Leslie Matrices - Modelling Complex Ecological Dynamics - page 128

piecewise constant values. Irrespective of the mode of evaluation (sum or integral),
the development is completed when the biological time reaches the value of 1. With
this concept, the real time t is transformed to another unit biol ( t ) which can be
thought of as the fraction of development which is completed up to time t .
A second drawback of the original Leslie model is that no density dependence is
incluced in the original model and all parameters are constant. A possible extension
of the original model is to formulate the fertilities, for example, as functions of the
number of individuals already present in the habitat under study (Sondgerath and
Schroder 2002). A first approach to do so may be a step function. Up to a specific
critical density the fertility is F 1 ; beyond it is F 2 (

< F 1 ). A more flexible approach is

a function of the Weibull-type:

Þ a F ð N Þ¼ F max 1 ð exp ð ð N = N crit Þ Þ;

with F max denoting the maximum fertility at low densities, N crit , the critical
population density and

-values this
function reduces to a step function. Of course, the survival rates can also be
formulated as density dependent in a similar manner (Pykh and Efremova 2000).
The original Leslie model deals with classes dependent on age or stage but not
both. For populations whose development depends on the environment (e.g.
insects) this is not sufficient, because age and stage are not linked in a straightfor-
ward way. For this reason, the extended Leslie model was set up (S

a , a steepness parameter. For very high a

ondgerath and
Richter 1990). This model coupled different Leslie models, one for each stage of
the life cycle. The coupling was done via time-dependent transition probabilities,
which reflected the development status of the individuals. These transition prob-
abilities were evaluated on the basis of the biological time defined above. First, the
biological time resulting from the stage-specific development rate for each stage
was evaluated. In Fig. 9.4b an artificial temperature curve is given. In Fig. 9.4c the
biological time resulting from integration of the time-dependent development rates
is shown. The latter can be reached by combining the O'Neill function with the time
course of temperature. In the first period the biological time increases slowly
because of the low temperature. Due to nearly optimal conditions, the increase is
higher after approximately 150 days. Temperatures above the optimum of 25 C
(between days 187 and 298) result in a flattening of the biological time curve in the
specified period. As outlined above, the development of one stage is completed
when the appropriate biological time reaches the value of 1. In the example shown
in Fig. 9.4 this is the case after 233 days. To take biological variation into account a
statistical distribution (e.g. a Weibull distribution) is applied subsequently to finally
reach at the transition probabilities for the model. The Weibull distribution has two
parameters: one scale parameter, which is 1 in this case, and a shape parameter,
which affects the steepness of the curve. The higher the shape parameter, the less
biological variation is included (see Fig. 9.4d ).
The general structure of the extended Leslie model for a population with three
development stages can be seen in Fig. 9.5 . This kind of model was successfully
used for different purposes, e.g. to forecast the dynamics of pest populations

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