Optimal control problem on insect pest populations

Applied Mathematics Letters 24 (2011) 1160–1164

Contents lists available at ScienceDirect

Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

Optimal control problem on insect pest populations Delphine Picart a,∗ , Bedr’Eddine Ainseba b , Fabio Milner a a

Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85287-1804, United States

b

University of Bordeaux, INRIA Bordeaux Sud Ouest EPI Anubis, 3 Place de la Victoire 33000 Bordeaux, France

article

abstract

info

Article history: Received 5 March 2010 Received in revised form 27 January 2011 Accepted 27 January 2011

In this article we present a model of insect infestation of grape vines and consider the optimal control of the pest through egg population removal. Existence and uniqueness of solutions are proved. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Optimal control Age-structured equations Population dynamics

1. Introduction The European Grapevine moth Lobesia botrana has been the most serious wine pest in Europe, North Africa and in many Asian countries since the end of the XVIII century [1–4]. Many interventions have been developed to control this pest, e.g. insecticides, insect growth regulators, and mating disruption [5,1,6,7]. A thorough description and modeling of this issue through age-structured deterministic population models was carried out in [8]. The well-posedness of the model was established and the three different control strategies just described were presented therein as optimal control mathematical problems. For best results egg pesticides are used to kill eggs as soon as they are laid by females. Their application must thus be planned during the egg laying dynamics [5,6]. Let ue , ul and uf denote, respectively, the age density of the egg, larval and female populations. Their dynamics under egg pesticide control is modeled by:

 ∂ ue ∂ ue  ( t , a) + (t , a) = −β e (a)ue (t , a) − me (a)ue (t , a), a ∈ [0, Le ]    ∂t ∂a   l ∂u ∂ ul (t , a) + (t , a) = −β l (a)ul (t , a) − ml (a)ul (t , a), a ∈ [0, Ll ]  ∂t ∂ a   f f    ∂ u (t , a) + ∂ u (t , a) = −mf (a)uf (t , a), a ∈ [0, Lf ] ∂t ∂a  ∫ Lf   ue (t , 0) = β f (a)uf (t , a)da − v(t ),    0  ∫ Le  ul (t , 0) = β e (a)ue (t , a)da,   0  ∫ Ll     uf ( t , 0 ) = β l (a)ul (t , a)da,

(1)

(2)

0

uk (0, a) = uk0 (a),

∗

a ∈ [0, Lk ], k = e, l, f ,

Corresponding author. Tel.: +1 480 727 7575. E-mail addresses: [email protected], [email protected] (D. Picart).

0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2011.01.043

(3)

D. Picart et al. / Applied Mathematics Letters 24 (2011) 1160–1164

1161

where v is the control function. Since the renewal condition of the egg population, the equation for ue (t , 0), models the egg laying dynamics in time, the control v then represents the quantity of newly-laid-eggs killed per unit time. This function must be chosen so that the boundary condition ue (t , 0) is non-negative, that is, v must be no larger than the number of newly-laid-eggs, as guaranteed by the definition of the compact set K (4) in which the optimal control is sought. The k stage mortality functions mk , for k equals to e, l, f , are non-negative, bounded functions which satisfy the condition a

∫

mk (s)ds = ∞,

lim

a→Lk

k = e, l, f .

0

The function β e models the transition function between the egg and the larval stages, whereas the function β l models the transition function between the larval and the female stages. The function β f corresponds to the birth function. The last three functions are non-negative and bounded, such that

β k ≤ β k (a) ≤ β¯ k ,

k = e, l, f .

We are looking for the solution of the following optimal control problem

[P] : Find min J (v), v∈K

where

 J (v) = η

T

∫

v 2 (t )dt + µ

T

∫

0

∫

0

2

Ll

ul (t , a)da

 dt  ,

0

with ul is given by (1)–(3), η and µ are two positive constants and the set of admissible solutions is defined by K = {g (t ) ∈ L∞ ([0, T ]), 0 < g ≤ g ≤ g¯ }.

(4)

The two terms in the functional in [P ] represent, respectively, the financial cost of performing the control and the financial loss due to the destruction of grapes by larvae developing from eggs that are not killed. Barbu and Iannelli studied in [9] the problem of optimal control with the nonlinear structured population model, the Gurtin–MacCamy model. The control was applied on the vital rate and not on the egg renewal condition as on the problem [P]. This article is concerned with the identification of the best possible rate of newly-laid-egg removal in order to minimize financial loss. Therefore, our model deals with the optimal control of the pest population by egg removal, and it finds the rate of removal that will lead to the lowest cost to the producer. In a forthcoming paper we shall address the issue of how close to this optimal protocol one can get with real-life products (e.g. ovicides and larvicides). In Section 2 we compute upper and lower bounds of the control and study the well-posedness of [P]. The optimality conditions are explicitly given in Section 3, and some numerical examples and conclusions are given in Section 4. 2. Existence of the control v In this section, we start by computing the bounds of the control and then we prove, using minimizing sequences, the well-posedness of [P]. To conserve the positivity of the biological systems (1)–(3), the control v has to satisfy some conditions. The total number of newborns defined in (2) cannot be larger than the total number of newborns without control 0 ≤ u (t , 0) ≤ e

