k\right) }\dfrac{I^{a_{1}+a_{2}}}{\theta _{\rm{rf}}^{a_{1}+a_{2}-k-1} }\dfrac{I_{a_{1}+a_{2}+k}\left( 2\theta _{\rm{rf}}\right) }{I_{1}(2\theta _{ \rm{rf}})} \quad \rm{for }n<J \\ \dfrac{1}{(I)_{J}}\sum_{a=1}^{J}\bar{s}(J,a)a\dfrac{I^{a}}{\theta _{\rm{rf} }^{a-1}}\dfrac{I_{a}(2\theta _{\rm{rf}})}{I_{1}(2\theta _{\rm{rf}})} \quad \rm{for }n=J \end{array}} \right. \) | (44) |
\(\mathbb{E}_{\rm{pm}}^{\rm{smp}}(S_{n}|I,\theta _{\rm{pm}},J)\) | \( \left\{{ \begin{array}{ll} \binom{J}{n}\dfrac{\theta _{\rm{pm}}}{\left( I\right) _{J}} \sum_{a_{1}=1}^{n}\sum_{a_{2}=1}^{J-n}\bar{s}(n,a_{1})\bar{s} (J-n,a_{2})\left( a_{1}-1\right) \dfrac{I^{a_{1}+a_{2}}}{\left( \theta _{ \rm{pm}}+a_{2}\right) _{a_{1}}} \quad \rm{for }n<J \\ \dfrac{\theta _{\rm{pm}}}{\left( I\right) _{J}}\sum_{a=1}^{n}\bar{s} (J,a)\left( a-1\right) \dfrac{I^{a}}{\left( \theta _{\rm{pm}}\right) _{a}} \quad \rm{for }n=J \end{array}} \right. \) | (45) |
When dispersal is not limited (i.e., I→ ∞), then we have the random fission counterpart of the Ewens sampling formula (Ewens 1972). This formula is provided as Eq. 38 in Table 2. The associated expressions for the expected number of species with abundance n in the dispersal-unlimited sample is given in Table 2, as Eq. 40. Derivations can be found in “Appendices 4 and Appendix 5”. For comparison, Table 2 also shows the results for the point mutation model (Vallade and Houchmandzadeh 2003; Etienne 2005; Etienne and Alonso 2005) which has θ _{pm} = ν _{pm} J _{M} where ν _{pm} is the point speciation rate that is comparable to ν _{rf}. Note that θ _{pm} is often written as \(\frac{\widetilde{\nu }_{ \rm{pm}}}{1-\widetilde{\nu }_{\rm{pm}}}(J_{\rm{M}}-1)\) in the literature. In “Appendix 7”, we explain this difference, but here, we note that ν _{pm} and \(\widetilde{\nu }_{\rm{pm}}\) are practically identical because they are very small and J _{M} is very large.
Results
Parameter estimates, maximum likelihood and model comparison for the fit of the two neutral models with different speciation modes to six local community abundance data sets
Site | J ^{a} | S ^{b} | Point mutation (pm)^{c} | Random fission (rf)^{c} | Comparison^{e} | |||||
---|---|---|---|---|---|---|---|---|---|---|
θ _{pm} | m | ML^{d} | θ _{rf} | m | ML^{d} | w _{pm} | w _{rf} | |||
BCI | 21,457 | 225 | 47.67 | 0.093 | − 308.73 | 595.1 | 0.0029 | − 311.92 | 0.96 | 0.04 |
Korup | 24,591 | 308 | 52.73 | 0.547 | − 317.04 | ∞ | 0.0020 | − 318.67 | 0.84 | 0.16 |
Pasoh | 26,554 | 678 | 190.9 | 0.093 | − 359.38 | 1,528 | 0.0098 | − 363.75 | 0.99 | 0.01 |
Sinharaja | 16,936 | 167 | 436.8 | 0.0019 | − 252.93 | 927.6 | 0.0019 | − 252.88 | 0.49 | 0.51 |
Yasuni | 17,546 | 821 | 204.2 | 0.429 | − 297.15 | 10,980 | 0.0111 | − 306.75 | 1.00 | 0.00 |
Lambir | 33,175 | 1,004 | 285.6 | 0.115 | − 386.38 | 2,500 | 0.0111 | − 402.32 | 1.00 | 0.00 |
Second, for Sinharaja, the point mutation and random fission models perform equally well, remarkably with almost identical m-values (but different θ-values). This suggests more strongly that there is extreme recruitment limitation and metacommunity diversity is high, in contrast to the values reported by Volkov et al. (2005), who missed the slightly higher likelihood optimum for high θ and low m (Etienne et al. 2007b).
