Q. 53 - Calculus

Question

Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish $q_k$ returning each year as $q_{k+1 }= (0.14{(-1)}^k + 0.36)(q_k + h)$, where h is the number of fish whose spawn she releases from the hatchery annually.

(a) Show that the sustained number of fish returning in even-numbered years approach approximately $q_e = 3h \sum _{k=1}^{\infty } 0.{11}^k.$

(Hint: Make a new recurrence by using two steps of the one given.)

(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately $q_o = \frac{61}{11}h\sum _{ k=1}^{\infty }0.{11}^k.$

(c) How should Leila choose h, the number of hatchery fish to breed in order to hold the minimum number of fish returning in each run near some constant P?

Step-by-Step Solution

Answer

Ans:

Step 1: Analyze the context

The problem involves understanding that multiple factors beyond just dams affect salmon populations at Redfish Lake, including good years and bad years (environmental variability).

Step 2: Modeling consideration

A model accounting for multiple factors would need stochastic elements or additional variables to capture year-to-year variability, rather than a simple deterministic model based solely on dam effects.