Ecological Archives A023-077-A5
Tarik C. Gouhier, Frédéric Guichard, Bruce A. Menge. 2013. Designing effective reserve networks for nonequilibrium metacommunities. Ecological Applications 23:1488–1503. http://dx.doi.org/10.1890/12-1801.1
Appendix E. The effects of reserve networks on stochastically forced equilibrium metacommunities.
Stochastically forced equilibrium metacommunities.
In the main text and Appendices A–D, we identified optimal reserve networks for nonequilibrium metacommunities characterized by spatially coupled endogenous fluctuations (limit cycles). However, nonequilibrium dynamics can also emerge when stochastic exogenous forcing perturbs a system away from its (point) equilibrium. We now focus on identifying the optimal reserve networks for such stochastically forced equilibrium metacommunities by using two different approaches: (1) sampling the model parameters from a log-normal probability distribution and (2) adding spatially uncorrelated (white) or correlated (red) noise directly to the abundance of the prey and predator. For the first approach, we sampled the prey carrying capacity K from a log-normal distribution with a variance of 2 and a mean of 7 for the type II functional response or a mean of 5 for the type III functional response. The rest of the parameter values were identical to those used in the main text and the appendices. For the second approach, we added stochastic noise directly to the abundance of the predator and the prey in the metacommunity model (Eq. E.1):
Where j=1 for the type II functional response and j=2 for the type III functional response. The stochastic noise was generated via the spectral synthesis method for each species i at each site x and time step t (Fontaine and Gonzalez 2005):
Where n=200 is the total number of sites in the metacommunity, determines the relationship between power and frequency , and is a uniform deviate in the interval [0, 2p). We set to generate spatially and temporally uncorrelated stochastic perturbations (i.e., white noise) with a mean of 0 and a variance of 0.1 to the abundance of the prey and the predator at each site and time step. We set to generate spatially correlated but temporally uncorrelated stochastic perturbations with a mean of 0 and a variance of 0.1 to the abundance of the prey and the predator at each site and time step.
Both approaches used to simulate stochastically forced equilibrium metacommunities generate qualitatively identical results to those obtained for equilibrium metacommunities (e.g., compare Fig. 2 to Fig. E1–E4). Networks of small and aggregated reserves promote the mean abundance of the predator and reduce that of the prey globally and within reserves because of trophic cascades (Fig. E1–E4 a–d). Conversely, networks of large and isolated reserves increase the global mean abundance of the prey and reduce that of the predator (Fig. E1–E4 a, b). Lower connectivity between reserves and unprotected sites leads to reduced predator spillover into unprotected areas and thus weaker trophic cascades. This allows prey abundance to build-up in unprotected areas faster than it decreases in reserves, and global mean prey abundance thus increases (Fig. E1–E4 c, e). Predator global mean abundance decreases because limited spillover from reserves to unprotected areas causes predator abundance to decrease faster outside reserves than it increases within reserves (Fig. E1–E4 d, f). Prey total yield is maximized for networks of large and isolated reserves because those networks promote the mean abundance of the prey outside reserves where it can be harvested by reducing predator spillover and thus trophic cascades (Fig. E1-E4 g). However, predator total yield never increases with reserves because the gains achieved in predator abundance occur primarily within reserves where harvesting is either limited or completely prohibited (Fig. E1–E4 h).
These simulations show that reserve networks based on the extent of patchiness are only optimal for nonequilibrium metacommunities that generate such patchiness via spatially coupled endogenous fluctuations. Indeed, when equilibrium dynamics are stochastically forced by either spatially correlated (red noise) or uncorrelated (white noise) exogenous perturbations, localized dispersal cannot generate an intermittent decoupling between regional immigration and local abundance dynamics, so no patchiness emerges at large spatial scales. Hence, the emergence of large-scale patchiness may serve as a signature of endogenously generated population fluctuations and guide the design of effective reserve networks.
Fontaine, C., and A. Gonzalez. 2005. Population synchrony induced by resource fluctuations and dispersal in an aquatic microcosm. Ecology 86:1463–1471.
Fig. E1. The effect of varying the level of protection and the distance between reserves (expressed as % of spatial extent) on prey and predator (a, b) global mean abundance, (c, d) mean abundance within reserves, (e, f) mean abundance outside reserves and (g, h) total yield for stochastically-forced equilibrium metacommunities with full dispersal and a type II functional response. Stochastic forcing was generated by selecting the carrying capacity K from a log-normal distribution with a mean of 7 and a variance of 2 for each site and time step. Each metric is measured in percent change relative to baseline scenarios without reserves (i.e., spatially uniform harvesting rate) represented by the semi-transparent black horizontal surface. The mean predator and prey harvesting rates are set to 0.1. The green, blue, and red axis tick labels represent respectively the extent of dispersal, the extent of patchiness, and the scale of coupling. Results represent means from 10 replicate simulations.