Consider a disease that propagates according to the systemdx357 — Зху — Зхdtdy= x²y – 13xy – 68ydtwhere X represents susceptible individuals and y represents infected individuals.(a) Find all biologically meaningful equilibria.(b) Show that J(x, y), the Jacobian matrix of this system, is given by-Зу — 3-3xJ =2ху — 13у х2 - 13х(c) For each of the biologically meaningful equilibria from (a), find the eigenvalues ofthe Jacobian matrix.(d) Classify each biologically meaningful equilibrium and determine its stability.(e) Draw the nullclines in the region X > 0, y > 0.(f) Draw the solution starting at(x0, yo) = (125, 1).(g) Describe in words what happens eventually.

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Consider a disease that propagates according to the system dx 3 357 — Зху — Зх dt dy Ex*у — 13ху — 68y dt where X represents susceptible individuals and y represents infected individuals. (a) Find all biologically meaningful equilibria. (b) Show that J(x, y), the Jacobian matrix of this system, is given by —Зу — 3 -3x J = 2ху — 13у х? — 13х — 68 - (c) For each of the biologically meaningful equilibria from (a), find the eigenvalues of the Jacobian matrix.

Consider a disease that propagates according to the system dx 3 357 — Зху — Зх dt dy Ex*у — 13ху — 68y dt where X represents susceptible individuals and y represents infected individuals. (a) Find all biologically meaningful equilibria. (b) Show that J(x, y), the Jacobian matrix of this system, is given by —Зу — 3 -3x J = 2ху — 13у х? — 13х — 68 - (c) For each of the biologically meaningful equilibria from (a), find the eigenvalues of the Jacobian matrix.

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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Question

Consider a disease that propagates according to the system
dx
357 — Зху — Зх
dt
dy
= x²y – 13xy – 68y
dt
where X represents susceptible individuals and y represents infected individuals.
(a) Find all biologically meaningful equilibria.
(b) Show that J(x, y), the Jacobian matrix of this system, is given by
-Зу — 3
-3x
J =
2ху — 13у х2 - 13х
(c) For each of the biologically meaningful equilibria from (a), find the eigenvalues of
the Jacobian matrix.
(d) Classify each biologically meaningful equilibrium and determine its stability.
(e) Draw the nullclines in the region X > 0, y > 0.
(f) Draw the solution starting at(x0, yo) = (125, 1).
(g) Describe in words what happens eventually.

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