Fusing spatial resource heterogeneity with a competition–colonization trade-off in model communities

Abstract

Two commonly cited mechanisms of multispecies coexistence in patchy environments are spatial heterogeneity in competitive abilities caused by variation in resources and a competition–colonization trade-off. In this paper, a model that fuses these mechanisms together is presented and analyzed. The model suggests that spatial variation in resource ratios can lead to multispecies coexistence, but this mechanism by itself is weak when the number of resources for which species compete is small. However, spatial resource heterogeneity is a powerful mechanism for multispecies coexistence when it acts synergistically with a competition–colonization trade-off. The model also shows how resource supply can control the competitive balance between species that are weak competitors but superior colonizers and strong competitors/inferior colonizers. This provides additional theoretical support for a possible explanation of empirically observed hump-shaped relationships between species diversity and ecological productivity.

Appendix

Additional details and analysis for the one- and two-resource models

I first show that the model in Eq. 5–7 behaves identically to the phenomenological model in Eq. 1. First, I show that if there is a fixed point of Eq. 7, then the fixed point is of the form $p^{ * }_{i} {\left( S \right)} = {\text{constant}}$ for $S > R^{ * }_{i}$. To keep the math simple, I present only the analysis for the competitively superior species, species 1. Altering the analysis for other species is straightforward. Suppose there exists a fixed point $p^{ * }_{1} {\left( S \right)}$ that may vary as a function of S. Let $N^{ * }_{1} = {\int\limits_S {p^{ * }_{1} {\left( S \right)}n^{ * }_{1} {\left( S \right)}\phi {\left( S \right)}dS} }$. Setting ${dp_{1} {\left( {S,t} \right)}} \mathord{\left/ {\vphantom {{dp_{1} {\left( {S,t} \right)}} {dt}}} \right. \kern-\nulldelimiterspace} {dt} = 0$ for all S (Eq. 7) implies that $p^{ * }_{1} {\left( S \right)} = {\alpha _{1} N^{ * }_{1} } \mathord{\left/ {\vphantom {{\alpha _{1} N^{ * }_{1} } {{\left( {\alpha _{1} N^{ * }_{1} + e_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\alpha _{1} N^{ * }_{1} + e_{1} } \right)}}$ for $S > R^{ * }_{1}$. Plugging this expression back into $N^{ * }_{1} = {\int\limits_S {p^{ * }_{1} {\left( S \right)}n^{ * }_{1} {\left( S \right)}\phi {\left( S \right)}dS} }$ yields $N^{ * }_{1} = p^{ * }_{1} {\left\langle {n^{ * }_{1} } \right\rangle }$, which in turn yields $p^{ * }_{1} {\left( S \right)} = 1 - {e_{1} } \mathord{\left/ {\vphantom {{e_{1} } {\alpha _{1} }}} \right. \kern-\nulldelimiterspace} {\alpha _{1} }{\left\langle {n^{ * }_{1} } \right\rangle }$ for $S > R^{ * }_{1}$.

I now use an invasibility analysis to show that if $p^{ * }_{1} {\left( S \right)} > 0$, then the boundary $p^{ * }_{1} {\left( S \right)} = 0$ is repelling. Again, extensions to competitively inferior species are straightforward. Without loss of generality, suppose $p_{1} {\left( S \right)} = \varepsilon > 0$ for $S > R^{ * }_{1}$. Then, the overall change of the population size N ₁ is

$$ \begin{array}{*{20}c} {\frac{{dN_{1} }} {{dt}} = \frac{d} {{dt}}{\int\limits_S {p_{1} {\left( S \right)}n^{ * }_{1} {\left( S \right)}\phi {\left( S \right)}dS} }} \\ { = {\int\limits_S {\frac{{dp_{1} {\left( S \right)}}} {{dt}}n^{ * }_{1} {\left( S \right)}\phi {\left( S \right)}dS} }} \\ { = {\int\limits_{S > R^{ * }_{1} } {{\left[ {\alpha _{1} \varepsilon {\left\langle {n^{ * }_{1} } \right\rangle }{\left( {1 - \varepsilon } \right)} - e_{1} \varepsilon } \right]}n^{ * }_{1} {\left( S \right)}\phi {\left( S \right)}dS} }} \\ { \approx \varepsilon {\left[ {\alpha _{1} {\left\langle {n^{ * }_{1} } \right\rangle } - e_{1} } \right]}{\left\langle {n^{ * }_{1} } \right\rangle }} \\ \end{array} $$

which is >0 if and only if $\alpha _{1} {\left\langle {n^{ * }_{1} } \right\rangle } > e_{1}$.

