Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Sminaire UMMISCO- 29 Avril 2009
STATE-DEPENDENT DELAYS ASSOCIATED
TO THRESHOLD PHENOMENA IN
STRUCTURED POPULATION DYNAMICS
Hassan HBID
[email protected]
LMDP et UMI-UMMISCO (Cadi Ayyad University, Marrakech,Morocco)
April 28, 2009
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Index
1. Introduction
2. Size Structured Models
3. Age-Structured Models with Moving Boundary Conditions
4. A short review on state dependent delay deferential equations
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
In this work we are interested in the role of state-dependent
delayed differential equations in the modelling of structured
population dynamics.
Our aim is to put in evidence the onset of state-dependent delays
in an important class of models: threshold models.
These are structured population dynamics models in which the
transition of individuals between different stages is due to some
quantity reaching a prescribed level.
An interesting presentation of this kind of equations can be found
in (H.L. Smith and Y. Kuang, Periodic solutions of differential delay equations with
threshold-type delays, Contemporary Math. Vol. 129 (1992) 153-176).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
Although the appearance of differential equations with
state-dependent delays goes back to the 19th century, the study of
them has recently been reactivated due to their frequent
occurrence in a wide variety of situations.
These equations can be found in the literature associated with a
wealth of applications in different fields, such as
!
Electrodynamics (R.D. Driver, 1969),
!
Economics (J. Bélair and J.M. Mahaffy, 2001),
!
Control (H. O. Walther, 2002)
!
Population dynamics (O. Arino, H. H. and R. Bravo de la
Parra, 1998)
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
From a mathematical point of view, it is well-known that
structured models of population dynamics which are formulated in
terms of hyperbolic partial differential equations can be reduced,
integrating along the characteristic lines, to a so-called renewal
equation for the function of births.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
This process, applied to the reduction of a structured model of a
population divided into several stages leads frequently to a delayed
equation with a time-dependent delay. This variable delay
corresponds to the time spent by the individuals of one stage in
passing to the next one.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
If this transition depends on the density of the population, then
the delay will be state-dependent. In many cases, as we will show
in the examples below, this condition of transition is formulated
like an equation which implicitly defines the delay and which has a
meaning in the model: the individuals change their stage after a
certain quantity reaches a predetermined threshold value.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Introduction
In some situations, the magnitude which determines the change of
stage depends on the state of a non-structured population x(t),
and therefore the threshold condition is formulated as
! t
k(x(σ)) dσ = W ∗
t−τ (t)
where k : R+ −→ R+ is a known function. There are also other
situations where the population is structured by some character in
all the stages. In this case, threshold condition should be expressed
in terms of a functional F :
! t
F (l (·, σ)) dσ = W ∗ .
t−τ (t)
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STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Size Structured Models
In (J.A.J. Metz and O. Diekmann, The Dynamics of Physiologically Structured
Populations. Lecture Notes in Biomath. 68 (Springer-Verlag, 1986)) the process
of modeling the dynamics of populations structured by some
physiological variable is described, along with a great amount of
examples and applications. With the aim of simplifying, and since
it is the context in which the following models are presented, we
stick to the one-dimensional case.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Size Structured Models
As is fully explained in (J.A.J. Metz and O. Diekmann, 1986) , the
model for the dynamics of this population is obtained through
making a balance of mass in [xb , xd ], giving rise to a transport
P.D.E.:
∂n
∂
(x, t) +
[g (x, t)n(x, t)] = Sources − Sinks
(1)
∂t
∂x
together with a boundary condition which takes into account the
entrance of individuals accross the boundary xb :
g (xb , t)n(xb , t) = Local arrival rate.
(2)
The function g (x, t) is the individual growth rate, so that the
individual state is determined by the O.D.E.:
x # (t) = g (x, t)
Hassan HBID [email protected]
;
x(0) = x0 .
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Size Structured Models
It is well known that solving along the characteristic lines
transforms the problem into a renewal equation (see Webb, 1985
and Iannelli, 1994). This equation involves a variable delay τ (x, t)
which is the time spent by an individual to reach the state x at
time t and, therefore, is determined by the equation
! t
g (x(σ), σ) dσ = x
xb +
t−τ (x,t)
where x(σ) is the characteristic line through (x, t).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Size Structured Models
Let us suppose that the population is organized in different stages,
say juvenile (J) and adult (A) to simplify, and the transition of an
individual from (J) to (A) is made when its size reaches a
predetermined threshold value W ∗ > xb . Then the entrance to the
adult stage will depend on n(W ∗ , t). In turn, calculating this value
for t big enough along the corresponding characteristic line, we get
an expression in terms of the density n(xb , t̃), for some t̃ < t.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Size Structured Models
As a consequence, a variable delay τ (t) = t − t̃ defined by
! t
xb +
g (x(σ), σ) dσ = W ∗
t−τ (t)
will appear in the model. This threshold condition, in those cases
where the growth rate g depends on the population produces in
the model a state-dependent delay.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Size Structured Models
In Hbid, Sanchez, Bravo de La Parra (M3AS, 2006), we illustrate
this idea in different situations. We present a classical model by
Nisbet and Gurney that develops completely a formalism to deal
with insect populations in which the duration of their different
instars is variable and is determined by obtaining a predetermined
weight. We also applies these ideas to the modeling of cell
populations.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
As we have seen in the previous section, the delay is the time
employed by each individual to reach the threshold value of a
certain variable, which means in age-structured models that the
age of passing from one stage to the next is time-dependent.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
To illustrate these ideas, let us consider a population divided into
two stages, the juveniles and the adults, each one being structured
by the age in the stage. Denote by l (a, t) (resp. L(a, t)) the
density of juveniles (resp. adults) of age a at time t.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
We have the following equations:

∂l
∂l



 ∂t (a, t) + ∂a (a, t) = −µl (a)l (a, t)



 ∂L (a, t) + ∂L (a, t) = −µL (a)L(a, t)
∂t
∂a
(3)
where µl (a) (resp. µL (a)) is the age-dependent mortality rate of
juveniles (resp. adults). We point out the fact that the variable a
represents the age of juveniles in the function l (a, t) while it is the
age of the adults in the function L(a, t).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
The above two equations have to be supplemented by boundary
conditions and initial values. For the juvenile we have
l (0, t) = B(t) ;
l (a, 0) = 0
where B(t) represents the time density of newborns and the initial
condition expresses the fact that no juvenile is supposed to exist at
the time t = 0 (or, no one is able to survive beyond a certain
period of time).
The moving boundary is caused by the expression of B(t) in each
model. Let us begin with a simple example (see O. Arino and E.
Sánchez, Delays induced in population dynamics, In Mathematical Modelling of
Population Dynamics, Banach Center Publications, Vol.63 (2004) 9-46).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
We define J(t) to be the biomass of juveniles at time t, that is
! aM
J(t) :=
l (a, t) da
0
aM (0 < aM < +∞) being the maximum age of juveniles (that is,
individuals which have not reached the threshold value past time
aM , will die or never reach the next stage) and let us suppose that
B(t) = kJ(t) ,
Hassan HBID [email protected]
(k > 0).
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
The passage from the juvenile to the adult stage is described in
terms of the weight function of the juvenile, namely we assume
that
∂w
∂w
K
(a, t) +
(a, t) =
∂a
∂t
J(t) + C
where w (a, t) is the quantity of food eaten until time t by an
individual entering the juvenile stage a time units ago.
K > 0 is the quantity of food entering the species habitat per unit
of volume and per unit of time, which for simplicity is considered
to be a constant.
The constant C > 0 represents the food (converted into a number
of individuals) taken per unit of volume by consumers other than
juvenile stage.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
Limitation of food due to density is modeled in a possible way,
assuming that the quantity of food available is shared in equal
parts by all the individuals occupying the same space at time t.
Assuming that w (0, t) = w (a, 0) = 0 (which corresponds to the
absence of&juveniles at time t = 0), we arrive at
t
K
dσ We now introduce the threshold
w (a, t) = t−a J(σ)+C
condition: juveniles turn adult when the food index reaches a
prescribed value W ∗ > 0, that is at each time t, a∗ (t) is the
unique solution to the equation
! t
K
dσ = W ∗
(4)
J(σ)
+
C
∗
t−a (t)
and so
J(t) =
!
a∗ (t)
l (a, t) da.
(5)
0
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
Solving the equation in l from system (3) along the characteristic
lines starting from a = 0, we obtain
l (a, t) = B(t − a)e −
Ra
0
µl (s) ds
= kJ(t − a)e −
Ra
0
µl (s) ds
which, bearing in mind (5) and accepting for simplicity that
µl = 0, provides the following equation for J(t):
! t
J(σ) dσ.
J(t) = k
(6)
t−a∗ (t)
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
Age-Structured Models with Moving Boundary Conditions
Differentiating in (4) and (6) we reduce the problem to the
following system of state-delayed differential equations

'
(
J(t − a∗ (t)) + C

∗
#

J (t) = k J(t) − J(t − a (t))



J(t) + C




 (a∗ )# (t) =
J(t − a∗ (t)) − J(t)
J(t) + C
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
In the sequel, we will consider the two following classes of
differential equations with state dependent delay:
1)
 #
 x (t) = f (x(t − τ (t)))

