0.75cm
Modelling and control Lobesia botrana
Bedr’Eddine AINSEBA
IMB
Bordeaux 2 University and INRIA Futures Bordeaux Sud Ouest
28 décembre 2008
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
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1 Introduction
2 Biological control
3 Biological description
4 The mathematical model
5 Existence and uniqueness
6 The Parameters estimation
7 Optimal Control Strategies
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Introduction
Lobesia botrana is a major grapevine pest in Europe
The larvas eat vine buds during spring , unripe and ripe berries during
summer. The grapes become unfit for human consumption and wine
making.
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Introduction
Lobesia botrana is a major grapevine pest in Europe
The larvas perforations favorize diseases
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Biological control
Several tools are used to reduce population size of Lobesia
botrana
3 acting levels :
butterflies population by interfering on the mating process (use of
pheromones).
larvas population using biological pesticides (Cascade (4 to 5 weeks)).
eggs using biological pesticides (Cascade (4 to 5 weeks), BT (2
weeks))
Constraints
These techniques must be very well applied (in time), are expensive and are
not eco-aware.
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Biological control
Main objective
Modeling population dynamics to predict and control butterflies and eggs
population
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Biological description
4 developmental stages : egg (6 days) larva (25 days) pupa (6 days)
butterfly (10 days).
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Biological description
A one year cycle
3, 4 or 5 generations by year : 1,5 months in spring 1 month in summer 6
months in winter
the insect growth is temperature dependent food (sort of cultivar)
dependent
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The mathematical model
To properly describe the reproductive cycle of the European grapevine
moth, we consider an age and stage structured population. We denote by
u e , u l , u m , and u f respectively the age density distribution of individuals at
time t of egg, larva, male butterfly and female butterfly populations. The
total population for the k stage is then defined by :
Z
k
P (t) =
Lk
u k (t, a)da,
t ≥ 0,
0
where Lk is the maximum age for the k stage and k takes the value egg,
larva, male moth and female moth.
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The mathematical model
Let us denote E the vector corresponding to the climatic and
environmental factors, it is written as E = (T , H, R) where T is the
temperature factor, H the humidity factor and R the grape variety factor.
The vector E is time dependent. We specify that R is not the quantity of
food eaten by the larva but the species of the vine.
The functions me , ml , mm and mf are the stage and age specific per
capita mortality rates. To model the food competition between larvae, we
suppose that ml depends on the total population of larvae.
The functions β k , k = e, l , m, f correspond to the k-stage age-specific
transition rates. In particular, β e models the physiological change between
egg and larva stages. It is called the hatching rate. The function β l is the
transition between the larva and butterfly (male and female) stages and is
named the flying rate. We suppose that it is the same for the male and
female individual. The transition rate between the butterfly and egg stages
is modelled by the function β f . The common name is the age-specific per
capita birth rate.
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The mathematical model
Observations carried out on the breeding of Lobesia botrana of UMR Santé
Végétale of INRA, indicate that the population does not grow exponentially
but up to a value threshold determined by the carrying capacity of the
medium. The growth of the population size is not restricted to the food
quantity but to the total number of butterflies per unit of volume.
Therefore, the birth rate is dependent on the density of individual
butterflies.
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The mathematical model
The study of laboratory data shows the difference in growth between
individuals of a cohort. This phenomenon is also observed for many other
species. We often model the variability of growth by introducing growth
rate that depends on the physiological age. In the Lobesia botrana model,
v k , k = e, l , p represent the k-stage age-specific per capita growth rates.
This function depending on age and coupled with the transition rate allows
one to model a greater variability of growth inside a cohort.
All these demographic functions vary with the environnemental conditions
and the cultivar.
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The mathematical model

∂ e
∂
e
e
e
e


 ∂t u (t, a) + ∂a [v (E (t), a)u (t, a)] = −m (E (t), a)u (t, a)



−β e (E (t), a)u e (t, a),



 ∂ u l (t, a) + ∂ £v l (E (t), a)u l (t, a)¤ = −ml (P l (t), E (t), a)u l (t, a)
∂t
∂a

