Appendix A.. The REEF simulation model.
We here describe in detail the main features of the population dynamics model central to the REEF program as we implemented it (Mapstone et al. 1996b). The population dynamics model itself has three main components:(1) recruitment to reefs,(2) post-recruitment mortality, and (3) adult migration. Recruitment and mortality (both natural and fishing) are the main sources of between-reef and between-year variation in population dynamics, while migration was treated as a spatially and temporally constant rate. Experimental treatment effects and the effects of background fishing on populations of coral trout were imposed by varying the mortality regimes applied at particular reefs. Estimates of existing fishing mortality were derived from existing data, while estimates of fishing mortality to be imposed during experiments were at our discretion. We believed they represented reasonable approximations of what could be achieved in practice, given prior experience and some knowledge of current fishing practices (Beinssen 1989, Gwynne 1990, Higgs 1993, 1996, Davies 1994, Mapstone et al. 1996a).
Each iteration of the model produced outputs of the number of 1-year-old, 2-year-old, and adult (3-year-old and older) fish on each hypothetical reef. These numbers represented the populations available for sampling, and included only variation arising from presumed biological and stochastic natural processes. These numbers were then multiplied by a randomly generated number to mimic the effects of sampling variation on field estimates of abundances.
Larval supply and recruitment
Much of the stochastic variation in the population dynamics model was introduced through the supply of larvae to reefs and the transition from larval stages, through settlement, to an age of 1 year post-settlement (which we defined as recruitment). Although we also allowed for variation in post-recruitment processes, especially mortality, the most likely genesis of large variations in populations of reef fish is the dispersive larval phase and the settlement and early juvenile stages (see Doherty and Williams 1988 for a review and Russ et al. 1998).
Larval supply to reefs
The gross index of larval supply (Lij) to the i-th reef in the j-th cluster (denoted as reef ij in the following) was assumed to be given by:
Lij = ssSLij + (1 - ss)Blij (A.1)
where SLij is the total larval production spawned on reef ij and surviving to settlement; BLij is the background larval supply to reef ij (from other reefs); and ss is a "self-seeding" coefficient, expressed as the proportion of larvae spawned on reef ij that subsequently settled on that reef, and was assumed to be constant among all reefs. The value of ss would be expected to diminish rapidly with increasing length of (planktonic) "larval" life beyond that interval within which particles would be expected to be retained in the vicinity of a reef by hydrodynamic features such as tidal eddies. We arbitrarily set ss to 0.05.
The larval production from each reef (SLij) was calculated from the index of adult spawning abundance, using a Beverton-Holt recruitment equation of the form:
where f is the product of the mean adult fecundity times the maximum rate of larval survival to settlement in the absence of density-dependent mortality of larvae; Aij is the number of adult females spawning on reef ij; and kL is the larval "carrying capacity", possibly reflecting the density dependence of larval survival before settlement. When kL is large, larval settlement is effectively proportional to adult abundance over the whole GBR; i.e., recruitment is not limited by processes during the larval stage. When kL is small, larval supply is effectively independent of adult abundance except when total adult spawning abundance is very low. The larval carrying capacity was scaled over a two-fold range from the north to the south of the GBR so that the observed north/south increase in abundance of coral trout would be at least partially reflected in differential local contributions to larval abundance.
The original REEF program simulated larval dispersal over the whole GBR in order to provide rough assessments of the links between reefs and estimates of background larval loading onto candidate reefs for experimental manipulation (Walters and Sainsbury 1990). In our implementation of the REEF program, we opted to set the larval supply onto each reef directly (which gave us greater control over the characteristics of the variation in recruitment) rather than to attempt to generate such effects by varying the simple dispersal model. By default, the background larval loading onto each experimental reef was assumed to be constant (BLav). We then set a (user definable) "scaling factor", which scaled this base value so that there was a linear n-fold variation in background larval supply across clusters. Hence the larval supply to reef ij was given by BLij = cjBLav where cj is the scaling factor for the j-th cluster. For p clusters to be scaled over an n-fold range, the scaling factor for the j-th cluster is:
For example, for two clusters with a three-fold scaling, the background larval loading for all reefs in a cluster will be BLav/2 and 3BLav/2 for clusters 1 and 2 respectively. The (unscaled) value for BLav was taken to be the average aggregate loading onto experimental reefs calculated from the original dispersal model of Walters and Sainsbury (1990), assuming that the larval supply from all reefs in the GBR was given by equation (A.2) above and that adult abundance was equal to the average across all experimental reefs.
