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[47] Each of the following problems can be interpreted as describing the interaction of two competing species with populations r₁(t) and r₂(t). In each of these problems, find the equilibrium of co-existence, give the linear approximating system near this equilibrium, and determine whether this equilibrium is stable, asymptotically stable, or unstable with respect to the nonlinear system. (a) { 12 = 729=21,-4522) x₁ = x₁(6—4x₁ - x₂) 1₂ = x₂(95x₁ - 2x₂) [47] (a) The equilibrium of co-existence is (7₁,72) = (2,1). The linear approximating system near (2, 1) is [₁]=[] dt 12 The equilibrium (2, 1) is unstable. d (b) The equilibrium of co-existence is (7₁, 7₂) = (1, 2). The linear approximating system near (1, 2) is The equilibrium (1, 2) is asymptotically stable. dt r -2 -2 [2]=[= -4 -10 $][22-1]. -8

[47] Each of the following problems can be interpreted as describing the interaction of two competing species with populations r₁(t) and r₂(t). In each of these problems, find the equilibrium of co-existence, give the linear approximating system near this equilibrium, and determine whether this equilibrium is stable, asymptotically stable, or unstable with respect to the nonlinear system. (a) { 12 = 729=21,-4522) x₁ = x₁(6—4x₁ - x₂) 1₂ = x₂(95x₁ - 2x₂) [47] (a) The equilibrium of co-existence is (7₁,72) = (2,1). The linear approximating system near (2, 1) is [₁]=[] dt 12 The equilibrium (2, 1) is unstable. d (b) The equilibrium of co-existence is (7₁, 7₂) = (1, 2). The linear approximating system near (1, 2) is The equilibrium (1, 2) is asymptotically stable. dt r -2 -2 [2]=[= -4 -10 $][22-1]. -8

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 65EQ
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[47] Each of the following problems can be interpreted as describing the interaction of two competing species with populations r₁(t) and r₂(t). In each of these problems, find the equilibrium of co-existence, give the linear approximating system near this equilibrium, and determine whether this equilibrium is stable, asymptotically stable, or unstable with respect to the nonlinear system. x₁ = x₁(6x₁4x2) - Jx₁ = x₁(6 - 4x1 - x₂) x₂ = x₂(95x₁2x₂) [47] (a) The equilibrium of co-existence is (₁, ₂) = (2, 1). The linear approximating system near (2, 1) is d dt I2 The equilibrium (2, 1) is unstable. (b) The equilibrium of co-existence is (₁, ₂) = (1,2). The linear approximating system near (1,2) is The equilibrium (1, 2) is asymptotically stable. -2 -8 [Q]-[333] -2 -5 = X2 -2 d -4 # [2] - [ + ][22] = dt -10
Transcribed Image Text:[47] Each of the following problems can be interpreted as describing the interaction of two competing species with populations r₁(t) and r₂(t). In each of these problems, find the equilibrium of co-existence, give the linear approximating system near this equilibrium, and determine whether this equilibrium is stable, asymptotically stable, or unstable with respect to the nonlinear system. x₁ = x₁(6x₁4x2) - Jx₁ = x₁(6 - 4x1 - x₂) x₂ = x₂(95x₁2x₂) [47] (a) The equilibrium of co-existence is (₁, ₂) = (2, 1). The linear approximating system near (2, 1) is d dt I2 The equilibrium (2, 1) is unstable. (b) The equilibrium of co-existence is (₁, ₂) = (1,2). The linear approximating system near (1,2) is The equilibrium (1, 2) is asymptotically stable. -2 -8 [Q]-[333] -2 -5 = X2 -2 d -4 # [2] - [ + ][22] = dt -10
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