−β l (E (t), a)u l (t, a),


£
¤


∂ f
∂
f
f
f
f


∂t u (t, a) + ∂a v (E (t), a)u (t, a) = −m (E (t), a)u (t, a),


 ∂ u m (t, a) + ∂ [v m (E (t), a)u m (t, a)] = −mm (E (t), a)u m (t, a),
∂t
∂a

R Lf

v e (E (t), 0)u e (t, 0) = 0 β f (P f (t), P m (t), E (t), s)u f (t, s)ds,



v l (E (t), 0)u l (t, 0) = R Le β e (E (t), s)u e (t, s)ds,
R0 l
v f (E (t), 0)u f (t, 0) = L τ β l (E (t), s)u l (t, s)ds,



R0 Ll
 m
m
v (E (t), 0)u (t, 0) = 0 (1 − τ )β l (E (t), s)u l (t, s)ds,
(2)
where τ denotes the sex ratio and t > 0. The model becomes complete
with the initial conditions :
u k (0, a) = u0k (a),
a ∈ [0, Lk ],
k = e, l , f , m.
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(3)
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Existence and uniqueness
Ωk ,
Let us denote
k = e, l , f , m, the domain [0, T ] × [0, Lk ]. Using the
characteristic method the solution u e , u f and u m of system (1)-(2)-(3) are
implicitly given by

Rt
k (X k (0; a, t))e − 0 hk (E (s),X k (s;a,t))ds ,

u
a > z k (t),

 0
u k (t, a) =
(4)
Rt

k
k

 u k (Z k (0;a,t),0) e − Z k (0;a,t) h (E (s),X (s;a,t))ds , a ≤ z k (t),
v k (E (Z k (0;a,t)),0)
where hk , for k = e, f , m is given by

e
e
e
e

h (E (t), a) = (m + β + ∂a v )(E (t), a),
hf (E (t), a) = (mf + ∂a v f )(E (t), a),

 m
h (E (t), a) = (mm + ∂a v m )(E (t), a),
and X k (t; a0 , t0 ) is the characteristic curve through (a0 , t0 ).
( k
∂X (t)
= v k (E (t), a(t)),
∂t
X k (t0 ) = a0 > 0.
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(5)
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Existence and uniqueness
Since v k > 0, it follows that X k (t; a0 , t0 ) is strictly increasing. Therefore, a
unique inverse function Z k (a; a0 , t0 ) exists. Let z k (t) = X k (t; 0, 0) be the
characteristic curve through the origin for k = e, f and m.
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Existence and uniqueness
Some hypotheses for existence of solutions
H
The growth rates v k , for k equal e, l , f , m are nonnegative and bounded
functions
∀(t, a) ∈ Ω v k < v k (E (t), a) < v¯k ,
and are of class C 1 with respect to a,
∃Cv k > 0,
k
∂v k (E (t), a)
k∞ ≤ Cv k
∂a
H
The hatching rate β e (E (t), a) and the flying rate β l (E (t), a) are
nonnegative and bounded functions
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Existence and uniqueness
H
The fertility rate β f (P f , E (t), a) is a nonnegative and bounded function,
Lipschitz with respect to its first variable.
H
The mortality functions me (E (t), a), mf (E (t), a) and mm (E (t), a) are
nonnegative, locally bounded and satisfy
Z t
lim
mk (E (t), X k (s; a, t)ds = ∞, a > Z k (t),
a→Lk 0
Z t
mk (E (t), X k (s; a, t)ds = ∞, a ≤ Z k (t),
lim
a→Lk
Z k (0;a,t)
for k equal e, f , m.
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Existence and uniqueness
H
The larva mortality ml (P l , E (t), a) is nonnegative, locally bounded and
locally Lipschitz with respect to its first variable and
Z t
lim
ml (P l (t), E (t), X l (s; a, t)ds = ∞, a > Z l (t),
a→Ll 0
Z t
lim
ml (E (t), X l (s; a, t)ds = ∞, a ≤ Z l (t),
a→Ll
Z l (0;a,t)
H
The initial data u0k , for k equal e, l , f , m are nonegative.
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Existence and uniqueness
Theorem
Under the hypotheses H 1 - H 6, system (1)-(2)-(3) admits a unique
solution.
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Existence and uniqueness
A sketch of proof in L2
Let λ be a nonnegative constant ; we will prove that for λ sufficiently large
there exist a unique solution for the system in
u e (t, a) = e λt û e (t, a),
u l (t, a) = e λt û l (t, a) et u f (t, a) = e λt û f (t, a).
(6)
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Existence and uniqueness
A sketch of proof in L2
Consider the system

