Environmental Engineering Reference
In-Depth Information
500
1000
0
Distance from the original stand (m)
Fig. 16.5 Simulation obtained with the pine invasion model. Each 25
25 m cell is coloured in
grey
shades according to tree density (
white
cells
¼
empty cells,
black
cells
¼
maximal tree
density). The simulation was run over 100 years
long- and short-distance dispersal, respectively. This function integrates to 1 when
summed over all directions and over all distances.
All components of the pine model are stochastic: demographic and environmen-
tal stochasticity is included in the demographic component and both the direction
and the distance of dispersal are randomly drawn (from a uniform distribution and
from the dispersal kernel, respectively). Simulations are necessary to explore the
dynamics of such a complex model, which combines a highly structured life-cycle
and stochasticity in every component. An example of a trajectory run over 100
years from an initial stand (one column of cells composed of 20% of dominant trees
and 80% of suppressed trees) located at the left edge of the rectangular landscape,
shows the patchy expansion of the population towards the opposite edge (Fig.
16.5
).
Invasion speed was computed from the linear regression of wave front location on
time (see Kawasaki et al. 2006 for mathematical details).
Elasticity analyses performed on the stochastic growth rate and the invasion
speed gave a better understanding of the interactions between the different factors
influencing seedling recruitment and notably revealed that grazing intensity was the
main factor limiting pine spread. This result has important consequences in terms of
management, as it suggests that grazing by domestic livestock could be an effective
tool to limit the invasion of open landscapes by pines, which is less expensive than
mechanical interventions.
16.4 Conclusions
Integro-differential models are a useful tool to model the spatio-temporal dynamics
of a population. They allow sufficient realism in both the life cycle (stage-structure)
and dispersal processes. Their flexible formalism permits to include intra-specific
