Spectral theory and time asymptotics of size-structured two-phase population models

This work provides a general spectral analysis of size-structured two-phase population models. Systematic functional analytic results are given. We deal first with the case of finite maximal size. We characterize the irreducibility of the corresponding $L^{1}$ semigroup in terms of properties of the different parameters of the system. We characterize also the spectral gap property of the semigroup. It turns out that the irreducibility of the semigroup implies the existence of the spectral gap. In particular, we provide a general criterion for asynchronous exponential growth. We show also how to deal with time asymptotics in case of lack of irreducibility. Finally, we extend the theory to the case of infinite maximal size.

The density of individuals in the active (resp. resting) stage of size s ∈ [0, m] at time t is denoted by u 1 (s, t) (resp. u 2 (s, t)) and m < ∞ is the maximal size that can be reached. For each stage, the individuals will grow respectively with the rate γ 1 and γ 2 . Furthermore, only proliferating individuals have a mortality rate denoted by µ and also can reproduce via the non-local integral recruitment term in (1). More precisely, β(s, y) gives the rate at which an individual of size y produces offspring of size s. Finally, the transition between the two lifestages is described by the size-dependent functions c 1 and c 2 .
In this paper, we deal also with the case of infinite maximal sizes m = ∞.
The natural functional space for such a system is X := L 1 (0, m) × L 1 (0, m).
Our approach of asynchronous exponential growth (see the definition below) of such a system is in the spirit of our previous work [19]. The analysis relies on two mathematical ingredients: (i) Check that the positive C 0 -semigroup {T (t)} t≥0 which governs this system has a spectral gap, i.e. ω ess < ω where ω and ω ess are respectively the type and the essential type of {T (t)} t≥0 . (Note that ω coincides with s(A), the spectral bound of its generator A).
(ii) Check that the C 0 -semigroup {T (t)} t≥0 is irreducible (see the different characterizations below).
Our assumptions are weaker than those given by J.Z Farkas and P. Hinow [10] and our construction is more systematic. We provide several new contributions. The most important ones are the following: 1. We show that the three conditions ∀ε ∈ (0, m), ε 0 m ε β(s, y)dyds > 0, inf supp c 1 = 0, and moreover, this characterizes the property that {T (t)} t≥0 has a spectral gap, (see Theorem 2.9 and Theorem 2.10). Note that here the irreducibility of {T (t)} t≥0 implies the presence of a spectral gap. It follows that under the conditions (3)-(4)-(5)) {T (t)} t≥0 has an asynchronous exponential growth, (see Theorem 2.14). 3. We show that once {T (t)} t≥0 has a spectral gap (i.e. once (6) is satisfied) the peripheral spectrum of A reduces to s(A), i.e. where D := (s(A) − A)P 0 , (see Theorem 2.15). A priori, if {T (t)} t≥0 is not irreducible then P 0 need not be one-dimensional and the nilpotent operator D need not be zero. 4. When {T (t)} t≥0 is not irreducible but has a spectral gap, it may happen that there exists a subspace of X which is invariant under {T (t)} t≥0 and on which {T (t)} t≥0 exhibits an asynchronous exponential growth, (see Theorem 2.16).
We deal also with the case m = ∞ which has never been dealt with before. Its analysis is quite different from the previous one: 5. The criterion of irreducibility is similar to the case m < +∞, (see Theorem 3.3). 6. However the criterion for the existence of a spectral gap is more involved. Indeed, {T (t)} t≥0 has a spectral gap provided that lim λ→s(B) (r σ refers to a spectral radius) where and B 3 u 1 u 2 = ∞ 0 β(·, y)u 1 (y)dy 0 , (see Theorem 3.4). Condition (7) is probably also necessary, see Remark 12. A priori, this condition is quite theoretical and not easy to check. But we also consider several situations of practical interest where the existence or the absence of the spectral gap property can be checked in an indirect way. Indeed: 7. We show first that the real spectrum of B is connected Some useful conjectures are also given, see Remark 11 and Remark 12.

