Mathematical Description of Trout-Stream Fisheries

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Abstract
A mathematical model of trout‐stream fisheries was developed that 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 (Salvelinus fontinalis) and three brown trout (Salmo trutta) populations studied in Michigan. Regression equations were used to describe the density‐dependent relationships for modeling purposes. Equations were developed that 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 recruited for any time period and age‐group. Also, addition of a length‐weight regression allowed equations to be 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. In addition, consequences of seasonal fluctuations in growth and fishing mortality, including shifts in length of fishing season or time frame, can be assessed. The equations were incorporated into a computer simulator, TROUT.DYNAMICS, and the brown trout fishery in a section of the Au Sable River, Michigan, was simulated for a period in the past and a period in the future to demonstrate applications of the model.