Environmental Engineering Reference
In-Depth Information
It yields a continuous line connecting successive states. Such a line is called a
trajectory
. A trajectory is a line in the phase space showing the fate of a system, i.e.
the successive sizes of the variables. The starting point is at the initial condition.
Can two-dimensional dynamic systems describe oscillatory processes? Yes, and
this implies that there are domains of the phase space where an increase of a
variable dominates and others where a decrease dominates. These domains can be
separated by a line where increase and decrease are balanced, i.e., the change of that
variable is zero for an infinitely small moment. This occurs whenever the size of a
variable transits a maximum or minimum value. If we consider all possible points of
the phase space on which zero growth for one of the variables occurs, we obtain a
line which is called the
zero growth isocline
, or just
isocline
, for that variable. There
is an isocline for each variable. An intersection of the isoclines represents an
equilibrium point of the system (where the rates of change of both variables are
zero). The area of the phase space from which an equilibrium point is reached is
called a
domain of attraction
or a
basin
. Points or areas (!) in the phase space that
are approached during the system dynamics are also referred to as
attractors
.
Before we proceed to a simulation of the Lotka-Volterra equations, we employ
the introduced concepts to anticipate some aspects of the dynamics of the system.
We first find equations for the isoclines. To do that we only need to set the rate of
change of each variable to zero and solve the resulting equation for one variable in
terms of the other. Using this procedure, the prey isocline is: (see 1.243)
0
¼
C
1
Prey
C
2
Prey
Pred
;
(6.12)
or
C
1
C
2
:
Pred
¼
When the size of the predator population equals
C
1/
C
2, the momentary change
of the prey population is zero (i.e. the prey population dynamics transits from
increase to decrease - or decrease to increase). The predator isocline is: (see l. 243)
calculated by the same procedure to obtain:
0
¼
C
2
b
Prey
Pred
C
3
Pred
;
(6.13)
or
C
3
Prey
¼
ð
b
C
2
Þ
It is apparent, that the isoclines for the Lotka-Volterra equations are both
constant. The change of all prey population sizes is zero when the size of the
predator population has a specific value. And the change of all predator values is
zero for a specific prey population size. The intersection of these lines is an
