Libro de Resúmenes / Book of Abstracts (Español/English)

Resumenes 143
phenomenon has been emphasized by recent developments in diverses
areas of ecology and conservation, and can be regarding as the basis of
animal sociability (Stephens and Sutherland 1999).
Mathematically, more simple models can show enough about this
dynamic (Stephens and Sutherland 1999), and it can implied the apparition
of new equilibrium points that changes the structural stability of system and
to occur changesin the stability of other equilibrium points.
Some authors estimates that the dense-dependient population
models for single specie that assume the Allee effect, must consider that
the population growth rate of this populations decreases if the population
size is below a threshold level m (Brauer and Castillo-Chávez 2001).
However, the Allee effect can be modeled by different forms as is presented
in
Sin embargo, el efecto Allee, que da cuenta del fenómeno que
aparece en ciertas poblaciones de animales a bajas densidades puede ser
modelado de otras formas como es presentado in (Thieme 2003, Wang et
al .1999, Zhou et al. 2005).
Model that we study is described by the Kolmogorov type differential
equation system (Freedman 1980).
⎧
2
dx ⎛ x ⎞ qx
⎪ = r⎜1
− ⎟(
x − m)
x − y 2
⎪ dt ⎝ K ⎠ x + a
⎪
X µ : ⎨
⎪
2
⎪dy
⎛ px ⎞
=
⎪ ⎜ − c ⎟ y
2
⎩ dt ⎝ x + a ⎠
where x = x(t) and y = y(t) indicate the predator and prey population size
respectively for t > 0 (number of individual, density or biomass), and the
parameters are all positives, having different biological meanings. By
biological reason a and m < K. If m = 0, it say that the population is affect
by a weak Allee effect (González-Olivares et al. 2005).
We obtain restrictions dependent of parameter values for:
a) To determine the quantity of equilibrium points at the first
quadrant.
b) To determine the nature of those equilibrium points.
c) To stablish the existente of separatrix curves and homoclinic or
heteroclinic curves .
d) To stablish the quantity of limit cycles.
Referencias
[1] Brauer, F., and Castillo-Chávez, C., 2001 Mathematical models in Population
Biology and Epidemiology, TAM 40, Springer.Verlag.
[2] Clark, C. W., 1990. Mathematical Bioeconomic: The optimal management of
renewable resources, (second edition). John Wiley and Sons.
[3] Conway E. D. and Smoller, J. A., 1986, Global Analysis of a System of
Predator-Prey Equations. SIAM J. Applied Mathematics, Vol. 46, No.4, 630-
642.
[4] Courchamp, F., Clutton-Brock, T., and Grenfell, B., 1999. Inverse
dependence and the Allee effect, Trends in Ecology and Evolution Vol 14, No.
10, 405-410.