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Lotka-Volterra Systems Consider the following example of a nonlinear two dimensional Lotka- Volterra predator-prey system r(t) =エ(t)-r'(t)-Air(t)y(t), where the constants λί > 0 and Az > 0 satisfy λιλ2 and we are interested in solutions of (1) that remain in the first quadrant of the ry-plane so that r(t) 20, and yt) 20, for all t0 1. Assume that either λί < 1CA2 or λ2 < 1<λι. (a) Find all critical points (à. ) of the system (1) located in the first quadrant of the (b) For each critical point in (o) compute the linearization of the system (1) about (e) For each critical point in (a) use your linearization in (b) to determine the local ry-plane, including on the boundary that critical point. stability (i.e. the stability in a neighborhood around the critical point) for the system (1)

Lotka-Volterra Systems Consider the following example of a nonlinear two dimensional Lotka- Volterra predator-prey system r(t) =エ(t)-r'(t)-Air(t)y(t), where the constants λί > 0 and Az > 0 satisfy λιλ2 and we are interested in solutions of (1) that remain in the first quadrant of the ry-plane so that r(t) 20, and yt) 20, for all t0 1. Assume that either λί < 1CA2 or λ2 < 1<λι. (a) Find all critical points (à. ) of the system (1) located in the first quadrant of the (b) For each critical point in (o) compute the linearization of the system (1) about (e) For each critical point in (a) use your linearization in (b) to determine the local ry-plane, including on the boundary that critical point. stability (i.e. the stability in a neighborhood around the critical point) for the system (1)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 17EQ
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Lotka-Volterra Systems Consider the following example of a nonlinear two dimensional Lotka- Volterra predator-prey system r(t) =エ(t)-r'(t)-Air(t)y(t), where the constants λί > 0 and Az > 0 satisfy λιλ2 and we are interested in solutions of (1) that remain in the first quadrant of the ry-plane so that r(t) 20, and yt) 20, for all t0 1. Assume that either λί < 1CA2 or λ2 < 1<λι. (a) Find all critical points (à. ) of the system (1) located in the first quadrant of the (b) For each critical point in (o) compute the linearization of the system (1) about (e) For each critical point in (a) use your linearization in (b) to determine the local ry-plane, including on the boundary that critical point. stability (i.e. the stability in a neighborhood around the critical point) for the system (1)
Transcribed Image Text:Lotka-Volterra Systems Consider the following example of a nonlinear two dimensional Lotka- Volterra predator-prey system r(t) =エ(t)-r'(t)-Air(t)y(t), where the constants λί > 0 and Az > 0 satisfy λιλ2 and we are interested in solutions of (1) that remain in the first quadrant of the ry-plane so that r(t) 20, and yt) 20, for all t0 1. Assume that either λί < 1CA2 or λ2 < 1<λι. (a) Find all critical points (à. ) of the system (1) located in the first quadrant of the (b) For each critical point in (o) compute the linearization of the system (1) about (e) For each critical point in (a) use your linearization in (b) to determine the local ry-plane, including on the boundary that critical point. stability (i.e. the stability in a neighborhood around the critical point) for the system (1)
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