38 Ecology and Applied Environmental Science
(α 12 N 1 N 2 ), with other more complex relationships might be useful when
appropriate experimental data are present. Otherwise, it is preferable to
remain with the linear relationship, which retains certain validity, at least
as an initial approach. The above equations also presuppose that predatorprey interactions generate instantaneous consequences for both populations. This may apply more for mortality than for natality, the increase of
which needs some time in order to materialise. Another possibility is the
introduction of population age structure through models based on the use
of matrices; however, this constitutes a structural change that results in
more complicated outcomes.
The above equations cannot be solved analytically, but with the application of numerical methods. The graphical representation of the system
demonstrates periodical oscillations. These oscillations have the same
period for both populations but exhibit a phase difference, i.e. the predator’s increase or decrease follows the respective prey’s change. Oscillation
amplitudes depend on the initial circumstances. These amplitudes remain
constant if the exponential model is applied (Figure 3.7), whereas they
decline if at least one of the populations exhibits self-limitation (Figure 3.8).
The system balances either at a specific point or at a range of values of N 1
and N 2 ; in the first case the equilibrium is indifferent (neutral) whereas in
the second case it is stable.
In the case of stable equilibrium, the prey’s environmental carrying
capacity must be large enough to support the predator population. In the
opposite case, the system is modified due to predator elimination, whereas
the prey population’s size tends to balance at the value of the environmental
carrying capacity. (See Figure 3.9.)
Elimination of one of the populations can occur in the case of indifferent equilibrium, when the initial values of population sizes are far enough
from the equilibrium point, so that their oscillations have large amplitudes,
i.e. are approaching the horizontal axis. This means that often, one of the
t
Prey
Predator
0
N
Figure 3.8 Predator-prey interaction with self-limitation. (From Hadjibiros 2007. Ecology.
Ecosystems and Environmental Protection, 3rd edition. Symmetria, Athens.
With permission.)
(α 12 N 1 N 2 ), with other more complex relationships might be useful when
appropriate experimental data are present. Otherwise, it is preferable to
remain with the linear relationship, which retains certain validity, at least
as an initial approach. The above equations also presuppose that predatorprey interactions generate instantaneous consequences for both populations. This may apply more for mortality than for natality, the increase of
which needs some time in order to materialise. Another possibility is the
introduction of population age structure through models based on the use
of matrices; however, this constitutes a structural change that results in
more complicated outcomes.
The above equations cannot be solved analytically, but with the application of numerical methods. The graphical representation of the system
demonstrates periodical oscillations. These oscillations have the same
period for both populations but exhibit a phase difference, i.e. the predator’s increase or decrease follows the respective prey’s change. Oscillation
amplitudes depend on the initial circumstances. These amplitudes remain
constant if the exponential model is applied (Figure 3.7), whereas they
decline if at least one of the populations exhibits self-limitation (Figure 3.8).
The system balances either at a specific point or at a range of values of N 1
and N 2 ; in the first case the equilibrium is indifferent (neutral) whereas in
the second case it is stable.
In the case of stable equilibrium, the prey’s environmental carrying
capacity must be large enough to support the predator population. In the
opposite case, the system is modified due to predator elimination, whereas
the prey population’s size tends to balance at the value of the environmental
carrying capacity. (See Figure 3.9.)
Elimination of one of the populations can occur in the case of indifferent equilibrium, when the initial values of population sizes are far enough
from the equilibrium point, so that their oscillations have large amplitudes,
i.e. are approaching the horizontal axis. This means that often, one of the
t
Prey
Predator
0
N
Figure 3.8 Predator-prey interaction with self-limitation. (From Hadjibiros 2007. Ecology.
Ecosystems and Environmental Protection, 3rd edition. Symmetria, Athens.
With permission.)