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Problem 20

In Project C of Chapter 4, it was shown that the …

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Problem 19

Predator-Prey Model. The Volterra-Lotka predatorprey model predicts some rather interesting behavior that is evident in certain biological systems. For example, suppose you fix the initial population of prey but increase the initial population of predators. Then the population cycle for the
prey becomes more severe in the sense that there is a long period of time with a reduced population of prey followed by a short period when the population of prey is very large.To demonstrate this behavior, use the vectorized Runge-Kutta algorithm for systems with $h=0.5$ 0.5 to approximate
the populations of prey $x$ and of predators $y$ over the period 30, 54 that satisfy the Volterra-Lotka system
$$\begin{aligned} x^{\prime} &=x(3-y) \\ y^{\prime} &=y(x-3) \end{aligned}$$
under each of the following initial conditions:
$$\begin{array}{ll}{\text { (a) } x(0)=2,} & {y(0)=4} \\ {\text { (b) } x(0)=2,} & {y(0)=5} \\ {\text { (c) } x(0)=2,} & {y(0)=7}\end{array}$$

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Problem 82

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation.
$$\left[\begin{array}{c}
j_{n+1} \\
s_{n+1} \\
a_{n+1}
\end{array}\right]=\left[\begin{array}{rrr}
0 & 0 & 0.33 \\
0.18 & 0 & 0 \\
0 & 0.71 & 0.94
\end{array}\right]\left[\begin{array}{c}
j_{n} \\
s_{n} \\
a_{n}
\end{array}\right]$$
The numbers in the column matrices give the numbers of females in the three age groups after $n$ years and $n+1$ years. Multiplying the matrices yields the following.
$$\begin{aligned}
&j_{n+1}=0.33 a_{n}\\
&s_{n+1}=0.18 j_{n}\\
&a_{n+1}=0.71 s_{n}+0.94 a_{n}
\end{aligned}$$
(Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and
C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. $6, \text { No. } 4 .)$
(a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years.
(b) Using advanced techniques from linear algebra, we can show that, in the long run,
$$
\left[\begin{array}{c}
j_{n+1} \\
s_{n+1} \\
a_{n+1}
\end{array}\right]=0.98359\left[\begin{array}{c}
j_{n} \\
s_{n} \\
a_{n}
\end{array}\right]
$$
What can we conclude about the long-term fate of the northern spotted owl?
(c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the $3 \times 3$ matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home.
Suppose that, due to better forest management, the number 0.18 can be increased to $0.3 .$ Rework part (a) under this new assumption.

Problem 158

The article "Statistical Behavior Modeling for Driver-Adaptive Precrash Systems" $(I E E E$ Trans. on Intelligent Transp. Systems, $2013 : 1-9$ ) proposed the following distribution for modeling the behavior of what the authors called "the criticality level of a situation" $X .$
$$f\left(x ; \lambda_{1}, \lambda_{2}, p\right)=\left\{\begin{array}{cc}{p \lambda_{1} e^{-\lambda_{1} x}+(1-p) \lambda_{2} e^{-\lambda_{2} x}} & {x \geq 0} \\ {0} & {\text { otherwise }}\end{array}\right.$$
This is often called the hyperexponential or mixed exponential distribution.
(a) What is the cdf $F\left(x ; \lambda_{1}, \lambda_{2}, p\right) ?$
(b) If $p=.5, \lambda_{1}=40, \lambda_{2}=200($ values of the $\lambda$ s suggested in the cited article $),$ calculate $P(X>.01) .$
(c) If $X$ has $f\left(x ; \lambda_{1}, \lambda_{2}, p\right)$ as its pdf, what is $E(X) ?$
(d) Using the fact that $E\left(X^{2}\right)=2 / \lambda^{2}$ when $X$ has an exponential distribution with parameter $\lambda,$ compute $E\left(X^{2}\right)$ when $X$ has pdf $f\left(x ; \lambda_{1}, \lambda_{2}, p\right) .$ Then compute $\operatorname{Var}(X) .$
(e) The coefficient of variation of a random variable (or distribution) is $\mathrm{CV}=\sigma / \mu .$ What is the $\mathrm{CV}$ for an exponential rv? What can you say about the value of $\mathrm{CV}$ when $X$ has a hyperexponential distribution?
(f) What is the CV for an Erlang distribution with parameters $\lambda$ and $n$ as defined in Sect. 3.4$?$ [Note: In applied work, the sample $\mathrm{CV}$ is used to decide which of the three distributions might be appropriate.
(g) For the parameter values given in (b), calculate the probability that $X$ is within one standard deviation of its mean value. Does this probability depend upon the values of the $\lambda$ s (it does not depend on $\lambda$ when $X$ has an exponential distribution)?

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