Simulating Invasion Growth Rates
Pairwise comparisons, as described above, are often used to evaluate conditions for coexistence (e.g. Adler et al. 2010; Wainwrightet al. 2019), but are likely to underestimate the total amount of interspecific competition in multispecies communities. Analytical solutions to this problem are notoriously complex, and lead to necessary circularity (Saavedra et al. 2017). Therefore, we repeated the above analysis comparing each species i with the averages of all other species in the relevant assemblage. A more complete approach, however, is to simulate population dynamics for all interacting species, treating each as an invader in turn, and evaluating emergent invasion growth rates (IGR) explicitly (Adler et al. 2010; Ellneret al. 2016). We employed the classical multispecies version of the Lotka-Volterra model:
\(N_{i,t+1}=N_{i,t}+N_{i,t}R_{max,i}\left(1-\frac{\sum_{j=1}^{n}{\alpha_{\text{ij}}N_{j,t}}}{K_{i}}\right)\)(Equation 2)
for each population i interacting with populations j = 1,2…n , where αij is a competition co-efficient (intraspecific when i = j , interspecific for all other interactions), and Ki is equilibrium density, or carrying capacity. The difference form of Equation 2 is preferred to differential equations because herbivore populations typically grow in discrete increments, e.g. producing young yearly, and reproductive events are often synchronized across species (Zerbeet al. 2012; Fryxell et al. 2014). To parameterize initial conditions, we assume that αij is equivalent toOij , and estimate Ki from the quantity \(\sum_{j=1}^{n}{\alpha_{\text{ij}}N_{j,t}}\), whereNi,t and Nj,t are mean population sizes. The latter is a simple analytical solution based on the observation that all populations in our data have mean growth rates ~0, and the assumption that current (average) densities are the result, at least in part, of interspecific competition.
For each simulation with a specific invader i , we setNi (0) = 0, allowed resident populations to develop over 500 time steps (sufficient for each to stabilize at a new equilibrium), and then re-introduced the invader at low density, i.e.Ni = 10-4. The model was then allowed to run a further 500 generations. IGRs were estimated as average growth rates \(\left(\ln\frac{N_{i,t}}{N_{i,t+1}}\right)\) over 200 generations, or for as long as Ni ≤ 10. The stabilizing effect of niche partitioning was inferred by comparing estimated IGRs to those from models assuming zero niche overlap, i.e. complete partitioning, and with those assuming complete overlap, i.e. all Oij = 1.