Contributions of high- and low-quality patches to a metapopulation with stochastic disturbance
Studies of time-invariant matrix metapopulation models indicate that metapopulation growth rate is usually more sensitive to the vital rates of individuals in high-quality (i.e., good) patches than in low-quality (i.e., bad) patches. This suggests that, given a choice, management efforts should focus on good rather than bad patches. Here, we examine the sensitivity of metapopulation growth rate for a two-patch matrix metapopulation model with and without stochastic disturbance and found cases where managers can more efficiently increase metapopulation growth rate by focusing efforts on the bad patch. In our model, net reproductive rate differs between the two patches so that in the absence of dispersal, one patch is high quality and the other low quality. Disturbance, when present, reduces net reproductive rate with equal frequency and intensity in both patches. The stochastic disturbance model gives qualitatively similar results to the deterministic model. In most cases, metapopulation growth rate was elastic to changes in net reproductive rate of individuals in the good patch than the bad patch. However, when the majority of individuals are located in the bad patch, metapopulation growth rate can be most elastic to net reproductive rate in the bad patch. We expand the model to include two stages and parameterize the patches using data for the softshell clam, Mya arenaria. With a two-stage demographic model, the elasticities of metapopulation growth rate to parameters in the bad patch increase, while elasticities to the same parameters in the good patch decrease. Metapopulation growth rate is most elastic to adult survival in the population of the good patch for all scenarios we examine. If the majority of the metapopulation is located in the bad patch, the elasticity to parameters of that population increase but do not surpass elasticity to parameters in the good patch. This model can be expanded to include additional patches, multiple stages, stochastic dispersal, and complex demography.
KeywordsMetapopulation Patch dynamics Disturbance Matrix population model Stage-structured Mya arenaria
Many populations of organisms are distributed among patches that vary in quality. When organisms move between these patches, thereby connecting them, the collection of patches is termed a “metapopulation.” When managing a metapopulation, effort must be allocated among patches and life cycle stages because of inevitable limitations on time, manpower, and money. One way to approach this allocation problem, and the motivation of this paper, is to develop a method for determining the potential impact of changes in patch-specific population parameters, particularly when the metapopulation in question is subject to stochastic disturbance.
The simpler problem of allocating effort among life cycle stages within a single population has been approached using elasticity analyses of the population growth rate (e.g., Crouse et al. 1987; Parker 2000; Aires-da Silva and Gallucci 2007; Kesler and Haig 2007; Raithel et al. 2007; Enneson and Litzgus 2008). The assumption is that, all else being equal, if population growth rate is very elastic to a parameter, then that parameter is a good target for management efforts. In this paper, we will show that the elasticities of the stochastic metapopulation growth rate can be used to extend this approach and identify both locations and stages that are good targets for management.
Previous theoretical studies of metapopulations (e.g., Pulliam 1988; Runge et al. 2006) have suggested that management should focus on patches where demographic rates (survival, reproduction, and growth) are most favorable. Using elasticity analyses of progressively more complex models, we examine when this rule of thumb is correct and when it is not. Firstly, we analyze a deterministic two-patch metapopulation without stage structure. We then add environmental stochasticity. Finally, we include stage structure and a life cycle model that conforms with the softshell clam, Mya arenaria.
One-stage, deterministic model
The transition from λ < 1 to λ > 1 indicates where the metapopulation switches from declining to growing. As one might expect, increasing net reproductive rate in either patch increases the area of parameter space where λ > 1 (shaded areas in Fig. 2). A proportional increase in R1, however, results in a greater proportion of space where λ > 1 than the same increase in R2.
