Matrix community models for ecology and evolution

Abstract

Ecological communities are shaped by biotic interactions as well as environmental forces, and both must be incorporated to obtain models capable of forecasting realistic community dynamics. Many community models first specify pairwise biotic interactions and then secondarily examine how extrinsic factors such as abiotic conditions affect species abundances. A disadvantage of this approach is that the species interactions themselves are often environment and context specific, making parameterization difficult. We propose an alternative approach, matrix community models (MCMs), which are sets of matrix population models linked by an assumption of aggregate density dependence. MCMs incorporate detailed species autecology but are neutral with respect to pairwise species interactions, instead allowing interactions to be revealed within the model structure. These model-revealed species interactions, including competitive exclusion, facilitation, and interference competition, shape the distribution and abundance of species within communities and generate empirically testable predictions about species interactions. We develop a framework for building MCMs using vital rates in a stochastic, multispecies framework. Single-species matrix population models are connected via an assumption of aggregate density dependence, pairwise species interactions are estimated with sensitivity analysis, and community trajectories are analyzed under different environmental regimes using standard statistical tools and network analysis. MCMs have the advantage that pairwise species interactions need not be specified a priori, and that mechanistic demographic-environment linkages permit forecasting of community dynamics under novel, non-stationary environmental regimes. A challenge is that species' autecological vital rates, such as fecundity, growth and survivorship, must be measured under a diverse range of environmental conditions to parameterize the models. We illustrate the approach with examples and discuss prospects for future theoretical and empirical developments.

Fig. 1. Building and projecting a matrix community model.

The right panel depicts that fecundity, in this case, is a function of aggregate density dependence. In models of riparian vegetation, the assumption that there is a finite amount of open space available for seedling recruitment serves as the dependency assumption.

Fig. 2. Latent interactions derived from cross-guild/stage sensitivity analysis (guilds were used in this specific study for model transferability).

The relevant vital rate (here, mortality due to flood events; scaled from 0 to 1) was changed by increments of 0.01 in the vicinity of its actual value and population size was recorded. At each increment, this was repeated independently up to 1000 times to achieve stable convergence on mean values. Sensitivity was calculated as the slope of the average population size of a guild regressed against the vital rate, which is analogous to taking the partial derivative of population size with respect to the vital rate of interest (Lytle & Merritt 2004). Only sensitivities with R² > 0.3 were retained for network analysis. Because each guild contains six stages, the matrix depicts both within-guild and among-guild effects. The diagonal depicts self-effects, where an increase in mortality causes a decrease in abundance of that guild/stage. Positive relationships suggest competition, where an increase in mortality rate in one guild/stage allows another one to increase. HT, XS, HS, MM, and DS represent riparian plant guilds that possess similar vital rates (respectively, hydroriparian tree, xerophytic shrub, hydroriparian shrub, mesic meadow, and desert shrub) and numbers indicate stage class.

Fig. 3. Network analysis of riparian vegetation showing how changes in a vital rate in one guild/stage affects other members of the community.

Edge thickness denotes the slope of the relationship and edge color denotes R² (colors and node labels as in Fig. 2). Note that a relationship can have a strong fit (high R²) but low magnitude of effect on abundance (shallow slope). The size of each node on the graph depicts the average abundance of that guild/stage. In this riparian community, which was modeled under a natural flow regime of flood and drought frequencies, mature hydroriparian trees (HT6) are a keystone of the ecosystem because of their high connectance to other nodes. In western U.S. riparian areas, HT6 often consists of cottonwood (Populus fremontii) gallery forest.