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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:

$$\begin{array}{ll}{\text { (a) } x(0)=2,} & {y(0)=4} \\ {\text { (b) } x(0)=2,} & {y(0)=5} \\ {\text { (c) } x(0)=2,} & {y(0)=7}\end{array}$$

Answer

1.13415

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## Discussion

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## Recommended Questions

In 1925 Lotka and Volterra introduced the predator-prey equations, a

system of equations that models the populations of two species, one of

which preys on the other. Let $x(t)$ represent the number of rabbits living in a region

at time $t,$ and $y(t)$ the number of foxes in the same

region. As time passes, the number of rabbits increases at a rate proportional to

their population, and decreases at a rate proportional to

the number of encounters between rabbits and foxes. The foxes, which

compete for food, increase in number at a rate proportional to the

number of encounters with rabbits but decrease at a rate proportional

to the number of foxes. The number of encounters between rabbits

and foxes is assumed to be proportional to the product of the two

populations. These assumptions lead to the autonomous system

$$

\begin{array}{l}{\frac{d x}{d t}=(a-b y) x} \\ {\frac{d y}{d t}=(-c+d x) y}\end{array}

$$

where $a, b, c, d$ are positive constants. The values of these constants

vary according to the specific situation being modeled. We can study

the nature of the population changes without setting these constants to

specific values.

What happens to the fox population if there are no rabbits

present?

Prey attracts predators Here is one way in which nature regulates the size of animal populations: high population density attracts predators, which remove a higher proportion of the population than when the density of the prey is low. One study looked at kelp perch and their common predator, the kelp bass. The researcher set up four large circular pens on sandy ocean bottoms off the coast of southern California. He chose young perch at random from a large group and placed 10, 20, 40, and 60 perch in the four pens. Then he dropped the nets protecting the pens, allowing bass to swarm in, and counted the perch left after two hours. Here are data on the proportions of perch eaten in four repetitions of this setup:6

The explanatory variable is the number of perch (the prey) in a confined area. The response variable is the proportion of perch killed by bass (the predator) in two hours when the bass are allowed access to the perch. A scatterplot of the data shows a linear relationship.

We used Minitab software to carry out a least-squares regression analysis for these data. A residual plot and a histogram of the residuals are shown below. Check whether the conditions for performing inference about the regression model are met.

Competition-colonization models In Exercise 7.Review.23 a metapopulation model for two species was introduced. The equations were

$$\begin{aligned} \frac{d p_{1}}{d t} &=c_{1} p_{1}\left(1-p_{1}\right)-m_{1} p_{1} \\ \frac{d p_{2}}{d t} &=c_{2} p_{2}\left(1-p_{1}-p_{2}\right)-m_{2} p_{2}-c_{1} p_{1} p_{2} \end{aligned}$$

$$\begin{array}{l}{\text { where } p_{i} \text { is the fraction of patches occupied by species } i} \\ {\text { and } c_{i} \text { and } m_{i} \text { are the species-specific rates of colonization }} \\ {\text { and extinction of patches, respectively. These equations }} \\ {\text { assume that any patch has at most one species, and spe- }} \\ {\text { cies } 2 \text { patches can be taken over by species } 1, \text { but not vice }} \\ {\text { versa. }}\end{array}$$

$$\begin{array}{l}{\text { (a) Suppose that } m_{1}=m_{2}=3, c_{1}=5, \text { and } c_{2}=30 . \text { Find }} \\ {\text { all equilibria. }} \\ {\text { (b) Calculate the Jacobian matrix. }}\end{array}$$

$$\begin{array}{l}{\text { (c) Determine the local stability properties of each equilib- }} \\ {\text { rium found in part (a) using the Jacobian from part (b). }} \\ {\text { (d) Are the species predicted to be able to coexist at a stable }} \\ {\text { equilibrium? }}\end{array}$$

In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by $ W(t), $ and caribou, given by $ C(t), $ in northern Canada. The interaction has been modeled by the equations

$ \frac {dC}{dt} = aC - bCW $ $ \frac {dW}{dt} = -cW + dCW $

(a) What values of $ dC/dt $ and $ dW/dt $ correspond to stable populations?

