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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 x² – 13x – 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 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 x² – 13x – 68 - (c) For each of the biologically meaningful equilibria from (a), find the eigenvalues of the Jacobian matrix.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 23EQ
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Consider a disease that propagates according to the system dx 357 — Зху — 3х 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 | -3y – 3 -3x J = 2ху — 13у х2 — 13х (c) For each of the biologically meaningful equilibria from (a), find the eigenvalues of the Jacobian matrix.
Transcribed Image Text:Consider a disease that propagates according to the system dx 357 — Зху — 3х 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 | -3y – 3 -3x J = 2ху — 13у х2 — 13х (c) For each of the biologically meaningful equilibria from (a), find the eigenvalues of the Jacobian matrix.
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