∫

Lf

β f (a)uf (t , a)da,

(5)

0

and is non-negative for all time. Let v and v¯ be the lower and the upper bounds of the control v , the first equation of (2) is bounded by

βf

∫

Lf

uf (t , a)da − v¯ ≤ ue (t , 0) ≤ β¯ f

0

∫

Lf

uf (t , a)da − v,

(6)

0

for all t. Let P be the total number of individuals in the population, P (t ) = P e (t ) + P l (t ) + P f (t ), where P k (t ) =

Lk

∫

uk (t , a)da,

k = e, l, f .

0

Integration of each differential equation in (1) over its age range gives the following equations

∂t P k (t ) + uk (t , Lk ) − uk (t , 0) = −

Lk

∫

β k (a)uk (t , a)da − 0

Lk

∫

mk (a)uk (t , a)da, 0

1162

D. Picart et al. / Applied Mathematics Letters 24 (2011) 1160–1164

for the egg and larval stages (k = e, l) and

∂t P f (t ) + uf (t , Lf ) − uf (t , 0) = −

Lf

∫

mf (a)uf (t , a)da. 0

By summing these three equations and neglecting the negative terms we get Lf

∫

∂t P (t ) ≤

β f (s)uf (t , s)ds ≤ β¯ f P (t ). 0

We apply the Gronwall Lemma on this last inequality to get an estimate on the total number of female P f (t ) ≤ P (t ) ≤ P (0)eβ T , ¯f

(7)

useful to deduce, with (5) and (6) inequalities, the upper and lower bounds of the control

v¯ = β f P (0)eβ T , ¯f

v = 0.

Now, let d be the lower bound of the cost function J (v) and {vn }n , with n a non-null integer, be a minimizing sequence of K such that 1

d < J (vn ) ≤ d +

n

.

The sequence {vn }n is bounded in L2 ([0, T ]) space, as a consequence there exists a subsequence, named {vnk }k , that converges weakly to the limit v ∗ of L2 ([0, T ]). The cost function defined in {vn } is dependent of the total number of larvae that is of the larval density function. Using the method of characteristics on the systems (1)–(2)–(3) with {vnk }k , we get explicit equations to compute the density functions of each populations,

 ujnk

j

u0 ( a − t ) e −

(t , a) =

t

u (t − a, 0)e j

β j (s)+mj (s)ds , a j − 0 β +mj (s)(s)ds

a>t

0

,

a ≤ t,

for a ∈ [0, Lj ], t ∈ [0, T ] and j = e, l, f . These sequences are bounded in L2 ([0, Lj ] × [0, T ]) space for j equals to e, l, f . f f We then can extract subsequences, respectively noted {˜uenk }k , {˜ulnk }k and {˜unk }k that converge weakly to ue∗ , ul∗ and u∗ in L2 ([0, Lj ] × [0, T ]) for j equals to e, l, f . The function of the total number of eggs, larvae and female defined by Pnj k

(t ) =

Lj

∫

u˜ jnk (t , s)ds,

0

j = e, l , f ,

is bounded. As computed in (7), the first derivative of the same functions satisfy the following inequalities

∂

j t Pnk

(t ) ≤

˜ Lj

∫ 0

˜

˜

β j (s)˜ujnk (t , s)ds,

j = e, l, f ,

˜j = f , e, l, f

f

which prove that are all bounded. As a consequence, the functions Pnek , Pnl k and Pnk converge uniformly to P∗e , P∗l , P∗ , that is, by the uniqueness of the limit, to P∗l =

Ll

∫

ul∗ (t , s)ds. 0

Finally, the cost function converges to

η

∫

−−−−→ η

∫

J (vnk ) =

T

(vnk )2 (t )dt + µ 0

k→+∞

T

(v ) (t )dt + µ ∗ 2

0

T

∫

∫

0 T

(P˜nl k (t ))2 dt (P∗l (t ))2 dt = J (v ∗ ) = d,

0

and the problem [P] admits at least one optimum.

D. Picart et al. / Applied Mathematics Letters 24 (2011) 1160–1164

1163

3. Optimality conditions Here, we give the optimality conditions for the problem [P] via the study of the Lagrangian. Let pe , pl and pf be the dual variables. These functions satisfy the adjoint problem

 ∂ ∂  − pe (t , a) − pe (t , a) + β e (a)pe (t , a) + me (a)pe − pl (t , 0)β e (a) = 0,   ∂t ∂a  ∫ Ll     l l −∂ p − ∂ p + β l pl + ml pl − pf (t , 0)β l (a) + 2µ ul da = 0, t

a

∂   − pf (t , a) −    ∂  k t   p (T , a) = 0, pk (t , Le ) = 0,

0

∂ f p (t , a) + mf (a)pf (t , a) − pe (t , 0)β f (a) = 0, ∂a k = e, l, f , k = e, l , f .