For Yasuni, the ratio of θ _{pm} and θ _{ rf} is the smallest of all data sets (approximately \(\frac{1}{54}\)) which means that ν _{rf} < ν _{pm} if ν _{pm} < 0.0003. As values of ν _{pm} larger than 0.0003 are highly unlikely, we can conclude that random fission must occur at a lower rate than point mutation to fit the observed SAD.
As a numerical example, assume that metacommunity size in the neotropics is of order \(J_{\rm{M}}\approx 10^{10}\) (Ricklefs 2003; Nee 2005). This means that, for Yasuni, the corresponding speciation rates are \(\nu _{\rm{pm}}\approx 10^{-8}\) and \(\nu _{\rm{rf} }\approx 10^{-12}\). The average species longevities are then (Ricklefs 2003) \(t_{\rm{pm}}\approx -2\ln (2\nu _{\rm{pm} })\approx 10^{2}\) and \(t_{\rm{rf}}\approx \nu _{\rm{rf}}^{-\frac{1}{2} }\approx 10^{6}\), and the expected total richness is \(\mathbb{E}_{\rm{rf} }^{\rm{meta}}(S_{\rm{M}})\approx 10^{4}\) (see Eq. 1) and \(\mathbb{E} _{\rm{pm}}^{\rm{meta}}(S_{\rm{M}})\approx \) 4·10^{3} (see Eq. 1). We discuss these results below. Note that all these speciation rates are community speciation rates divided by community death–birth events (remember we assumed the latter to be μ = 1). For an estimate of the species-level speciation rates measured in inverse generations, these numbers need to be multiplied by the average population size \(\frac{J_{\rm{ M}}}{\mathbb{E}^{\rm{meta}}(S_{\rm{M}})}\).
Parameter estimates, maximum likelihood, and model comparison for the fit of the two neutral models with different speciation modes to two metacommunity abundance data sets
Metacommunity | J ^{a} | S ^{b} | Point mutation (pm)^{c} | Random fission (rf)^{c} | Comparison^{e} | |||
---|---|---|---|---|---|---|---|---|
θ _{pm} | ML^{d} | θ _{rf} | ML^{d} | w _{pm} | w _{rf} | |||
Panama (pooled sample) | 16,292 | 759 | 164.7 | − 311.09 | 795.8 | − 795.51 | 1.00 | 0.00 |
Panama (repeated samples) | 41 | 35.4 | 119.8 | − 3.4644 | 252.3 | − 3.5348 | 0.48 | 0.52 |
Western Ghats (pooled sample) | 19,555 | 304 | 50.99 | − 298.65 | 308.5 | − 524.03 | 1.00 | 0.00 |
Western Ghats (repeated samples) | 50 | 30.4 | 51.45 | − 5.9220 | 116.3 | − 6.1143 | 0.53 | 0.47 |
Discussion
In this paper, we have presented a full sampling formula for the SAD in the neutral model with random fission speciation, with and without recruitment limitation. In contrast to Hubbell’s (2001) conjecture, the metacommunity abundance distribution (i.e., without recruitment limitation) is not a zero-sum multinomial. In fact, we have shown that the random fission mode of speciation produces a SAD that is identical to the broken-stick model of MacArthur (1957), when conditioned on total metacommunity size and total metacommunity species richness. An expression for the SAD (i.e., \({\mathbb{P}}\)) for the discrete broken-stick (DBS) model was given in Etienne and Olff (2005), and this expression is mathematically identical to our Eq. 14. Our Eq. 18c is the continuous form of the broken-stick model (CBS), that is, in the limit of J _{M}→ ∞. Thus, random fission speciation provides a mechanistic explanation of this classic broken-stick model. Indeed, random fission and random stick-breaking are equivalent processes mathematically, but is not immediately obvious that they lead to the same distribution because the random fission model includes ecological drift (random birth and death processes) whereas the broken-stick model is not clearly linked to biological processes.
