Lotka-Volterra Systems Consider the following example of a nonlinear two dimensional Lotka-Volterra predator-prey system where the constants λί > 0 and λ2 > 0 satisfy λίλ2 71, and we are interested in solutions of (1) that remain in the first quadrant of the ry-plane so that rt) 0, and y(t) 0, 2. Assume that either λι < 1 and λ2 < 1, or Ai > 1 and A2 > 1. (a) Find all critical points (a. ý) of the system (1) located in the first quadrant of the ry-plane, including on the boundary (b) For each critical point in (a) compute the linearization of the system (1) about that critical point. (c) For each critical point in (a) determine the local stability
Lotka-Volterra Systems Consider the following example of a nonlinear two dimensional Lotka-Volterra predator-prey system where the constants λί > 0 and λ2 > 0 satisfy λίλ2 71, and we are interested in solutions of (1) that remain in the first quadrant of the ry-plane so that rt) 0, and y(t) 0, 2. Assume that either λι < 1 and λ2 < 1, or Ai > 1 and A2 > 1. (a) Find all critical points (a. ý) of the system (1) located in the first quadrant of the ry-plane, including on the boundary (b) For each critical point in (a) compute the linearization of the system (1) about that critical point. (c) For each critical point in (a) determine the local stability
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:Lotka-Volterra Systems
Consider the following example of a nonlinear two dimensional Lotka-Volterra predator-prey
system
where the constants λί > 0 and λ2 > 0 satisfy λίλ2 71, and we are interested in solutions
of (1) that remain in the first quadrant of the ry-plane so that
rt) 0, and y(t) 0,
2. Assume that either λι < 1 and λ2 < 1, or Ai > 1 and A2 > 1.
(a) Find all critical points (a. ý) of the system (1) located in the first quadrant of the
ry-plane, including on the boundary
(b) For each critical point in (a) compute the linearization of the system (1) about
that critical point.
(c) For each critical point in (a) determine the local stability
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