6 Ecology and Applied Environmental Science
course of time, forming an equal number of paths in the space of n dimensions. The temporal evolution of the system can be represented by the set
of those paths. In order to fully determine this evolution, appropriate variables must be chosen, which will be sufficient to describe the variations of
the system with no uncertainty. If the change in the variables is made under
certain conditions, then it can be represented by a mathematical equation,
usually a differential one. In this case, a set of equations describes the temporal evolution of the state of the system; they are called state equations.
When the state of a system is stagnant, then the system is at an equilibrium point. If the system is at that point at some moment in time and no
significant disturbance occurs, it is expected to remain at the same point
at the next moment in time. The state of equilibrium is expressed mathematically by the assumption that the temporal derivative of all the system’s
variables is equal to zero:
dx
dt
i
n
i
=
=
…
0
1 2
,
, ,
The temporal evolution of a system can be represented graphically in two
ways. In the first case, the change in each variable can be represented by a
two-dimensional graph, where the value of the variable is measured on the
vertical axis and time on the horizontal axis. Thus the temporal evolution
of the system is depicted in a set of such diagrams, which are equal in number with the system variables. The following equations apply:
x f t
i
n
i
i
= ( ) = …
,
, ,
1 2
In the second case, the change in each variable is represented by a curve
in a space of n dimensions, the phase space. The evolution of the system is
depicted by a line in a hypersurface of n dimensions where each axis corresponds to one variable, and time t has been eliminated.
Stability is a property of the specific equilibrium point and is defined
in relation to it. If the system moves away from the equilibrium point,
i.e. undergoes a disturbance and has the tendency to return to the same
point, then the specific equilibrium point is stable. By contrast, if a system
that has been disturbed continues to move away from the equilibrium point,
then that point is unstable.
There are varying degrees of stability, depending on how great a disturbance the equilibrium point can withstand. If the system can return to the
equilibrium point only when there have been small deviations from it, then
we have local stability. In this case, in the phase space around the equilibrium point A, there will be a specific area, from all points of which the
course of time, forming an equal number of paths in the space of n dimensions. The temporal evolution of the system can be represented by the set
of those paths. In order to fully determine this evolution, appropriate variables must be chosen, which will be sufficient to describe the variations of
the system with no uncertainty. If the change in the variables is made under
certain conditions, then it can be represented by a mathematical equation,
usually a differential one. In this case, a set of equations describes the temporal evolution of the state of the system; they are called state equations.
When the state of a system is stagnant, then the system is at an equilibrium point. If the system is at that point at some moment in time and no
significant disturbance occurs, it is expected to remain at the same point
at the next moment in time. The state of equilibrium is expressed mathematically by the assumption that the temporal derivative of all the system’s
variables is equal to zero:
dx
dt
i
n
i
=
=
…
0
1 2
,
, ,
The temporal evolution of a system can be represented graphically in two
ways. In the first case, the change in each variable can be represented by a
two-dimensional graph, where the value of the variable is measured on the
vertical axis and time on the horizontal axis. Thus the temporal evolution
of the system is depicted in a set of such diagrams, which are equal in number with the system variables. The following equations apply:
x f t
i
n
i
i
= ( ) = …
,
, ,
1 2
In the second case, the change in each variable is represented by a curve
in a space of n dimensions, the phase space. The evolution of the system is
depicted by a line in a hypersurface of n dimensions where each axis corresponds to one variable, and time t has been eliminated.
Stability is a property of the specific equilibrium point and is defined
in relation to it. If the system moves away from the equilibrium point,
i.e. undergoes a disturbance and has the tendency to return to the same
point, then the specific equilibrium point is stable. By contrast, if a system
that has been disturbed continues to move away from the equilibrium point,
then that point is unstable.
There are varying degrees of stability, depending on how great a disturbance the equilibrium point can withstand. If the system can return to the
equilibrium point only when there have been small deviations from it, then
we have local stability. In this case, in the phase space around the equilibrium point A, there will be a specific area, from all points of which the