Once germinated, native and/ or alien species can colonize the empty site with a
probability
p c ¼ 1 À 1 À p g
n seeds
,
(14)
where n seeds is the total number of seeds of the corresponding species (native or
invader) present in the site. The empty site will be colonized by the plant species
having the greatest p c value. If eventually both native and alien plants have the same
value of p c , then one of them is selected with equal chance and wins the competition.
CA Simulation protocol. The CA simulations were implemented through the
following procedure. Initially, a single invader plant with an age randomly chosen
between t m and t max was introduced in the center of the lattice. At each time step,
(i) plants can die with probability p d ; (ii) any survival plant ages, and the mature ones
can produce and disperse seeds with probability p s ; (iii) the empty sites can be
colonized, with probability p c , by native or alien plants; (iv) the non-stationary
amount of phytotoxin is determined according to Eq. (8) for each lattice site;
(v) the quantities of interest, namely, evolution in time of plant populations, gyration
radius, and roughness of the border of the invaded region, are determined. At the end
of this sequence of actions, a new time step (Monte Carlo step-MCS) begins, and the
entire procedure is iterated. Additional details can be found in reference [44].
4.3
Simulational Results
In order to investigate the controversial role of allelopathy in plant invasion,
simulations were performed using basically the same values of the parameters for
native and invader species. So, both native and alien plants have exactly the same
skews in the competition for resources if allelopathic suppression is neglected. These
values, listed in Table 1 of reference [44], were inspired in typical herb species with
annual life cycle such as Euphorbia heterophylla L. It must be noticed that the model
parameters are rather arbitrary since the details of most invasion processes are
largely imprecise. This apparent handicap really represents the strength of the
model able to investigate the dynamics of invasion in a broad range of parameter
values.
Figure 15a shows the invasion probability as a function of the ratio between the
seed productions of invasive and native plants. In turn, the invasion probability as
a function of the initial fraction of resistant plants in the native community is shown
in Fig. 15b. Here, resistant plants have a phytotoxic threshold 100 times greater
than the “normal” native plants. As expected, the chance of a successful invasion
increases if the alien plants produce more seeds than the native species. Also, the
invasion probability decreases as the fraction of resistant plants increases in the
native community. Finally, Fig. 15c and d show the invasion probability as
a function of the phytotoxic threshold and the maximum dispersion radius of the
alien seeds, respectively.
496
S. A. Carvalho and M. L. Martins
probability
p c ¼ 1 À 1 À p g
n seeds
,
(14)
where n seeds is the total number of seeds of the corresponding species (native or
invader) present in the site. The empty site will be colonized by the plant species
having the greatest p c value. If eventually both native and alien plants have the same
value of p c , then one of them is selected with equal chance and wins the competition.
CA Simulation protocol. The CA simulations were implemented through the
following procedure. Initially, a single invader plant with an age randomly chosen
between t m and t max was introduced in the center of the lattice. At each time step,
(i) plants can die with probability p d ; (ii) any survival plant ages, and the mature ones
can produce and disperse seeds with probability p s ; (iii) the empty sites can be
colonized, with probability p c , by native or alien plants; (iv) the non-stationary
amount of phytotoxin is determined according to Eq. (8) for each lattice site;
(v) the quantities of interest, namely, evolution in time of plant populations, gyration
radius, and roughness of the border of the invaded region, are determined. At the end
of this sequence of actions, a new time step (Monte Carlo step-MCS) begins, and the
entire procedure is iterated. Additional details can be found in reference [44].
4.3
Simulational Results
In order to investigate the controversial role of allelopathy in plant invasion,
simulations were performed using basically the same values of the parameters for
native and invader species. So, both native and alien plants have exactly the same
skews in the competition for resources if allelopathic suppression is neglected. These
values, listed in Table 1 of reference [44], were inspired in typical herb species with
annual life cycle such as Euphorbia heterophylla L. It must be noticed that the model
parameters are rather arbitrary since the details of most invasion processes are
largely imprecise. This apparent handicap really represents the strength of the
model able to investigate the dynamics of invasion in a broad range of parameter
values.
Figure 15a shows the invasion probability as a function of the ratio between the
seed productions of invasive and native plants. In turn, the invasion probability as
a function of the initial fraction of resistant plants in the native community is shown
in Fig. 15b. Here, resistant plants have a phytotoxic threshold 100 times greater
than the “normal” native plants. As expected, the chance of a successful invasion
increases if the alien plants produce more seeds than the native species. Also, the
invasion probability decreases as the fraction of resistant plants increases in the
native community. Finally, Fig. 15c and d show the invasion probability as
a function of the phytotoxic threshold and the maximum dispersion radius of the
alien seeds, respectively.
496
S. A. Carvalho and M. L. Martins