The population dynamics of a species is governed by the discrete model
where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population for , where
. Since a species becomes extinct if
for any n > 1, show using iterations, that irrespective of the size of r > 1 the species could become extinct if the carrying capacity
.
The model is the Ricker map x_{n+1}=f(x_n)=x_n exp[r(1−x_n/K)], with r>0 and K>0. Steady states satisfy x=f(x). Either x=0 or exp[r(1−x/K)]=1, which gives r(1−x/K)=0 and hence x=K. Thus the equilibria are x*=0 and x*=K. Stability follows from f′(x)=exp[r(1−x/K)](1−%28r/K%29x). At x*=0, f′(0)=exp(r)>1 for all r>0, so x=0 is unstable. At x*=K, f′(K)=1−r. Hence |1−r|<1 gives local asymptotic stability, i.e. 0
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Maximize , subject to the constraints
and
and are integers.
Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:
| Customer Orders (in thousands) | Add-delete items (in thousands) | CPU time (in hours) |
| 123.5 | 2.108 | 141.5 |
| 146.1 | 9.213 | 168.9 |
| 133.9 | 1.905 | 154.8 |
| 128.5 | 0.815 | 146.5 |
| 151.5 | 1.061 | 172.8 |
| 136.2 | 8.603 | 160.1 |
| 92.0 | 1.125 | 108.5 |
The population dynamics of a species is governed by the discrete model
where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population for , where
. Since a species becomes extinct if
for any n > 1, show using iterations, that irrespective of the size of r > 1 the species could become extinct if the carrying capacity
.
Ships arrive at a port at the rate of one in every 6 hours with exponential distribution of inter-arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 12 hours. If the average delay of ships waiting for berths is to be kept below 15 hours, how many berths should be provided at the port?
Return distributions of the two securities are given below:
| Return | Probabilities | |
| X | Y |
|
| 0.20 | 0.15 | 0.30 |
| 0.15 | 0.08 | 0.25 |
| 0.10 | 0.05 | 0.15 |
| 0.11 | 0.09 | 0.25 |
Let be a portfolio of two securities X and Y. Find the values of w1 and w2 in the following situations:
i) and P is risk free.
ii) and variance P is minimum.
iii) Variance P is minimum and and
.
Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?
| Portfolio | Expected return | Standard deviation |
| W | 10% | 25% |
| X | 5% | 7% |
| Y | 17% | 37% |
| Z | 12% | 13% |