Q1 Life Science - The Ricker Salmon model

In the Pacific Northwest, where salmon is king, the economic survival of fishermen and many others depends on the success of the salmon fisheries. A useful mathematical model developed by W.E. Ricker in 1954 predicts how much harvesting of salmon should be done in order to achieve a maximum sustainable yield. We will examine the simplified version of Ricker’s model presented by Raymond N Greenwell and Ho Kuen Ng. The model is based on the follow six assumptions:

1. The number of eggs laid by the salmon in any generation is proportional to the number of adult salmon.
2. If N(t) is the salmon population in the year t, then the next year’s population N(t+1) is proportional to the number of eggs laid.
3. When the baby salmon ( called "recruits" ) reach a certain size, they eaten by predators at a rate proportional to their number.
4. After a certain time T, the recruits become too big for most predators to swallow and their population stops decreasing.
5. The "cut-off" T is proportional to the number of eggs laid.
6. The number of adults in the next generation is proportional to the number of recruits.

Problems

1. One way to satisfy these conditions is to require
   \[N\left( t+1 \right)=N\left( t \right){{e}^{r\left[ 1-\frac{N\left( t \right)}{P} \right]}}\]
   where r and P are positive constants. Suppose N(t) happens to equal P at some time. What can be said about N(t+1)? What about N(t+2)? Why is it reasonable to call P the equilibrium population ?
2. If N is the population of the present generation of salmon, let the population of the next generation be given by the function
   \[f\left( N \right)=N{{e}^{r\left( 1-\frac{N}{P} \right)}}\]
   Using EXCEL, giving suitable data, draw the graph of f(N) with N on the horizontal axis. You may assume that r > 1 and that f(0) = 0. Where does the graph intersect the line y = N? What happens as N → ∞?
3. For what value of N is f(N) maximized? ( This is called the maximum recruitment level )
4. One harvesting ( fishing ) strategy is to catch all the surplus population Y(N) = f(N) – N, so, in effect, the salmon population will remain constant. The fishing industry wants the yield Y(N) to be large as possible, and this so called maximum sustainable yield occurs when Y’(N) = 0. Show that this occurs at a population N = N*, where
   \[\ln \left( 1-\frac{rN*}{P} \right)=r\left( \frac{N*}{P}-1 \right)\]
   Then, find an expression for the maximum sustainable yield Y(N*) in terms of r, P, and N*.
5. Suppose r = 1.03 and the equilibrium population is P = 3000. Use EXCEL and Newton-Raphson Method to solve the equation in (d) for N* and then find the corresponding maximum sustainable yield.

Q2) Economics

The Evans Price Adjustment model is a dynamic model in which price p denote the price of a particular commodity, S(p) and D(p) denote the supply and demand functions of that commodity respectively. These 3 parameters are regarded as function of time t. The time rate of change of price is assumed to be proportional to the shortage D – S, so that

\[\frac{dp}{dt}=k\left( D-S \right)\]

where k is a positive constant.

Suppose the price p(t) of a particular commodity varies in such a way that its rate of change with respect to time is proportional to the shortage D – S, where D(p) and S(p) are the linear demand and supply functions D = 8 - 2p and S = 2 + p.

Problems

1. If the price is $5 when t = 0 and $3 when t = 2, find p(t)
2. Determine what happen to p(t) in the long run ( as t → ∞ ).

Q3 Linguistics

Linguists and psychologists who are interested in the evolution of language have noticed an interesting pattern in the frequency of so-called "rare words" in certain literary works. According to one classic model, " if a book contain a total of T different words, then approximately \[\frac{T}{\left( 1 \right)\left( 2 \right)}\] words appear exactly once, \[\frac{T}{\left( 2 \right)\left( 3 \right)}\] words appear exactly twice, and in general, \[\frac{T}{\left( k \right)\left( k+1 \right)}\] words appear exactly k times.

Problems

1. Assuming the word frequency pattern in the model is accurate, why should the series \[\sum\limits_{k=1}^{\infty }{\frac{T}{k\left( k+1 \right)}}\] expected to converge? What do you think its sum should be? Use EXCEL to show that this series converges.
2. Verify your conjecture in part (a) by actually summing the series.

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