Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

Competing Species. Let pi(t) denote, respectively, the populations of three competing species $S_{i}, i = 1, 2, 3 .$ Suppose these species have the same growth rates, and the maximum population that the habitat can support is the same for each species. (We assume it to be one unit.) Also, suppose the competitive advantage that $S_{1}$ has over $S_{2}$ is the same as that of $S_{2}$ over $S_{3}$ and over. This situation is modeled by the system
$p'_{1} = p_{1} (1 - p_{1} - {ap}_{2} - {bp}_{3}) p'_{2} = p_{2} (1 - b p_{1} - p_{2} - {ap}_{3}) p'_{3} = p_{3} (1 - a p_{1} - {bp}_{2} - p_{3})$
where a and b are positive constants. To demonstrate the population dynamics of this system when a = b = 0.5, use the Runge–Kutta algorithm for systems with h = 0.1 to approximate the populations over the time interval [0, 10] under each of the following initial conditions:
$(a) p_{1} (0) = 1 . 0, p_{2} = 0.1, p_{3} = 0.1 (b) p_{1} (0) = 0 . 1, p_{2} = 1.0, p_{3} = 0.1 (c) p_{1} (0) = 0 . 1, p_{2} = 0.1, p_{3} = 1.0$

In all cases, the population approaches to 0.5.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Given conditions

Given that the system is:

$p'_{1} = p_{1} (1 - p_{1} - {ap}_{2} - {bp}_{3}) p'_{2} = p_{2} (1 - b p_{1} - p_{2} - {ap}_{3}) p'_{3} = p_{3} (1 - a p_{1} - {bp}_{2} - p_{3})$

And

The initial conditions are:

$p_{1} (0) = 1 . 0, p_{2} = 0.1, p_{3} = 0.1$

T	$p_{1}$	$p_{2}$	$p_{3}$
0	1	0.1	0.1
0.1	0.99035	0.103	0.1035
0.5	0.9574	0.1189	0.1189
1	0.9245	0.1406	0.140
1.5	0.8960	0.1647	0.164
3	0.817	0.245	0.245
4	0.766	0.298	0.298
5	0.7187	0.349	0.349
9	0.583	0.452	0.452
10.1	0.565	0.463	0.4638

Solve for part (b)

The initial conditions are $p_{1} (0) = 0 . 1, p_{2} = 1.0, p_{3} = 0.1 .$

T	$p_{1}$	$p_{2}$	$p_{3}$
0	0.1	1	0.1
0.1	0.103	0.990	0.1035
0.5	0.118	0.957	0.1189
1	0.1406	0.9245	0.1406
1.5	0.1647	0.8960	0.164
3	0.245	0.8177	0.245
4	0.298	0.7668	0.298
5	0.344	0.7187	0.3449
9	0.452	0.583	0.452
10.1	0.463	0.565	0.4638

Find the result of part (c)

The initials conditions are $p_{1} (0) = 0 . 1, p_{2} = 0.1, p_{3} = 1.0 .$

T	$p_{1}$	$p_{2}$	$p_{3}$
0	0.1	0.1	1
0.1	0.103	0.103	0.990
0.5	0.118	0.118	0.957
1	0.1406	0.1406	0.9245
1.5	0.1647	0.1647	0.8960
3	0.2452	0.2453	0.8117
4	0.2982	0.2982	0.7618
5	0.344	0.3449	0.7187
9	0.4521	0.4521	0.5834
10.1	0.463	0.4638	0.565

In all cases, the population approaches 0.5.

This is the required result.

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Fundamentals Of Differential Equations And Boundary Value Problems

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Fundamentals Of Differential Equations And Boundary Value Problems

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