Ordinary Differential Equations - Modelling Complex Ecological Dynamics - page 74

- b is a conversion factor. It specifies what fraction of the captured prey biomass is
transformed into predator biomass.
- C 3 is a mortality factor specifying what fraction of predators is lost per unit of
time.

To understand the dynamics of the model we look at the expressions which are
summed up in the function:

- C 1* Prey is an exponential growth term. The prey (in absence of predators, i.e.
Pred ¼

0) would exponentially increase
- C 2* Prey * Pred is the product of both populations multiplied with a constant.
It represents the frequency that predator and prey meet. C 2 is typically chosen to
be small. In the predator equation we find the same term multiplied with an
additional constant b . This represents the amount of caught prey which is
converted to predator biomass.
- C 3* Pred describes an exponential decay. Per unit of time a certain fraction of
the predator population is lost. This is how the model describes death on the
population level. We see the implication of the pool approach: The age of
individuals does not play a role. Depending on the size of the pool always the
same proportion is subtracted.

Now we look at the equations as a whole and can conclude which properties are
represented in the equations and which typical properties of real organisms are
ignored. We can see that, without predators, prey would grow infinitely. There is no
capacity limitation. Without prey, on the other hand, the predators would decline at
a constant rate. The implication is that the smaller the population becomes, the
longer each remaining predator will survive. There is no limitation of life span.
Another unrealistic feature of the model is that an increase in the amount of prey
will always lead to an increase in the rate of prey capture and rate of predator
increase. The predators have no saturation limit and thus no limit how fast they can
grow. These aspects could easily be modelled more realistically, but this would
make the model less simple. Now we take a look at its dynamic features and then
add model properties to show how the equations can be extended.
An overview of the system dynamics can be obtained if we display the system
using one axis to show the size of the prey population and the other axis for the
predator population. Each combination of predator and prey values is marked by a
point in the plane. This is why the number of the variables is said to indicate the
dimension of the system . A two-dimensional system can be easily displayed in the
plane. This form of display is called state space or phase space . We can get a coarse
overview of what is happening in the phase space by calculating the resulting
dynamics for a set of grid points and draw them as vectors. Such a graphic is called
direction field (see Fig. 6.5 ).
Any calculation of the dynamics has to start with initial values for the predator
and prey populations ( initial conditions) . Displayed in the phase space, the combi-
nation of the two initial values will appear as a point. The differential equations
describe the successive fate of systems states emerging from this starting point.