2.1.1 Bistability
A basic feature of the spatially homogeneous system – Eq. 1 – is that everywhere in
the models’ parameter space, two distinct fixed points, either associated with species
coexistence or a single species survival, are simultaneously stable. Indeed, according
to Table 1, the stability of x
! Ã
1 implies in the instability of x
! Ã
3 , because ν 2 > 1, and vice
versa, because in this case ν 2 < 1. The same holds true for x
! Ã
2 and x
! Ã
4 . In fact, since
Φ(P) < μ, 8 P, then ν 1 þ
μ βÀδ
ð
Þ
δ cÀ1
ð
Þþβ ¼ ν 1 þ Φ β=δ
ð Þ
Á > 1 implies in ν 1 + μ > 1, which
results in x
! Ã
2 stable but x
! Ã
4 unstable. Conversely, if ν 1 + μ < 1 and x
! Ã
4 is stable, then
ν 1 + Φ(β/δ) < 1 also and x
! Ã
2 is unstable.
So, based on the previous analysis, the following scenarios are possible in the
models parameter space: (I) only x
! Ã
1 and x
! Ã
2 are stable and consequently one species
will be extinct; (II) only x
! Ã
1 and x
! Ã
4 are stable, leading to either the extinction of the
invader or its coexistence with the native species; (III) only x
! Ã
2 and x
! Ã
3 are stable,
resulting on either the extinction of the native or the coexistence of both species; (IV)
only x
! Ã
3 and x
! Ã
4 are stable and the coexistence is the rule; (V) only x
! Ã
1 is stable and
the invader species is extinct; and (VI) only x
! Ã
2 is stable and the native plant is
extinct.
Therefore, excepting in the scenarios V and VI, there are always two, and only
two, attractors in the phase space, and the system exhibits bistability. The competition outcome will depend on the initial populations N 0 and I 0 and allelochemical
concentration P 0 , as shown in Fig. 1. In particular, as shown in Fig. 1a, the invasion
leads to the extinction of the native plant only if the initial density of the alien species
is greater than that of the native one. So, from the ecological viewpoint, if the alien
plant is introduced at low level in a native plant community only slightly disturbed,
the invasion will fail. Hence, under strong competition, the role of environmental
disturbance, here thought as a reduced native population density, is central in
determining invasive success, supporting the claim that “there is no invasion of
natural communities without disturbance” [27]. However, taking into account the
other possible scenarios shown in Fig. 1, coexistence is the rule except in situations
where the invader plant is introduced at a large enough number relative to the native
species. Again, the main role of environmental disturbance is observed jointly with
the chemical stability of the phytotoxin. Under weak competition alien species can
invade, but genetic diversity can be sustained.
2.1.2 Invasion from a Single Focus
In reference [24], the invasion spreading starting from a single focus was studied
through numerical integration of a spatially explicit version of the model 1. The
major prediction of this model version is that an invasion focus spreads as an
expanding wave with constant speed. Behind the invasion front, the population
densities evolve to stationary and spatially uniform values corresponding to the
fixed points of the spatially homogeneous system. These nontrivial fixed points
imply on either the extinction of one or the coexistence of both species. Specifically,
if the native species is a strong interspecific competitor (ν 2 > 1), it extincts the
invader. Instead, if the invader armed with its allelochemical is a strong competitor
472
S. A. Carvalho and M. L. Martins
A basic feature of the spatially homogeneous system – Eq. 1 – is that everywhere in
the models’ parameter space, two distinct fixed points, either associated with species
coexistence or a single species survival, are simultaneously stable. Indeed, according
to Table 1, the stability of x
! Ã
1 implies in the instability of x
! Ã
3 , because ν 2 > 1, and vice
versa, because in this case ν 2 < 1. The same holds true for x
! Ã
2 and x
! Ã
4 . In fact, since
Φ(P) < μ, 8 P, then ν 1 þ
μ βÀδ
ð
Þ
δ cÀ1
ð
Þþβ ¼ ν 1 þ Φ β=δ
ð Þ
Á > 1 implies in ν 1 + μ > 1, which
results in x
! Ã
2 stable but x
! Ã
4 unstable. Conversely, if ν 1 + μ < 1 and x
! Ã
4 is stable, then
ν 1 + Φ(β/δ) < 1 also and x
! Ã
2 is unstable.
So, based on the previous analysis, the following scenarios are possible in the
models parameter space: (I) only x
! Ã
1 and x
! Ã
2 are stable and consequently one species
will be extinct; (II) only x
! Ã
1 and x
! Ã
4 are stable, leading to either the extinction of the
invader or its coexistence with the native species; (III) only x
! Ã
2 and x
! Ã
3 are stable,
resulting on either the extinction of the native or the coexistence of both species; (IV)
only x
! Ã
3 and x
! Ã
4 are stable and the coexistence is the rule; (V) only x
! Ã
1 is stable and
the invader species is extinct; and (VI) only x
! Ã
2 is stable and the native plant is
extinct.
Therefore, excepting in the scenarios V and VI, there are always two, and only
two, attractors in the phase space, and the system exhibits bistability. The competition outcome will depend on the initial populations N 0 and I 0 and allelochemical
concentration P 0 , as shown in Fig. 1. In particular, as shown in Fig. 1a, the invasion
leads to the extinction of the native plant only if the initial density of the alien species
is greater than that of the native one. So, from the ecological viewpoint, if the alien
plant is introduced at low level in a native plant community only slightly disturbed,
the invasion will fail. Hence, under strong competition, the role of environmental
disturbance, here thought as a reduced native population density, is central in
determining invasive success, supporting the claim that “there is no invasion of
natural communities without disturbance” [27]. However, taking into account the
other possible scenarios shown in Fig. 1, coexistence is the rule except in situations
where the invader plant is introduced at a large enough number relative to the native
species. Again, the main role of environmental disturbance is observed jointly with
the chemical stability of the phytotoxin. Under weak competition alien species can
invade, but genetic diversity can be sustained.
2.1.2 Invasion from a Single Focus
In reference [24], the invasion spreading starting from a single focus was studied
through numerical integration of a spatially explicit version of the model 1. The
major prediction of this model version is that an invasion focus spreads as an
expanding wave with constant speed. Behind the invasion front, the population
densities evolve to stationary and spatially uniform values corresponding to the
fixed points of the spatially homogeneous system. These nontrivial fixed points
imply on either the extinction of one or the coexistence of both species. Specifically,
if the native species is a strong interspecific competitor (ν 2 > 1), it extincts the
invader. Instead, if the invader armed with its allelochemical is a strong competitor
472
S. A. Carvalho and M. L. Martins