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.