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Linear Cooperative Systems of ODE´s

Guo, Ziyi LU (2023) In Bachelor’s Theses in Mathematical Sciences MATK11 20232
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
A solution to an ordinary differential equation may be stable or unstable.
Let x = φ(t) be a solution of an autonomous differential equation x′ = Ax.
The solution x = φ(t) with φ(0) = t0 is stable if every other solution ψ(t)
with initial condition sufficiently close to φ(t) will remain close to φ(t) for
all t > 0. For an autonomous system of linear differential equations, it is
possible to determine the stability/instability of a solution by inspecting the
sign of eigenvalues. The system is stable if all the eigenvalues have negative
real part, the system is unstable if there exist at least one eigenvalue with
positive real part.
However regarding non-autonomous linear ordinary differential equations
x′ = A(t)x, the criteria... (More)
A solution to an ordinary differential equation may be stable or unstable.
Let x = φ(t) be a solution of an autonomous differential equation x′ = Ax.
The solution x = φ(t) with φ(0) = t0 is stable if every other solution ψ(t)
with initial condition sufficiently close to φ(t) will remain close to φ(t) for
all t > 0. For an autonomous system of linear differential equations, it is
possible to determine the stability/instability of a solution by inspecting the
sign of eigenvalues. The system is stable if all the eigenvalues have negative
real part, the system is unstable if there exist at least one eigenvalue with
positive real part.
However regarding non-autonomous linear ordinary differential equations
x′ = A(t)x, the criteria discussed above is no longer applicable since the
sign of the real parts of the eigenvalues of A(t) does not have any correlation
with the stability. The solution could grow to infinity even if the real parts
of all eigenvalues are negative and bounded away from zero.
In this thesis we present a method from of constructing unstable solutions when the matrix A(t) have constant negative eigenvalues and positive
off-diagonal entries, a so called a strongly cooperative system. We will also
examine some open problems as Proposition 6.1 and Proposition 6.2. (Less)
Popular Abstract
When studying dynamical systems it is important to determine the behavior
of its trajectories. However it is very hard to determine the long time behavior of any but the simplest systems. In consequence a lot of research has
been dedicated to determine the long time behavior of two common types
of systems, so called competitive and cooperative systems. They are commonly used in the fields of ecology, epidemiology and economics, in particular,
they’ve had some remarkable applications to the biological sciences. These
two systems are characterized by the dependence between its components. A
cooperative system is one in which both species benefit from interacting with
one another, for example flowers and bees, whereas a competitive... (More)
When studying dynamical systems it is important to determine the behavior
of its trajectories. However it is very hard to determine the long time behavior of any but the simplest systems. In consequence a lot of research has
been dedicated to determine the long time behavior of two common types
of systems, so called competitive and cooperative systems. They are commonly used in the fields of ecology, epidemiology and economics, in particular,
they’ve had some remarkable applications to the biological sciences. These
two systems are characterized by the dependence between its components. A
cooperative system is one in which both species benefit from interacting with
one another, for example flowers and bees, whereas a competitive system is
one in which interaction between the two species is detrimental to both, there
are several competitive interactions between the different components in the
natural immune system. Non-autonomous cooperative systems will be the
main focus in this thesis. There has been considerable progress made in the
study of the asymptotic behavior of solutions of cooperative systems, and it
is easy to determine the stability of an autonomous system by inspecting the
sign of the eigenvalues, but analyzing a non-autonomous cooperative system
can be much more challenging. (Less)
Please use this url to cite or link to this publication:
author
Guo, Ziyi LU
supervisor
organization
alternative title
Lineära kooperativa system av ordinära differentialekvationer
course
MATK11 20232
year
type
M2 - Bachelor Degree
subject
keywords
Ordinary differential equations, Transition matrix, Autonomous systems, cooperative systems, stability.
publication/series
Bachelor’s Theses in Mathematical Sciences
report number
LUNFMA-4153-2023
ISSN
1654-6229
other publication id
2023:K30
language
English
id
9144907
date added to LUP
2025-06-27 15:55:22
date last changed
2025-06-27 15:55:22
@misc{9144907,
  abstract     = {{A solution to an ordinary differential equation may be stable or unstable.
Let x = φ(t) be a solution of an autonomous differential equation x′ = Ax.
The solution x = φ(t) with φ(0) = t0 is stable if every other solution ψ(t)
with initial condition sufficiently close to φ(t) will remain close to φ(t) for
all t > 0. For an autonomous system of linear differential equations, it is
possible to determine the stability/instability of a solution by inspecting the
sign of eigenvalues. The system is stable if all the eigenvalues have negative
real part, the system is unstable if there exist at least one eigenvalue with
positive real part.
However regarding non-autonomous linear ordinary differential equations
x′ = A(t)x, the criteria discussed above is no longer applicable since the
sign of the real parts of the eigenvalues of A(t) does not have any correlation
with the stability. The solution could grow to infinity even if the real parts
of all eigenvalues are negative and bounded away from zero.
In this thesis we present a method from of constructing unstable solutions when the matrix A(t) have constant negative eigenvalues and positive
off-diagonal entries, a so called a strongly cooperative system. We will also
examine some open problems as Proposition 6.1 and Proposition 6.2.}},
  author       = {{Guo, Ziyi}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor’s Theses in Mathematical Sciences}},
  title        = {{Linear Cooperative Systems of ODE´s}},
  year         = {{2023}},
}