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A stochastic biodiversity model with overlapping niche structure

Abstract

The niche is a fundamental ecological concept that underpins many explanations of patterns of biodiversity. The complexity of niche processes in ecological systems, however, means that it is difficult to capture them accurately in theoretical models of community assembly. In this study, we build upon simple neutral biodiversity models by adding the important ingredient of overlapping niche structure. Our model is spatially implicit and contains a fixed number of equal-sized habitats. Each species in the metacommunity arises through a speciation event; at which time, it is randomly assigned a fundamental niche or set of environments/habitats in which it can persist. Within each habitat, species compete with other species that have different but overlapping fundamental niches. Species abundances then change through ecological drift; each, however, is constrained by its maximum niche breadth and by the presence of other species in its habitats. Using our model, we derive analytical expressions for steady-state species abundance distributions, steady-state distributions of niche breadth across individuals and across species, and dynamic distributions of niche breadth across species. With this framework, we identify the conditions that produce the log-series species abundance distribution familiar from neutral models. We then identify how overlapping niche structure can lead to other species abundance distributions and, in particular, ask whether these new distributions differ significantly from species abundance distributions predicted by non-overlapping niche models. Finally, we extend our analysis to consider additional distributions associated with realized niche breadths. Overall, our results show that models with overlapping niches can exhibit behavior similar to neutral models, with the caveat that species with narrow fundamental niche breadths will be very rare. If narrow-niche species are common, it must be because they are in a non-overlapping niche or have countervailing advantages over broad-niche species. This result highlights the role that niches can play in establishing demographic neutrality.

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Acknowledgments

The authors thank Stephen Pacala, Paul Armsworth, Joe Hughes, and Nathan Sanders for helpful discussions. RAC acknowledges the financial support of the Smithsonian Institution Global Earth Observatories and the HSBC Climate Partnership. SB, EA, and WG conducted this work while Postdoctoral Fellows at the National Institute for Mathematical and Biological Synthesis (NIMBioS), an Institute sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #EF-0832858, with additional support from the University of Tennessee, Knoxville. RAC was assisted by attendance as a Short-term Visitor at NIMBioS. We acknowledge the support of the University of Tennessee, Division of Biology Field Station.

Author information

Correspondence to Sharon Bewick.

Appendices

Appendices

Appendix 1: Numerical simulations of metacommunity SAD

We ran numerical simulations to assess the accuracy of the approximation made in the derivation of Eq. (6) for the metacommunity SAD. First, we verified the approximation that ξ j , the number of individuals potentially capable of living in a particular habitat, is roughly constant and equal to \( {\overline{\xi}}_M \) (see Appendix 2, Eq. (29)) and found this to hold (Fig. 6 shows this for just one set of parameter values). We then ran SAD simulations (Figs. 7, 8, 9, and 10). We parameterized these with a composite parameter \( {\nu}_{\max }=\frac{\nu }{h_{\max }} \), which describes the rate at which species of maximum niche breadth arise. Each simulation was repeated 50 times, and results were averaged over the simulations. The simulations confirm that our approximation is generally good but that deviations occur at low diversity (e.g., Fig. 7). Further simulations at lower diversity (not shown) confirmed that the deviations from the theoretical approximation in Eq. (6) occur because the number of individuals in the community able to survive in a given habitat (ξ j ) varies across habitats in these scenarios.

Fig. 6
figure6

The number of individuals ξ j potentially capable of living in a particular habitat in a simulation with K = 4, h max = 4, J M  = 10, 000, ν max = 0.005, and κ(h) = 1/4. Results are averaged across 50 repeats of the same simulation. The left panel shows the mean value of ξ j across all K = 4 habitats (black) along with a 95 % confidence interval (gray). The red line is the theoretical value of \( {\overline{\xi}}_M \) (see Appendix 2, Eq. (29)). The right panel shows the coefficient of variation of ξ j across all K = 4 habitats (black), along with a 95 % confidence interval (gray)

Fig. 7
figure7

SADs for the theoretical neutral model (thick gray lines) and the basic theoretical (thin black line, open circles) and simulated (thin black line, closed circles) overlapping niche models assuming a uniform distribution of niche breadths at speciation. Simulation results are presented as averages over 50 trials. Parameters are K = 4, J M  = 10, 000, and ν max = 0.0005. Values of h max vary across panels: a h max = 1, b h max = 2, c h max = 3, d h max = 4

Fig. 8
figure8

SADs for the theoretical neutral model (thick gray lines) and the basic theoretical (thin black line, open circles) and simulated (thin black line, closed circles) overlapping niche models assuming a uniform distribution of niche breadths at speciation. Simulation results are presented as averages over 50 trials. Parameters are K = 4, J M  = 10, 000, and ν max = 0.005. Values of h max vary across panels: a h max = 1, b h max = 2, c h max = 3, d h max = 4

Fig. 9
figure9

SADs for the theoretical neutral model (thick gray lines) and the basic theoretical (thin black line, open circles) and simulated (thin black line, closed circles) overlapping niche models assuming a uniform distribution of niche breadths at speciation. Simulation results are presented as averages over 50 trials. Parameters are K = 16, J M  = 10, 000, and ν max = 0.0005. Values of h max vary across panels: a h max = 1, b h max = 6, c h max = 11, d h max = 16

Fig. 10
figure10

SADs for the theoretical neutral model (thick gray lines) and the basic theoretical (thin black line, open circles) and simulated (thin black line, closed circles) overlapping niche models assuming a uniform distribution of niche breadths at speciation. Simulation results are presented as averages over 50 trials. Parameters are K = 16, J M  = 10, 000, and ν max = 0.005. Values of h max vary across panels: a h max = 1, b h max = 6, c h max = 11, d h max = 16

Appendix 2: Basic model SADs

Metacommunity SADs

The expression for the expected number of species with abundance n and niche breadth h in the metacommunity at equilibrium is (following Vallade and Houchmandzadeh 2003):

$$ \left\langle {\phi}_M\left(n,h\right)\right\rangle \approx {\displaystyle \prod_{i=0}^{n-1}}\frac{\ {\overline{b}}_M\left(i,h\right)}{{\overline{d}}_M\left(i+1,h\right)} $$
(18)

Substituting Eqs. (3) and (4) from the main text into Eq. (18), we find

$$ \left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{\nu }{{\overline{d}}_M\left(n,h\right)}\kappa (h){\displaystyle \prod_{i=1}^{n-1}}\frac{\left(\frac{h{J}_M}{K}-i\right)}{\left({\overline{\xi}}_M-i\right)+\frac{\left({\overline{\xi}}_M-1\right)\nu }{1-\nu }} $$
(19)

Simplifying gives

$$ \begin{array}{l}\left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{\nu }{{\overline{d}}_M\left(n,h\right)}\kappa (h)\frac{\Gamma \left(\frac{h{J}_M}{K}\right)}{\Gamma \left(\frac{h{J}_M}{K}-n+1\right)}\frac{\Gamma \left({\overline{\xi}}_M+\frac{\left({\overline{\xi}}_M-1\right)\nu }{1-\nu }-n+1\right)}{\Gamma \left({\overline{\xi}}_M+\frac{\left({\overline{\xi}}_M-1\right)\nu }{1-\nu}\right)}\\ {}=\frac{\left({\overline{\xi}}_M-1\right){J}_M\nu }{n\left(1-\nu \right)}\kappa (h)\frac{\Gamma \left(\frac{h{J}_M}{K}\right)}{\Gamma \left(\frac{h{J}_M}{K}-n+1\right)}\frac{\Gamma \left({\overline{\xi}}_M+\frac{\left({\overline{\xi}}_M-1\right)\nu }{1-\nu }-n\right)}{\Gamma \left({\overline{\xi}}_M+\frac{\left({\overline{\xi}}_M-1\right)\nu }{1-\nu}\right)}\\ {}=\frac{\theta^{\ast }K}{nh\ }\kappa (h)\frac{\Gamma \left(\frac{h{J}_M}{K}+1\right)}{\Gamma \left(\frac{h{J}_M}{K}-n+1\right)}\frac{\Gamma \left({\overline{\xi}}_M+{\theta}^{\ast }-n\right)}{\Gamma \left({\overline{\xi}}_M+{\theta}^{\ast}\right)}\end{array} $$
(20)

where \( {\theta}^{*}=\left({\overline{\xi}}_M-1\right)\nu /\left(1-\nu \right) \). Equation (20) is Eq. (6) in the main text. Taking the limit of large J M , Eq. (20) reduces to