Stochastic variation in larval supply
The value of Lij generated from (A.1) was taken as a "long-run" mean level of larval supply for each reef. We then allowed for stochastic interannual variation about this mean, generated by three effects: (1) sG, the global mean interannual variation common to all reefs, (2) VCj, a cluster-specific component of interannual variation, and (3) VRij, a reef-specific component of interannual variation. Explicitly, stochastic variation for reef ij was modelled as a log-normally distributed effect, resulting in:
Lij = [ssSLij + (1 - ss) cj BLav] eVRij+VCj (A.4)
where VRij = z1(sG +z2sC) and VCj = z3sR and zn, n = 1,2,3 are random normal deviates, sR is the standard deviation (within clusters) in sG, and s C is the standard deviation among clusters in sG. The values used for these three parameters are given in Table 2. Further, the variation introduced here can be set to be independent for each cluster and/or reef within a cluster or similar for all clusters and/or reefs in each year. For example, if the normal deviates zn are independent in each year, then each reef and cluster behaves independently. If z3 is set the same for each cluster in a given year, then the cluster-scale variation from the deterministic value of Lij is the same for all clusters in that year; i.e., clusters act coherently. Similarly, all reefs in a cluster can be made to act somewhat coherently within a year by setting z2 to be the same. If both z2 and z3 are the same for all years, then larval supply is largely synchronous at both reef and cluster scales; i.e., a given year is either "good", "bad" or "mediocre" for all reefs. Some variation among reefs is retained in all years, however, by setting independently for each reef in each year.
Recruitment to reefs
Larval supply to each reef, Lij, is next translated into abundance of juveniles at age 1 by means of a second Beverton-Holt (recruitment) relationship:
where Jij,1 is the number of juveniles reaching age 1 on reef ij; s0 is the average rate of survival from settlement to age 1, Jij is the total number of juveniles on reef ij before the new cohort is added; and kJ is the juvenile "carrying capacity". Setting kJ large results in a "recruitment-limited" population in which the relative abundances of cohorts are proportional to their relative abundances at larval settlement. Setting kJ small results in populations in which abundances are limited by post-settlement density-dependent processes, which increase in importance as recruitment increases. We used an (arbitrary) value of 0.4 for s0 in our simulations, but a component of interannual variation in that rate was allowed (see below).
As before, there was assumed to be a two-fold increase in the carrying capacity from the north to the south of the GBR, producing similar carrying capacities within clusters but differences between clusters. Furthermore, the juvenile carrying capacity on reefs within clusters can be scaled so that there is a linear m-fold variation across these reefs (using the same mechanism as that used to scale background larval supply among clusters). For example, for six reefs with a three-fold scaling, the carrying capacities of the reefs would vary as follows: 5kav/10, 7kav/10, 9kav/10, 11kav/10, 13kav/10, 15kav/10, where kav is the mean recruitment capacity of the cluster. This control allowed us to model explicitly potential systematic differences among reefs locally, without assuming a relationship between dispersal pattern and such systematic variation.
Equations (A.1-A.5) allow simulation of a great range of population dynamics for reef species. For example, varying the larval retention parameter (ss) in equation (A.1) allows examination of the effects of local vs. global stock-recruitment relationships; decreasing the fecundity*survival parameter (f) reflects increasing risks of recruitment overfishing; and changing kL and kJ changes how variation in larval abundance influences the subsequent abundances of juveniles and adults.
Age structure and mortality of juveniles
Cohorts of juveniles were aged through their 1+ and 2+ years through the following survival model:
Jij,a+1,y+1 = sij,a (1 - mj) Jij,a,y (A.6)
where Jij,a,y is the number of juveniles of age a present on reef ij in year y; sij,a is the age-specific annual survival rate on reef ij in the absence of trawling; and mj is the mean annual mortality due to the effects of trawling near cluster j. The effects of trawling parameter was set to zero in all our simulations. Survival at ages 1+ and 2+ were set at 0.6 and 0.7 respectively. Fish were assumed to be adult after their third year post settlement, and we applied the average annual survival rate (sij,A) of 0.83 to the adult population. This figure was estimated from field data (Mapstone et al. 1996b). The values for the reef-specific survival rates (sij,a) were derived from applying a normally distributed interannual variation to the mean survival rate at each age, as follows:
where z4 is a random normal deviate, and rsa is the coefficient of variation for mean annual survival rate at age a and is set by the user. Setting rsa = 0.05 results in a 95 percent confidence interval for annual adult survival from 0.73 to 0.93.