(b) How would the statement "The caribou go extinct" be represented mathematically?

(c) Suppose that $ a = 0.05, b = 0.001, c = 0.05, $ and $ d = 0.0001. $ Find all population pairs (C, W) that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?

Prey attracts predators Refer to Exercise 3. Computer output from the least-squares regression

analysis on the perch data is shown below.

The model for regression inference has three parameters: $\alpha, \beta,$ and $\sigma .$ Explain what each parameter represents in context. Then provide an estimate for each.

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.

Consider the predator-prey model

$$\frac{d x}{d t}=x(2-y), \frac{d y}{d t}=y(x-2)$$

Sketch the phase plane for $0 \leq x \leq 10,0 \leq y \leq 10$ Compare the behavior of the two specific cases corresponding to the initial conditions $x(0)=1, y(0)=$

$0.1,$ and $x(0)=1, y(0)=1$

Using the vectorized Runge-Kutta algorithm for systems with $h=0.125$ approximate the solution to the initial value problem

$$\begin{array}{ll}{x^{\prime}=2 x-y ;} & {x(0)=0} \\ {y^{\prime}=3 x+6 y ;} & {y(0)=-2}\end{array}$$

at $t=1$ Compare this approximation to the actual solution

$$x(t)=e^{5 t}-e^{3 t}, \quad y(t)=e^{3 t}-3 e^{5 t}$$

Combat Model. A simplified mathematical model for conventional versus guerrilla combat is given by the system

$$\begin{array}{ll}{x_{1}^{\prime}=-(0.1) x_{1} x_{2} ;} & {x_{1}(0)=10} \\ {x_{2}^{\prime}=-x_{1} ;} & {x_{2}(0)=15}\end{array}$$

where $x_{1}$ and $x_{2}$ are the strengths of guerrilla and conventional troops, respectively, and 0.1 and 1 are the $combat$ $effectiveness$ $coefficients$ Who will win the conflict: the conventional troops or the guerrillas? [Hint: Use the vectorized Runge-Kutta algorithm for systems with $h=0.1$ to approximate the solutions.]

Controlling a population The fish and game department in a

certain state is planning to issue hunting permits to control the

deer population (one deer per permit). It is known that if the deer

population falls below a certain level $m,$ the deer will become extinct. It is also known that if the deer population rises above the carrying capacity $M,$ the population will decrease back to $M$

through disease and malnutrition.

\begin{equation}\begin{array}{l}{\text { a. Discuss the reasonableness of the following model for the }} \\ {\text { growth rate of the deer population as a function of time: }}\end{array}\end{equation}

$$\frac{d P}{d t}=r P(M-P)(P-m)$$

\begin{equation}\begin{array}{l}{\text { where } P \text { is the population of the deer and } r \text { is a positive con- }} \\ {\text { stant of proportionality. Include a phase line. }}\end{array}\end{equation}

\begin{equation}\begin{array}{l}{\text { c. Show that if } P > M \text { for all } t \text { , then } \lim _{t \rightarrow \infty} P(t)=M .} \\ {\text { d. What happens if } P < m \text { for all } t ?} \\ {\text { e. Discuss the solutions to the differential equation. What are }} \\ {\text { the equilibrium points of the model? Explain the dependence }} \\ {\text { of the steady-state value of } P \text { on the initial values of } P . \text { About }} \\ {\text { how many permits should be issued? }}\end{array}\end{equation}

[T] The populations of the snowshoe hare (in thousands) and the lynx (in hundreds) collected over 7

years from 1937 to 1943 are shown in the following table. The snowshoe hare is the primary prey of the lynx.

a. Graph the data points and determine which Holling-type function fits the data best.

b. Using the meanings of the parameters a and n, determine values for those parameters by examining a graph of the data. Recall that n measures what prey value results in the half- maximum of the predator value.

c. Plot the resulting Holling-type I, II, and III functions on top of the data. Was the result from part a. correct?