$$ \begin{array}{c}\hfill \left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{\theta^{\ast }K}{nh\ }\kappa (h)\frac{\Gamma \left(\frac{h{J}_M}{K}+1\right)}{\Gamma \left(\frac{h{J}_M}{K}-n+1\right)}\frac{\Gamma \left({\overline{\xi}}_M+{\theta}^{\ast }-n\right)}{\Gamma \left({\overline{\xi}}_M+{\theta}^{\ast}\right)}\hfill \\ {}\hfill \approx \frac{\theta^{\ast }K}{nh\ }\kappa (h)\frac{\left(\frac{h{J}_M}{K}+1\right)\left(\frac{h{J}_M}{K}\right)\dots \left(\frac{h{J}_M}{K}-n+2\right)\Gamma \left(\frac{h{J}_M}{K}-n+1\right)\Gamma \left({\overline{\xi}}_M+{\theta}^{\ast }-n\right)}{\Gamma \left(\frac{h{J}_M}{K}-n+1\right)\left({\overline{\xi}}_M+{\theta}^{\ast}\right)\left({\overline{\xi}}_M+{\theta}^{\ast }-1\right)\dots \left({\overline{\xi}}_M+{\theta}^{\ast }-n+1\right)\Gamma \left({\overline{\xi}}_M+{\theta}^{\ast }-n\right)}\hfill \\ {}\hfill \approx \frac{\theta^{\ast }K}{nh\ }\kappa (h)\frac{\left(\frac{h{J}_M}{K}+1\right)\left(\frac{h{J}_M}{K}\right)\dots \left(\frac{h{J}_M}{K}-n+2\right)}{\left({\overline{\xi}}_M+{\theta}^{\ast}\right)\left({\overline{\xi}}_M+{\theta}^{\ast }-1\right)\dots \left({\overline{\xi}}_M+{\theta}^{\ast }-n+1\right)}\approx \frac{\theta^{\ast }K}{nh\ }\kappa (h){\left(\frac{h{J}_M}{K\left({\overline{\xi}}_M+{\theta}^{\ast}\right)}\right)}^n\hfill \\ {}\hfill \left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \kappa (h)\frac{{\overline{\xi}}_M\nu K}{nh\left(1-\nu \right)}{\left(\frac{J_M\left(1-\nu \right)h}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n\hfill \end{array} $$
(21)

which is Fisher’s log-series distribution and is Eq. (7) in the main text. For Eq. (21) to behave sensibly, we require:

$$ \frac{J_M\left(1-\nu \right){h}_{\max }}{K\left({\overline{\xi}}_M-\nu \right)}<1 $$
(22)

Rearranging Eq. (22) gives

$$ \frac{J_M\left(1-\nu \right){h}_{\max }}{K}+\nu <{\overline{\xi}}_M $$
(23)

Following Eq. (2) in the main text, Eq. (23) can be rewritten

$$ \frac{J_M\left(1-\nu \right){h}_{\max }}{K}+\nu <{J}_M{\displaystyle \sum_{h=1}^K}\frac{\left\langle {\lambda}_M(h)\right\rangle h}{K}\le {J}_M\frac{h_{\max }}{K} $$
(24)

where the second inequality follows from the fact that 〈λ M (h)〉 is a probability distribution. Dividing through by J M /K and rearranging gives

$$ {h}_{\max }+\nu \left(\frac{K}{J_M}-{h}_{\max}\right)<{\displaystyle \sum_{h=1}^K}\left\langle {\lambda}_M(h)\right\rangle h\le {h}_{\max } $$
(25)

Note that \( \frac{K}{J_M}-{h}_{\max }<0 \) (for large J M , there will always be at least one individual per habitat) and ν is very small, so that the summation term in Eq. (25) is restricted to a narrow range of values. The condition is satisfied for \( \left\langle {\lambda}_M(h)\right\rangle =\left\{\begin{array}{cc}\hfill 1\hfill & \hfill h={h}_{\max}\hfill \\ {}\hfill 0\hfill & \hfill h<{h}_{\max}\hfill \end{array}\right. \) or for extremely small perturbations of this. In practice, this means that Eq. (22) is only sensibly defined if the suite of species with the broadest niche breadth almost completely dominate the system at equilibrium. This fact will be used in the derivations that follow.

From Eq. (21) and the definition of 〈λ M (h)〉, (the probability that a randomly selected individual has niche breadth h at the speciation–extinction equilibrium), we know that:

$$ \left\langle {\lambda}_M(h)\right\rangle =\frac{1}{J_M}{\displaystyle \sum_{n=1}^{J_M}}n\left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{1}{J_M}{\displaystyle \sum_{n=1}^{J_M}}\ \kappa (h)\frac{{\overline{\xi}}_M\nu K}{h\left(1-\nu \right)}{\left(\frac{J_M\left(1-\nu \right)h}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n $$
(26)

Equation (26), in combination with Eq. (2) from the main text, allows us to write the following approximate expression involving \( {\overline{\xi}}_M \):

$$ {\overline{\xi}}_M={J}_M{\displaystyle \sum_{h=1}^K}\frac{\left\langle {\lambda}_M(h)\right\rangle h}{K}\approx {\displaystyle \sum_{h=1}^{h_{\max }}}{\displaystyle \sum_{n=1}^{J_M}}\left\{\kappa (h)\frac{{\overline{\xi}}_M\nu }{1-\nu }{\left(\frac{J_M\left(1-\nu \right)h}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n\right\} $$
(27)

Now, we seek an explicit expression approximating \( {\overline{\xi}}_M \) for the special case that κ(h) = 1/h max. In this case, Eq. (27) can be rearranged to give:

$$ \begin{array}{l}{h}_{\max}\frac{\left(1-\nu \right)}{\nu}\approx {\displaystyle \sum_{h=1}^{h_{\max }}}{\displaystyle \sum_{n=1}^{J_M}}{\left(\frac{J_M\left(1-\nu \right)h}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n\approx {\displaystyle \sum_{n=1}^{J_M}}{\left(\frac{J_M\left(1-\nu \right){h}_{\max }}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n\\ {}=\frac{c{h}_{\max}\left(1-{\left(c{h}_{\max}\right)}^{J_M}\right)\ }{1-c{h}_{\max }}\approx \kern0.5em \frac{c{h}_{\max }\ }{1-c{h}_{\max }}\end{array} $$
(28)

where \( c=\frac{J_M\left(1-\nu \right)}{K\left({\overline{\xi}}_M-\nu \right)} \) and we have dropped the summation over h since 〈λ M (h)〉 ≈ 0 for h < h max. Solving Eq. (28) for c and then for \( {\overline{\xi}}_M \) gives:

$$ {\overline{\xi}}_M\approx \frac{J_M}{K}\left({h}_{\max}\left(1-\nu \right)+\nu \right)+\nu \approx \kern0.5em \frac{J_M}{K}\left({h}_{\max}\left(1-\nu \right)+\nu \right) $$
(29)

Note that this satisfies inequality (22). Equation (29) is used in the main text to estimate \( {\overline{\xi}}_M \) when evaluating Eq. (6) to produce Fig. 1 curves that assume speciation into a distribution of niche breadths.

Next, we seek an explicit expression approximating \( {\overline{\xi}}_M \) for the special case that \( \kappa (h)=\left\{\begin{array}{c}\hfill 1\kern1.25em h={h}_S\hfill \\ {}\hfill 0\kern1.25em h\ne {h}_S\hfill \end{array}\right. \). In this case, Eq. (27) can be rearranged to give:

$$ \left(\frac{1-\nu }{\nu}\right)\approx {\displaystyle \sum_{n=1}^{J_M}}{\left(\frac{J_M\left(1-\nu \right){h}_S}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n=\frac{c{h}_S\left(1-{\left(c{h}_S\right)}^{J_M}\right)\ }{1-c{h}_S}\approx \kern0.5em \frac{c{h}_S\ }{1-c{h}_S} $$
(30)

where, as before, \( c=\frac{J_M\left(1-\nu \right)}{K\left({\overline{\xi}}_M-\nu \right)} \). Solving (30) for c and then for \( {\overline{\xi}}_M \) gives:

$$ {\overline{\xi}}_M\approx \frac{J_M}{K}{h}_S+\nu \approx \frac{J_M}{K}{h}_S $$
(31)

which again satisfies inequality (22). Equation (31) is used in the main text to estimate \( {\overline{\xi}}_M \) when evaluating Eq. (6) to produce Fig. 1 curves that assume speciation into a single niche breadth.

Lastly, we show the equivalence of metacommunity SADs predicted by the overlapping niche model with speciation into a single niche breadth and metacommunity SADs predicted by the non-overlapping niche model with equivalent species carrying capacities. For the overlapping niche model with speciation into a single niche breadth h = h S , Eqs. (6) and (8) in the main text give the following expression for the SAD

$$ \left\langle {\phi}_M(n)\right\rangle \approx \frac{\theta^{\ast }K}{n{h}_S\ }\frac{\Gamma \left(\frac{h_S{J}_M}{K}+1\right)}{\Gamma \left(\frac{h_S{J}_M}{K}-n+1\right)}\frac{\Gamma \left(\frac{h_S{J}_M}{K}+{\theta}^{\ast }-n\right)}{\Gamma \left(\frac{h_S{J}_M}{K}+{\theta}^{\ast}\right)} $$
(32)

where we have used \( \kappa (h)=\left\{\begin{array}{c}\hfill 1\kern1.25em h={h}_S\hfill \\ {}\hfill 0\kern1.25em h\ne {h}_S\hfill \end{array}\right. \) to simplify the sum in Eq. (8) and have substituted the approximation \( {\overline{\xi}}_M\approx \frac{J_M}{K}{h}_S \) from Eq. (31). A similar expression for the SAD in the non-overlapping model can be recovered from Eqs. (6) and (8) by taking h max = 1, which gives