Framework and hypotheses.
In order to analyse the problem described by (1)-(2), we define the Banach space We denote by X + the nonnegative cone of X and we introduce some hypotheses on the different parameters: Using (1), we define the operator We decompose B into three bounded operators: We are then concerned with the following Cauchy problem

Semigroup generation.
It is easy to prove: for every s ∈ [0, m]. In particular, s(A) = −∞ and for every (h 1 , h 2 ) ∈ X + , where supp (f ) refers to the support of a function f and inf supp (f ) is its lower bound.
Proof. Since B is bounded, it suffices to prove that A generates a contraction semigroup. We easily see that D(A) is densely defined in X . Moreover, for λ ∈ R, the range condition (λI − A)U = H, with U = (u 1 , u 2 ) and H = (h 1 , h 2 ) ∈ X , is straightforward since (u 1 , u 2 ) is given by (8), so . By definition, we have u i (0) = 0 and We multiply the latter equation by sign(u i (s)) then integrate between 0 and m. We get Any nonempty open set of the real line is a finite or countable union of disjoints open intervals (see [1] Theorem 3.11, p. 51) so and we get the dissipativity of A. Thus A generates a contraction C 0 -semigroup {T A (t)} t≥0 by Lumer-Phillips Theorem (see [22] Theorem 4.3, p. 14). Finally, as bounded perturbations of A, the 2.3. On positivity. The time asymptotics of {T A (t)} t≥0 is related to irreducibility arguments. We remind first some definitions and results about positive and irreducible operators. We denote by ·, · the duality pairing between X and X ′ . 1. For f ∈ X , the notation f > 0 means f ∈ X + and f = 0.
2. An operator O ∈ L(X ) is said to be positive if it leaves the positive cone X + invariant. We note this by O ≥ 0. 3. A C 0 -semigroup {T (t)} t≥0 on X is said to be positive if each operator T (t) is positive.

4.
A positive operator O ∈ L(X ) is said to be positivity improving if for every f ∈ X , f > 0 and every where | · | denotes the absolute value.
We recall that a C 0 -semigroup {T (t)} t≥0 on X with generator A is positive if and only if, for λ large enough, the resolvent operator (λI − A) −1 is positive (see e.g. [5], p. 165). We recall also that a positive C 0 -semigroup {T (t)} t≥0 on X with generator A is irreducible if and only if, for λ large enough, the resolvent operator (λI − A) −1 is positivity improving, if and only if, for λ large enough, there is no closed ideal of X (except X and {0}) which is invariant under (λ − A) −1 (see [20] C-III, Definition 3.1, p. 306).
Definition 2.4. For a closed operator A : D(A) ⊂ X → X , we denote by σ(A) its spectrum, ρ(A) its resolvent set and s(A) its spectral bound defined by We recall the following result which is a particular version of [27], Theorem 1.1.
Lemma 2.5. Let A be a resolvent positive operator in X and B ∈ L(X ) a positive operator. We have for every λ > s(A + B) and Here r σ (·) refers to the spectral radius. We introduce the following assumptions inf supp c 1 = 0, sup supp c 2 = m. 2. Now we suppose that the assumptions (11)-(12)- (13) are satisfied and we prove that (λI − A) −1 is positivity improving for λ large enough. Actually, since Using (9), we first see that Consequently we have Step 1: we start by proving that If H := (h 1 , 0), then it is clear that (15) is satisfied, by taking h = h 1 . If H := (0, h 1 ), then, using Lemma 2.1, we get By assumption (13), we have where |I| denotes the Lebesgue measure of an interval I. Thus and (15) is satisfied with h = c 2 h 2 . In any case it suffices to show that Step 2: now we prove that for every H :

Suppose by contradiction that
Using Lemma 2.1, we get holds, then we get a contradiction by definition of k and (16) is satisfied. So it remains to prove (17). Suppose by contradiction thatk ≥ k, then we get Moreover, since h 3 (y) > 0 a.e. y ∈ (k, m], we would get m k β(s, y)dy = 0, a.e. s ∈ [0, k] which contradicts Assumption (11).
We identify L 1 (ε, m) to the closed subspace of L 1 (0, m) of functions vanishing a.e. on (0, ε). Let λ > s(A), we want to prove that where the latter resolvent is given by (14).
It is also clear that Y is invariant under B 2 and consequently also under (λI −(A+B 2 )) −1 by using (9). It remains to prove that Y is invariant under B 3 . Let by Assumption (19). Thus Y is invariant under B 3 and consequently under (λI − (A + B 2 + B 3 )) −1 by using (9). Finally, Y is invariant under (λI − A) −1 by using (20).
(b) Suppose that (12) does not hold. Let λ > s(A) and We want to prove that We then get We want to prove that It remains to prove that it is also invariant under B 3 . But this is obvious since Remark 3. We note that in [10], the irreducibility is obtained under the assumptions (12)-(13) and the following one: In the continuous case, this latter assumption implies β(0, m) > 0, so active cells of maximal size can produce offspring of minimal size. This is not necessary in our statement. The biological meaning of (12)-(13) is the following: active cells of minimal size can become quiescent, and quiescent cells of maximal size can become active.