∂
∂ e
e
e
e
e
e


∂t ûi (t, a) + ∂a [v (E (t), a)ûi (t, a)] + λûi (t, a) = −m (E (t), a)ûi (t, a)




−β e (E (t), a)ûie (t, a),


£
¤

∂ l
∂
l
l
l
l
l
l



∂t ûi (t, a) + ∂a v (E (t), a)ûi (t, a) + λûi (t, a) = −m (Pi , E (t), a)ûi (t, a)




−β l (E (t), a)ûil (t, a),


£
¤

∂ f
∂
f
f
f



∂t ûi (t, a) + ∂a v (E (t), a)ûi (t, a) + λûi (t, a) =



−mf (E (t), a)ûif (t, a),
R
f
L
e
e
f
f
f

v (E (t), 0)ûi (t, 0) = 0 β (Pi (t), E (t), s)ûi (t, s)ds,

R
e

L


v l (E (t), 0)ûil (t, 0) = 0 β e (E (t), s)ûie (t, s)ds,



R

Ll

v f (E (t), 0)ûif (t, 0) = 0 β l (E (t), s)φi (t, s)ds,




e (a),

ûie (0, a) = û0,i




l (a),

ûil (0, a) = û0,i



 f
f (a).
ûi (0, a) = û0,i
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Existence and uniqueness
A sketch of proof in L2
where φi , for i = 1, 2 are given in L2 (Ωl ). By the characteristic method,
one can prove that this system admits a unique solution (û e , û l , û f ).
Lemma
Let ∧ be the operator
∧ : L2 (Ωl ) → L2 (Ωl )
φ → û l
where û l is the unique solution of (7). If λ is s.t.
(λ − Cv e )(λ − Cv l )(λ − Cv f ) >
Le kβ e kL∞ (Ωe ) Lf kβ f kL∞ (Ωf ) Ll kβ l kL∞ (Ωl )
8v l v e v f
,
then ∧ is a contraction.
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Existence and uniqueness
Proof 1 : Let φ = φ1 − φ2 and û l = û1l − û2l .
We multiply the first equation in (7) by û e , the second by û l and the third
one by û f and we integrate on Ωe , Ωl , Ωf . After some computations we
obtain,
Z
Z
1
( ∂t û e (t, a)2 + (λ + me (E (t), a) + β e (E (t), a))û e (t, a)2 )dadt =
Ωe 2
Z
Z
1
1 T e
e
2
v (E (t), 0)(û (t, 0)) dt −
∂a v e (E (t), a)û e (t, a)2 dadt,
2 0
2
e
Ω
1
( ∂t û l (t, a)2 + (λ + ml (P l (t), E (t), a) + β l (E (t), a))û l (t, a)2 )dadt =
Ωl 2
Z
Z
1 T l
1
l
2
v (E (t), 0)(û (t, 0)) dt −
∂a v l (E (t), a)û l (t, a)2 dadt,
2 0
2
l
Ω
Z
1
( ∂t û f (t, a)2 + (λ + mf (E (t), a))û f (t, a)2 )dadt =
Ωf 2
Z
Z
1
1 T f
v (E (t), 0)(û f (t, 0))2 dt −
∂a v f (E (t), a)û f (t, a)2 dadt,
2 0
2
e
Ω
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Existence and uniqueness
Thus
Z
Z
e
λ
1
2
Z
0
Ωe
T
2
(û (t, a)) dadt ≤ Cv e
(û e (t, a))2 dadt+
Ωe
ÃZ f
!2
L
1
β f (P f (t), E (t), s)û f (t, s)ds dt
v e (E (t), 0)
0
Z
Z
l
2
λ
(û (t, a)) dadt ≤ Cv l
(û l (t, a))2 dadt+
1
2
Ωl
Z
1
2
T
1
l
v (E (t), 0)
µZ
Ωl
Le
e
¶2
e
β (E (t), s)û (t, s)ds dt
Z
Z
f
2
λ
(û (t, a)) dadt ≤ Cv f
(û f (t, a))2 dadt+
0
Z
0
Ωf
T
1
f
v (E (t), 0)
0
ÃZ
Ωf
Ll
l
β (E (t), s)φ(t, s)ds
!2
dt
0
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Existence and uniqueness
Under H 1 - H 6 and using Cauchy Schwartz inequality we get,
Z
(λ −
Cve )
e 2
(û ) (t, a)dadt ≤
Ωe
Z
(λ − Cvl )
(û l )2 (t, a)dadt ≤
Ωl
(λ −
Lf kβ f k2L∞ (Ωf ) Z
2v e
Ωf
Z
e
e
2
L kβ kL∞ (Ωe )
2v l
Z
Cvf )
f 2
(û ) (t, a)dadt ≤
Ωf
(û f )2 (t, s)dsdt,
(û e )2 (t, s)dsdt,
Ωe
Ll kβ l k2L∞ (Ωl )
2v f
kφk2L2 (Ωl ) .
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Existence and uniqueness
(λ − Cve )(λ − Cvl )(λ − Cvf )kû l k2L2 (Ωl ) ≤ K kφk2L2 (Ωl ) ,
K=
Le kβ e k2L∞ (Ωe ) Lf kβ f k2L∞ (Ωf ) Ll kβ l k2L∞ (Ωl )
8v l v e v f
·
∧ is a contracting map if λ is large enough.
¤
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The Parameters estimation
We consider in this section experimental population dynamics obtained in
laboratory conditions. Thus we have,


