ON A MATHEMATICAL MODEL WITH NON-COMPACT BOUNDARY CONDITIONS DESCRIBING BACTERIAL POPULATION (II)

. This work is a natural continuation of an earlier one [1] in which a mathematical model has been studied. This model is based on maturation- velocity structured bacterial population. The bacterial mitosis is mathematically described by a non-compact boundary condition. We investigate the spectral properties of the generated semigroup and we give an explicit estimation of the bound of its inﬁnitesimal generator.


1.
Introduction. In this work, we continue our investigation already started in [1]. This study concerns a mathematical model describing a maturation-velocity structured bacterial population. Each bacterium is distinguished by its degree of maturity µ and its maturation velocity v. The degree of maturity of a daughter bacterium is µ = 0 while that of a mother bacterium is µ = 1. Between birth and division, the degree of maturity of each bacterium is 0 < µ < 1. As each bacterium may not become less mature, its maturation velocity v must be positive (i.e., 0 < v < ∞). The bacterial density f = f (t, µ, v), with respect to the degree of maturity µ and the maturation velocity v, fulfils ∂f ∂t where, r(µ, v, v ) (resp. s(µ, v , v)) stands for the transition rate at which bacteria change their velocities from v to v (resp. from v to v ). During mitosis, there may be a correlation τ := τ (v, v ) between the maturation velocity v of a mother bacterium and that of its daughter v. If α 0 denotes the average number of bacteria daughter viable per mitotic, then this correlation (called Transition Rule) is mathematically described by The model (1)- (2) was proposed in [9] and numerically treated in [5]. In our knowledge, the first theoretical approach of the model (1)-(2) is due to [6]. One have proved that this model is governed by a contractive strongly continuous semigroup provided that the average number of daughter bacteria is less than 1 (i.e., α < 1).
The most interesting case, α 1 (corresponding to an increasing number of daughter bacteria), was studied in [2]. We have then proved that the model (1)-(2) is governed by a strongly continuous semigroup.
During mitosis, there also may be a total inheritance of the maturation velocity between a mother bacterium and its daughters (i.e., τ (v, v ) = δ v (v )). If β 0 denotes the average number of bacteria daughter viable per mitotic, then this inheritance (called Perfect Memory Rule) is mathematically described by We have proved (see [3]) that the model (1),-(3) is governed by a contractive strongly continuous semigroup if and only if β < 1. In other words, there are no solutions for the most interesting case β 1 corresponding to an increasing number of daughter bacteria.
In most observed mitosis, the bacterial population is divided into two subpopulations. The first one obeys to Transition Rule described by (2) while the second one obeys to Perfect Memory Rule described by (3). We are then facing a third biological rule mathematically described by The full model (1),-(4) has recently been the subject of a mathematical study (see [1]). We have then proved that this model is governed by a strongly continuous semigroup U α,β = (U α,β (t)) t 0 ; this was our first challenge. For more explanations, we refer to [1] and the references therein. (See also [7] for another mathematical background slightly different than that used in [1]). Now, our new challenge is to prove that the bacterial population, described by the full model (1),-(4), possesses "Asynchronous Growth Property". This property describes the bacterial profile whose privileged direction is mathematically interpreted by an eigenvector corresponding to the leading eigenvalue. This is what the biologist observes in his laboratory.
It is well known that strict inequalities, such as (5), are too hard to prove and therefore adequate strategies are needed.
Henceforth, we are looking for δ such that Our computations have shown that the most suitable δ is and for which (6) can be considered instead of (5) ; that is our strategy.
In this work, we focus our attention only on the first inequality of (6), that is to say that, ω 0 (U α,β ) > −s. We then start by preparing the necessary matter and putting all the relevant hypotheses on the kernel of correlation τ := τ (·, ·) and on both transition rates s and r. Next, we characterize the type ω 0 (T α,β ) of the unperturbed semigroup T α,β = (T α,β (t)) t 0 (i.e., s = r = 0) as the unique solution of a certain characteristic equation. We also prove that both semigroups T α,β = (T α,β (t)) t 0 and U α,β = (U α,β (t)) t 0 are ordered. At the end of this work, we prove the desired strict inequality (see Theorem 4). We end this work by some useful Remarks.
Regarding to the second inequality of (6) (i.e., −s ω ess (U α,β )), this one is clearly hardest than the first one and needs to be proved separately. Its proof needs different technics than those used in this works and it will be the aim of [4]. Finally, note the novelty of this work. 2. Full model (1)-(4). This section deals with useful results for the sequel. Let then Y 1 be the following trace space Until the end of this work, both average numbers α 0 and β 0, of daughter bacteria viable per mitosis, are assumed to be fixed unless stated otherwise. The correlation kernel τ := τ (·, ·), between the maturation velocity v 0 of a mother bacterium and that of its daughter bacteria v 0, is also assumed to be fixed and likely to fulfill the following hypotheses Let L α,β be the following linear operator whose useful properties are given by Lemma 1. Let α 0 and β 0. If (H 1 τ ) holds, then L α,β is a bounded linear operator from Y 1 into itself satisfying, where, I ω denotes the characteristic operator of the subset (ω, ∞).
3. Stability properties. The aim of this section is to prove two stability results of the unperturbed semigroup T α,β = (T α,β (t)) t 0 . The first one concerns the case α = 0 while the second one concerns the case α > 0. The case α = 0 can be formulated as follows.
Proof. This follows from Theorem 1.
Next, suppose that τ = 0. Let ϕ ∈ L 1 and let λ 0. Since (18), we can write that where, For convenience, we divide the rest of the proof into several steps.
Step I. (Computation of A λ ).
Using the explicit form (20), it follows that Obviously, L α,β,λ is a bounded linear operator, from Y 1 into itself, because of the boundedness of the linear operator L α,β (Lemma 1). Hence, B λ becomes In order to improve (30), the norm L α,β,λ L(Y 1 ) needs to be estimated.
Step III. (Estimation of L α,β,λ ). Let ψ ∈ Y 1 . Observe that (29) can, by virtue of (8), be written as As the norm of L 0,β,λ is obviously less than β, then On the other hand, using (31) we can write That is to say that, However, for almost all v ∈ (0, ∞), we have which we put into (32), to finally get the desired estimation Step IV. (Conclusion).
Corollary 2. Suppose that (H 1 τ ) holds and let α > 0 and 0 β < 1 be such that where, C t is a finite constant.

