9. The following system has been proposed as a model of the population dynamics of two species of animal that compete for the same resource: dx dt dy dt = ax - by, x(0) = xo, =-cx+dy, y(0) = yo. Here a, b, c, d are positive constants, x(t) is the population of the first species at time t, and y(t) is the corresponding population of the second species (x and y are measured in some convenient units, say thousands or millions of animals). The equations are easy to understand: either species increases (exponentially) if the other is not present, but, since the two species compete for the same resource, the presence of one species contributes negatively to the growth rate of the other. (a) Solve the IVP with b = c = 2, a = d = 1, x(0) = 2, and y(0) = 1, and explain (in words) what happens to the populations of the two species over the long term. (b) With the values of a, b, c, d given in part (a), is there an initial condition which will lead to a different (qualitative) outcome?
9. The following system has been proposed as a model of the population dynamics of two species of animal that compete for the same resource: dx dt dy dt = ax - by, x(0) = xo, =-cx+dy, y(0) = yo. Here a, b, c, d are positive constants, x(t) is the population of the first species at time t, and y(t) is the corresponding population of the second species (x and y are measured in some convenient units, say thousands or millions of animals). The equations are easy to understand: either species increases (exponentially) if the other is not present, but, since the two species compete for the same resource, the presence of one species contributes negatively to the growth rate of the other. (a) Solve the IVP with b = c = 2, a = d = 1, x(0) = 2, and y(0) = 1, and explain (in words) what happens to the populations of the two species over the long term. (b) With the values of a, b, c, d given in part (a), is there an initial condition which will lead to a different (qualitative) outcome?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:9. The following system has been proposed as a model of the population dynamics of
two species of animal that compete for the same resource:
dx
dt
dy = -cx+dy, y(0) = yo.
dt
= ax - by, x(0) = xo,
Here a, b, c, d are positive constants, x(t) is the population of the first species
at time t, and y(t) is the corresponding population of the second species (x and
y are measured in some convenient units, say thousands or millions of animals).
The equations are easy to understand: either species increases (exponentially) if the
other is not present, but, since the two species compete for the same resource, the
presence of one species contributes negatively to the growth rate of the other.
(a) Solve the IVP with b = c = 2, a = d = 1, x(0) = 2, and y(0) = 1, and explain (in
words) what happens to the populations of the two species over the long term.
(b) With the values of a, b, c, d given in part (a), is there an initial condition which
will lead to a different (qualitative) outcome?
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