Spatial heterogeneity matters in the two-resource model but not the one-resource model because in the two-resource model the competitive hierarchy can vary among patches depending on the resource ratio. The simplest setting in which to see this is a two-resource, two-species model where all patches have sufficient resources to support either species, and there are no patches where both species can coexist simultaneously. Let S ₁ denote the set of patches in which species 1 outcompetes species 2, and let S ₂ denote the set of patches in which species 2 outcompetes species 1. In the absence of species 2, an argument similar to the one above shows that species 1 comes to an equilibrium occupancy $p^{ * }_{1}$ that is constant for all ${\overrightarrow{S}} = {\left( {S_{1} ,S_{2} } \right)}$ pairs and an equilibrium density $N^{ * }_{1} = p^{ * }_{1} {\left\langle {n^{ * }_{1} } \right\rangle }$. Now suppose $p_{2} {\left( {{\overrightarrow{S}} } \right)} = \varepsilon > 0$ everywhere. The invasion criterion that must be satisfied in order for species 2 to invade is:

$$ {\int\limits_{{\overrightarrow{S}} \in \mathcal{S}_{1} } {{\left[ {\alpha _{2} {\left\langle {n^{ * }_{2} } \right\rangle }{\left( {1 - p^{ * }_{1} } \right)} - {\left( {e_{2} + \alpha _{1} N^{ * }_{1} } \right)}} \right]}n^{ * }_{2} {\left( {{\overrightarrow{S}} } \right)}\phi {\left( {{\overrightarrow{S}} } \right)}d{\overrightarrow{S}} } } + {\int\limits_{{\overrightarrow{S}} \in \mathcal{S}_{2} } {{\left[ {\alpha _{2} {\left\langle {n^{ * }_{2} } \right\rangle } - e_{2} } \right]}n^{ * }_{2} {\left( {{\overrightarrow{S}} } \right)}\phi {\left( {{\overrightarrow{S}} } \right)}d{\overrightarrow{S}} } } > 0. $$

Clearly, this invasion criterion depends on the distribution $\phi {\left( {{\overrightarrow{S}} } \right)}$.

Simulation details for the one-resource model

R* values were assigned to species by independent random draws from a normal distribution with mean 5 and variance 1. The α values were drawn from a lognormal distribution with parameters μ = 0 and σ = 1 (thus the average value of α was $e^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \approx 1.65$). The species with the smallest R* value was assigned the smallest α value, and so on. Other parameter values that were identical for all species were m = 1 and b = 1 (Eq. 4). The resource turnover rate a was equal to 1 (Eq. 4). The patch extinction rate e was 1 in the low disturbance simulations, and 5 in the high disturbance simulations. The range of ϕ(S) was 10 in the spatially heterogeneous simulations and 0 in the spatially homogeneous simulations.

Simulation details for the two-resource model

$R^{ * }_{1}$ and $R^{ * }_{2}$ values were assigned to species by independent draws from a normal distribution with mean 0 and variance 1. The α values were drawn from a lognormal distribution with parameters μ = 0 and σ = 1. Each species’ competitive ability was measured by the probability that the species’ $R^{ * }_{1}$ and $R^{ * }_{2}$ values would be smaller than the $R^{ * }_{1}$ and $R^{ * }_{2}$ values, respectively, of a new, randomly chosen species. The species with the greatest competitive ability (measured in this way) was assigned the smallest colonization rate, and so on.

At the beginning of each simulation, each of the 10⁵ patches were assigned S ₁ and S ₂ values by randomly drawing S ₁ and S ₂ from uniform probability distributions (spatially heterogeneous simulations) or by assigning each patch the same S ₁ and S ₂ values (spatially homogeneous simulations; to generate Fig. 6, the limits of the distribution of S ₁ were 0.75 times the limits of the distribution of S ₂). Patches were populated by drawing species randomly and with equal probability from the species pool. Thus, each species occupied approximately 10³ patches at the beginning of the simulation. Simulations were run as a generalized birth-death process, with the following types of events: either a species could attempt to colonize another patch, or a disturbance would strike a patch. Attempted colonizations happened with rate α _i N _i , where N _i was the total density of species i summed over all patches. Here, rates are stochastic rates in the sense that if an event occurs with rate β, then the probability that event occurs in an infinitesimal time interval Δt is $\beta \Delta t + o{\left( {\Delta t} \right)}$, and the probability of more than one event occurring in Δt is o(Δt) (Ross 2003). Disturbances occurred with rate e times the total number of patches, 10⁵. If species i attempted to colonize a patch, one target patch was picked at random from all the patches in the landscape. If species i already occupied the target patch, no changes occurred. Otherwise, the consumer-resource model (Eq. 11) was used to determine the outcome of resource competition within that patch. If a disturbance event occurred, one patch was selected at random from all the patches in the landscape. If the selected patch was already vacant, no changes occurred. Otherwise, any species present in that patch were eliminated from the patch.

Time was recorded in units of epochs, were one epoch was equal to a total of 10⁵ attempted colonizations or disturbances striking occupied patches (note that disturbances striking unoccupied patches were not included in the calculation of epochs). Thus, on average, each patch in the landscape would be the target of one attempted colonization or disturbance per epoch. After 500 epochs, the average per patch density of each species was recorded.