AN ADAPTATIVE MODEL FOR A MULTISTAGE STRUCTURED POPULATION UNDER FLUCTUATING ENVIRONMENT

. We consider a modiﬁed version of a mathematical model describing the dynamics of the European Grapevine Moth, studied by Ainseba, Picart and Thiery. The improvment consists in including adaptation at the larval stage. We establish well-posedness of the model under suitable hypothesis.

1. Introduction. Models with continuous age structure represent a special case of physiologically structured population models.These latter may incorprate other state variables like size , genetic trait and level of maturation of individuals.
Age structured models were first introduced by Sharp and Lotka [25], then Mc Kendrick and Von Foester [21] studied the following linear case Webb [31] analyzed this class of models using linear semigroup theory. Later on, Gurtin and Mac Camy [12] turned their attention to the nonlinear case. They considered the case where b and d become functions of the total population P (t) = +∞ 0 n(a, t)da.
We refer the reader to Iannelli [13] for mathematical analysis of nonlinear age structured models.
Slobodkin [26] found that age or size alone was not sufficient to characterize the physiological behavior of Daphhnia Obtuse population. Then a major change in populations analysis was introduced by Sinko and Streifer [27] with age and size structure.This class of models has been formulated and analyzed by Calsina, Saldana [7], Metz and Diekmann [22], to cite a few.
The goal of this paper is to present a modified version of a system describing the life cycle of the European Grapevine Moth, introduced in [2], and [1]. In contrast to the works in [2], and [1], where the trait is assumed to be constant, we investigate a model with phenotypical trait and age structure. The trait can vary at the pupae stage when insect changes its morphology, and adapt its strategies to maximize the fitness.
The paper is organized as follows:In the next section , we give the formulation of the model. Section 3 is devoted to existence of solutions and continuous dependence on initial values. In section 4, we discuss the achievements of the paper.
2. Description of the model. We divide the population of insects into five classes: eggs, larvae, pupae, female and male. In the first stage, females lay eggs. Eggs remain until natural death or emerge into larvae. After a few days, larvae die by natural death or evolves into pupae. Individuals stay approximately seven days before encloding into adult. For each category , we consider the chronological age of the individuals that we denotes a. The population changes over time t, and a physiologically trait denoted x.
The following model describes the dynamics of these populations, for a ∈ (0, L i ), t ≥ 0, x ∈ Ω, and i = e, l, f, m, c.
(1) The boundary conditions are defined by where σ denotes the sex ratio and t > 0.
The system is completed with initial conditions and The first equation of (1 ) gives the evolution of the eggs density. The second equation of (1 ) describes the temporal variation of the larvae density. The third equation of (1 ) gives the density of individuals at the pupae stage and takes into account the changes and the adaptation of the new generation. The fourth and last equations of (1 ) describes respectively the evolution of female and male densities.
Here L k is the maximum age for the k−stage. The quantity E corresponds to the climatic and environmental factors and it is time-dependent. The function µ k is the k-stage specific per capita mortality function. The total population at time t and trait xfor the k-stage is then defined by Because of the inter-individual competition among larvaes for food, we suppose that µ l depends on the total population P l (t, x) .The function β k denotes the kstage age specific transition function. The function v k represents the k-stage specific per capita growth function which depends on the physiological age. The inflow of the mutant newborns at the pupae stage is given by the integral operator where Ω is a bounded domain of R n .The quantity γ (x, y) , represents the probability that a larvae with trait y emerges with a new trait x, see [5], [8], and [9] for similar representation of mutation.
The main idea behind the model is that individuals of the pest population change their behavior , and adapt their dynamic at the pupae stage, where the insect changes morphologically and adapt its abilities and strategy to maximize its fitness, to delay emergence and resist to hostile environment.
Remark 1. Note that in the integral giving the newborns (2), we use 0 as a minimal value only for mathematical convenience. It may be replaced by a positive value.
3. Main result. We develop a solution determined by the method of characteristics. We formulate the problem as a vector integral equation. Then, we prove the existence and uniqueness of solutions using the Banach Contraction Principle. First, we introduce some notations. For every t > 0, and x ∈ Ω, we define  For k = e, l, f, m, Let where G e (u e (t, ., x)) (a) = −µ e (E(t), a) u e (t, a, x) − β e (E(t), a) u e (t, a, x) , For every t > 0, and x ∈ Ω, let H t, x, u l (t, ., . ) = Ω L l 0 β l (E(t), a) γ (x, y) u l (t, a, y) dady.

3.1.
Assumptions. To obtain the main results , we require the following assumptions is strictly positive and continuously differentiable with respect to a. In addition, there exists a positive constant L k v such that ∂v k ∂a uniformly with respect to t and a.Let are bounded, nonnegative. The function β f is lipschitz continuous with constant L f β with respect to the variables P f (t, x) , P m (t, x) .
(M) The mortality function µ l P l (t, x) , E(t), a is non negative and locally bounded. The mortality function µ k (E(t), a), is non negative and bounded for each k = e, f, m.
The mortality function of larvae stage is nonnegative, locally bounded , Lipschitz continuous with respect to the first variable.
The Pupae mortality µ c is nonnegative, continuous and bounded.
(K) the kernel γ(x, y) is a probability density function , it satisfies and is uniformly bounded in Ω × Ω by γ ∞ .