This system admits a unique solution [10] which is given below

∫ T t e e   β e (a)pl (s, 0)e− s (β +m )(τ )dτ ds, a ≤ t ,  e t p (t , a) = ∫ Le a e  e   β e (s)pl (t , 0)e− s (β +m )(τ )dτ ds, a > t ,

(8)

 ∫  ∫ Ll T t l  l l f l   β (a)p (s, 0) + 2µ u (s, x)dx e− s (β +m )(τ )dτ ds, a ≤ t ,   t 0  pl (t , a) = ∫ l  ∫ L Ll  a l  l l f l   β (s)p (t , 0) + 2µ u (s + t − a, x)dx e− s (β +m )(τ )dτ ds, a > t , 

(9)

a

a

0

∫ T t f   β f (a)pe (s, 0)e− s m (τ )dτ ds, a ≤ t ,  pf (t , a) = ∫t Lf a f    β f (s)pe (t , 0)e− s m (τ )dτ ds, a > t .

(10)

a

We remark that the dual variables are independent of the control v making the proof of existence of a unique solution to [P] is somewhat easier. Theorem 1. The problem [P] admits a unique optimum given by 2ηv ∗ (t ) = −pe (t , 0), where pe (t , 0) is given by (8)–(10) and satisfies

−2ηβ f P (0)eβ

¯f T

≤ pe (t , 0) ≤ 0,

thus guaranteeing that v ∗ makes the lower bound in (6) non-negative. 4. Conclusions In this paper, we give proofs of existence and uniqueness of solution for problem [P ], with quadratic dependence of the cost functional on both the control and the larval population size. The reason for the squares is technical, but the existence and uniqueness of an optimal control can be established using the same techniques for linear dependence on the larval population size,

J ′ (v) =

∫ 0

T

(v(t ))2 dt +

T

∫ 0

Ll

∫

ul (t , a)dadt , 0

except that the dual problem becomes

 ∂ ∂  − pe (t , a) − pe (t , a) + β e (a)pe (t , a) + me (a)pe − pl (t , 0)β e (a) = 0,   ∂a  ∂t l   −∂t p − ∂a pl + β l pl + ml pl − pf (t , 0)β l (a) + µ = 0, ∂ ∂ − pf (t , a) − pf (t , a) + mf (a)pf (t , a) − pe (t , 0)β f (a) = 0,   ∂a  k ∂t    pk (T , ae) = 0, k = e, l, f , p (t , L ) = 0, k = e, l, f .

1164

D. Picart et al. / Applied Mathematics Letters 24 (2011) 1160–1164

However, the theory used does require quadratic dependence on the control function and we will need to give a meaningful relation between that quadratic term and the cost of the ovicide or the other real-life product used. In this paper we are not modeling explicitly the dependence of the financial cost on the amount of ovicide used. References [1] D. Esmenjaud, S. Kreiter, M. Martinez, R. Sforza, D. Thiéry, M. Van Helden, M. Yvon, Ravageurs de la vigne, Éditions Féret, Bordeaux, 2008. [2] J. Moreau, A. Richard, B. Benrey, D. Thiéry, Host plant cultivar of the grapevine moth Lobesia botrana affects the life history traits of an egg parasitoid, Biological Control 50 (2009) 117–122. [3] D. Thiéry, J. Moreau, Relative performance of European grapevine moth (Lobesia botrana) on grapes and other hosts, Oecologia 143 (2005) 548–557. [4] M.E. Tzanakakis, B.I. Katsoyiannos, Insect pests of fruit trees and vines, Agrotypos, AE, Athens, Greece, 2003, pp. 38–44. [5] P.J. Charmillot, D. Pasquier, C. Salamin, F. Briande, Efficacité larvicide et ovicide sur les vers de la grappe Lobesia botrana et Eupoecilia ambiguella de différents insecticides appliqués par trempage des grappes, Revue Suisse de Viticulture Arboriculture Horticulture 38 (2006) 289–295. [6] X. Sun, B.A. Barrett, Fecundity and fertility changes in adult codling moth (Lepidoptera: Tortricidae) exposed to surfaces treated with tebufenozide and methoxyfenozide, Journal of Economic Entomology 92 (1999) 1039–1044. [7] V.A. Vassiliou, Control of Lobesia botrana (Lepidoptera: Tortricidae) in vineyards in Cyprus using the mating disruption technique, Crop Protection 28 (2009) 145–150. [8] D. Picart, Modélisation et estimation des paramètres liés au succès reproducteur d’un ravageur de la vigne (Lobesia botrana DEN. & SCHIFF), Ph.D. Thesis, No. 3772 University of Bordeaux, 2009. [9] V. Barbu, M. Iannelli, Optimal control of population dynamics, Journal of Optimization Theory and Applications 102 (1999) 1–14. [10] B. Ainseba, D. Picart, D. Thiéry, An innovative multistage physiologically structured population model to understand the European Grapevine moth dynamics, Preprint.