E (t) = E0 , t > 0,





v k (E (t), a) = 1, k = e, l , f , m, (t, a) ∈ Ω,



mk (E (t), a) = 0, k = e, f , m, (t, a) ∈ Ω,

ml (P l (t), E (t), a) = 0, (t, a) ∈ Ω,





β k (E (t), a) = β k (a), k = e, l , (t, a) ∈ Ω,



β f (P f (t), P m (t), E (t), a) = β f (a), (t, a) ∈ Ω.
The vector E0 is (T0 , H0 , R0 ) which means that the environmental variables
are constant with time. The unit of age is the chronological age because
the measurements are taken at a one day regular frequency.
As referred in the introduction, we formulate three parameter estimation
problems. Then, an analysis is performed from both a continuous and a
discrete approach.
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The Parameters estimation
The hatching rate
The objective is to identify the hatching rate β e (a), ∀a ∈ [0, L]. We use
experimental data relating to the hatching dynamics obtained in climatic
chambers. This experiment starts with an identically old egg population
and consists to note the date of first appearance of the larva. We deduce
the spreading out of the hatching dynamics and the egg mean development
at 60% of moisture and 20˚C.
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The Parameters estimation
The hatching rate
This experiment can be modelled by the following PDE
 e
∂u
∂u e
e
e

