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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
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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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.
Transcribed Image Text: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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