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- Spawner-Recruit Model In fish management it is important to know the relationship between the abundance of the spawners also called the parent stock and the abundance of the recruitsthat is, those hatchlings surviving to maturity. According to the Ricker model, the number of recruits R as a function of the number of spawners P has the form R=APeBp for some positive constants A and B. This model describes well a phenomenon observed in some fisheries: A large spawning group can actually lead to a small group of recruits. In a study of the sockeye salmon, it was determined that A=4 and B=0.7. Here we measure P and R in thousands of salmon. a. Make a graph of R versus P for the sockeye salmon. Assume there are at most 3000 spawners. b. Find the maximum number of salmon recruits possible. c. If the number of recruits R is greater than the number of spawners P, then the difference R-P of the recruits can be removed by fishing, and next season there will once again be P spawners surviving to renew the cycle. What value of P gives the maximum value of R-P, the number of fish available for removal by fishing?Resale Value: The resale value V, in dollars, of a certain car is a function of the number of year t since the year 2012. In the year 2012, the resale ale is 18,000 and each year thereafter the resale value decreases by 1700. a. What is the resale value in the year 2013? b. Find a formula for V as a function of t. c. Make a graph of V versus t covering the first 4 years since the year 2012. d. Use functional notation to express the resale value in the year 2015, and then calculate that value.Planet Growth The amount of growth of plants in an ungrazed pasture is a function of the amount of plant biomass already present and the amount of rainfall. For a pasture in the arid zone of Australia the formula Y=55.120.01535N0.00056N2+3.946R gives an approximation of the growth. Here R is the amount of rainfall, in millimeters, over a 3 month period; N is the plant biomass, in kilograms per hectare, at the beginning of that period; and Y is the growth, in kilograms per hectare, of the biomass over that period. For comparison, 100 millimeters is about 3.9 inches, and 100 kilograms per hectare is about 89 pounds per acre. For this exercise, assume that the amount of plant biomass initially present is 400 kilograms per hectare, so N=400. a. Find a formula for the growth Y as a function of the amount R of rainfall. b. Make a graph of Y versus r. Include values of R from 40 to 160 millimeters. c. What happens to Y as R increases? Explain your answer in practical terms. d. How much growth will there be over a 3 month period if initially there are 400 kilograms per hectare of plant biomass and the amount of rainfall is 100 millimeters?