$$ \left\langle {\phi}_M(n)\right\rangle \approx \frac{\theta^{\ast }{K}_{\mathrm{no}}}{n}\frac{\Gamma \left(\frac{J_M}{K_{\mathrm{no}}}+1\right)}{\Gamma \left(\frac{J_M}{K_{\mathrm{no}}}-n+1\right)}\frac{\Gamma \left(\frac{J_M}{K_{\mathrm{no}}}+{\theta}^{\ast }-n\right)}{\Gamma \left(\frac{J_M}{K_{\mathrm{no}}}+{\theta}^{\ast}\right)} $$
(33)

where we have used he fact that \( {\overline{\xi}}_M\approx \kern1em \frac{J_M}{K} \) for the case that h max = 1 and have set K = K no to denote habitat size in the non-overlapping niche model. In order for species to have equivalent carrying capacities in the overlapping and non-overlapping niche models, respectively, it is necessary to set K no = K/h S , where K and h S are the habitat size and niche breadth in the overlapping niche model and K no is the habitat size in the non-overlapping niche model. Substituting this relationship into Eq. (33) gives

$$ \left\langle {\phi}_M(n)\right\rangle \approx \frac{\theta^{\ast }K}{n{h}_S}\frac{\Gamma \left(\frac{J_M{h}_S}{K}+1\right)}{\Gamma \left(\frac{J_M{h}_S}{K}-n+1\right)}\frac{\Gamma \left(\frac{J_M{h}_S}{K}+{\theta}^{*}-n\right)}{\Gamma \left(\frac{J_M{h}_S}{K}+{\theta}^{\ast}\right)} $$
(34)

which is identical to the expression in (32) for the overlapping niche model. Thus, to the extent that approximations used to derive Eq. (32) are correct, overlapping and non-overlapping niche models with identical species carrying capacities give identical SADs. This suggests that it is carrying capacity restrictions, rather than the details of niche structure itself that gives rise to deviations in the SAD as compared to purely neutral theory predictions.

Local community SADs

We extended the metacommunity model, described in the main text, to include a coupled local community of size J ≪ J M . This is in keeping with the local versus metacommunity paradigm of the spatially implicit neutral model (Hubbell 2001). The local community is semi-isolated from the metacommunity, and the metacommunity is assumed to be static on the timescale of the local community. To include overlapping niches, we consider one of two possible limiting scenarios: (1) the local community consists of a single habitat (while the metacommunity continues to be composed of K equally represented habitats) and (2) the local community, like the metacommunity, consists of K equally represented habitats.

The dynamics in the local community operate as follows. At each timestep a randomly selected individual is killed. With probability 1 − m, the parent of a replacement individual is drawn at random from the remaining individuals in the local community who can potentially survive in the habitat corresponding to the location vacated by the killed individual. With probability m, the replacement individual is drawn at random from those individuals in the metacommunity who can potentially survive in the vacated habitat.

A local community with a single habitat, K L = 1

In the case that the local community consists of a single habitat, the local community birth and death rates for our overlapping niche model are as follows:

$$ {b}_L\left(n,h,k\right)\approx \left(\frac{J-n}{J}\right)\left(\frac{m{\mu}_k}{{\overline{\xi}}_M}+\frac{\left(1-m\right)n}{J-1}\right) $$
(35)
$$ {d}_L\left(n,h,k\right)\approx \left(\frac{n}{J}\right)\left(\frac{m\left({\overline{\xi}}_M-{\mu}_k\right)}{{\overline{\xi}}_M}+\frac{\left(1-m\right)\left(J-n\right)}{J-1}\right) $$
(36)

where n is now used to denote the number of individuals of species k in the local community, while μ k represents the number of individuals of species k in the metacommunity. In addition, J is the total number of individuals in the local community and m is a parameter that determines the relative rate of immigration from the metacommunity (Chisholm and Lichstein 2009). Equations (35) and (36) are expressed as asymptotics because they assume the metacommunity is large and static relative to the local community so that \( {\overline{\xi}}_M \) and μ k are constant.

To derive an expression for the local community SAD under this model, we begin from Eq. (7), which is the large J M limit for the metacommunity SAD. Substituting \( x=n/{\overline{\xi}}_M \) and assuming that \( {\overline{\xi}}_M\gg \nu \), we find

$$ \left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \kappa (h)\frac{\nu K}{xh\left(1-\nu \right)}{\left(\frac{J_M\left(1-\nu \right)h}{K{\overline{\xi}}_M}\right)}^{{\overline{\xi}}_Mx} $$
(37)

For large \( {\overline{\xi}}_M \), x can be approximated as continuous. Accordingly, from Eq. (37), we write

$$ {\rho}_{M,h}(x) dx\equiv \left\langle {\phi}_M\left(n,h\right)\right\rangle dn\approx \kappa (h)\frac{\nu K}{xh\left(1-\nu \right)}{\left(\frac{J_M\left(1-\nu \right)h}{K{\overline{\xi}}_M}\right)}^{x{\overline{\xi}}_M}{\overline{\xi}}_M dx $$
(38)

Following Volkov et al. (2007), the local community SAD is then given by

$$ \left\langle \phi (n)\right\rangle ={\displaystyle \underset{0}{\overset{1}{\int }}}{P}_n\left(J,m,x\right){\displaystyle \sum_{h=1}^{h_{\max }}}\left[\frac{h}{K}{\rho}_{M,h}(x)\right] dx $$
(39)

where P n (J, m, x) is the probability that a species with relative abundance x in the metacommunity is represented by n individuals in a local community of size J, assuming that the migration rate is m. The additional factor of h/K in the summation term in Eq. (39) comes from the fact that a species with niche breadth h has an h/K chance of being adapted to the habitat comprising to the local community in question. An expression for P n (J, m, x) can be derived from Eqs. (35) and (36), following a standard method (Volkov et al. 2003; Volkov et al. 2007; Alonso and McKane 2004), yielding:

$$ {P}_n\left(J,m,x\right)=\left(\begin{array}{c}\hfill J\hfill \\ {}\hfill n\hfill \end{array}\right)\frac{\Gamma \left(n+\gamma x\right)}{\Gamma \left(\gamma x\right)}\frac{\Gamma \left(J-n+\gamma -\gamma x\right)}{\Gamma \left(\gamma -\gamma x\right)}\frac{\Gamma \left(\gamma \right)}{\Gamma \left(\gamma +J\right)} $$
(40)

where \( \gamma =\frac{m\left(J-1\right)}{\left(1-m\right)} \) is a rescaled immigration parameter. Substituting Eqs. (40) and (38) into Eq. (39) and looking at the special case where κ(h) = 1/h max gives

$$ \begin{array}{l}\left\langle \phi (n)\right\rangle \approx \frac{{\overline{\xi}}_M\nu }{h_{\max}\left(1-\nu \right)}\left(\begin{array}{c}\hfill J\hfill \\ {}\hfill n\hfill \end{array}\right)\frac{\Gamma \left(\gamma \right)}{\Gamma \left(\gamma +J\right)}\\ {}\times {\displaystyle \underset{0}{\overset{1}{\int }}}\frac{\Gamma \left(n+\gamma x\right)}{\Gamma \left(\gamma x\right)}\frac{\Gamma \left(J-n+\gamma -\gamma x\right)}{\Gamma \left(\gamma -\gamma x\right)}\frac{1}{x}{\left(\frac{J_M\left(1-\nu \right)}{K{\overline{\xi}}_M}\right)}^{x{\overline{\xi}}_M}{\displaystyle \sum_{h=1}^{h_{\max }}}{h}^{x{\overline{\xi}}_M} dx\end{array} $$
(41)

Equation (41) gives the expression for the SAD of a local community with a single habitat. Figure 11 (a,b) shows plots of local community SADs based on Eq. (41).

Fig. 11
figure11

Local community SADs assuming that the metacommunity has either K = 4 habitats (a, c) or K = 16 habitats (b, d) and that the local community has either a single habitat, K L = 1 (a, b) or else the same number of habitats as are present in the metacommunity, K L = 4 (c) or K L = 16 (d). In all panels, we assume v max = 0.0005, J M  = 1,000,000, J = 21,000, and m = 0.075. In panels with K = 4 (a, c), we assume maximum niche breadths of h max = 1 (dotted), 2 (short dashes), 3 (long dashes), or 4 (solid). In panels with K = 16 (b, d), we assume maximum niche breadths of h max = 1 (dotted), 6 (short dashes), 11 (long dashes), or 16 (solid)

A local community with K equal-sized habitats, K L = K

We now consider the case that the local community contains all K habitats found in the metacommunity. While the processes governing birth and death rates are the same as in the previous example, birth and death rates themselves must be slightly modified as a result of the niche structure present at the local scale. Specifically:

$$ {b}_L\left(n,h,k\right)\approx \left(\frac{hJ/K-n}{hJ/K}\right)\left(\frac{h}{K}\right)\left(\frac{m{\mu}_k}{{\overline{\xi}}_M}+\frac{\left(1-m\right)n}{\left({\overline{\xi}}_L-1\right)}\right) $$
(42)
$$ {d}_L\left(n,h,k\right)\approx \left(\frac{h}{K}\right)\left(\frac{n}{hJ/K}\right)\left(m\left(1-\frac{\mu_k}{{\overline{\xi}}_M}\right)+\frac{\left(1-m\right)\left(\overline{\xi_L}-n\right)}{\left({\overline{\xi}}_L-1\right)}\right) $$
(43)

where \( {\overline{\xi}}_L \) is the local community equivalent to \( {\overline{\xi}}_M \), implying that we have, again, assumed that stochastic fluctuations in \( {\overline{\xi}}_L \) are negligible for the systems that we model. An approximate expression for \( {\overline{\xi}}_L \) is derived in Appendix 3. Simulations suggest that this approximation is valid, at least for communities of similar size to the local communities that we consider here.

