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:

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

## Discussion

## Video Transcript

No transcript available

## Recommended Questions

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

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

Using the vectorized RungeKutta algorithm with $h=0.5$, approximate the solution to the initial value problem

$$\begin{array}{l}{3 t^{2} y^{n}-5 t y^{\prime}+5 y=0} \\ {y(1)=0, \quad y^{\prime}(1)=\frac{2}{3}}\end{array}$$

at $t=8$. Compare this approximation to the actual solution $y(t)=t^{3 / 3}-t$

Use the fourth-order Runge-Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem

$$ y^{\prime}=2 y-6, \quad y(0)=1 $$

at x = 1. (Thus, input N = 4.) Compare this approximation to the actual solution $ y=3-2 e^{2 x} $ evaluated at x = 1.

Use the fourth-order Runge-Kutta algorithm to approximate the solution to the initial value problem

$$ y^{\prime}=1-x y, \quad y(1)=1 $$

at x = 2. For a tolerance of e = 0.001, use a stopping procedure based on the absolute error.

Use the fourth-order Runge-Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem

$$ y^{\prime}=1-y, \quad y(0)=0 $$

at x = 1. Compare this approximation with the one obtained in Problem 6 using the Taylor method of order 4.

Use Euler's method to approximate the solution to the given initial value problem at the points $x=0.1,0.2,0.3,0.4,$ and $0.5,$ using steps of size 0.1 $(h=0.1)$

$d y / d x=x / y, \quad y(0)=-1$

Use the fourth-order Runge-Kutta algorithm to approximate the solution to the initial value problem

$$ y^{\prime}=y \cos x, \quad y(0)=1 $$

at x = p. For a tolerance of e = 0.01, use a stopping procedure based on the absolute error.

Use Euler's method with step size 0.5 to compute the approximate $y$ -values $y_{1}, y_{2}, y_{3},$ and $y_{4}$ of the solution of the initial-value problem $y^{\prime}=y-2 x, y(1)=0$

Use Euler's method with step size 0.5 to compute the approximate y-values $ y_1, y_2, y_3, $ and $ y_4 $ of the solution of the initial-value problem $ y' = y - 2x, y(1) = 0. $

Use Euler's method to approximate the solution to the given initial value problem at the points $x=0.1,0.2,0.3,0.4,$ and $0.5,$ using steps of size 0.1 $(h=0.1)$

$d y / d x=y(2-y), \quad y(0)=3$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

$y^{\prime}=\frac{2 y}{x}, \quad y(1)=-1, \quad d x=0.5$

Use Euler's method with step size 0.2 to estimate $y(1),$ where $y(x)$ is the solution of the initial-value problem $y^{\prime}=x y-x^{2}, y(0)=1$

By experimenting with the fourth-order Runge-Kutta subroutine, find the maximum value over the interval 31, 24 of the solution to the initial value problem

$$ y^{\prime}=\frac{1.8}{x^{4}}-y^{2}, \quad y(1)=-1 $$

Where does this maximum occur? Give your answers to two decimal places.

Use the fourth-order Runge-Kutta subroutine with h = 0.1 to approximate the solution to

$$ y^{\prime}=3 \cos (y-5 x), \quad y(0)=0 $$

at the points x = 0, 0.1, 0.2, . . . , 4.0. Use your answers to make a rough sketch of the solution on [0, 4].

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

$y^{\prime}=2 x e^{x^{2}}, \quad y(0)=2, \quad d x=0.1$

Use Euler's method with step size 0.2 to estimate $ y(1), $ where $ y(x) $ is the solution of the initial- value problem $ y' = 1 - xy, y(0) = 0. $

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

$y^{\prime}=x(1-y), \quad y(1)=0, \quad d x=0.2$

Use the strategy of Example 3 to find a value of $h$ for Euler's method such that $y(1)$ is approximated to within$\pm 0.01,$ if $y(x)$ satisfies the initial value problem

$$y^{\prime}=x-y, \quad y(0)=0$$

Also find, to within $\pm 0.05,$ the value of $x_{0}$ such that $y\left(x_{0}\right)=0.2 .$ Compare your answers with those given by the actual solution $y=e^{-x}+x-1$ (verify!). Graph the polygonal-line approximation and the actual solution on the same coordinate system.

Use Euler's method with step size 0.1 to estimate $y(0.5),$ where $y(x)$ is the solution of the initial-value problem $y^{\prime}=y+x y, y(0)=1$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

$y^{\prime}=2 x y+2 y, \quad y(0)=3, \quad d x=0.2$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

$y^{\prime}=y e^{x}, \quad y(0)=2, \quad d x=0.5$

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

$y^{\prime}=y^{2}(1+2 x), \quad y(-1)=1, \quad d x=0.5$

a) Use Euler's method with step size 0.2 to estimate $ y(1.4), $ where $ y(x) $ is the solution of the initial-value problem $ y' = x - xy, y(1) = 0. $

(b) Repeat part (a) with step size 0.1.

Use Euler's method with step size 0.1 to estimate $ y(0.5), $ where $ y(x) $ is the solution of the initial-value problem $ y' = y + xy, y(0) = 1. $

Two steps of Euler's method For the following initial value problems, compute the first two approximations $u_{1}$and$u_{2}$ given by Euler's method using the given time step.

$$y^{\prime}(t)=-y, y(0)=-1 ; \Delta t=0.2$$

Use numerical integration (such as Simpson's rule, Appendix C) to approximate the solution, at $ x=1 $, to the initial value problem

$$ \frac{d y}{d x}+\frac{\sin 2 x}{2\left(1+\sin ^{2} x\right)} y=1, \quad y(0)=0 $$.

Ensure your approximation is accurate to three decimal places.

Two steps of Euler's method For the following initial value problems, compute the first two approximations $u_{1}$and$u_{2}$ given by Euler's method using the given time step.

$$y^{\prime}(t)=2-y, y(0)=1 ; \Delta t=0.1$$

Euler's Method In Exercises $73-78,$ use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use $n$ steps of size $h$ .

$$y^{\prime}=e^{x}, y(0)=1, \quad n=10, \quad h=0.1$$

Show that when Euler's method is used to approximate

the solution of the initial value problem

$$y ^ { \prime } = - \frac { 1 } { 2 } y , \quad y ( 0 ) = 3$$

at $$x = 2$$, then the approximation with step size h is

$$3 \left( 1 - \frac { h } { 2 } \right) ^ { 2 / h }$$.

Euler's Method In Exercises $73-78,$ use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use $n$ steps of size $h$ .

$$y^{\prime}=x+y, \quad y(0)=2, \quad n=20, \quad h=0.05$$

(a) Use Euler's method with step size 0.2 to estimate $y(0.4),$ where $y(x)$ is the solution of the initial-value problem $y^{\prime}=x+y^{2}, y(0)=0$

(b) Repeat part (a) with step size 0.1 .

Two steps of Euler's method For the following initial value problems, compute the first two approximations $u_{1}$and$u_{2}$ given by Euler's method using the given time step.

$$y^{\prime}(t)=t+y, y(0)=4 ; \Delta t=0.5$$