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Fishery Bulletin 97(2), 1999 



cated model, or one designed for a nongeneric salmo- 

 nid, the model would be indexed by generation time 

 instead of year, but the potential problems described 

 in this paper would still apply. 



We assumed s(J,i,t) is an exponential function of 

 the distance between stream ; and j (see Hanski, 

 1994). We assumed that the streams are evenly 

 spaced along the waterway and numbered consecu- 

 tively, so that the distance between stream / and 

 streamy is proportional to | i-j\ . Therefore, the num- 

 ber of individuals straying from stream j to stream / is 



s(j,i,t)- 



N(j,t)fe 



-m\i-j\ 



1' 



-m\k-j\ 



(2) 



where m defines the rate at which straying decreases 

 with distance, and the denominator is a normaliza- 

 tion so that all strays end up in a stream (i.e. none 

 are lost to the system). 



We assumed that per-capita reproduction rate of 

 each deme is a function of some baseline rate of per- 

 capita reproduction that is equal for all streams (e.g. 

 ocean conditions and harvest) plus a function of a 

 component of the habitat that contributes positively 

 towards per-capita reproduction (e.g. width of the 

 riparian zone) and a function of a component of the 

 habitat that contributes negatively to per-capita re- 

 production (e.g. road density). If Zq is the baseline 

 per-capita rate of reproduction and z^{i,t) and z^^i.t) 

 are the amounts of the beneficial and detrimental 

 habitat components for stream / in year t, then the 

 per-capita rate of reproduction for stream / in year t 

 is modeled as 



/•(,.n = zo + g{i-e--"'"}-6{i-e--"-"} 



(3) 



where g = the maximum increase in the per-capita 

 reproduction due to the beneficial habi- 

 tat component; and 

 b = the maximum decrease in the per-capita 

 reproduction due to the detrimental 

 habitat component. 



As z^(i,t) 



mcreases, e 



-zhi.t) 



0, so that the effect of 



the beneficial habitat component approaches an as- 

 ymptote at g. Similarly, the detrimental habitat 

 component approaches an asymptote at b. There- 

 fore, r{i,t] is constrained to lie between z^-b and 

 Zf^+g. When r{i,t)il-P >1, the deme is a source popu- 

 lation (rate of change due to per-capita reproduction 

 counteracts the rate of change due to emigration); 

 when r(i,t) ( 1-/1 <1, the deme is a sink population ( it 



can not sustain itself without immigration from other 

 streams). 



To include the effects of temporally varying envi- 

 ronments, we added 



'27tt' 



qsm 



to the per-capita reproduction. This causes per-capita 

 reproduction to change sinusoidally with a maximum 

 change o{2q with a ti'-year period. Such oscillations 

 could be due to events such as El Nino (Pearcy, 1992) 

 and essentially represent changes in the baseline 

 conditions (z^) over time. 



Simulations 



We set the baseline per-capita rate of reproduction 

 with ^Q=l,g=0.2, and 6=0.15. Thus per-capita repro- 

 duction was constrained to 0.85 < r(i,t) <1.2. We drew 

 the Zj(/,0) and z.,(z.O) from a gamma distribution with 

 parameters 1 and 1 (Hilborn and Mangel, 1997). 



We set f- 0.05 for all populations. This value lies 

 within the ranges found by most of the previously 

 mentioned research on salmonid straying rates. 

 Streams were labeled as sources and sinks on the 

 basis of their initial per-capita rate of reproduction 

 given this straying rate. Therefore, a source popula- 

 tion was one with an initial per-capita rate of repro- 

 duction greater than or equal to 1.05, and a sink 

 population was one whose rate was less than 1.05. 

 With these parameters, about 40% of the streams were 

 sources and 60*^ were sinks, as would be the case for a 

 heavily impacted region (Fig. 1 ). We set m =0.1. 



The initial deme abundance for each stream was 

 assumed to be proportional to the initial per-capita 

 reproduction rate for that stream, even though such 

 relationships may not hold over time (van Home, 

 1983). As such, the initial deme abundance, Nii.O), 

 was set equal to 100r(;,0). We simulated each 

 metapopulation over a 100-year period, using four sce- 

 narios with at least 150 replications for each scenario: 



1 All parameters were constant over the 100-year 

 period; 



2 Starting in year 5, for all initial source popula- 

 tions {r{i,0)> 1.05), the good habitat component 

 (2j) decreased by 5'^ of its value from the year 

 before, and the bad habitat component (z^) in- 

 creased by 57r of its value from the year before. 



3 Same as scenario 2, except that all habitats with 

 /•((,0)>1 were degraded. 



4 Temporally varying environment was incorpo- 

 rated into scenario 2. In this case, the baseline 

 per-capita rate of reproduction (z,,) oscillated be- 

 tween 0.95 and 1.05 over a 20-year period. 