The derivation for the local community SAD with multiple habitats is similar to the derivation for the local community SAD with a single habitat. In place of Eq. (39), however, we write

$$ \left\langle \phi (n)\right\rangle ={\displaystyle \underset{0}{\overset{1}{\int }}}{\displaystyle \sum_{h=1}^{h_{\max }}}{P}_{n,h}\left(J,m,x\right){\rho}_{M,h}(x) dx $$
(44)

Notice that the probability of a species with relative abundance x in the metacommunity being represented by  n individuals in a local community of size J now depends on the niche breadth of the species. This is a direct result of the niche structure at the local scale. In addition, because the birth and death rates in Eqs. (42) and (43) already contain the h/K factor ensuring that an individual, selected at random from the metacommunity, is adapted to fill the available site for which it is selected, there is no additional factor of h/K in Eq. (44). From Eqs. (42) and (43), we can again derive an expression for P n,h (J, m, x):

$$ {P}_{n,h}\left(J,m,x\right)=\left(\begin{array}{c}\hfill hJ/K\hfill \\ {}\hfill n\hfill \end{array}\right)\frac{\Gamma \left(\gamma x+n\right)\Gamma \left({\overline{\xi}}_L-n+\gamma -\gamma x\right)\Gamma \left(\gamma +{\overline{\xi}}_L- hJ/K\right)}{\Gamma \left(\gamma x\right)\Gamma \left({\overline{\xi}}_L+\gamma \right)\Gamma \left(\gamma -\gamma x+{\overline{\xi}}_L- hJ/K\right)} $$
(45)

Substituting Eqs. (45) and (38) into Eq. (44) gives

$$ \left\langle \phi (n)\right\rangle \approx \frac{{\overline{\xi}}_M\nu K}{h_{\max}\left(1-\nu \right)}\times {\displaystyle \underset{0}{\overset{1}{\int }}}\frac{1}{x}{\displaystyle \sum_{h=1}^{h_{\max }}}\left\{\left(\begin{array}{c}\hfill hJ/K\hfill \\ {}\hfill n\hfill \end{array}\right)\frac{\Gamma \left(\gamma x+n\right)\Gamma \left({\overline{\xi}}_L-n+\gamma -\gamma x\right)\Gamma \left(\gamma +{\overline{\xi}}_L- hJ/K\right)}{\Gamma \left(\gamma x\right)\Gamma \left({\overline{\xi}}_L+\gamma \right)\Gamma \left(\gamma -\gamma x+{\overline{\xi}}_L- hJ/K\right)}\frac{1}{h}{\left(\frac{J_M\left(1-\nu \right)h}{K{\overline{\xi}}_M}\right)}^{x{\overline{\xi}}_M}\right\} dx $$
(46)

Equation (46) gives the expression for the SAD of a local community with K equal-sized habitats. Figure 11 (c,d) shows plots of local community SADs based on Eq. (46).

Appendix 3: Basic model niche breadth distribution functions

Metacommunity

Fundamental niche breadth distribution by individuals

The goal is to derive an analytical expression for the proportion of individuals with a given niche breadth in the metacommunity at the speciation–extinction steady state (the equilibrium). Beginning from Eq. (26), we have

$$ \begin{array}{l}\left\langle {\lambda}_M(h)\right\rangle =\frac{1}{J_M}{\displaystyle \sum_{n=1}^{J_M}}n\left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{1}{J_M}{\displaystyle \sum_{n=1}^{J_M}}\ \kappa (h)\frac{{\overline{\xi}}_M\nu K}{h\left(1-\nu \right)}{\left(\frac{J_M\left(1-\nu \right)h}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n\\ {}=\frac{1}{J_M}\ \kappa (h)\frac{{\overline{\xi}}_M\nu K}{h\left(1-\nu \right)}\frac{(ch)^{J_M+1}- ch}{ch-1}\end{array} $$
(47)

where \( c=\frac{J_M\left(1-\nu \right)}{K\left({\overline{\xi}}_M-\nu \right)} \) is as defined in Eq. (28). As noted earlier, ch < 1 for all h. For h  <  h max and for the special case κ(h) = 1/h max, Eq. (47) can be rewritten as

$$ \left\langle {\lambda}_M(h)\right\rangle \approx \frac{1}{J_M}\kappa (h)\frac{{\overline{\xi}}_M\nu K}{h\left(1-\nu \right)}\frac{ch}{1- ch}=\frac{\left(1-\nu \right)+\frac{\nu }{h_{\max }}}{1+\frac{1-\nu }{\nu}\left({h}_{\max }-h\right)} $$
(48)

where the second equality follows from Eq. (29). Numerical investigations show that for h = h max, Eq. (48) is a poor approximation to Eq. (47) because \( \begin{array}{l}c{h}_{\max}\\ {}\end{array} \) is closer to one. To get a better approximation for 〈λ(h max)〉, we use:

$$ \left\langle {\lambda}_M\left({h}_{\max}\right)\right\rangle =1-{\displaystyle \sum_{h=1}^{h_{\max }-1}}\left\langle {\lambda}_M(h)\right\rangle $$
(49)

The summation can be evaluated (with no further approximations) using digamma functions ψ 0(x) to give:

$$ \left\langle {\lambda}_M\left({h}_{\max}\right)\right\rangle \approx 1-\left(1+\frac{\nu }{h_{\max}\left(1-\nu \right)}\right)\nu \left({\psi}_0\left(1-{h}_{\max }-\frac{\nu }{1-\nu}\right)-{\psi}_0\left(-\frac{\nu }{1-\nu}\right)\right) $$
(50)

Equations (48) and (50) give Eq. (9) in the main text.

Fundamental niche breadth distribution by species

In this case, the goal is to derive an analytical expression for the proportion of species with a given niche breadth,  〈ς M (h)〉, in the metacommunity at the speciation–extinction–immigration steady state (the equilibrium). Using the approximation for the SAD given by Eq. (7), we write

$$ \left\langle {\varsigma}_M(h)\right\rangle =\frac{1}{S_{M,\mathrm{tot}}}{\displaystyle \sum_{n=1}^{h{J}_M/K}}\left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{1}{S_{M,\mathrm{tot}}}{\displaystyle \sum_{n=1}^{h{J}_M/K}}\frac{{\overline{\xi}}_M\nu K}{h_{\max } nh\left(1-\nu \right)}{\left(\frac{J_M\left(1-\nu \right)h}{K\left({\overline{\xi}}_M-\nu \right)}\right)}^n $$
(51)

where we have again assumed that κ(h) = 1/h max, and where S M,tot is the expected total number of species in the metacommunity at equilibrium. Using \( c=\frac{J_M\left(1-\nu \right)}{K\left({\overline{\xi}}_M-\nu \right)} \), in the limit as hJ M /K → ∞, we can approximate Eq. (51) as:

$$ \left\langle {\varsigma}_M(h)\right\rangle \approx \frac{{\overline{\xi}}_M\nu K}{S_{M,\mathrm{tot}}{h}_{\max }h\left(1-\nu \right)}{\displaystyle \sum_{n=1}^{h{J}_M/K}}\frac{(ch)^n}{n}\approx -\frac{{\overline{\xi}}_M\nu K}{S_{M,\mathrm{tot}}{h}_{\max }h\left(1-\nu \right)} \ln \left(1- ch\right) $$
(52)

Applying the normalization condition to Eq. (52) gives an asymptotic expression for S M,tot:

$$ {S}_{M,\mathrm{tot}}\approx -\frac{{\overline{\xi}}_M\nu K}{h_{\max}\left(1-\nu \right)}{\displaystyle \sum_{h=1}^{h_{\max }}}\frac{ \ln \left(1- ch\right)}{h} $$
(53)

Combining Eqs. (52) and (53) leads to Eq. (10) in the main text.