(7)
τ # (t) = h(x(t), τ (t))
where f : R → R, h : R × R → R, and the delay is determined by a
differential equation.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Convenient assumptions are made on f and h in order to have
t − τ (t) increasing and the function τ (t) bounded, that is,
τ (t) ∈ [τ , τ ], where τ is the small delay and τ is the maximum
delay.
2)
(8)
x # (t) = g (x(t), x(t − r1 (xt )), ...., x(t − rn (xt )))
where g : Rn+1 → R, ri : C ([−M, 0]; R) → [0, M], for i = 1, ..., n,
and the delays are explicit functions of the solution x(t).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Equation (7) was discussed by :
! (O. Arino, M. L. Hbid and R. Bravo de la Parra.
A Mathematical Model of
Growth of Population of Fish in the Larval Stage: Density-Dependence Effects.
Math. Biosci. 150: 1-20,1998.)
!
to describe a renewal equation of a
population of fish.
(O. Arino, K. P. Hadeler, and M. L. Hbid.
Existence of Periodic Solutions for
Delay Differential Equation with State Dependent Delay. Journal of differential
equations, 144: 263-301, 1998.)
!
to show the existence of oscillating
periodic solutions.
Equation (8) is studied by Mallet-Paret et al. in (J. Mallet-Paret,
R. D. Nussbaum et P. Paraskevopoulos. Periodic Solutions for Functional
Differential Equations with Multiple State-Dependent Time Lags. Topological
Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, 3:
Hassan)HBID
STATE-DEPENDENT
DELAYSperiodic
ASSOCIATED TO THRESHO
to [email protected]
show the existence
of oscillating
101-162, 1994.
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Equations (7) and (8) can be written in the following “general
generic” formulation:
)
ẋ (t) = F (xt ) , t ≥ 0
x0 = ϕ ∈ C
(9)
where
F (ψ1 , ψ2 ) = (f (ψ1 (−ρ(ψ2 (0)))), h(ψ1 (0), ρ(ψ2 (0)))),
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
for all (ψ1 , ψ2 ) ∈ C ([−τ , 0]; R2 ), and ρ is the retraction
ρ : R → [τ , τ ] defined by:

if a ≥ τ
 τ
ρ(a) =
a
if a ∈ [τ , τ ]

τ
if a ≤ τ .
for equation (7) and
F (ϕ) = g (ϕ(0), ϕ(−r1 (ϕ)), ..., ϕ(−rn (ϕ))),
for equation (8).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
The purpose of this section is to study the aspect of semigroup
associated with the solutions of the following delay differential
equation:
 #
 x (t) = f (x (t − τ (t)))
τ # (t) = h (x (t) , τ (t))