 ∂t (t, a) + ∂a (t, a) = −β (a)u (t, a),
(3.1) u e (0, a) = u0e (a),

 e
u (t, 0) = 0,
where a ∈ [0, L] and t > 0. The hatching dynamic is taken into account by
the relation :
Z L
(3.2) qexp (t) =
β e (a)u e (t, a)da,
t > 0,
0
where qexp is the observed data. The boundary condition is null because
there is no introduction of new individuals.
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The Parameters estimation
The hatching rate, no regularisation vs regularisation
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1
2
3
4
5
6
7
8
9
10
Figure: These figures show the approximate (solid line) and the exact (dashed
dotted line (a), cross line (b)) hatching rate with respect to the age.
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
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The Parameters estimation
The flying rate
We would like to identify the function β l for a ∈ [0, L]. So, we use
experimental data on emergence dynamics. The experiment starts with an
identically old egg population and finishes when all individuals have became
butterflies. We then know the date of first appearance of the butterfly and
its sex. We deduce the spreading out of the flying dynamics and the mean
development of an egg to become a larva. This experiment is made under
stable conditions of 60% of moisture and 20˚C.
This information on the experiment enables to formulate the evolution
problem associated to the egg population growth until the end of the larva
stage.
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2008
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The Parameters estimation
The flying rate
 e
∂u e
∂u
e
e


∂t (t, a) + ∂a (t, a) = −β (a)u (t, a),


l
l
 ∂u (t, a) + ∂u (t, a) = −β l (a)u l (t, a),
(a, t) ∈ [0, L] × [0, T ],
(a, t) ∈ [0, L] × [0, T ],
∂a
(3.3) ∂te
e
l

u (0, a) = u0 (a), and u (0, a) = 0,
a ∈ [0, L],



u e (t, 0) = 0, and u l (t, 0) = R L β e (s)u e (t, s)ds, t > 0,
0
and the relation describing the emergence dynamic :
Z
(3.4) qexp (t) =
L
β l (s)u l (t, a)da,
t > 0,
0
where qexp is the experimental data.
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2008
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The Parameters estimation
Hatching and Flying Rates
1.0
1.6
0.9
1.4
0.8
1.2
0.7
1.0
0.6
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0.0
0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Figure: These figures show the approximate (solid line) and the exact (dashed
dotted line) transition rates with respect to the age.
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
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2008
33 / 51
The Parameters estimation
The flying rate when the hatching rate is given
(a)
(b)
1.0
0.8
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1.0
0.8
0.6
0.4
0.2
0.0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
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2008
34 / 51
The Parameters estimation
The hatching rate when the flying rate is given
(a)
(b)
1.0
0.8
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1.0
0.8
0.6
0.4
0.2
0.0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
35 / 51
The Parameters estimation
The birth rate
In this part, we determine the laying rate of a female according to her age.
We then consider the dynamics of laying obtained in laboratory fixed
conditions. These experiments use a female population which has the same
age. The butterflies are mated during 48h with a male. After that, the
experimenter notes the number of eggs laid by a female until her death.
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
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2008
36 / 51
The Parameters estimation
The birth rate
With this information, we deduce the mathematical problem for the
estimation of β f . The female population dynamics are modelled by the
equation :

∂ f
∂ f

(t, a) ∈ [0, L] × [0, T ],
 ∂t u (t, a) + ∂a u (t, a) = 0,
f
f
(3.7) u (0, a) = u0 (a),

 f
u (t, 0) = 0,
and the laying dynamics by :
Z
(3.8) qexp (t) =
L
β f (a)u f (t, a)da,
t > 0,
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
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2008
37 / 51
The Parameters estimation
The flying rate
The problem consists in identifying the birth rate β f satisfying the equation
(3.8). The explicit solution of the system (3.7) is u f (t, a) = u0f (a − t) for
all a > t. So, we seek the birth function satisfying the equation :
Z
(3.8) qexp (t) =
t
L
β f (a)u0f (a − t)da,
t > 0.
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
38 / 51
The Parameters estimation
The birth rate
(a)
(b)
1.0
2.5
0.8
2.0
0.6
1.5
0.4
1.0
0.2
0.5
0.0
0.0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Figure: These figures show the approximate (dotted line) and the exact (solid
line) birth rate with respect to the age.
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
39 / 51
Optimal Control Strategies
Birth Control
Reduce the size of the larvae population.
· Z T
Z T Z
2
Minv η
v (t)dt + µ
(
0
(2)
0
L
¸
l
2
u (t, a)da) dt
0
 e
e
∂u

(t, a) + ∂u
(t, a) = −β e (a)u e (t, a), t > 0 a ∈]0, L]