Realized niche breadth distribution by species

We begin by defining \( \left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle \) to be the fraction of the total species with fundamental niche breadth h and realized niche breadth \( \widehat{h} \). From this definition and Eqs. (7) and (11), we write

$$ \begin{array}{l}\left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle =\frac{1}{S_{M,\mathrm{tot}}}{\displaystyle \sum_{n=1}^{J_M}}\left\langle {\phi}_M\left(n,h\right)\right\rangle\ q\left(\widehat{h};h,n\right)\\ {}\approx \frac{{\overline{\xi}}_M\nu K}{S_{M,\mathrm{tot}}h{h}_{\max}\left(1-\nu \right)}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}}{\displaystyle \sum_{n=1}^{J_M}}\frac{1}{n}{(ch)}^n{\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right){\left(\frac{\widehat{h}-l}{h}\right)}^n\\ {}=\frac{{\overline{\xi}}_M\nu K}{S_{M,\mathrm{tot}}h{h}_{\max}\left(1-\nu \right)}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}}{\displaystyle \sum_{n=1}^{J_M}}\frac{{\left(c\left(\widehat{h}-l\right)\right)}^n}{n}{\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right)\end{array} $$
(54)

where, again, \( c=\frac{J_M\left(1-\nu \right)}{K\left({\overline{\xi}}_M-\nu \right)} \). Because \( \widehat{h}-l\le h \), we know that \( c\left(\widehat{h}-l\right)\le ch<1 \); thus in the limit of large J M , Eq. (54) can be approximated

$$ \left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle \approx -\frac{{\overline{\xi}}_M\nu K}{S_{M,\mathrm{tot}}h{h}_{\max}\left(1-\nu \right)}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}} \ln \left(1-c\left(\widehat{h}-l\right)\right){\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right) $$
(55)

Substituting Eqs. (29) and (53) into Eq. (55) and simplifying gives

$$ \left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle \approx \frac{\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle {\sum}_{l=0}^{\widehat{h}} \ln}\left(\frac{\left({h}_{\max }-\widehat{h}+l\right)\left(1-\nu \right)+\nu }{h_{\max}\left(1-\nu \right)+\nu}\right){\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right)}{h{\displaystyle {\sum}_{l=1}^{h_{\max }}\frac{ \ln \left(\frac{\left({h}_{\max }-l\right)\left(1-\nu \right)+\nu }{h_{\max}\left(1-\nu \right)+\nu}\right)}{l}}} $$
(56)

The realized niche breadth distribution across the entire metacommunity is then given by

$$ \begin{array}{l}\left\langle {\widehat{\varsigma}}_M\left(\widehat{h}\right)\right\rangle \approx \frac{1}{{\displaystyle {\sum}_{h=1}^{h_{\max }}}\frac{1}{h} \ln \left(\frac{\left({h}_{\max }-h\right)\left(1-\nu \right)+\nu }{h_{\max}\left(1-\nu \right)+\nu}\right)}\\ {}\kern6em \times {\displaystyle \sum_{h=\widehat{h}}^{h_{\max }}}\frac{1}{h}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}} \ln \left(\frac{\left({h}_{\max }-\widehat{h}+l\right)\left(1-\nu \right)+\nu }{h_{\max}\left(1-\nu \right)+\nu}\right){\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right)\\ {}=\frac{\left({h}_{\max }-\widehat{h}+1\right)\left(\begin{array}{c}\hfill {h}_{\max }+1\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right)}{\widehat{h}\left(1+{h}_{\max}\right)}\frac{{\displaystyle {\sum}_{h=0}^{\widehat{h}}} \ln \left(\frac{\left({h}_{\max }-\widehat{h}+h\right)\left(1-\nu \right)+\nu }{h_{\max}\left(1-\nu \right)+\nu}\right){\left(-1\right)}^h\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill h\hfill \end{array}\right)}{{\displaystyle {\sum}_{h=1}^{h_{\max }}}\frac{1}{h} \ln \left(\frac{\left({h}_{\max }-h\right)\left(1-\nu \right)+\nu }{h_{\max}\left(1-\nu \right)+\nu}\right)}\end{array} $$
(57)

Local community

Here, we derive fundamental niche breadth distributions for individuals in a local community of J individuals with immigration parameter m, as described in Appendix 2. We investigate both scenarios described in Appendix 2, namely that in which the local community is comprised of a single habitat and that in which the local community is comprised of K equal-sized habitats. In this case, the direct approach for determining niche breadth distributions used for the metacommunity is too difficult, so we adopt a different strategy. At equilibrium, we must have:

$$ \left\langle {\lambda}_L(h)\right\rangle =m{\mathrm{P}}_{\mathrm{I}}(h)+\left(1-m\right){\mathrm{P}}_{\mathrm{B}}(h) $$
(58)

where PI(h) is the probability that an immigrant has niche breadth h, and PB(h) is the probability that a newly born individual has niche breadth h. To obtain PI(h), note that the probability that the first propagule arriving at a site is suited to the site and has niche breadth h is 〈λ M (h)〉h/K. The probability that the second propagule to arrive at the site is suited to the site and has niche breadth h, given that the first propagule was not suited to the site is \( \left(1-{\overline{\xi}}_M/{J}_M\right)\left\langle {\lambda}_M(h)\right\rangle h/K \), and so on. Thus, the overall probability that the immigrant has niche breadth h is:

$$ {\mathrm{P}}_{\mathrm{I}}(h)\approx \left\langle {\lambda}_M(h)\right\rangle \frac{h}{K}+\left(1-\frac{{\overline{\xi}}_M}{J_M}\right)\left\langle {\lambda}_M(h)\right\rangle \frac{h}{K}\kern0.5em +{\left(1-\frac{{\overline{\xi}}_M}{J_M}\right)}^2\left\langle {\lambda}_M(h)\right\rangle \frac{h}{K}+\dots =\left\langle {\lambda}_M(h)\right\rangle \frac{h}{K}\frac{1}{{\overline{\xi}}_M/{J}_M}\approx \frac{\left\langle {\lambda}_M(h)\right\rangle h}{h_{\max}\left(1-\nu \right)+\nu } $$
(59)

where the asymptotic approximation comes in because we have used expression (29), which is valid for large J M . From Eq. (9) in the main text, we then know that

$$ {\mathrm{P}}_{\mathrm{I}}(h)\approx \left\{\begin{array}{c}\hfill \frac{1}{1+\frac{\left(1-\nu \right)}{\nu}\left({h}_{\max }-h\right)}\frac{h}{h_{\max }}\kern5em h<{h}_{\max}\hfill \\ {}\hfill 1-{\displaystyle {\sum}_{k=1}^{h_{\max }-1}}{\mathrm{P}}_{\mathrm{I}}(k)\kern8.5em h={h}_{\max}\hfill \end{array}\right. $$
(60)

To obtain PB(h), we must first distinguish between our two different local community scenarios. In the case that the local community is comprised of a single habitat, all propagules from within the community will be adapted to living at all sites within the community, thus PB(h) = 〈λ L (h)〉. Substitution into Eq. (58) then gives 〈λ L (h)〉 = PI(h), and we can refer to Eq. (59) for the result.

In the scenario with K habitats per local community, the probability of a newly born individual being of niche breadth h is, by a similar argument used in the derivation of (59):

$$ {\mathrm{P}}_{\mathrm{B}}(h)=\left\langle {\lambda}_L(h)\right\rangle \frac{h}{K}\frac{1}{{\overline{\xi}}_L/J} $$
(61)

Again, substituting into Eq. (58) gives

$$ \left\langle {\lambda}_L(h)\right\rangle =\left\{\begin{array}{c}\hfill \frac{mh}{h_{\max}\left(1-\left(1-m\right)\frac{Jh}{K}\frac{1}{{\overline{\xi}}_L}\right)}\frac{1}{\left(1+\frac{\left(1-\nu \right)}{\nu}\left({h}_{\max }-h\right)\right)}\kern1.3em h<{h}_{\max}\hfill \\ {}\hfill 1-{\displaystyle {\sum}_{k=1}^{h_{\max }-1}}\left\langle {\lambda}_L(k)\right\rangle \kern14em h={h}_{\max}\hfill \end{array}\right. $$
(62)

where we have made use of the relationship \( {\displaystyle {\sum}_{k=1}^{h_{\max }}}\left\langle {\lambda}_L(k)\right\rangle =1 \) for the case that h = h max.

To simplify this further, we assume that \( {\overline{\xi}}_L\approx J{h}_{\max }/K \), which would be exactly true if all of the individuals in the local community had maximum niche breadth but in practice is only approximately true because, although such species dominate the community, species with h < h max do also occur. Numerically, this approximation works well when calculating 〈λ L (h)〉 for h < h max, but for h = h max, it gives (unsurprisingly) 〈λ L (h max)〉 = 1, so we obtain 〈λ L (h max)〉 from the normalization condition. Therefore, the general approximate formula for 〈λ L (h)〉 is:

$$ \left\langle {\lambda}_L(h)\right\rangle \approx \left\{\begin{array}{c}\hfill \frac{h}{\left({h}_{\max }+\frac{1-m}{m}\left({h}_{\max }-h\right)\right)\left(1+\frac{1-\nu }{\nu}\left({h}_{\max }-h\right)\right)\ }\kern2em h<{h}_{\max}\hfill \\ {}\hfill 1-{\displaystyle {\sum}_{k=1}^{h_{\max }-1}}\left\langle {\lambda}_L(k)\right\rangle \kern14em h={h}_{\max}\hfill \end{array}\right. $$
(63)

From Eq. (63), we can calculate a more accurate approximation \( {\overline{\xi}}_L=\frac{J{\displaystyle {\sum}_{k=1}^{h_{\max }}}k\left\langle {\lambda}_L(k)\right\rangle }{K} \), which can be used to evaluate the local SAD given by Eq. (44).

Appendix 4: Numerical simulations of metacommunity niche breadth distributions

We ran numerical simulations to test the accuracy of the approximate metacommunity niche breadth distributions derived in the main text. Specifically, we tested the fundamental niche breadth distribution by individual 〈λ M (h)〉 (Eq. (9)), the fundamental niche breadth distribution by species 〈ς M (h)〉 (Eq. (10)), and the realized niche breadth distribution by species \( \left\langle {\widehat{\varsigma}}_M\left(\widehat{h}\right)\right\rangle \) (Eq. (12)) for the basic (i.e., no trade-off) overlapping niche model with a uniform distribution of niche breadths at speciation. The model parameters used were the same as those used in Appendix 1 to test the SAD approximations. Each simulation was repeated 50 times and results were averaged over simulations. The results show that the approximations are generally very good (Figs. 12 and 13).

