Environmental Engineering Reference
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by mowers ( k mow ) and a large proportion (1
p 1 ) of ballistic dispersal ( k bal ):
k 1 ( x )
(1 - p 1 ) k bal ( x ). Ballistic dispersal is bidirectional in the one-
dimensioned road verges (Fig. 16.3 ) whereas dispersal by mowers is expected to be
unidirectional. The corresponding kernel functions are respectively a bidirectional
Weibull function for ballistic dispersal [ k bal ( x )
¼
p 1 k mow ( x )
þ
m | x | c exp (- d | x | 2 ) estimated by
Colbach et al. (2001)] and a unidirectional exponential function for mowers
( k mow ( x )
¼
0 - hypothetical).
The secondary seed dispersal kernel ( k 2 ), that includes the combined (and
unidentified) effect of dispersal by wind and by vehicles is a unidirectional mixture
of two Gaussian functions that was fitted from a seed dispersal experiment, using a
maximum-likelihood method (Garnier et al. 2008).
The invasion model was run with contrasting values of selective advantage (with
an effect on both seed germination and plant survival) due to herbicide tolerance.
We performed elasticity analyses, which are similar to sensitivity analyses (see
Neubert and Caswell 2000 for their application to integro-differential models) to
identify the key-parameters of the model. Results showed that the invasion speed
was primarily determined by long-distance dispersal (whatever its low occurrence
rate). The selective advantage noticeably increased invasion speed, provided some
long-distance dispersal events were included.
This study underlined the necessity to obtain relevant estimates of long-distance
dispersal of feral oilseed seeds along road verges to make reliable predictions of the
spread of (GM) feral populations. However, long-distance dispersal events are
known to be very difficult to detect and to quantify because they are rare and highly
stochastic (Clobert et al. 2001; Bullock et al. 2002).
¼
exp (- x / a )/ a if x
>
0 and k mow ( x )
¼
0if x
<
16.3.2
Invasion of Grasslands by Pines
The invasion of open habitats by native and introduced tree species is a growing
concern in many regions of the world (e.g. Richardson et al. 1994; Dovciak et al.
2005) and is favoured by changes in natural and human environmental factors: fire
regime, climate, farming practices (grazing) and forestry practices (large planta-
tions of exotic trees). Seedling recruitment is known to have a key-impact on the
population dynamics of several tree species (Harper 1977) and is thus expected to
have also a predominant effect on their invasion dynamics. Because seedling
recruitment itself depends highly on different environmental factors, simulation
models are a useful tool to better understand how these factors interact and
influence the spread of trees.
Stage-structured integro-differential models are particularly useful to model the
spread of invasive tree species. This is because in slow growing tree species, where
maturity occurs after several years, it is crucial to distinguish several age-classes in
their life-cycle. Moreover seed dispersal kernels estimated from field data can
easily be implemented in this type of models. Stochastic environmental, demo-
graphic and dispersal variations are known to have a critical impact on the spread
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