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

Resumenes 137
the Allee effect are those related to decreasing functional and numerical
responses as population size drops to very low levels.
The most simple model which incorporates Allee effect is the equation
derived from the logistic growth adding a new parameter:
a)
dx
dt
⎛ x ⎞
= r ⎜1
− ⎟ ( x − m ) x ,
⎝ K ⎠
[1, 3, 4, 7]
where m represents the minimum viable population. With m > 0, the
population exhibits strong Allee effect, and with m = 0 it exhibits a weak
Allee effect. Another equations presented in the literature are:
b)
dx ⎛ ⎛ x ⎞⎛
x ⎞ ⎞
= ⎜ r⎜1
− ⎟⎜
⎟ − D⎟
x [19]
dt ⎝ ⎝ R ⎠⎝
x + C ⎠ ⎠
c)
dx ⎛ x ⎞ ⎛ b + C ⎞
= r ⎜1
− ⎟ ⎜1
− ⎟ x
dt ⎝ K ⎠ ⎝ x + C ⎠
[2]
d)
dx ⎛ rx b ⎞
= ⎜ r − − ⎟ x
dt ⎝ K x + C ⎠
[16, 18]
e)
dx
dt
⎛ x ⎞
= r x ⎜ ⎟
⎝ x + C ⎠
[20]
In order to discuss the different biological reason that leads to an
Allee effect, we are going to consider the following baseline model with
linear self-limitation
dx
dt
= x g φ
2
( x,
( x)
) − bx
where φ represents the extraction rate of resources per unit consumer and
g represents the conversion rate resource to per capita population growth
rate. Various functional forms can be found in [9, 11, 15, 18]. Let us
assume that at low population density, the reproduction or the survival are
diminished or metabolic losses are increased. Then, the population requires
to consume more resources in order to balance the extra losses or the
diminished gain. Considering the simplest and more widely used (linear)
conversion function g depending on x we have:
g( x)
= ε ( φ − κ ( x)
)
The conversion function g describes the basal growth of population as
a function of the feeding rate φ and the parameter κ which represents the
consumption level required for zero growth. We study the mathematical
equivalences derived from this modeling approach respect to the ones
already presented in the ecological and biomathematical literature.
References
[1] Brauer, F., and Castillo-Chávez, C., 2001 Mathematical models in Population
Biology and Epidemiology, TAM 40, Springer.Verlag.
[2] Boukal, D. S. and L. Berec, 2002, Single-species models and the Allee
effect: Extinction boundaries, sex ratios and mate encounters, Journal of
Theoretical Biology, Vol 218, 375-394.
[3] Clark, C. W., 1990. Mathematical Bioeconomic: The optimal management of
renewable resources, (second edition). John Wiley and Sons.