Fig. 12
figure12

Fundamental niche breadth distribution by individual λ M (h) (green), fundamental niche breadth distribution by species ς M (h) (blue), and realized niche breadth distribution by species \( {\widehat{\varsigma}}_M\left(\widehat{h}\right) \) (red) for the basic (i.e., no trade-off) overlapping niche model with a uniform distribution of niche breadths at speciation. Parameters are K = 4, J M  = 10, 000, h max = 2 (top panels), 3 (middle panels), or 4 (bottom panels), and ν max = 0.0005 (left panels) or 0.005 (right panels). Solid lines show analytical predictions, and dashed lines show means of 50 simulations. In most cases, the analytical and simulated results are nearly indistinguishable

Fig. 13
figure13

Fundamental niche breadth distribution by individual λ M (h) (green), fundamental niche breadth distribution by species ς M (h) (blue), and realized niche breadth distribution by species \( {\widehat{\varsigma}}_M\left(\widehat{h}\right) \) (red) for the basic (i.e., no trade-off) overlapping niche model with a uniform distribution of niche breadths at speciation. Parameters are K = 16, J M  = 10, 000, h max = 6 (top panels), 11 (middle panels), or 16 (bottom panels), and ν max = 0.0005 (left panels) or 0.005 (right panels). Solid lines show analytical predictions, and dashed lines show means of 50 simulations. In most cases, the analytical and simulated results are nearly indistinguishable

Appendix 5: Metacommunity time-dependent solutions

The probability that a species has a realized niche breadth \( \widehat{h}(t)=\widehat{i} \) at time t, given that it has fundamental niche breadth h = i, can be expressed as follows:

$$ {P}_M\left(\widehat{h}(t)=\widehat{i}\left|\ h=i\right.\right)={\displaystyle \sum_{k=1}^{J_Mh/K}}{P}_M\left(n(t)=k\right)P\left(\widehat{h}(t)=\widehat{i}\left|n=k,h=i\right.\right) $$
(64)

where n is the species’ abundance. The first factor in Eq. (64), P M (n(t) = k), is the probability that the species has abundance k in the metacommunity at time t and is given by the kth element of the following vector:

$$ {\mathbf{p}}_{M,t}={\mathbf{V}}_M{\Lambda}_M^t{\mathbf{V}}_M^{-1}{\mathbf{e}}_1 $$
(65)

where V M is a matrix with right eigenvectors of the metacommunity transition matrix, A M , in columns, Λ M is a matrix of eigenvalues of A M (with eigenvalues non-increasing down the diagonal), and e 1 = {0, 1, 0, 0 … 0} is a unit vector representing the initial state at speciation (in which the species has only one individual, with probability one). The second factor inside the summation in Eq. (64) is the probability that a species has realized niche breadth \( \widehat{i} \) given that it has abundance k and fundamental niche breadth i and is given by Eq. (11) in the main text. Writing \( {\mathbf{q}}_{\widehat{i},i} \) as the vector with kth element \( q\left(\widehat{i};i,k\right) \), Eq. (64) becomes:

$$ {P}_M\left(\widehat{h}(t)=\widehat{i}\left|h=i\right.\right)=\left({\mathbf{V}}_M{\Lambda}_M^t{\mathbf{V}}_M^{-1}{\mathbf{e}}_1\right)\cdot {\mathbf{q}}_{\widehat{i},i} $$
(66)

Equation (66) describes the probability that a species has \( \widehat{h}=\widehat{i} \) at time t. For larger times, however, this will be a very small number, not because the species is likely to have a different realized niche breadth but because there is a high probability that n(t) = 0 meaning that the species has gone extinct (it is straightforward to show that Eq. (66) tends to e 0 as t → ∞). More interesting than Eq. (66) itself, then, is the probability that a species has realized niche \( \widehat{i} \) given that the species has not yet gone extinct. From Eq. (66), this quantity is given by:

$$ {P}_M\left(\widehat{h}(t)=\widehat{i}\left|\ n(t)>0,h=i\right.\right)=\frac{P_M\left(\widehat{h}(t)=\widehat{i}\left|h=i\right.\right)}{1-{P}_M\left(n(t)=0\right)}=\frac{\left({\mathbf{V}}_M{\Lambda}_M^t{\mathbf{V}}_M^{-1}{\mathbf{e}}_1\right)\cdot {\mathbf{q}}_{\widehat{i},i}}{1-\left({\mathbf{V}}_M{\Lambda}_M^t{\mathbf{V}}_M^{-1}{\mathbf{e}}_1\right)\cdot {\mathbf{e}}_0} $$
(67)

Comparing of predictions from Eq. (67) with simulation results (Appendix 7, Fig. 14) suggests that Eq. (67) is accurate, except in the limit of extremely low diversity.

We now turn to the special case that a species has maximum fundamental niche breadth h = h max, which is of primary ecological interest because, as shown earlier, species with h = h max tend to dominate the community. In this case, it is possible to evaluate Eqs. (66) and (67) by approximating the population dynamics with a neutral model and using analytical expressions for V M , Λ M , and V M − 1 (based on results in Chisholm 2011). The justification for this is that species with h = h max behave neutrally with respect to one another and that the influence of species with h < h max is small because they are rare.

This neutral approximation approach is particularly useful for simplifying the realized niche breadth distribution conditional on a species’ persistence (Eq. (67)). In the asymptotic limit as t becomes large, we obtain an expression that involves only the first and second eigenvalues λ 0 and λ 1 and eigenvectors v M,0 and v M,1:

$$ {P}_M\left(\widehat{h}(t)=\widehat{i}\left|n(t)>0,h={h}_{\max}\right.\right)=\frac{P_t\left(\widehat{h}=\widehat{i}\left|h={h}_{\max}\right.\right)}{1-{P}_t\left(n=0\right)}\sim \frac{\left({\alpha}_0{\mathbf{v}}_{M,0}{\lambda}_0^t+{\alpha}_1{\mathbf{v}}_{M,1}{\lambda}_1^t\right)\cdot {\mathbf{q}}_{\widehat{i},i}}{1-\left({\alpha}_0{\mathbf{v}}_{M,0}{\lambda}_0^t+{\alpha}_1{\mathbf{v}}_{M,1}{\lambda}_1^t\right)\cdot {\mathbf{e}}_0} $$
(68)

where α 0 = 1, α 1 is determined by the initial conditions, λ 0 = 1, λ 1 = 1 − ν/J , v M,0 = e 0, and the first element of v M,1 is v M,1,0 = 1 and the other elements of v M,1 are (k ≥ 1):

$$ {v}_{M,1,k}=-{\theta}^{*}\frac{{\left({\theta}^{*}+{J}_M\right)}_{-k}}{{\left({J}_M\right)}_{1-k}} $$

Simplifying and taking the asymptotic limit of large J M and large t gives:

$$ \begin{array}{l}\frac{\left({\alpha}_0{\mathbf{v}}_{M,0}{\lambda}_0^t+{\alpha}_1{\mathbf{v}}_{M,1}{\lambda}_1^t\right)\cdot {\mathbf{q}}_{\widehat{i},i}}{1-\left({\alpha}_0{\mathbf{v}}_{M,0}{\lambda}_0^t+{\alpha}_1{\mathbf{v}}_{M,1}{\lambda}_1^t\right)\cdot {\mathbf{e}}_0}=\left(-\frac{{\mathbf{e}}_0}{\alpha_1{\lambda}_1^t}-{\mathbf{v}}_{M,1}\right)\cdot {\mathbf{q}}_{\widehat{i},i}\sim -{\mathbf{v}}_{M,1}\cdot {\mathbf{q}}_{\widehat{i},{h}_{\max }}\\ {}={\theta}^{*}\left(\begin{array}{c}\hfill {h}_{\max}\hfill \\ {}\hfill \widehat{i}\hfill \end{array}\right)\widehat{i}!\frac{\Gamma \left({J}_M\right)}{\Gamma \left({\theta}^{*}+{J}_M\right)}{\displaystyle \sum_{k=0}^{J_M}}\frac{\Gamma \left({\theta}^{*}+{J}_M-k\right)}{\Gamma \left({J}_M+1-k\right)}\frac{S\left(k,\widehat{i}\right)}{{h_{\max}}^k}\sim \frac{\nu }{1-\nu}\left(\begin{array}{c}\hfill {h}_{\max}\hfill \\ {}\hfill \widehat{i}\hfill \end{array}\right)\widehat{i}!\frac{{\left(-1\right)}^{\widehat{i}}}{{\left(1-\frac{h_{\max }}{1-\nu}\right)}_{\widehat{i}}}\end{array} $$

So:

$$ {P}_M\left(\widehat{h}(t)=\widehat{i}\left|n(t)>0,h={h}_{\max}\right.\right)\sim \frac{\nu }{1-\nu}\left(\begin{array}{c}\hfill {h}_{\max}\hfill \\ {}\hfill \widehat{i}\hfill \end{array}\right)\widehat{i}!\frac{{\left(-1\right)}^{\widehat{i}}}{{\left(1-\frac{h_{\max }}{1-\nu}\right)}_{\widehat{i}}} $$

This expression closely approximates the probability distribution of realized niche breadths of an h = h max species that has persisted in the community for a long time. We also have:

$$ \underset{\nu \to 0}{ \lim}\left\{\frac{\nu }{1-\nu}\left(\begin{array}{c}\hfill {h}_{\max}\hfill \\ {}\hfill \widehat{i}\hfill \end{array}\right)\widehat{i}!\frac{{\left(-1\right)}^{\widehat{i}}}{{\left(1-\frac{h_{\max }}{1-\nu}\right)}_{\widehat{i}}}\right\}=1 $$

meaning that when the speciation rate is sufficiently low, every species will tend eventually to realize its full potential niche, if it has not already gone extinct.

