ones and is affected by k
in
nþ1 of them. So, k
out
nþ1 elements μ n + 1, i in the line n + 1 of the
enlarged allelochemical interaction matrix are randomly chosen and assigned to nonnull values, and the remaining are set to 0. Analogously, k
in
nþ1 randomly chosen
elements μ i, n + 1 in the column n + 1 of the enlarged interaction matrix are assigned
to non-null values, and the remaining are fixed in 0. (See reference [35] for details.)
Finally, the initial toxin concentration of the alien species is B n + 1 = 0, and its
population density is N n + 1 = 0.01N i , with N i corresponding to the stationary
population density of one species chosen at random between the n current members
of the community. Regarding the initial community structure, the SIE evolutionary
dynamics starts from a single species.
3.2
Numerical Results
Since our primary interest relied on how allelopathic suppression affects the community structure, ν i, j = ν = 0.1 was fixed in order to ensure equal competition
coefficients for every species in a regime of interspecific coexistence.
The scenario of equal (or homogeneous) allelopathic traits was investigated.
Thus, each species has fixed toxin sensibility, c i = c = 0.1, release, degradation,
and uptaken rates, β i = β = 0.2, δ i = δ = 0.2, and γ j, i = γ = 0.1, respectively, 8i, j.
In turn, two mortality rates induced by allelochemicals were considered, namely,
weak (μ = 0.1) and strong (μ = 0.5) 8 i.
In Fig. 10, the average diversity is shown as a function of the number n SIE of SIE.
The diversity or species richness is defined as the fraction of species that survive at
the community stationary state. As expected, weak allelopathic suppression allows
the assembly of communities exhibiting large diversities. This is true for all response
functions tested, and, as expected, the diversity decreases as the response to toxins
increases. For instance, in our simulations, Φ
(1) (x) < Φ
(5) (x) < Φ
(3) (x) except for
small (x < 0.11) or large (x > 0.89) toxin concentrations. In contrast, community
diversity is drastically reduced at strong allelopathy for all response functions. As an
example, the number of surviving species decreases from $100, at weak, to $8, at
strong allelopathic suppression and response function Φ
1 . In this strong regime,
diversity seems to decrease slowly after it reaches a maximum as the number of
invasion events increases. Also, the effect of toxins uptaken is significant as revealed
by the right column in Fig. 10. In these graphs the response functions depend on the
absorbed fraction of toxins, not on their total concentration present in the homogeneous environment. So, even the regime of strong allelopathic suppression (μ = 0.5)
at low toxins’ absorption can become effectively equivalent to the weak (μ = 0.1)
regime.
In Fig. 11, the average connectivity of allelochemical networks is illustrated as a
function of the number n of surviving species observed at the stationary state reached
after a SIE. In a network of size n, the connectivity C(n) is defined as the fraction of
non-null elements in its n  n interaction matrix. Our results indicate that the average
connectivity is essentially the same at weak (μ = 0.1) and strong (μ = 0.5)
486
S. A. Carvalho and M. L. Martins
in
nþ1 of them. So, k
out
nþ1 elements μ n + 1, i in the line n + 1 of the
enlarged allelochemical interaction matrix are randomly chosen and assigned to nonnull values, and the remaining are set to 0. Analogously, k
in
nþ1 randomly chosen
elements μ i, n + 1 in the column n + 1 of the enlarged interaction matrix are assigned
to non-null values, and the remaining are fixed in 0. (See reference [35] for details.)
Finally, the initial toxin concentration of the alien species is B n + 1 = 0, and its
population density is N n + 1 = 0.01N i , with N i corresponding to the stationary
population density of one species chosen at random between the n current members
of the community. Regarding the initial community structure, the SIE evolutionary
dynamics starts from a single species.
3.2
Numerical Results
Since our primary interest relied on how allelopathic suppression affects the community structure, ν i, j = ν = 0.1 was fixed in order to ensure equal competition
coefficients for every species in a regime of interspecific coexistence.
The scenario of equal (or homogeneous) allelopathic traits was investigated.
Thus, each species has fixed toxin sensibility, c i = c = 0.1, release, degradation,
and uptaken rates, β i = β = 0.2, δ i = δ = 0.2, and γ j, i = γ = 0.1, respectively, 8i, j.
In turn, two mortality rates induced by allelochemicals were considered, namely,
weak (μ = 0.1) and strong (μ = 0.5) 8 i.
In Fig. 10, the average diversity is shown as a function of the number n SIE of SIE.
The diversity or species richness is defined as the fraction of species that survive at
the community stationary state. As expected, weak allelopathic suppression allows
the assembly of communities exhibiting large diversities. This is true for all response
functions tested, and, as expected, the diversity decreases as the response to toxins
increases. For instance, in our simulations, Φ
(1) (x) < Φ
(5) (x) < Φ
(3) (x) except for
small (x < 0.11) or large (x > 0.89) toxin concentrations. In contrast, community
diversity is drastically reduced at strong allelopathy for all response functions. As an
example, the number of surviving species decreases from $100, at weak, to $8, at
strong allelopathic suppression and response function Φ
1 . In this strong regime,
diversity seems to decrease slowly after it reaches a maximum as the number of
invasion events increases. Also, the effect of toxins uptaken is significant as revealed
by the right column in Fig. 10. In these graphs the response functions depend on the
absorbed fraction of toxins, not on their total concentration present in the homogeneous environment. So, even the regime of strong allelopathic suppression (μ = 0.5)
at low toxins’ absorption can become effectively equivalent to the weak (μ = 0.1)
regime.
In Fig. 11, the average connectivity of allelochemical networks is illustrated as a
function of the number n of surviving species observed at the stationary state reached
after a SIE. In a network of size n, the connectivity C(n) is defined as the fraction of
non-null elements in its n  n interaction matrix. Our results indicate that the average
connectivity is essentially the same at weak (μ = 0.1) and strong (μ = 0.5)
486
S. A. Carvalho and M. L. Martins