Similarly, our dispersal-limited sampling formula can be interpreted as a dispersal-limited broken-stick model. Although the fits to data are not better than the point mutation model (which produces the logseries), adding dispersal limitation makes the fits much better than found by Etienne and Olff (2005) for the pure DBS model. We might therefore say that the broken-stick model has been resurrected to some extent from the natural death, so longed for by its inventor (MacArthur 1966). Cohen (1968) also showed that the broken stick could be produced by alternative models, but none of these were dynamical such as our birth–death-speciation model. The random fission model is also more general than these models because it also predicts an SAD without conditioning on total species number.
In contrast to another conjecture by Hubbell (2001), the metacommunity abundance distribution is not governed by two parameters (ν _{rf} and J _{M}), but by only one just like in the point mutation model. This parameter, the random fission fundamental biodiversity number, is defined as \(\theta _{\rm{rf}}=\sqrt{\nu _{\rm{rf}}}J_{\rm{M}}\). This definition of θ _{rf} is different than that of θ _{ pm} by a factor of \(\sqrt{\nu _{\rm{rf}}}\). This explains why Hubbell (2001) thought that two independent parameters govern the random fission model: He found with his simulations that fixing ν _{rf} J _{ M} did not fully determine the abundance distribution. Had he fixed \( \sqrt{\nu _{\rm{rf}}}J_{\rm{M}}\), then he would have found that the SAD was completely specified. Strictly, this is only true in the limit of large J _{M} that we are considering, but in practice, this limit is always approximating the system extremely well.
We have compared the fit of the random fission model against the point mutation model in three different ways. First, we confronted six local community data sets with the dispersal-limited sampling formulas. We found that the random fission model never performed significantly better than the point mutation model and did much worse in several cases. However, this may be due to nonneutral recruitment limitation (Jabot et al. 2008) obscuring the comparison of speciation modes. We therefore also compared fits of the models to metacommunity data directly. Unfortunately, large metacommunity data sets, where individuals are randomly sampled across a large spatial scale, are scarce. At best, there are a limited number of local samples, e.g., 41 and 50 in the data sets we used here. Although we recognize that the sampling effort for these data sets is appreciable, these data sets require some restrictive assumptions in their analysis. Our second comparison consisted of pooling all the samples as if the individuals were randomly sampled rather than sampled in local clusters, thereby ignoring spatial structure. Here, the random fission model performed miserably. Our third comparison consisted of repeatedly sampling one individual from each plot and averaging results over all these small samples (of sizes 41 and 50), thereby ignoring that the data showed an actual metacommunity richness that was much higher than that of each of those samples. In this case, random fission and point mutation performed equally well. Remarkably, the maximum likelihood estimates for θ _{pm} are roughly of the same order of magnitude for the pooled sample and repeated samples, whereas the estimates for θ _{rf} are three-fold smaller in the repeated samples than in the pooled samples. This suggests that the random fission model does not produce the correct scaling with sample size and even more so because the sample size in the pooled sample is larger than a true metacommunity sample with the same number of species, due to recruitment limitation. Thus, we conclude that the point mutation model really seems a much better description of the metacommunity than random fission model. The relatively reasonable fit of the random fission model to some of the local community data sets is only achieved by allowing for strong (and arguably unrealistic) dispersal limitation which eradicates the signature of the speciation mode (recall Fig. 4).
The performance of a model in producing realistic SADs depends on all ingredients of the model (Etienne 2007), not just the one under consideration, such as the speciation mode in this paper. We have studied the speciation mode in a neutral context, so the performance is also affected by the neutrality assumption. Random fission or point mutation or both may perform better or worse in a nonneutral setting, or even have negligible influence. Zillio and Condit (2007) have, however, found that many communities, neutral or non-neutral, are primarily driven by the process of how new species enter the community. In our opinion, this justifies our study of the influence of speciation in the simplest context where species differences do not play a role, but we do not rule out the possibility that the mode of speciation may lead to more or less realistic predictions under nonneutral conditions. Only by formulating different models and comparing their predictions to data can we get a feel for the behavior of these models and for how informative SADs are for inferring mechanism.