Appendix 6: Trade-off model

Metacommunity SADs

Rearranging Eqs. (13) and (14) in the main text gives:

$$ {\overline{b}}_F\left(n,h\right)=\frac{\left(\frac{h{J}_M}{K}-n\right)}{J_M}\frac{\left(1-\nu \right)f(h)n}{{\overline{\zeta}}_F-g\left(h,n\right)}=\frac{\left(\frac{h{J}_M}{K}-n\right)}{J_M}\frac{f(h)n\left(1-\nu \right)}{\left({\overline{\zeta}}_F-g\left(h,n\right)\right)} $$
(69)
$$ {\overline{d}}_F\left(n,h\right)=\frac{n}{J_M}\left\{\left(1-\nu \right)\frac{{\overline{\zeta}}_F-f(h)n}{{\overline{\zeta}}_F-f(h)}+\nu \right\}=\frac{n\left(1-\nu \right)}{J_M\left({\overline{\zeta}}_F-f(h)\right)}\left({\overline{\zeta}}_F+{\theta}_f(h)- nf(h)\right) $$
(70)

where \( {\theta}_f(h)=\frac{\nu \left({\overline{\zeta}}_F-f(h)\right)}{1-\nu } \). The expression for the expected number of species with abundance n and niche breadth h in the metacommunity at equilibrium is (following Vallade and Houchmandzadeh 2003):

$$ \left\langle {\phi}_M\left(n,h\right)\right\rangle \approx {\displaystyle \prod_{i=0}^{n-1}}\frac{\ {\overline{b}}_F\left(i,h\right)}{{\overline{d}}_F\left(i+1,h\right)} $$
(71)

Substituting Eqs. (69) and (70) into Eq. (71) and simplifying gives:

$$ \left\langle {\phi}_M\left(n,h\right)\right\rangle ={\displaystyle \prod_{i=0}^{n-1}}\frac{\ {\overline{b}}_F\left(i,h\right)}{{\overline{d}}_F\left(i+1,h\right)}=\frac{J_M{\theta}_f(h)\kappa (h)}{n\left({\overline{\zeta}}_F+{\theta}_f(h)- nf(h)\right)}\times {\displaystyle \prod_{i=1}^{n-1}}\frac{\left(\frac{h{J}_M}{K}-i\right)f(h)\left({\overline{\zeta}}_F-f(h)\right)}{\left({\overline{\zeta}}_F+{\theta}_f(h)- if(h)\right)\left({\overline{\zeta}}_F-g\left(h,i\right)\right)} $$
(72)

When J M  → ∞, the number of propagules from a single individual will be much smaller than the number of propagules from all individuals, so \( {\overline{\zeta}}_F\gg f(h) \) and \( {\overline{\zeta}}_F\gg g\left(h,n\right) \). We can then use Eq. (72) to derive the following asymptotic approximation:

$$ \begin{array}{l}\left\langle {\phi}_M\left(n,h\right)\right\rangle \sim \frac{J_M{\theta}_f(h)\kappa (h)}{n\left({\overline{\zeta}}_F+{\theta}_f(h)- nf(h)\right)}{\displaystyle \prod_{i=1}^{n-1}}\frac{\left(\frac{h{J}_M}{K}-i\right)f(h)}{\left({\overline{\zeta}}_F+{\theta}_f(h)- if(h)\right)}\\ {}=\frac{K{\theta}_f(h)\kappa (h)\Gamma \left(\frac{h{J}_M}{K}+1\right)\Gamma \left(\frac{{\overline{\zeta}}_F+{\theta}_f(h)}{f(h)}-n\right)}{hnf(h)\Gamma \left(\frac{h{J}_M}{K}-n+1\right)\Gamma \left(\frac{{\overline{\zeta}}_F+{\theta}_f(h)}{f(h)}\right)}\end{array} $$
(73)

which is Eq. (15) in the main text.

Again in the limit that J M  → ∞, Eq. (73) reduces to

$$ \left\langle {\phi}_M\left(n,h\right)\right\rangle \sim \frac{K{\theta}_f(h)\kappa (h)}{hnf(h)}{\left(\frac{h{J}_Mf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f(h)\right)}\right)}^n $$
(74)

In order for Eq. (74) to give a finite number of species, we require that

$$ \frac{h{J}_Mf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f(h)\right)}<1 $$
(75)

for all h ≤ h max. Defining h* as the value of h that maximizes hf(h), and thus the left-hand side of Eq. (75), we focus on the condition that Eq. (75) is satisfied for h = h*. This guarantees that Eq. (75) will be satisfied for all other values of h as well. Following the derivation in Appendix 2, we rearrange Eq. (75). This gives

$$ \left({h}^{\ast }-\nu \left({h}^{\ast }-\frac{K}{J_M}\right)\right)f\left({h}^{\ast}\right)<{\displaystyle {\sum}_{h=1}^K} hf(h)\left\langle {\lambda}_M(h)\right\rangle \le {h}^{\ast }f\left({h}^{\ast}\right),\kern0.4em h\le {h}_{\max } $$
(76)

where the second inequality comes from the fact that 〈λ M (h)〉 is a probability distribution.

Because h* − K/J M  > 0 (in the limit of large J M , there will be more than one individual in each habitat) and ν is very small, the summation term in Eq. (76) is restricted to a narrow range of values, meaning that the vast majority of non-rare species must have hf(h) close to h*f(h*). The interpretation, then, is that while trade-offs do allow species with differing niche breadths to coexist, this occurs only when such trade-offs almost exactly compensate for niche breadth effects, i.e., h ∝ 1/f(h). Thus, a species that can survive in half as many habitat types as its competitor must have approximately twice the propagule pressure to be represented at any significant abundance in the metacommunity.

Next, we derive an approximate expression for \( {\overline{\zeta}}_F \). Beginning from our definition for \( {\overline{\zeta}}_F \) from the main text, we have

$$ {\overline{\zeta}}_F=\frac{J_M}{K}{\displaystyle {\sum}_{h=1}^{h_{\max }}} hf(h)\left\langle {\lambda}_M(h)\right\rangle =\frac{1}{K}{\displaystyle {\sum}_{h=1}^{h_{\max }}} hf(h){\displaystyle \sum_{n=1}^{J_M}}n\left\langle {\phi}_M\left(n,h\right)\right\rangle $$
(77)

where the second equality follows from the definition \( \left\langle {\lambda}_M(h)\right\rangle =\frac{1}{J_M}{\displaystyle {\sum}_{n=1}^{J_M}}n\left\langle {\phi}_M\left(n,h\right)\right\rangle \). Substituting Eq. (74) into Eq. (77) and assuming \( {\theta}_f(h)\approx \frac{\nu {\overline{\zeta}}_F}{\left(1-\nu \right)}\equiv {\theta}_f^{\ast } \) gives

$$ {\overline{\zeta}}_F\approx {\displaystyle {\sum}_{h=1}^{h_{\max }}}{\theta}_f^{\ast}\kappa (h){\displaystyle \sum_{n=1}^{J_M}}{\left(\frac{J_M hf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)}\right)}^n $$
(78)

In the limit that J M  → ∞ and keeping in mind that \( \frac{J_M hf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f^{*}\right)}<1 \), Eq. (78) reduces to

$$ {\overline{\zeta}}_F\approx {\displaystyle {\sum}_{h=1}^{h_{\max }}}{\theta}_f^{\ast}\kappa (h)\frac{Ch\left(1-{(Ch)}^{J_M}\right)}{1- Ch}\approx {\displaystyle {\sum}_{h=1}^{h_{\max }}}{\theta}_f^{\ast}\kappa (h)\frac{Ch}{1- Ch} $$
(79)

where \( C=\frac{J_Mf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)} \). Substituting in this expression for C and simplifying gives

$$ {\overline{\zeta}}_F\approx {\displaystyle {\sum}_{h=1}^{h_{\max }}}{\theta}_f^{*}\kappa (h)\frac{J_M hf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)-{J}_M hf(h)} $$
(80)

For a perfect trade-off with f(h max) = 1, the relationship between niche breadth and fecundity can be expressed as hf(h) = h max. Substituting this relationship, along with κ(h) = 1/h max (a uniform niche breadth distribution at speciation) and \( {\theta}_f^{\ast }=\frac{\nu {\overline{\zeta}}_F}{\left(1-\nu \right)} \) into Eq. (80) gives

$$ 1\approx {\displaystyle {\sum}_{h=1}^{h_{\max }}}\frac{J_M\nu }{K{\overline{\zeta}}_F-{J}_M{h}_{\max}\left(1-\nu \right)} $$
(81)

Simplifying gives

$$ {\overline{\zeta}}_F\approx \frac{J_M{h}_{\max }}{K} $$
(82)

which we use as our approximation for \( {\overline{\zeta}}_F \) when deriving niche breadth distributions below. Note that Eq. (80) also implies C ≈ 1 − ν, which will be important in the derivations below.