2.4.
On the spectral bound. We start with a useful Lemma 2.7. Let k > 0 a positive constant and define the so-called Volterra operator Then r σ (V ) = 0 and σ(V ) = {0}.
Proof. By induction, we can show that for every s ∈ [0, m], n ≥ 0 and h ∈ L 1 (0, m). We then get Consequently, We need also Proof. It is clear that by using Gelfand's formula.
Remark 4. Note that A has a compact resolvent (and consequently the spectrum of A is composed (at most) of isolated eigenvalues with finite algebraic multiplicity). This follows from the fact that the canonical injection i : We are ready to show Theorem 2.9. The spectrum of A + B 1 + B 2 is empty and consequently s( Proof. Let λ > −∞ and define the operators for every u ∈ D( where k 1 and k 2 are some positive constants and V 1 , V 2 are Volterra operators. We see that is a positive operator. The fact thatB 2 and (V 1 , V 2 ) T commute implies that using Lemma 2.8. Since V 1 and V 2 are Volterra operators, then Consequently, we have (10) and Lemma 2.1. Finally, since B 1 ≤ 0, then we get which ends the proof.
On the other hand, σ(A) need not be empty. Indeed: Proof. (24) is satisfied. By continuity argument, we can find

Suppose that
Let λ > s(A) then and
2. Now to prove the converse, we use the contraposition. Suppose that the assumption (24) is not satisfied, that is i.e. β(s, y) = 0, a.e. s < y. Suppose momentarily that there exists a Volterra operator V in L 1 (0, m) such that for every λ > −∞, where A 1 0 is given by (21). We would have r σ (λ − (A 1 0 + K)) −1 ≤ r σ (V ) = 0 and then By assumption, we know that where V 2 andB 2 are respectively defined by (22) and (23). The fact thatB 2 and (V, V 2 ) T commute implies that using Lemma 2.8 and since V and V 2 are Volterra operators. Consequently, we have for every λ > −∞ and by using (10). Finally we have Consequently it remains to prove (29). First, we know that MUSTAPHA MOKHTAR-KHARROUBI AND QUENTIN RICHARD using (9) for λ large enough. Let v ∈ L 1 + (0, m), then we have using (28), where k 1 is defined in (22). We then get and We then show by induction that where C > 0, for every v ∈ L 1 + (0, m), which proves (29).
Remark 5. Note that Assumption (24) which characterizes that s(A) > −∞ is much weaker than the assumptions in Theorem 2.6 which characterize the irreducibility of the semigroup.
Remark 6. Theorem 2.10 provides us with the existence of a real leading eigenvalue since s(A) ∈ σ(A) (see e.g. [5] Theorem 8.7, p. 202). In [10], the spectral gap is obtained under the assumption It is clear that (31) implies that (24) is satisfied.

2.5.
On asynchronous exponential growth. Let us remind some definitions and results about asynchronous exponential growth (see [9], [20] and [28] for the details).
Definition 2.11. Let L(X ) be the space of bounded linear operators on X and let K(X ) be the subspace of compact operators on X . The essential norm L ess of L ∈ L(X ) is given by Let {T (t)} t≥0 be a C 0 -semigroup on X with generator A : D(A) ⊂ X → X . The growth bound (or type) of {T (t)} t≥0 is given by and the essential growth bound (or essential type) of {T (t)} t≥0 is given by

Definition 2.12 (Asynchronous Exponential Growth). [28, Definition 2.2]
Let {T (t)} t≥0 be a C 0 -semigroup with infinitesimal generator A in the Banach space X . We say that {T (t)} t≥0 has asynchronous exponential growth with intrinsic growth constant λ 0 ∈ R if there exists a nonzero finite rank projection P 0 in X such that lim t→∞ e −λ0t T (t) = P 0 .
We are ready to give the main result of this subsection.
Proof. The semigroups {T A (t)} t≥0 and {T A+B1+B2 (t)} t≥0 are related by the Duhamel equation Since B 3 is a weakly compact operator then so is T A+B1+B2 (t − s)B 3 T A (s) for all s ≥ 0. It follows that the strong integral Proof. It follows from [5], Theorem 9.10, p. 223 and Theorem 9.11, p. 224.