Adult abundance and dispersal among reefs
The dynamics of the adult population on reef i in cluster j in year y (Aij,y), allowing for migration of adults among reefs, was modelled with the relationship:
where Hij,y is the annual mortality rate of adults on reef ij caused by line fishing; Pij,y is the proportion of adults on reef ij susceptible to line fishing; d is the average dispersal rate (as a proportion) of adults from each reef; is the average abundance of adults on other reefs that might immigrate to reef ij; and Jij,m,y is the number of juveniles entering the adult stock on reef ij in year y. Thus the first term in the above equation represents adults on reef ij surviving from the previous year, while the second term gives the net change in abundance on reef ij as a result of expected immigration and emigration. Hence, when fishing depletes the population on a reef, it is expected that immigration will outweigh emigration, but in the absence of fishing on a reef, net migration should be off-reef. Since the abundance on all reefs in the GBR is unknown, is approximated by the average over all experimental reefs.
Line-fishing exploitation rates for each reef (Hij,y) were derived after applying a normally distributed interannual variation to the exploitation rate for that year, (i.e., the fishing treatment set by the user), as follows:
where z5 is a random normal deviate and rH is the expected inter-annual coefficient of variation for the mean harvest rate. Setting rH = 0.25 results in 95 percent of reef-specific harvest rates lying within a three-fold range. We did not consider any effects of temporal autocorrelation within reefs or spatial correlation among reefs in harvest rates. These omissions mean that we implicitly assumed that fishing on the experimental reefs would be deliberately controlled to target levels, though with randomly varying success.
Sampling of reef populations
The number of juveniles of each age (Jij,a) and adults (Aij) on each reef in each year represent indices of abundance for the whole of each reef. Such estimates would usually be derived from field sampling and would inevitably involve taking numerous samples from each reef, usually in some hierarchical sampling design, and averaging those sample values within each reef. They are therefore analogous to estimates of mean per-reef abundances from any field-sampling scheme and those estimated means would introduce additional uncertainty in the representation of the effects of fishing. We depicted this additional "sampling variation" as an independent, normally distributed effect applied to each abundance index for juveniles and adults. Thus, the sampling variation inherent in the observations of adults on reef ij in each year was modelled as follows:
where is the estimated abundance of adults on reef ij; z6 is a random normal deviate; and robs is the precision (standard error/mean) expected from taking nobs observations per reef.
Choice and use of normal distribution
We assume that the mean survival rates on reefs has a normal distribution rather the usual practice of using the log-normal when one is considering the survival of an individual animal or population. In the REEF model, the parameters of interest relate to the means across a range of populations. For example, if Si is the survival rate of coral trout on reef i, then , the mean survival rate across all n reefs, by the Central Limit Theorem, is expected to be approximately normally distributed. We are here mimicking not variation in the survival (or mortality) process but uncertainty or real variation in the mean value for survival. Hence, a normal distribution for this variate (being the mean survival rate) is appropriate. The same is also true for the distribution of mean harvest rates.
We also modelled the estimated abundance from each reef as normally distributed, rather than as log-normally distributed, as might be appropriate for the raw-count data from subsamples. We did this because we expected that each effective replicate data point would be a mean for each reef, calculated from the n observations (or subsamples) from that reef. Again, by the Central Limit Theorem, those means would be expected to be approximately normally distributed, with variance MS/n, where MS is the highest level Mean Square derived from any hierarchy of subsampling within reefs, and n is the total number of data points from which the reef mean is calculated. The value for was derived from empirical data as , where is the estimated overall per-reef mean abundance. For the experiment n will always be greater than 30 (and possibly greater than 60). Furthermore, one of us (B. Mapstone) has found using simulations with real data similar to that which is being obtained from the experiment that means of quite small sample sizes (<10) are approximately normally distributed. This would suggest that the underlying data also were probably not too non-normal, but certainly suggests that the distributions of the means of 30-60 data (or more) would closely resemble normal. However, it would be relatively straightforward to substitute an asymmetric distribution function if the survey data is so skewed that the sample size is inadequate to invoke the Central Limit Theorem.
Finally, all normal random deviates, zi, were truncated at -2 or +2. Given the values of r (= s /m ) used in the REEF model (0.05 for survival rate, 0.25 for harvest rate, and 0.20 for sampling variation) this prevents the values of the random variables ever being outside the range (0,1).
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