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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

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.4: Combining And Decomposing Functions
Problem 8E
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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: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
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 ; .)
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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