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This model can explain the shape of Gompertz's mortality law [60]. If one
imposes that reproduction occurs only after a certain age, one can easily explain
the accumulation of bad genes after that age, as exhibited by the catastrophic
senescence of many semelparous animals (like the pacic salmon) [61]. By inserting
maternal cares, one can also give a motivation for the appearance of menopause in
women [62]: after a certain age, the tness may be more increased by stopping giving
birth to more sons, with the risk of loosing own life and that of not-yet-independent
ospring.
15.5. Dynamic Ecosystems
15.5.1. Speciation in the phenotypic space
We are here referring to the formation of species in a spatially homogeneous en-
vironment, i.e. to sympatric speciation. In this frame of reference, a niche is a
phenotypic realization of relatively high tness. Species have obviously to do with
niches, but one cannot assume that the coexistence of species simply reects the
presence of \pre-existing" niches; on the contrary, what appears as a niche to a
given individual is co-determined by the presence of other individuals (of the same
or of dierent species). In other words, niches are the product of co-evolution.
In this section we introduce a new factor in our model ecosystem: a short-range
(in phenotypic space) competition among individuals. As usual, we start the study
of its consequences by considering the evolution in phenotypic space [63, 64].
We assume that the static tness V (u), when not at, is a linear decreasing
function of the phenotype u except in the vicinity of u = 0, where it has a quadratic
maximum:
1
u
r
1
V (u) = V
0
+ b
(15.27)
1 + u=r
so that close to u = 0 one has V (u)'V
0
bu
2
=r
2
and for u!1, V (u)'
V
0
+ b(1u=r). The master sequence is located at u = 0.
We have checked numerically that the results are qualitatively independent on
the exact form of the static tness, providing that it is a smooth decreasing func-
tion. We have introduced this particular form because it is suitable for analytical
computation, but a more classic Gaussian form can be used.
For the interaction matrix W we have chosen the following kernel K
uv
R
1
uv
R
K
= exp
:
The parameter J and control the intensity and the steepness of the intra-species
competition, respectively. We use a Gaussian ( = 2) kernel, for the motivations
illustrated in Section 15.5.1.