Fundamental niche breadth distribution by individuals

The goal is to derive an analytical expression for the proportion of individuals with a given niche breadth in the metacommunity at the speciation–extinction steady state (the equilibrium). Beginning from Eq. (74) and our definition for 〈λ M (h)〉, we have

$$ \left\langle {\lambda}_M(h)\right\rangle =\frac{1}{J_M}{\displaystyle \sum_{n=1}^{J_M}}n\left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{1}{J_M}{\displaystyle \sum_{n=1}^{J_M}}\ \frac{K{\theta}_f^{\ast}\kappa (h)}{hf(h)}{\left(\frac{h{J}_Mf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)}\right)}^n $$
(83)

Simplifying (83) gives

$$ \left\langle {\lambda}_M(h)\right\rangle =\frac{1}{J_M}\ \frac{K{\theta}_f^{\ast}\kappa (h)}{h_{\max }}\frac{C-{C}^{J_M+1}}{1-C} $$
(84)

where \( C=\frac{h{J}_Mf(h)}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)} \). For the special case of a perfect trade-off (hf(h) = h max) and assuming a uniform niche breadth distribution at speciation (κ(h) = 1/h max), we know that C < 1. This allows us to rewrite Eq. (84) as

$$ \left\langle {\lambda}_M(h)\right\rangle \approx \frac{1}{h_{\max }} $$
(85)

where we have substituted Eq. (82) for \( {\overline{\zeta}}_F \). Thus, as expected, under the trade-off model, all niche breadths are equally represented in the pool of individuals.

Fundamental niche breadth distribution by species

In this case, the goal is to derive an analytical expression for the proportion of species with a given niche breadth,  〈ς M (h)〉, in the metacommunity at the speciation–extinction steady state (the equilibrium). Using the approximation for the SAD given by Eq. (74), we write

$$ \left\langle {\varsigma}_M(h)\right\rangle =\frac{1}{S_{M,\mathrm{tot}}}{\displaystyle \sum_{n=1}^{h{J}_M/K}}\left\langle {\phi}_M\left(n,h\right)\right\rangle \approx \frac{K{\theta}_f^{\ast }}{S_{M,\mathrm{tot}}{\left({h}_{\max}\right)}^2}\times {\displaystyle \sum_{n=1}^{h{J}_M/K}}\frac{1}{n}{\left(\frac{J_M{h}_{\max }}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)}\right)}^n $$
(86)

where we have again assumed that hf(h) = h max and κ(h) = 1/h max, and where S M,tot is the expected total number of species in the metacommunity at equilibrium. In the limit as hJ M /K → ∞ (there are a large number of individuals with niche breadth h suited to each habitat), we can approximate (86) as:

$$ \left\langle {\varsigma}_M(h)\right\rangle \approx \frac{K{\theta}_f^{\ast }}{S_{M,\mathrm{tot}}{\left({h}_{\max}\right)}^2}{\displaystyle \sum_{n=1}^{h{J}_M/K}}\frac{(C)^n}{n}\approx -\frac{K{\theta}_f^{\ast }}{S_{M,\mathrm{tot}}{\left({h}_{\max}\right)}^2} \ln \left(1-C\right) $$
(87)

where, as before, \( C=\frac{J_M{h}_{\max }}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)} \). Recalling that 〈ς M (h)〉 must sum to one, Eq. (87) gives the following asymptotic expression for S M,tot:

$$ {S}_{M,\mathrm{tot}}\approx -\frac{K{\theta}_f^{\ast }}{h_{\max }} \ln \left(1-C\right) $$
(88)

Combining Eqs. (87) and (88) gives

$$ \left\langle {\varsigma}_M(h)\right\rangle \approx \frac{1}{h_{\max }} $$
(89)

Thus, under the trade-off model, all niche breadths are represented equally in the species pool, just as we found that they are represented equally in the individual pool in the previous subsection.

Realized niche breadth distribution by species

We begin by defining \( \left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle \) to be the fraction of the total species with fundamental niche breadth h and realized niche breadth \( \widehat{h} \). From this definition and Eqs. (74) and (11), we write

$$ \begin{array}{l}\left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle =\frac{1}{S_{M,\mathrm{tot}}}{\displaystyle \sum_{n=1}^{J_M}}\left\langle {\phi}_M\left(n,h\right)\ \right\rangle q\left(\widehat{h};h,n\right)\\ {}\approx \frac{K{\theta}_f^{\ast }}{S_{M,\mathrm{tot}}{\left({h}_{\max}\right)}^2}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}}{\displaystyle \sum_{n=1}^{J_M}}\frac{1}{n}{(C)}^n{\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right){\left(\frac{\widehat{h}-l}{h}\right)}^n\\ {}=\frac{K{\theta}_f^{\ast }}{S_{M,\mathrm{tot}}{\left({h}_{\max}\right)}^2}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}}{\displaystyle \sum_{n=1}^{J_M}}\frac{{\left(C\left(\frac{\widehat{h}-l}{h}\right)\right)}^n}{n}{\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right)\end{array} $$
(90)

where, again, \( q\left(\widehat{h};h,n\right) \) is the probability that a species with abundance n and fundamental niche breadth h will have realized niche breadth \( \widehat{h} \), and we have assumed a perfect trade-off (hf(h) = h max) and a uniform distribution of niche breadths at speciation (κ(h) = 1/h max), giving \( C=\frac{J_M{h}_{\max }}{K\left({\overline{\zeta}}_F+{\theta}_f^{\ast}\right)} \). Because \( \frac{\widehat{h}-l}{h}\le 1 \), we know that \( C\left(\frac{\widehat{h}-l}{h}\right)\le C<1 \); thus, in the limit of large J M , Eq. (90) can be approximated

$$ \left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle \approx -\frac{K{\theta}_f^{*}}{S_{M,\mathrm{tot}}{\left({h}_{\max}\right)}^2}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}} \ln \left(1-C\left(\frac{\widehat{h}-l}{h}\right)\right){\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right) $$
(91)

From the previous subsection, we know that C ≈ 1 − ν. Substituting in this and the approximation (88) for S M,tot and then simplifying gives

$$ \left\langle {\widehat{\varsigma}}_M\left(\widehat{h},h\right)\right\rangle \approx \frac{1}{h_{\max } \ln \left(\nu \right)}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right){\displaystyle \sum_{l=0}^{\widehat{h}}} \ln \left(1-\left(1-\nu \right)\left(\frac{\widehat{h}-l}{h}\right)\right){\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right) $$
(92)

The realized niche breadth distribution across the entire metacommunity is then given by

$$ \left\langle {\widehat{\varsigma}}_M\left(\widehat{h}\right)\right\rangle \approx \frac{1}{h_{\max } \ln \left(\nu \right)}{\displaystyle \sum_{h=\widehat{h}}^{h_{\max }}}{\displaystyle \sum_{l=0}^{\widehat{h}}}\left(\begin{array}{c}\hfill h\hfill \\ {}\hfill \widehat{h}\hfill \end{array}\right) \ln \left(1-\left(1-\nu \right)\left(\frac{\widehat{h}-l}{h}\right)\right){\left(-1\right)}^l\left(\begin{array}{c}\hfill \widehat{h}\hfill \\ {}\hfill l\hfill \end{array}\right) $$
(93)

Appendix 7: Numerical simulations of metacommunity time-dependent behavior

We ran numerical simulations to test the accuracy of the time-dependent realized niche breadth distributions derived in Appendix 5. Specifically, we looked at the probability that a species has maximum niche breadth (i.e., its realized niche breadth is equal to its fundamental niche breadth) given that it has not gone extinct (Eq. (67)). We found that Eq. (67) is accurate for a sufficiently diverse community, i.e., a community with large J M ν, where J M is the metacommunity size and ν is the speciation rate, but underestimates the probability of a species achieving maximum niche breadth for smaller J M ν. This is because the theoretical model assumes that the individuals belonging to any given species are assigned to particular habitats independently of one another. This is very close to true for relatively rare species, which is why the approximation works well in diverse communities. In the extreme low-diversity case in which a single species dominates the whole community, it is clear that the independence assumptions fail completely because the distribution of habitats assigned to individuals of the species is constrained by the distribution of habitats in the landscape.

Fig. 14
figure14

Probability that a species’ realized niche breadth \( \widehat{h} \) is equal to its fundamental niche breadth h over time, for a species with h = h max = 3 and initial abundance n = 1, in a community with K = 3 habitats and a speciation rate ν = 0.1. The left panel has J M  = 21 individuals while the right panel has J M  = 99 individuals. The pink curves show the theoretical prediction from Eq. (67), and the blue points show the means of 10,000 simulations

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Bewick, S., Chisholm, R.A., Akçay, E. et al. A stochastic biodiversity model with overlapping niche structure. Theor Ecol 8, 81–109 (2015). https://doi.org/10.1007/s12080-014-0227-7

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Keywords

  • Niche overlap
  • Habitat heterogeneity
  • Biodiversity
  • Species abundance distribution
  • Niche breadth distribution
  • Neutral theory