∂t
∂a
 l
l

∂u
∂u
l
l


t > 0 a ∈]0, L]

∂t (t, a) + ∂a (t, a) = −β (a)u (t, a),

f
f

∂u
∂u


 ∂t (t, a) + ∂a (t, a) = 0, t > 0 a ∈]0, L]
u i (0, a) = u0i (a), a ∈]0, L]

RL


u e (t, 0) = 0 f β f (a)u f (t, a)da − v (t), t > 0



R

u l (t, 0) = Le β e (a)u e (t, a)da, t > 0



R0L
 f
u (t, 0) = 0 l β l (a)u l (t, a)da, t > 0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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28 décembre
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2008
40 / 51
Optimal Control Strategies
The Optimality system
Z
T
Z
L
L(S) = J (v ) +
[∂t u e + ∂a u e + β e (a)e u] p e (t, a)dadt
0
0
Z T Z Lh
i
+
∂t u l + ∂a u l + β l (a)u l p l (t, a)dadt
0
0
Z T Z Lh
i
+
∂t u f + ∂a u f p f (t, a)dadt
0
0
¸
Z T·
Z L
e
f
f
+
u (t, 0) + v (t) −
β (s)u (t, s)ds he (t)dt
0
0
¸
Z T·
Z L
+
u l (t, 0) −
β e (s)u e (t, s)ds hl (t)dt
0
0
¸
Z L
Z T·
f
l
l
u (t, 0) −
+
β (s)u (t, s)ds hf (t)dt
0
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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28 décembre
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2008
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Optimal Control Strategies
The Optimality system

∂ e
∂ e

− ∂t
p (t, a) − ∂a
p (t, a) + β e (a)p e (t, a) = p l (t, 0)β e (a)


R


l (t, a) − ∂ p l (t, a) + β l (a)p l (t, a) = β l (a)p f (t, 0) − 2µ L u l (t, a)da

−∂
p
t
a

0


−∂ p f − ∂ p f = β f (a) p e (t,0)
t
a
1−v (t)
e (T , a) = p e (t, L) = 0

p




p l (T , a) = p l (t, L) = 0



p f (T , a) = p f (t, L) = 0
2ηv ∗ (t) = −p∗e (t, 0)
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
42 / 51
Optimal Control Strategies
Numerical simulations, One generation
1200
1000
800
600
400
200
0
0
2
4
6
pop oeuf ss traitement
avec traitement
larve
8
10
12
14
16
18
0
2
4
6
naissance oeuf ss traitement
avec traitement
control
8
10
12
14
16
18
1400
1200
1000
800
600
400
200
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
43 / 51
Optimal Control Strategies
Numerical simulations, Two generations
3500
3000
2500
2000
1500
1000
500
0
0
5
10
pop oeuf ss traitement
avec traitement
larve
15
20
25
30
35
40
0
5
10
15
naissance oeuf ss traitement
avec traitement
control
20
25
30
35
40
3000
2500
2000
1500
1000
500
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
44 / 51
Optimal Control Strategies
Pesticides degradation
∂v
= −rv + w
∂t
" Z
Minw η
0
T
Z
w 2 (t)dt + η
0
T
µZ
L
#
¶2
u l (t, a)da
dt
0

∂v


∂t = −rv + w


e

∂u
∂u e
e
e

t > 0 a ∈ (0, L)

∂t (t, a) + ∂a (t, a) = −β (a)u (t, a),


l
l

∂u
∂u
l
l


 ∂tf (t, a) + ∂af (t, a) = −β (a)u (t, a), t > 0 a ∈ (0, L)

 ∂u (t, a) + ∂u (t, a) = 0, t > 0 a ∈ (0, L)
∂t
∂a
i
i

u (0, a) = u0 (a), a ∈ (0, L)