Exponential model The following table shows the time of useful consciousness at various altitudes in the situation where a pressurized airplane suddenly loses pressure. The change in pressure drastically reduces available oxygen, and hypoxia sets in. The upper value of each time interval is roughly modeled by $T=10 \cdot 2^{-0.274 a},$ where $T$ measures time in minutes and $a$ is the altitude over 22,000 in thousands of feet $(a=0$ corresponds to $22,000 \mathrm{ft})$

$$\begin{array}{cc}

\text { Altitude (in }\mathrm{ft}) & \text { Time of useful consciousness } \\

\hline 22,000 & 5 \text { to } 10 \mathrm{min} \\

25,000 & 3 \text { to } 5 \mathrm{min} \\

28,000 & 2.5 \text { to } 3 \mathrm{min} \\

30,000 & 1 \text { to } 2 \mathrm{min} \\

35,000 & 30 \text { to } 60 \mathrm{s} \\

40,000 & 15 \text { to } 20 \mathrm{s} \\

45,000 & 9 \text { to } 15 \mathrm{s}

\end{array}$$

a. A Learjet flying at $38,000 \mathrm{ft}(a=16)$ suddenly loses pressure when the seal on a window fails. According to this model, how long do the pilot and passengers have to deploy oxygen masks before they become incapacitated?

b. What is the average rate of change of $T$ with respect to $a$ over the interval from 24,000 to 30,000 ft (include units)?

c. Find the instantaneous rate of change $d T / d a$, compute it at $30,000 \mathrm{ft},$ and interpret its meaning.

Predator Population Model In a predator/prey model, the predator population is modeled by the function

$$

y=900 \cos 2 t+8000

$$

$$

\begin{array}{l}{\text { where } t \text { is measured in years. }} \\ {\text { (a) What is the maximum population? }} \\ {\text { (b) Find the length of time between successive periods of }} \\ {\text { maximum population. }}\end{array}

$$

$$\begin{array}{l}{\text { Cell cycle In Exercise } 7 \text { . Review. } 25 \text { a model for the cell }} \\ {\text { cycle was introduced. It modeled the concentrations of a }} \\ {\text { molecule called MPF (maturation promoting factor) and }} \\ {\text { another molecule called cyclin. MPF production is stimu- }} \\ {\text { lated by cyclin, but the presence of MPF also inhibits its }}\end{array}$$ $$

\begin{array}{l}{\text { own production. Using } M \text { and } C \text { to denote the concentra- }} \\ {\text { tions of these two biomolecules (in } \mathrm{mg} / \mathrm{mL} ), \text { the model for }} \\ {\text { their interaction is }}\end{array}

$$$$\begin{aligned} \frac{d M}{d t} &=\alpha C+\beta C M^{2}-\frac{\gamma M}{1+M} \\ \frac{d C}{d t} &=\delta-M \end{aligned}$$

$$\begin{array}{l}{\text { (a) Suppose that } \alpha=2, \beta=1, \gamma=10, \text { and } \delta=1 . \text { Find }} \\ {\text { the only equilibrium. }}\end{array}$$.

(b) Calculate the Jacobian matrix.

$$\begin{array}{l}{\text { (c) Determine the local stability properties of the equilib- }} \\ {\text { rium found in part (a) using the Jacobian from part (b). }}\end{array}$$

$$\begin{array}{l}{\text { (d) Describe how } M \text { and } C \text { change near the equilibrium }} \\ {\text { point. }}\end{array}$$.

