Consider a system of differential equations describing the progress of a disease in a population, given by In our particular case, this is: x=3-3xy-1x y' = 3xy - 2y where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y = The y-nullclines are y = and x = b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1, y₁) and (x2, y2), where we order them so that x1 < x2. The equilibrium with the smaller x-coordinate is (x₁, y₁) = The equilibrium with the larger x-coordinate is (x2,42) where A = c) The linearization of the system of differential equations at the equilibrium (1,1) gives a system of the form (+)-^(+). = A ab 0 sin (a) a əx = a F(x, y) for a vector-valued function F. Ω
Consider a system of differential equations describing the progress of a disease in a population, given by In our particular case, this is: x=3-3xy-1x y' = 3xy - 2y where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y = The y-nullclines are y = and x = b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1, y₁) and (x2, y2), where we order them so that x1 < x2. The equilibrium with the smaller x-coordinate is (x₁, y₁) = The equilibrium with the larger x-coordinate is (x2,42) where A = c) The linearization of the system of differential equations at the equilibrium (1,1) gives a system of the form (+)-^(+). = A ab 0 sin (a) a əx = a F(x, y) for a vector-valued function F. Ω
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:Consider a system of differential equations describing the progress of a disease in a population, given by
In our particular case, this is:
a) Find the nullclines (simplest form) of this system of differential equations.
The x-nullcline is y =
x = 3-3xy - 1x
y = 3xy - 2y
where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in
units of 1,000 individuals.
The y-nullclines are y =
and x =
b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1,9₁) and (x2, y2), where we
order them so that x1 < x2.
The equilibrium with the smaller x-coordinate is (x₁, y₁) :
The equilibrium with the larger x-coordinate is (x2, y₂) =
where A =
0
sin (a)
c) The linearization of the system of differential equations at the equilibrium (ï1, Y₁) gives a system of the form
(+)
¹ (~),
f
əx
∞
a
Ω
= A
x'
G
E
= F(x, y) for a vector-valued function F.
P
Transcribed Image Text:The eigenvalues of A are
We conclude that the equilibrium (x₁, y₁) is
where B
d) The linearization of the system of differential equations at the equilibrium (x2, y₂) gives a system of the form
(+¹)
ab
sin (a)
f
əx
The eigenvalues of B are
∞
asymptotically stable (a sink)
(List them separated by semicolons ; .)
a Ω
.
= B
(₁),
X
(List them separated by semicolons ; .)
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