[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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
47 solution provided by instructor for practice therefore not graded work
![[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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c89c353-252e-4eca-b673-064be33ffdb0%2F4c469f25-5766-4c9d-90c0-73ba772979ac%2Fsi8gfib_processed.png&w=3840&q=75)
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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