Speciation is not the only way new species can enter the community: Immigration is another. Interestingly, the sampling formula resulting from point mutation speciation also describes long-distance dispersal (see, e.g., Etienne et al. 2007a) which may leave a more profound signature than speciation. Only a model that includes both long-distance dispersal and speciation will be able to distinguish the relative contribution of these processes. An analytically tractable spatially implicit model with long-distance dispersal and random fission speciation is not yet available, let alone a spatially explicit model. Nevertheless, a spatially explicit neutral model with long-distance (fat-tailed) dispersal and point mutation speciation has been studied (Rosindell and Cornell 2009) using coalescence techniques (Rosindell et al. 2008). It has been shown that speciation rates required to yield predictions on species–area relationships that are in agreement with observations are much lower (i.e., more realistic) than in models with Gaussian dispersal kernels (Rosindell and Cornell 2007). This suggests that long-distance dispersal is an important force shaping ecological communities and can mimic the effect of speciation, but does not rule out that speciation still leaves a signature on community structure. Unfortunately, coalescence does not seem compatible with random fission speciation because random fission is not time reversible, and therefore, this powerful simulation method cannot be employed to further study the effect of random fission speciation on community structure in a spatially explicit context. Perhaps new analytical techniques based on functional differential equations (O’Dwyer et al. 2009) will prove useful.
Random fission assigns a very low probability to samples with a few very abundant species, even more so than point mutation. This explains why the fit to data is worse for the former speciation mode and the fact that the best fitted model does not seem to follow unimodal SAD data in some cases (even though the unimodal shape is more characteristic for random fission than for point mutation, see Figs. 3 and 4). Etienne et al. (2007b) pointed out that a visual goodness-of-fit estimate may be deceiving and logtransformed the data and predictions to make this clear. Here, logtransformation supports the same conclusion (not shown). At the same time, random fission predicts fewer rare species in the metacommunity than point mutation and therefore produces a more lognormal-like shaped SAD which is believed to be a better description by many ecologists (Preston 1948, 1962; McGill 2003). The fit to the metacommunity data sets suggests that this advantage does not offset the disadvantage at large abundances. In fact, this advantage may not be so advantageous after all because the random fission model even seems to predict too few rare species.
For Hubbell’s neutral models ν is a constant speciation rate per individual (for both point mutation and random fission), in contrast to common practice in speciation research where the speciation rate is usually a rate per species (Stanley 1979; Etienne and Apol 2009). The results in this paper apply to Hubbell’s model. Assuming a constant speciation rate per species in the neutral model with point mutation has also been studied (Etienne et al. 2007b), and it was found that this assumption can make a large difference. Therefore, the remaining combination of random fission speciation with a constant speciation rate at the species level appears to be a necessary last step to get a complete picture. In fact, while a constant speciation rate per individual seems a logical first choice for the individual-level process of point mutation, a constant speciation rate per species seems more obvious for the species-level process of random fission. The community will be less diverse than in the individual-level case because abundant species will be less likely to undergo fission. For the same reason, the species-level point mutation model produced less diversity than the individual-level point mutation model (Etienne et al. 2007a). Figure 7 shows a numerical comparison of all four speciation modes. One observes that species-level random fission is similar to individual-level random fission but has a bit more abundant and rare species. Because very abundant species are not only more likely than in individual-level random fission model but also than in the individual-level point mutation model (this can only be seen after logtransforming Fig. 7), it might be that a model with species-level random fission speciation provides a good fit to SAD data as well as reasonable predictions for speciation rates and species longevities. We have not been able to find a sampling formula for this model, so this remains an open question.
We have presented analytical results for the spatially implicit model with the random fission mode of speciation. These are now on a par with analytical results for point mutation, allowing future studies of community models to compare both alternatives.
Notes
Acknowledgements
We thank Franck Jabot and a few anonymous reviewers for their constructive comments. RSE thanks the financial support of The Netherlands Organization for Scientific Research (NWO).
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