Remark 7.
Note that, if {T A (t)} t≥0 is irreducible, then it has also a spectral gap, whence the asynchronous exponential growth of the semigroup. In this case, the spectral bound s(A) is algebraically simple (see e.g. [5], Theorem 9.10, p.223) and the nilpotent operator D that appears in (32) is actually zero. Whether the spectral bound could be semi-simple (i.e. a simple pole of the resolvent) when {T A (t)} t≥0 is not irreducible, is an open problem.
It may happen that {T A (t)} t≥0 is not irreducible but leaves invariant a subspace on which it is irreducible. This is our second result. Let Then Y is invariant under {T A (t)} t≥0 , and there exists a projectionP 0 of rank one, in Y such that lim Proof. Define the operator and As in Theorem 2.2, A Y generates a C 0 -semigroup {T AY (t)} t≥0 . Using the point 3.(a) of the proof of Theorem 2.6, with ε = b 1 , we know that is a closed ideal of X that is invariant under (λI − A) −1 for every λ > s(A). Then, using the point 3.(b) of the proof of Theorem 2.6, with k = b 2 , we can prove that Y is a closed ideal of X that is invariant under (λI − A) −1 for every λ > s(A). Consequently By means of (34) and by definition of b 1 , we see that Consequently, as for Theorem 2.6, A Y is irreducible and Therefore, as in Theorem 2.14, the semigroup {T AY (t)} t≥0 has the property of asynchronous exponential growth. Thus we get whereP 0 is a projection of rank one in Y.

Remark 8. Note that s(A Y ) ≤ s(A).
It is unclear whether the inequality is strict.
3. Models with unbounded sizes. In this section we consider the following model for s, t ≥ 0, with the Dirichlet boundary conditions (2). Let the Banach space We denote by X + the nonnegative cone of X . We now introduce some hypotheses on the different parameters: is weakly compact.

Using (35), we define
We decompose B into three operators: We are then concerned with the following Cauchy problem

Semigroup generation.
Lemma 3.1. Let H := (h 1 , h 2 ) ∈ X and λ ∈ R. The solution of is given by for every s ≥ 0. In particular, U := (u 1 , u 2 ) ∈ D(A) if and only if U ∈ X . Moreover, if H ∈ X + , then Remark 9. In all the sequel, for the simplicity of notations, we write symbolically (λ − A)U = H instead of (36) even if U need not belong to the domain of A. We will also use similar symbolic abbreviations in similar contexts.

An integration then leads to
for every finite m > 0. Hence  Proof. The proof is similar to that of Theorem 2.6.

3.3.
Asynchronous exponential growth. In contrast to the finite case, the asynchronous exponential growth needs an additional condition.
then the semigroup {T A (t)} t≥0 has asynchronous exponential growth.

s(A) > s(B).
As for the finite case, the weak compactness of B 3 implies that {T A (t)} t≥0 and {T B (t)} t≥0 have the same essential spectrum, and consequently the same essential type: ω ess (A) = ω ess (B) . Since then Thus {T A (t)} t≥0 exhibits a spectral gap and has asynchronous exponential growth since it is irreducible.

3.4.
Further spectral results. The object of this subsection is to show that the real spectrum of the differential operators appearing in B is connected and to estimate their spectral bounds. This is a useful step to check the existence or the absence of a spectral gap in some situations of practical interest without relying on the tricky condition (42), (see Subsection 3.5).

Now, define the operators
We give now more information on the spectrum of A 1 µ . Theorem 3.6. We have In particular

MUSTAPHA MOKHTAR-KHARROUBI AND QUENTIN RICHARD
if the latter exists.
Remark 11. We suspect that the spectra of A 1 µ , A 2 c2 and B are invariant by translation along the imaginary axis (and therefore are half-spaces), in the spirit of [15]. We conjecture also that their spectrum consist of essential spectrum only.
Remark 12. If σ(B) = σ ess (B) (see Remark 11), then the stability of the essential spectrum given in the proof of Theorem 3.4 implies that the essential type of {T A (t)} t≥0 is equal to s(B). In this case, the sufficient condition (42) for the existence of a spectral gap for {T A (t)} t≥0 is also necessary.
3.5. On the existence of the spectral gap. This subsection deals with different cases where one can check directly the existence or not of a spectral gap.
3.5.1. Sub (resp. super) conservative systems. We start with: Proof. The fact that s(B) < 0 is given by Theorem 3.7. To prove that s(A) ≥ 0, let the initial condition (u 0 1 , u 0 2 ) ∈ D(A) ∩ X + . An integration of (35) gives us d dt whence the result.
Remark 13. We note that in contrast to the case m < ∞, the irreducibility of the semigroup does not imply the existence of spectral gap since (50) and (51) are compatible with the irreducibility of the semigroup.