There are instances when E < 1 and λ > 1 (Fig. 2b, d). That is, there are times when the metapopulation growth rate is positive and is more sensitive to net reproductive rate in the bad patch. In these instances, migration rates are such that the majority of the metapopulation is found in the bad patch. The dotted lines on Fig. 2b–d are where the population is evenly distributed among patches. To the left of this line, there are more individuals in patch 1; to the right of this line, there are more individuals in patch 2. Cases where E < 1 and λ > 1 are always below the line. By solving for the stable structure from the matrix (Eq. 2), it can be shown that E < 1 only if the majority of the population is in the bad patch; i.e., that the dashed line in Fig. 2 is always to the right of the dotted line. One might assume that when the majority of individuals are in the bad patch, then metapopulation growth rate would be negative; however, this clearly is not always the case. The above result indicates that increasing R2 has a larger effect because it impacts more individuals than increasing R1. It also suggests that not only should individual patch growth rates be considered when determining where management efforts should be focused, but also the distribution of individuals among patches within the metapopulation.
One-stage, stochastic model
In the case where migration rates are equal (Fig. 4b, c), the majority of metapopulation individuals are located where most individuals originating from the good patch ultimately settle (patch 2 in the case where m1, m2 = 0.9, Fig. 5b; patch 1 in the case where m1, m2 = 0.1, Fig. 5c). Again, the elasticity ratio is greater than 1.
The result changes if emigration from patch 1 is high and emigration from patch 2 is low (Fig. 4d). Migration rates in this case result in more individuals, on average, in patch 2 (Fig. 5d). Consequently, Es < 1 for all combinations of disturbance parameters p and c. These results parallel those of the deterministic model: Es < 1 only if there are a lot more individuals on average in patch 2 than in patch 1. Changes to parameters in patch 2 thus will affect more individuals of the metapopulation, resulting in a proportionally greater increase in metapopulation growth rate.
Es also is affected by disturbance parameters, although to a lesser extent when compared with the effects of migration rates. In general, c has a greater effect on Es than p; contours of Es tend to be more horizontal than vertical in all panels of Fig. 2. That is, the temporal relationship between disturbance at the two patches is more influential on their relative elasticities than the probability that disturbance occurs over any given time period.
Although there is only a small amount of variability in Es within a given panel, Es varies dramatically among the four panels (note the log scale). Migration rates appear to affect Es to a much greater extent than disturbance parameters. Es < 1 only when emigration was high from patch 1 and low from patch 2 (Fig. 4d), and in that migration scenario logλs < 0 for most combinations of p and c. There is a small set of disturbance parameter values (p < 0.05, c ≈ 0) where both logλs > 0 and Es < 1; this occurs at the lowest values of p, when disturbance is so improbable that results from the stochastic model are comparable to those of the deterministic model (Fig. 2b).
For the reproductive rate and disturbance intensity used for Fig. 4, the stochastic metapopulation growth rate is positive only when the probability of disturbance is low (shaded areas, Fig. 4). The set of disturbance parameters p and c for which logλs > 0 is largest when on average population 1 is larger than population 2. The set shrinks as individuals become more evenly distributed between the two patches. When population 2 is larger, as in Fig. 4d, logλs > 0 only for the smallest values of the disturbance probability p.
Two-stage, stochastic model
For many species, especially marine invertebrates, the model of the previous section lacks the appropriate level of demographic detail to adequately summarize the animal’s life history. An example of one such invertebrate is the softshell clam, M. arenaria, a commercially important bivalve commonly found in New England estuaries. M. arenaria’s life cycle is typical of nearshore marine benthic invertebrates (Thorson 1950; Abraham and Dillon 1986). It is characterized by a relatively sedentary adult stage, with adults highly aggregated into patches of suitable habitat. Migration between populations occurs via dispersal of the short-lived larval stage. Like many marine invertebrate species, M. arenaria larvae are produced in vast quantities during a short reproductive season and typically most larvae die before recruiting to the adult phase.
Since M. arenaria adults are sedentary, we set M2 = I. To simplify notation, we will set m11 = m1 and m21 = m2 from here on.
