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

Resumenes 175
los parámetros son todos positivos teniendo diferentes significados
biológicos y los sistemas son analizados en el primer cuadrante. Se
demuestra que:
1) El punto de equilibrio (0,0) siempre es punto silla y que la naturaleza de
(K, 0) depende de la existencia de puntos de equilibrio al interior del primer
cuadrante.
2) Existen condiciones en los parámetros para los cuales hay dos puntos de
equilibrio al interior del primer cuadrante (positivos), o bien uno de
multiplicidad dos, o ninguno.
3) Cuando colapsan los puntos críticos positivos, este único punto puede ser
un punto cúspide de codimensión 2 (bifurcación de Bogdanovic-Takens).
4) Existe una región en el espacio de parámetros donde dos ciclos límites
pueden coexistir.
La extinción de la población de depredadores es posible porque el
punto (K,0) es atractor cuando existen dos puntos de equilibrio positivos.
Comparative study of Gause type predator-prey models with
nonmonotonic functional response of rational type
In this work we make a comparative analyze of two deterministic
continuous-time predator-prey models of Gause type considering a Holling
type IV functional response which is nonmonotonic [10].
The Gause type models [4] are characterized by the predator
numerical response which is a function of functional response, correspond to
a compartimental models or mass action, and are described by a differential
equations system of the form:
⎧ dx
⎪ = F(
x)
− y p ( x)
dt
X : ⎨
⎪ dy
= ( c p(
x)
− d)
y
⎪⎩
dt
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). Function
F(x) represents the prey growth rate and function p(x) is the predator
functional response or consumption rate, that is, it indicates the change in
the density prey attached per unit time per predator as the prey density
changes.
This function describes a type of antipredator behavior called group
defense or aggregation, that is, a phenomenon whereby predators
decrease, or even prevented altogether, due to the increased ability of the
prey to better defend or disguise themselves when their number are large
qx
enough [2, 4, 9, 12, 13, 14, 15]. Most employed function is h(
x)
= 2
x + a
[5, 6, 7, 8, 9], deduced in [2]. However, in [15], the more general form
qx
h(
x)
= is used.
2
x + bx + a
In above works [1, 11] we have demonstrate that using different
form to nonmonotonic functional response change the bifurcation diagram
of system and in this work we pretend find a generalization of this results.
Models that we will study are described by the following Kolmogorov type
differential equations systems [3]: