Ecological Archives A020-033-A2

Douglas P. Peterson, Bruce E. Rieman, Michael K. Young, and James A. Brammer. 2010. Modeling predicts that redd trampling by cattle may contribute to population declines of native trout. Ecological Applications 20:954–966.

Appendix B. Matrix population model for stream-resident cutthroat trout in southwestern Montana, USA, used to estimate effects of redd trampling by cattle.

We used literature values to estimate survival for trout ages-0 and older, estimated female reproductive output from region-specific maturity and fecundity data, and selected an egg-to-fry mortality rate that by default produced a stable population. The final matrix population for westslope cutthroat trout (Oncorhynchus clarkii lewisi) had six stages (Tables B1 and B2) that were based on length-frequency data from cutthroat trout in southwestern Montana (J. Brammer, unpublished data) and age- and size-at-maturity from similar populations (Downs et al. 1997). In this female-only model we assumed that fish were age-3 (≥ 150 mm, FL) at age of first reproduction, and spawned in alternate years after initial maturity. We defined three size classes of adults based on length-frequency data assuming a maximum adult size of 250 mm, and size ranges of 25, 25, and 50 mm for small, medium, and large adults, respectively (Table B1). The model assumes that small and medium adults will grow 25 mm per year, on average, and transition (survive) to the next stage. Trout reaching the large adult stage (> 200 mm) remain in that stage until they die.

Survival of trout age 0 and older was estimated as the midpoint in the range of values used in other models for similar species and environments. These midpoint values were similar to empirical estimates for riverine or stream-resident cutthroat trout populations (Peterson et al. 2004, McHugh and Budy 2006, summarized by Carlson and Rahel 2007), and to those used to build general matrix population models for stream-resident populations of cutthroat trout (Hilderbrand 2002, 2003; Peterson et al. 2008). Mean maturity at age is based on headwater populations of westslope cutthroat trout in the upper Missouri River system in Montana (Downs et al. 1997) with mean fecundity-at-length estimated as: No. eggs = -494.4 + (4.4)(FL), where FL is fork length in mm. Conceptually, stage-specific reproductive output for a pre-breeding census is calculated as the product of fecundity, the proportion mature, the proportion of females (assumed 0.5), and egg-to-fry survival. Females were assumed to spawn in alternate years following the initial spawning, so we calculated a correction factor that we applied to the calculation for medium and large adult females that accounts for individuals spawning for the first time or those that may have spawned in the previous year (Table B1). Egg-to-fry survival of 0.19 (mortality 0.81) was estimated by default, given the mean value of all other vital rates, as the value necessary to produce a population growth rate characteristic of a stable population (i.e., λ ≈ 1.0). The final version of the deterministic model had λ = 1.008 (Table B2, part a). Sensitivity analyses based on elasticity values indicated population growth rate is particularly sensitive to changes in survival before reproduction (ages 0, 1, and 2; Table B2, part b).


TABLE B1. Size range, maturity schedule, fecundity, and reproductive output for three size classes of adult female westslope cutthroat trout.

 

Mean (range)

Stage or size class

Length (FL, mm)

Proportion females
mature

Fecundity

Raw reproductive
output

Spawning frequency
correction

Corrected reproductive
output

Small adult

162.5 (150–175)

0.25 (0.1–0.4)

220 (165–275)

5.2 (3.9–22.0)

-

5.2 (3.9–22.0)

Medium adult

187.5 (175–200)

0.7 (0.6–0.8)

330 (275–385)

22.0 (18.3–25.6)

0.82

18.0 (15.0–21.0)

Large adult

225 (200–250)

0.9 (0.8–1.0)

495 (385–605)

42.3 (32.9–51.7)

0.58

24.6 (19.1–24.6)

Note: The reproductive output calculations assume egg-to-fry mortality is 0.81. Maturity and fecundities are based on Downs et al. (1997), using length frequency data from cutthroat trout populations in southwestern Montana.

†  A correction factor was applied to reproductive output calculation to account for alternate year spawning after initial maturity, and for incomplete maturity within each size class.

‡ Mean corrected reproductive output for each adult stage is calculated as: mean proportion mature × mean fecundity × proportion of females (0.5) × egg-to-fry survival (0.19) × spawning frequency correction.

 

TABLE B2. Matrix depiction of the deterministic population model (a) and matrix element elasticities (b) for the pre-breeding census, birth-pulse population model for resident westslope cutthroat trout. In the matrix model (a) reproductive output is in the first row, and survival probabilities are in rows 2–6. Elasticity values (b) represent the relative sensitivity of population growth rate to a proportional change in the vital rate. By definition, elasticity values sum to 1 for a given matrix. Results show that population growth rate is particularly sensitive to changes in survival before reproduction (ages 0, 1, and 2), and relatively insensitive to reproductive output.

(a) MATRIX VALUES

Age-3

Age-4

Age-5+

 

0

0

0

5.23

18.00

24.55

Age-0

0.277

0

0

0

0

0

Age-1

0

0.3405

0

0

0

0

Age-2

0

0

0.4675

0

0

0

Age-3

0

0

0

0.4675

0

0

Age-4+

0

0

0

0

0.4675

0.4675

 

(b) ELASTICITIES

 

Age-3

Age-4

Age-5+

 

0

0

0

0.0401

0.0641

0.0756

Age-0

0.1798

0

0

0

0

0

Age-1

0

0.1798

0

0

0

0

Age-2

0

0

0.1798

0

0

0

Age-3

0

0

0

0.1397

0

0

Age-4+

0

0

0

0

0.0756

0.0654


LITERATURE CITED

Carlson, A. C., and F. J. Rahel. 2007. A basinwide perspective on entrainment of fish in irrigation canals. Transactions of the American Fisheries Society 136:1335–1343.

Downs, C. C., R. G. White, and B. B. Shepard. 1997. Age at sexual maturity, sex ratio, fecundity, and longevity of isolated headwater populations of westslope cutthroat trout. North American Journal of Fisheries Management 17:85–92.

Hilderbrand, R. H. 2002. Simulating supplementation strategies for restoring and maintaining stream resident cutthroat trout populations. North American Journal of Fisheries Management 22:879–887.

Hilderbrand, R. H. 2003. The roles of carrying capacity, immigration, and population synchrony on persistence of stream-resident cutthroat trout. Biological Conservation 110:257–266.

McHugh, P., and P. Budy. 2006. Experimental effects of nonnative brown trout on the individual- and population-level performance of native Bonneville cutthroat trout. Transactions of the American Fisheries Society 135:1441–1455.

Peterson, D. P., K. D. Fausch, and G. C. White. 2004. Population ecology of an invasion: effects of brook trout on native cutthroat trout. Ecological Applications 14:754–772.

Peterson, D. P., K. D. Fausch, J. Watmough, and R. A. Cunjak. 2008. When eradication is not an option: modeling strategies for electrofishing suppression of nonnative brook trout to foster persistence of sympatric native cutthroat trout in small streams. North American Journal of Fisheries Management 28:1847–1867.


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