Environmental Engineering Reference
In-Depth Information
largest of the eigenvalues of the transition matrix A at small densities (i.e. in
absence of intra-specific competition). Values of
l
greater than unity guarantee
that population size will grow when small.
In its deterministic form described above, and under some assumptions (no Allee
effects, thin-tailed kernel, population growth rate greater than unity) the integro-
differential population model generates travelling waves which move at a constant
speed
c
* (Weinberger 1978; Neubert and Caswell 2000). The invasion speed
c
*is
determined by the low-density leading edge of the travelling wave and is calculated
as a function of the transition matrix A for small densities and the kernel matrix K
(see Neubert and Caswell 2000 for mathematical expressions). Perturbation ana-
lyses, i.e. sensitivity (see Chap. 23) and elasticity analyses (see Chap. 9), on
and
c
* can be analytically performed for deterministic integro-differential models [see
Neubert and Caswell (2000) for mathematical details]. These analyses generally
require simulations when the model is stochastic (however, see Lewis and Pacala
2000 for stochastic unstructured integro-differential models).
l
16.3 Application Examples
16.3.1
Invasion of Road Verges by Feral Oilseed Rape
Oilseed rape (
Brassica napus
L.) is an annual cultivated species for which feral
populations (i.e. populations escaped from crops and established in uncultivated
areas) are a common feature along road verges in European and North American
farming landscapes (Pivard et al. 2008a, b; Knispel and McLachlan 2009). The
persistence and spread of these feral populations are expected to raise both agro-
nomical and ecological issues in the case of genetically modified (GM) cultivars
(Hancock et al. 1996; Pessel et al. 2001). Indeed, if the transgene confers a selective
advantage to the plant (e.g. herbicide tolerance), and if efficient dispersal vectors
move feral seeds across long distances, the GM feral plants could invade unculti-
vated habitats and thus modify the composition of semi-natural plant communities.
Moreover, feral-to-crop gene flow and feral seed surviving within soil seed banks
could make the (spatial and temporal) isolation between GM and conventional
crops less feasible.
Stage-structured integro-differential models are well adapted to model the
spread of GM feral oilseed rape populations along road verges because: (1) the
stage-structured dynamics component of the model can account for the different
stages of oilseed rape life cycle (seeds in the seed bank and reproductive plants) and
for the impact of a selective advantage on specific parts of the life-cycle and (2) the
dispersal component can be used to explicitly model the successive action of the
different seed dispersal vectors occurring along road verges.
The invasion model we developed for GM feral oilseed rape in a previous study
is deterministic and is one-dimensional to mimic the linearity of road verges. The
first version of the model (Garnier and Lecomte 2006) was thereafter refined to