R


e (t, 0) = Lf β f (a)u f (t, a)da − v (t),

u
t>0


R 0Le e


l
e

u (t, 0) = 0 β (a)u (t, a)da, t > 0



u f (t, 0) = R Ll β l (a)u l (t, a)da, t > 0
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Bordeaux Sud Ouest
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2008
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Optimal Control Strategies
Numerical results, Pesticides degradation
1600
1400
1200
1000
800
600
400
200
0
0
2
4
6
naissance oeuf ss traitement
avec traitement
control
8
10
12
14
16
18
0
2
4
quantite W
controle v
8
10
12
14
16
18
3000
2500
2000
1500
1000
500
0
6
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
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2008
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Optimal Control Strategies
Numerical results, Pesticides degradation, Two generations
12000
900
800
10000
700
8000
600
500
6000
400
4000
300
200
2000
100
0
0
0
5
10
15
naissance oeuf ss traitement
avec traitement
control
20
25
30
35
40
0
5
quantite W
controle v
10
15
20
25
30
35
40
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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andcontrol
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Bordeaux Sud Ouest
28 décembre
()
2008
47 / 51
Optimal Control Strategies
Structured Eggs Control
 e
∂u
∂u e
e
e
e

t > 0 a ∈ (0, L)

∂t (t, a) + ∂a (t, a) = −β (a)u (t, a) − v (t)u (t, a),

l
l

∂u
∂u
l
l


t > 0 a ∈ (0, L)

∂t (t, a) + ∂a (t, a) = −β (a)u (t, a),

f
f

∂u
∂u


 ∂t (t, a) + ∂a (t, a) = 0, t > 0 a ∈ (0, L)
u i (0, a) = u0i (a), a ∈ (0, L)

R Lf f

e
f


u (t, 0) = R 0 β (a)u (t, a)da, t > 0


L

u l (t, 0) = 0 e β e (a)u e (t, a)da, t > 0



RL
 f
u (t, 0) = 0 l β l (a)u l (t, a)da, t > 0
" Z
Minv η
0
T
Z
v 2 (t)dt + η
0
T
µZ
L
#
¶2
u l (t, a)da
dt
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
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Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
48 / 51
Optimal Control Strategies
Numerical results
1200
1000
800
600
400
200
0
0
2
4
6
pop oeuf ss traitement
avec traitement
larve
8
10
12
14
16
18
0
2
4
6
naissance oeuf ss traitement
avec traitement
control
8
10
12
14
16
18
1400
1200
1000
800
600
400
200
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
()
2008
49 / 51
Optimal Control Strategies
Numerical results, with pesticide degradation
1200
1000
800
600
400
200
0
0
2
4
6
pop oeuf ss traitement
avec traitement
larve
8
10
12
14
16
18
0
2
4
quantite W
controle v
8
10
12
14
16
18
6
5
4
3
2
1
0
6
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
Universityand
andcontrol
INRIA Lobesia
Futures botrana
Bordeaux Sud Ouest
28 décembre
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2008
50 / 51
Optimal Control Strategies
Butterfly control
Goal : Reduce Lobesia Botrana population by using pheromone diffusers.
The control act on the fecondity rate.
 e
e
∂u

(t, a) + ∂u
(t, a) = −β e (a)u e (t, a), t > 0 a ∈ (0, L)

∂t
∂a

l
l

∂u
∂u
l
l


t > 0 a ∈ (0, L)

∂t (t, a) + ∂a (t, a) = −β (a)u (t, a),

 ∂uf
∂u f
f
f


 ∂t (t, a) + ∂a (t, a) = −β (a)u (t, a), t > 0 a ∈ (0, L)
u i (0, a) = u0i (a), a ∈ (0, L)

RL


u e (t, 0) = 0 f (β f (a) + v (t))u f (t, a)da, t > 0



RL


u l (t, 0) = 0 e β e (a)u e (t, a)da, t > 0



RL
 f
u (t, 0) = 0 l β l (a)u l (t, a)da, t > 0
· Z
Minv η
0
T
Z
2
v (t)dt + µ
0
T
Z
(
L
¸
l
2
u (t, a)da) dt
0
Bedr’Eddine AINSEBA IMB Bordeaux 2Modelling
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Bordeaux Sud Ouest
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2008
51 / 51