Predator Population Model In a predator/prey model, the

predator population is modeled by the function

$$y=900 \cos 2 t+8000$$

where $t$ is measured in years.

$$

\begin{array}{l}{\text { (a) What is the maximum population? }} \\ {\text { (b) Find the length of time between successive periods of }} \\ {\text { maximum population. }}\end{array}

$$

Consider the predator-prey model

$$\frac{d x}{d t}=x(3-x-y), \frac{d y}{d t}=y(x-1)$$

Sketch the phase plane for $0 \leq x \leq 4,0 \leq y \leq 4$ What happens to the populations of both species as $t \rightarrow+\infty ?$

In each of the applied situations in Exercises $61-64$, find an appropriate viewing window for the equation (that is, a window that includes all the points relevant to the problem but does not include large regions that are not relevant to the prob. lem, and has easily readable tick marks on the axes). Explain why you chose this window. See the Hint in Exercise $60(a)$

Beginning in 1905 the deer population in a region of Arizona rapidly increased because of a lack of natural predators. Eventually food resources were depleted to such a degree that the deer population completely died out. In the equation $y=-125 x^{5}+3.125 x^{4}+4000, y$ is the number of deer in year $x,$ where $x=0$ corresponds to 1905

42. Predator-prey dynamics In Chapter 7 we study a model

for the population sizes of a predator and its prey species. If

$u(t)$ and $v(t)$ denote the prey and predator population sizes at

time $t,$ an equation relating the two is

$$v e^{-t} u^{a} e^{-a u}=c$$

where $c$ and $\alpha$ are positive constants. Use logarithmic

differentiation to obtain an equation relating the relative (per capita) rate of change of predator (that is, $v^{\prime} / v )$ to that of

prey (that is, $u^{\prime} / u ) .$

(a) Use Euler's method with each of the following step sizes to estimate the value of $ y(0.4), $ where $ y $ is the solution of the initial-value problem $ y' = y, y(0) = 1. $

(i) $ h = 0.4 $ (ii) $ h = 0.2 $ (iii) $ h = 0.1 $

(b) We know that the exact solution of the initial-value problem in part (a) is $ y = e^x. $ Draw, as accurately as you can, the graph of $ y = e^x, 0 \le x \le 0.4, $ together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble Figure 12, 13, and 14.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.

(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of $ y(0.4), $ namely $ e^{0.4}. $ What happens to the errors each time the steps size is halved?

Squirrels and their food supply $(3.2)$ Animal species produce more offspring when their supply of food goes up. Some animals appear able to anticipate unusual food abundance. Red squirrels eat seeds from pinecones, a food source that sometimes has very large crops. Researchers collected data on an index of the abundance of pinecones and the average number of offspring per female over 16 years. $^{3}$ Computer output from a least-squares regression on these data and a residual plot are shown on the next page.

(a) Give the equation for the least-squares regression line. Define any variables you use.

(b) Explain what the residual plot tells you about how well the linear model fits the data.

(c) Interpret the values of $r^{2}$ and $s$ in context.

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)?

Leslie matrices In Exercise 8.7 .29 we modeled an age-structured population using the recursion $\mathbf{n}_{t+1}=L \mathbf{n}_{t}$ where

$$L=\left[ \begin{array}{cc}{b} & {2} \\ {\frac{1}{2}} & {0}\end{array}\right] \quad b>0$$

(a) Verify that the Perron-Frobenius Theorem can be applied to this model.

(b) Calculate the eigenvalues of $L$ .

(c) Given your answers to parts (a) and (b), what is the long-term behavior of $\mathbf{n}_{\text {r}}$ as a function of $b ?$

(d) Using the initial condition $\mathbf{n}_{0}=\left[ \begin{array}{l}{1} \\ {0}\end{array}\right],$ express the solution to the recursion in terms of the eigenvectors and eigenvalues of $L .$

(e) Verify your answer to part (c) using the solution you obtained in part (d).

Modeling yeast populations (cont.) Verify that

$N(t)=\frac{42 e^{0.55 t}}{209.8+0.2 e^{0.55 t}}$

is an approximate solution of the differential equation

$$\frac{d N}{d t}=(0.55-0.0026 N) N$$