(x0 , τ (0)) = (ϕ, τ ) ∈ C 0,1 ([−τ , 0]) × [τ , τ ] := C 0,1 × K
(10)
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
where f and h are two functions defined respectively from R into
R and from R × R into R, C 0,1 is the Banach space of Lipschitz
continuous functions defined from [−M, 0] to R, τ and τ are two
positive constants, and as usual xt denotes the history segment of
the solution x(t) at time t, xt (θ) = x(t + θ) for all θ ∈ [−τ , 0].
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
The functions f and h satisfy the following assumptions:
H1 : ”f and h are two functions locally Lipschitzians,”
H2 : ” there exists constants α and β such that, for each x ∈ R
|f (x)| ! α |x| + β, ”
H3 : ” i: |h (x, τ )| ! 1; ii: h (x, τ ) ≤ 0 and h (x, τ ) ≥ 0 , for each
(x, τ ) ∈ R × K .”
Hassan HBID [email protected]
(11)
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Assumptions H1 and H2 , which we will use throughout the paper
without further repetition, are introduced for the next reasons:
- H1 guaranteed the existence and the uniqueness of the solution of
the equation (10) (O. Arino, K. P. Hadeler, and M. L. Hbid. Existence of
Periodic Solutions for Delay Differential Equation with State Dependent Delay.
Journal of differential equations, 144: 263-301, 1998.),
- H2 serves to obtain an estimation of the solution of the equation
(10),
- H3 is imposed on h in view to obtain a frame of the delay τ (t)
between the two values τ and τ (ii), and to insure that the
function t (−→ t − τ (t) is increasing (i).
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Put T (t) (ϕ, τ ) = (xt , τ (t)) , for each (ϕ, τ ) ∈ C 0,1 × K , where
(x(·), τ (·)) is the solution of (10) with initial data (ϕ, τ ).
Remark : From the existence and the uniqueness of the solution
of (10) one deduces that the family (T (t))t"0 constitutes an
algebraic nonlinear semigroup on C 0,1 × K . However, the example
given in the proposition 3.1 of (M. Louihi, M. L. Hbid and O. Arino.
Semigroup Properties and the Crandall Liggett Approximation for a Class of
Differential Equations with Satate Dependent-Delays. Journal of Differential
Equations 181: 1-30, 2002. ) remains valid to prove that
t (→ xt (ϕ0 , τ ) for all τ ∈ K is not strongly continuous from [0, ∞]
into C 0,1 .
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Thus we are brought to consider the set E of strong continuity of
T (t), it is to say
E :=
+
*
(ϕ, τ ) ∈ C 0,1 × K : t (→ T (t) (ϕ, τ ) is cont. from R+ into C 0,1 × K
(12)
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Theorem
Suppose that H1 , H2 and H3 hold. Then, we have:
1.
*
+
E = (ϕ, τ ) ∈ C 1 × K : ϕ# (0) = f (ϕ (−τ ))
(13)
2. E is closed, invariable by T (t), for each t ≥ 0, and the
restriction of family of operators (T (t))t"0 to the set E,
noted again by (T (t))t"0 , is a strongly continuous semigroup.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Proposition
Assume that H1 , H2 and H3 are satisfied. If moreover f is of class
C 1 , then
a) the infinitesimal generator of the semigroup T (t) is the
operator Λ defined by:
,
D(Λ) = {(ϕ, τ ) ∈ C 2 × K : ϕ#− (0) = f (ϕ(−τ )) and ϕ##− (0) = L(ϕ, τ )}
Λ(ϕ, τ ) = (ϕ# , h(ϕ(0), τ )).
(14)
b) the closure of D(Λ) in C 1 × R, is equal to the set E.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
In this section we consider the next notations: ) · )1,γ is the norm,
equivalent to ) · )1 in C 1 , defined by:
)ϕ)1,γ = )ϕ)∞ +
1-ϕ# - , for each ϕ ∈ C 1
∞
γ
Vγ is the retraction on the ball of center 0 and radius 1, in relation
to the norm ) · )1,γ , defined by:
)
ϕ
if )ϕ)1,γ ! 1
Vγ =
ϕ
if not,
%ϕ%
1,γ
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Gk and Hk are the functions defined respectively on C 1 × K and
R × K by:
.
.ϕ/
/
Gk (ϕ, τ ) = f kVγ
(−τ ) , for each ϕ ∈ C 1 and all τ ∈ K .
k
(15)
/
.
x
(16)
Hk (x, y ) = h kv ( ), y , for each (x, y ) ∈ R × K .
k
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STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
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A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Ak is the operator defined by:
*
+
D(Ak ) = (ϕ, τ ) ∈ C 2 × K : ϕ#− (0) = Gk (ϕ, τ )
Ak (ϕ, τ ) = (ϕ# , H(ϕ(0), τ ).
(17)
E k designates the set
*
+
E k = (ϕ, τ ) ∈ C 1 × K : ϕ#− (0) = Gk (ϕ, τ ) .
(18)
,
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Jλ,k designates the operator Jλ,k = (I − λAk )−1 .
R is the function defined on C 1 × [−τ , 0] by
0
0 1
exp( θ ) &
R (ψ, θ) = λ λ
exp − λs ψ (θ) ds.
θ
Ultimately, one considers the set Bρ defined by
Bρ = {(ϕ, τ ) ∈ E : )ϕ)1 ≤ ρ}
(19)
.
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Theorem
Suppose that H1 , H2 and H3 hold. Then, for all T > 0, ρ > 0 and
t ∈ [0, T ], one has
Lim -Jnλ,k (ϕ, τ ) − T (t) (ϕ, τ )-1 = 0, for all (ϕ, τ ) ∈ Bρ
nλ→t
where Bρ is the set defined by (19), k = γ0 (ρ, T ) +
defined by: γ1 = max {α(ρ + βt) exp (αt) + β, ρ} .
Hassan HBID [email protected]
γ1 (ρ,T )
,
γ
γ1 are
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
A short review on state dependent delay deferential
equation
Theorem
Moreover:
)T (t) (ϕ1 , τ1 ) − T (t) (ϕ2 , τ2 ))1(ω) ! exp (ω0 t) )(ϕ1 , τ1 ) − (ϕ2 , τ2 ))1(ω)
for each (ϕ1 , τ1 ) , (ϕ2 , τ2 ) ∈ Bρ × K . With ω =
ω0 = 2lipk (h) + ω.
Hassan HBID [email protected]
max{α1 ,α2 }
1−α3
and
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO
Introduction
Size Structured Models
Age-Structured Models with Moving Boundary Conditions
A short review on state dependent delay deferential equation
References
!
Arino, Hbid, Bravo de La Parra, 1998
!
Arino, Hadeler, Hbid, 1998
!
Louihi, Hbid, Arino, 2001
!
Louihi, Hbid, 2006 and 2007
!
Hbid, Sanchez, Bravo de La Parra, 2006
!
Hbid, Louihi, Sanchez (in press)
Hassan HBID [email protected]
STATE-DEPENDENT DELAYS ASSOCIATED TO THRESHO