We analyzed the model in Eqs. 38 and 39 for σ1 = 0.9, σ2 = 0.3, γ1 = 0.8, γ2 = 0.24, β1 = 5.6, and β2 = 7.5. These parameter values (1) produce individual patch growth rates similar to the net reproductive rates for the one-stage case, i.e., R1 = 2.5 and R2 = 0.9; and (2) roughly comport with estimated demographic parameters for M. arenaria from field studies (Ripley and Caswell 2006). First we set migration rates to be equal and low, at m1 = m2 = 0.1. Migration rates are difficult to obtain for benthic invertebrates with pelagic larvae, and were not available for M. arenaria, but these values are arguably realistic. We assumed that disturbance affects all patches and stages with intensity D = 0.9, and set the probability of disturbance at the two patches to p = 0.5.
When the model complexity was increased by adding a second stage, the elasticities of stochastic metapopulation growth rate to population 1 parameters increased, while those same elasticities to population 2 parameters decreased. Even when the majority of individuals were in the bad patch, growth rate elasticity was still greatest to parameters in the good patch for the scenarios we examined.
Discussion & conclusions
Good and bad patches, as defined in this paper, are also referred to as “sources” and “sinks”. The source/sink literature (Pulliam 1988; Howe and Davis 1991; Runge et al. 2006) suggests that if one must choose between focusing management efforts on a source or a sink, one should always choose the source. In most cases, our results agree, however we found that under some conditions sink populations are important to long-term metapopulation persistence.
For two patches without demographic structure, the distribution of individuals among patches is important in addition to individual patch growth rates. Improving the source population is best except when the majority of individuals are in the sink population. When most of the individuals are in the sink, changes to parameters in that population affect more individuals and can therefore have a proportionally larger effect on overall metapopulation growth rate.
There is no way to know a priori how likely such patterns are, or the ecological factors that might produce them. The distribution of individuals among patches may account for previous suggestions of the importance of sink populations, for example, higher overall metapopulation size resulting when individuals unable to settle at source populations due to high population densities migrate to sinks (Pulliam 1988) or when emigration from the source patch is suggested to increase population size by offering “insurance” against catastrophe (Levin et al. 1984). This is especially true if the environmental variability is spatially negatively correlated (Wiener and Tuljapurkar 1994).
When conditions are stochastic the correlation of disturbance among patches also influences the relative impacts of the good and bad patch. In cases of positive covariance, disturbance reduces conditions in the good and patches in concert, so results are the same as for the deterministic case. However, when the covariance of disturbance is low, disturbance affecting the good patch can cause the good patch to become worse than the bad patch, so focusing management on the bad patch becomes more important (Figs. 4, 7 and 8).
When demographic structure is considered, the relative importance of the bad patch to overall metapopulation growth rate is reduced. The additional stages act as a buffer for stochastic disturbance and the distribution of individuals is less influential on metapopulation growth rate.
We have focused here on elasticities, which indicate where management efforts should be directed if changes in all parameters of the same magnitude can be made and all else is equal. Other factors also play a role in the implementation of management actions; including cost, feasibility, and inherent parameter variability. Such factors, which can vary both among parameters and among patches, greatly influence the efficacy of management efforts in concert with elasticities. Such constraints can easily be incorporated into a modified metric within the framework of elasticities, and then be used to guide management decisions.
These results offer no rules of thumb for allocating management efforts. Rather, they suggest the importance of conducting elasticity or other perturbation analyses to determine the contributions of individual patches to overall metapopulation growth rate. The analyses presented here can be extended to include additional patches, stages, types of stochasticity, etc. as well as additional constraints on the implementation of management actions.
The authors would like to thank Lauren Mullineaux for her insightful comments and contributions to the manuscript. Financial support was provided by the Woods Hole Oceanographic Institution Academic Programs Office; National Science Foundation grants OCE-0326734, OCE-0215905, OCE-0349177, DEB-0235692, DEB-0816514, DMS-0532378, OCE-1031256, and ATM-0428122; and by National Oceanic and Atmospheric Administration National Sea Grant College Program Office, Department of Commerce, under Grant No. NA86RG0075 (Woods Hole Oceanographic Institution Sea Grant Project No. R/0-32), and Grant No. NA16RG2273 (Woods Hole Oceanographic Institution Sea Grant Project No. R/0-35).
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