Abstract
Abstract
For scalar equations of population dynamics with an infinite distributed delay
x
′
(
t
)
=
r
(
t
)
[
∫
−
∞
t
f
(
x
(
s
)
)
d
s
R
(
t
,
s
)
−
x
(
t
)
]
,
x
(
t
)
=
φ
(
t
)
,
t
⩽
t
0
,
where f is the delayed production function, we consider asymptotic stability of the zero and a positive equilibrium K. It is assumed that the initial distribution is an arbitrary continuous function. Introducing conditions on the memory decay, we characterize functions f such that any solution with nonnegative nontrivial initial conditions tends to a positive equilibrium. The differences between finite and infinite delays are outlined, in particular, we present an example when the weak Allee effect (meaning that
f
′
(
0
)
=
1
together with
f
(
x
)
>
x
,
x
∈
(
0
,
K
)
) which has no effect in the finite delay case (all solutions are persistent) can lead to extinction in the case of an infinite delay.
Funder
Natural Sciences and Engineering Research Council of Canada