8. Consider the following model for the populations of two species in competition in a given environment at time t (for instance, lions and hyenas). Here, r(t) represents the number of hundreds of species X at time t, and y(t) represents the number of hundreds of species Y. =x-x²_ =y-y² dx dt dy dt 1 2 ty 1 2xy (a) Use the framework of #5 to do a phase plane analysis of this system. ii. x(0) = 1, y(0) = 1. - (b) For each of the initial conditions given below, describe how the sizes of the populations of X and Y change with time and what happens in the long run. i. x(0) = 2, y(0) = 0. iii. x(0) = 1, y(0) = 1. (From the phase plane analysis only, we cannot determine the exact shape of the trajectory. Draw two possibilities on your sketch. Is there more than one possibility
8. Consider the following model for the populations of two species in competition in a given environment at time t (for instance, lions and hyenas). Here, r(t) represents the number of hundreds of species X at time t, and y(t) represents the number of hundreds of species Y. =x-x²_ =y-y² dx dt dy dt 1 2 ty 1 2xy (a) Use the framework of #5 to do a phase plane analysis of this system. ii. x(0) = 1, y(0) = 1. - (b) For each of the initial conditions given below, describe how the sizes of the populations of X and Y change with time and what happens in the long run. i. x(0) = 2, y(0) = 0. iii. x(0) = 1, y(0) = 1. (From the phase plane analysis only, we cannot determine the exact shape of the trajectory. Draw two possibilities on your sketch. Is there more than one possibility
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:8. Consider the following model for the populations of two species in competition in a given environment
at time t (for instance, lions and hyenas). Here, r(t) represents the number of hundreds of species X
at time t, and y(t) represents the number of hundreds of species Y.
=x-x²_
=y-y²
dx
dt
dy
dt
1
2 ty
1
2xy
(a) Use the framework of #5 to do a phase plane analysis of this system.
ii. x(0)= ¹, y(0) = 1.
-
(b) For each of the initial conditions given below, describe how the sizes of the populations of X and
Y change with time and what happens in the long run.
i. x(0) = 2, y (0) = 0.
iii. x(0) = 1, y(0) = 1. (From the phase plane analysis only, we cannot determine the exact shape
of the trajectory. Draw two possibilities on your sketch. Is there more than one possibility
for the limiting behavior, i.e., what happens as t → ∞o?)
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