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

Resumenes 20
donde x = x(t) e y = y(t) representan los tamaños de las poblaciones de
presas y depredadores respectivamente para t > 0; los parámetros son
6
todos positivos, esto es, ( , , , , , ) + ℜ ∈
= c p a q K r µ , y tienen diferentes
significados biológicos.
Multiples Hopf bifurcations in Gause type predator-prey
models with nonmonotonic functional response.
In this work, we show a method for determine the number of limit
cycles surrounding a equilibrium point that can bifurcate from a weak or
fine focus on a family of predator prey model described by two-order
differential equations systems.
The characteristic of Gause type models [4] is that the predator
numerical responses is a function of functional response and correspond to
a compartimental models or mass action. The predator functional response
or consumption rate, measures the prey change density rate per predator
and per time unit [13].
The type IV functional response describe the effect of defence group
or aggregation [1, 4, 5, 6, 16, 17, 18], which is a form of antipredator
qx
behavior [12]. Most usual form is described by the function h(
x)
= 2
x + a
[7, 8, 9, 10, 14, 15], deduced in [2], where x = x(t) indicates the prey
population size.
The determination of the number of limit cycles surrounding a
equilibrium point it is based in the calculus of Liapunov quantities [2, 3, 11].
For obtain this quantities the system must be expressed in a normal form
[3]. Denoting λ = ( λ1,
λ2,......,
λN
) the parameters vector of system, we assume
that the coefficients of the Taylor series at the origin for the component
functions of system are polynomial in the components of l..
The Liapunov quantities are computed when the trace of Jacobian
(community) matrix evaluated at the equilibrium point is zero, that is, l1 =
0. We denote for Λ = ( 0,
λ2,......,
λN
) the vector of variable parameters that
determuines a hypersurface on the parameter space of system, such as the
Liapunov quantities are functions of those parameter named li with i =
2,3,....., N . Of this manner, those quantities are obtained as functions of
polynomial coefficients, which at once are dependents of the parameters of
original system.
This methodology it will be applied to the Gause type predator-prey
model described by the ordinary differential equations system:
3
⎧dx
x q x y
⎪ = r(
1 − ) x − 4
⎪ dt k x + a
X µ
: ⎨
3
⎪dy
⎛ p x ⎞
=
⎪ ⎜ − c ⎟ y
4
⎩ dt ⎝ x + a ⎠