MOHAMED BOULANOUAR
Proof. Let t 0. For all ϕ ∈ L 1 , we can write that 4. Lattice property. This section is devoted to some lattice properties of the semigroups T α,β = (T α,β (t)) t 0 and V α,β = (V α,β (t)) t 0 and U α,β = (U α,β (t)) t 0 whose existence is already proved in Lemma 2. So, let us put and let us start with the following result.
5. Spectral properties. The aim of this section is to estimate the type ω 0 (U α,β ), of the full semigroup U α,β = (U α,β (t)) t 0 , defined by ω 0 (U α,β ) := lim Obviously, (52) needs the explicit form of U α,β = (U α,β (t)) t 0 which is unfortunately unavailable. In order to overcome this difficulty, we firstly prove that the spectral bound s (T α,β ), of the generator T α,β defined by is the unique solution of a certain characteristic equation (see the proof of Theorem 4). We then conclude the desired estimation of the type ω 0 (U α,β ) through both (43) and (47). Notice, by the way, that we have Until now, our functional framework L 1 was always assumed to be a Banach space over the real field. However, in this section, L 1 will sometimes be assumed to be over the complex field. The transition between L 1 over the real field and its complexification is well known. Therefore, in the sequel we will not distinguish between L 1 over the real field and its complexification.
Let L α,β,λ be the following linear operator As we are going to see, the operator L α,β,λ will play a crucial role in this spectral study and therefore, it is natural to prove all its useful properties. We let, in the sequel, C + := λ ∈ C : Re(λ) 0 . The first useful property of L α,β,λ is given as follows.

ON A MATHEMATICAL MODEL 263
Firstly, both Steps I and II obviously yield that ϕ ∈ D (T α,β ). Next, from (59), we can write that which completes the proof of the desired (60).
To prove more interesting properties of the operator L α,β,λ , we put Remark 2. The hypothesis ( H 1 τ ) can also be formulated as follows So, if ( H 1 τ ) holds then Hence, all results of [1], related to (H 1 τ ), still true whenever ( H 1 τ ) holds true. The first consequence of ( H 1 τ ) is given by the following result. Proof. Let λ 0 and let ψ ∈ Y 1 . Using (55), we can write that where, τ ∞ is defined by (70). However,
The second consequence of ( H 1 τ ) is given by the following result.
Step I. Let η 0 and let ψ ∈ B 0 (B 0 denotes the unit closed ball of Y 1 ). Obviously, (55) shows that L α,β,η is a positive operator because of (H 2 τ ). Accordingly, for almost all v ∈ (0, ∞), we can write that which can be written as where, τ ∞ is defined by (70). Therefore, for all T > 0.
As we have pointed out in the beginning of this section, our aim is to estimate the type ω 0 (U α,β ) of the full semigroup U α,β = (U α,β (t)) t 0 . So, this one can now be announced as follows then ω 0 (U α,β ) > −s (87) where, L α,β, s−s and s and s are respectively defined by (55), (17) and (7).
Proof. Firstly, let us put λ s := s − s. So, due to (H 1 s ) we get that 0 λ s < ∞. For convenience, we divide the rest of the proof into several steps.
Remark 4. The unique role of the hypothesis (H 2 τ ) is to insure the irreducibility of the integral operator L 1,0 defined by (55). Accordingly, this hypothesis can be weakened without losing any results of this work. which is obviously less strong than (H 2 s ). Remark 6. Suppose that the kernel τ can be decomposed as follows That is to say that, both hypotheses (H 1 τ ) and ( H 1 τ ) are the same. Furthermore, (55) becomes which is obviously a rank one operator into L 1 .

Remark 7.
We point out that an addendum was attached to [2]. denotes the number of all bacteria at time t 0. Even so, we claim that all results of this work can easily be extended to L p (Ω) (p > 1). It then suffices to update all hypotheses to the desired context L p (Ω) (p > 1).