Existence of solutions.
For k ∈ {e, l, f, m} ,we define the characteristic curve through (t 0 , a 0 ) by the solution of the differential equation Assumptions on the function v k imply that there exists a unique solution be the characteristic curve through the origin. This curve separates the trajectories of individuals that were present at the initial time t = 0 from the trajectories of those individuals born after the initial time.
We define implicitly by the relation

Integrating this last equation, gives
for some initial time t * ≥ 0 . We distinguish two cases: i) If a 0 ≤ z(t 0 ), then τ k = τ (t 0 , a 0 ) > 0 and the solution is given by ii)If a 0 ≥ z(t 0 ), then we take τ k = 0 as the initial time and the solution is given by Definition 2. A vector u e , u l , u f , u m , u c is a solution of the system (1) up to time T > 0, with boundary and initial conditions (2), (3) and (4) , if For k = e, l, f, m, let L 1 0, L k ; R n be the Banach space of Lebesgue integrable function with the norm Let M > 0, be chosen such that We denote u = u e , u l , u f , u m , u c and define the closed subset in E by The space X is complete since it is a closed subset of E.
Remark 3. For simplicity of notations, except precision, we shall write We define the map K : X → X, Hs, x, u l (s, ., .) ds.
A fixed point to the map K corresponds a solution to the system (1) in the sense of definition (2).
Lemma 6. Let a = X k (t; τ, η), then i) the function a is differentiable with respect to τ , and ii) the function a is differentiable with respect to η, and A proof of this Lemma, can be found in [16] and [4] p.114.

3.2.2.
A priori estimates. We need some a priori estimates on u e , u l , u f , u m , u c .
Proof. a) We give the proof only for k = e, since the other cases are quite similar with the obviously similar estimates.
Similarly, to estimate I 2 + I 4 , we make the change of variables η = X e (s, t, a) = X e (s, τ e , 0) .
τe G e (u e (s, ., x)) (η) dηds + Therefore, we obtain that To estimate I 3 , we use the change of variables ζ = X e (0, t, a) , this gives that Hence Similarly, we get the other inequalities. b) since µ c is nonnegative, we have Let K i u e , u l , u f , u m , u c = u i for i ∈ {e, l, f, m, c} .
To show that K is a strict contraction, we derive some estimates.

Note that
Using the change of variables η = X e (s, t, a) = X e (s, τ e , 0) , we obtain that 0 G e (s, u e 1 (s, ., x)) (η) − G e (s, u e 2 (s, ., x)) (η) dηda+ Consequently The other cases are similar, so we omit the proof. By lemma 5, we obtain Proof. a) For T small enough, the operator K maps X into X. Indeed, from Lemma 7 , it follows that for T small enough. Similarly, we obtain that for T small enough. b) The operator K : X → X is a contraction operator. It follows from Lemma 8, that where γ is a positive constant. Note that T γ ∈ (0, 1) provided that T is small enough. Hence This proves that K is a contraction operator on X. u e 0 (X e (0, t, a) , x) − u e 0 (X e (0, t, a) , x) da Beginning by R 1 , from Lemma 5 and Lemma 7, we obtain that For R 4 , we have To estimate R 2 + R 3 , we make the change of variables η = X e (s, t, a) = X e (s, τ e , a). Lemma 5 yields We get Analogously to (a), we obtain the estimates We conclude that It follows that where Γ 1 is a positive constant depending on L v , T, and c i , i = 1.2.3.
According to Gronwall's inequality we conclude that It follows that   The damages are caused in larval stage where caterpillar may provoke entire loss of vinetage. The caterpillar enter its pupation and stays about seven days before emergence.
Chemical control has been extensively used to reduce the proliferation of the pest population. However, Insecticides can be a factor driving insects to acquire resistance, interpreted by behavioral, physiological change or genetic mutation. To avoid unfavorable conditions and limit the risk of mortality, the insect develop at the pupal stage one or several phenotypes , leading to appropriate timing of emergence. Note,this behavioral change is observed in Malaria Infection where mosquitoes have adapted survival strategies by shifting their bitting behavior, see [11].
In warmer regions, four generations may occur instead of three. Based on insecticides only, it is much harder to control the pest, and the use of intensive insecticides can produce serious human health consequences, destruction of predator-prey relationships and polluting disasters.
Hence, a need to get better understanding of the mechanisms of insects resistance is crucial.
The advantage of the present study is that it includes in the model endogeneous and exogeneous factors that can explain the observed differences within a cohort.
The validation of the model with experimental data is still in progress.