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

Resumenes 84
⎧dx
⎛ x ⎞ qx
⎪ = r⎜1
− ⎟(
x − m)
x − y
⎪ dt ⎝ K ⎠ x + a
⎪
X µ : ⎨
⎪
⎪dy
⎛ y ⎞
= s
⎪ ⎜
⎜1−
⎟ y
⎩ dt ⎝ n x ⎠
7
where ( , , , , , , ) + ℜ ∈
= n s a b q K r µ and the parameters have different biological
meanings. By means of a diffeomorphism (Chicone 1999, Guckenheimer
and Holmes 1983), we make the study in the topologically equivalent
system given by:
⎧du
2
⎪ = ( ( 1−
u)(
u − M )( u + A)
− v)
u
dτ
⎪
Yη
: ⎨
⎪ dv
⎪ = B(
u − v)(
u + A)
v
⎩dτ
We prove that the Allee effect produces a great modification in the behavior
of May-Holling-Tanner model, analysed in (Sáez and González-Olivares
1999), where it was demonstrated the existence of two limit cycles
surrounding the unique equilibrium point at the interior of the first
quadrant, for a set of parameter values, showing that local stability does
not imply global stability of the equilibrium point.
In the model studied here varied dynamics appear that are not
topologically equivalent to the original one, for some we give the respective
ecological interpretation. Among other results we have that:
1. For any parameter values the singularity (0, 0) is a non-hyperbolic
attractor (Chicone 1999, Guckenheimer and Holmes 1983), and (1,0) is an
hyperbolic saddle point.
2. There exists a set of parameter for which two equilibrium point in the
interior of the first quadrant exists; also an unique or one having multiplicity
two can exist.
3. When we have an unique positive equilibrium point of multiplicity two, it
occurs the codimension 2 saddle-node bifurcation (Chicone 1999).
4. When two positive equilibrium points exist one of them is always a
hyperbolic saddle point and the nature of the other depends on sign of
trace; in this case there exists an unique limit cycle obtained via Hopf
bifurcation.
System Yn is highly sensitive to initial conditions and as the origin is
always an attractor the possibility of extinction of both populations is large
depending mainly on the predator-prey ratio.
Referencias
[1] Arrowsmith, D. K., Place, C. M. 1992. Dynamical System. Differential
equations, maps and chaotic behaviour. Chapman and Hall.
[2] Chicone C. 1999. Ordinary differential equations with applications, Text in
Applied Mathematics 34, Springer.
[3] Clark, C. W. 1990. Mathematical Bioeconomic: The optimal management of
renewable resources, (second edition). John Wiley and Sons.