Michigan Department of Natural Resources
Fisheries Research Report No. 1869, 1979
Mathematical Description of Trout Stream Fisheries
Richard D. Clark Jr., Gaylord R. Alexander, and Howard Gowing
Abstract-A mathematical model of trout stream fisheries was developed which can be used to
evaluate a variety of fishing regulations.
Density-dependent mortality was found in the first 2 years of life for each of the two brook trout
and three brown trout populations studied. Regression equations were used to describe the density-dependent relationships for modeling purposes.
Equations were developed which used mortality, growth, and length-frequency information to calculate
the number of fish in a population, number caught and harvested, number caught and released,
number of deaths due to hooking mortality, number of natural deaths, and number recruiting for
any time period and age group. Also, by adding a length-weight regression relationship, equations
were developed for calculating yield in weight harvested, yield in weight caught and released,
and gross biomass production for any time period and age group.
Effects of imposing different types of length limits, including minimum, inverted, or slot limits,
can be analyzed with this mathematical technique. Fishing mortality and hooking mortality can be
adjusted to simulate values typical for different gear types (e.g., artificial flies or live bait).
Also, consequences of seasonal fluctuations in growth and fishing mortality can be assessed,
including shifts in fishing season length or time frame.
The equations were incorporated into a computer simulator, TROUT.DYNAMICS. The brown trout fishery
of the Main Au Sable River was simulated for a period in the past and a period in the future,
to demonstrate application of the model.