TIME ASYMPTOTICS OF STRUCTURED POPULATIONS WITH DIFFUSION AND DYNAMIC BOUNDARY CONDITIONS

. This work revisits and extends in various directions a work by J.Z. Farkas and P. Hinow (Math. Biosc and Eng, 8 (2011) 503-513) on struc- tured populations models (with bounded sizes) with diﬀusion and generalized Wentzell boundary conditions. In particular, we provide ﬁrst a self-contained L 1 generation theory making explicit the domain of the generator. By using Hopf maximum principle, we show that the semigroup is always irreducible re- gardless of the reproduction function. By using weak compactness arguments, we show ﬁrst a stability result of the essential type and then deduce that the semigroup has a spectral gap and consequently the asynchronous exponential growth property. Finally, we show how to extend this theory to models with arbitrary sizes and point out an open problem pertaining to this extension.


1.
Introduction. Structured population models are widely discussed in the literature on population dynamics (see e.g. [17,19]). A model with size-structure appeared in a work by J.W. Sinko and W. Streifer [30] (see also [36] and the references therein). The introduction of spatial diffusion in population biology goes back to A. Kolmogorov I. Petrovskii and N. Piscunov [16] and J.G. Skellam [31]. We refer to the book by J.D. Murray [23] for a survey of reaction-diffusion equations in biology. Later, R. Waldstätter, K.P. Hadeler and G. Greiner [34] introduced diffusion in structure variable other than space. In [15], K.P. Hadeler introduced diffusion in a size-structured model where the main concern is the understanding of relevant boundary conditions for realistic models. In this context, some special cases of general Robin boundary condition were considered. Other developments for more general boundary conditions are due to J.Z. Farkas and P. Hinow [11], J.Z. Farkas and A. Calsina [6,7] and A. Bart lomiejczyk and H. Leszczyński [3,4].
The goal of the present work is to provide a systematic spectral analysis of the diffusive and linear structured population model considered by J.Z. Farkas and P. Hinow [11] u t (s, t) + (γ(s)u(s, t)) s = (d(s)u s (s, t)) s − µ(s)u(s, t) + [(d(s)u s (s, t)) s ] s=m + b m u s (m, t) + c m u(m, t) = 0, The different parameters will be defined thereafter. We note that there exists an important literature on second order equations with Wentzell boundary conditions which goes back to W. Feller [13] and A.D. Wentzell [38] (see e.g. A. Favini G.R. Goldstein J.A. Goldstein and S. Romanelli [12] and the references therein). We refer to [4] and to the book by A. Bobrowski [5] Chapter 3 for a biological interpretation of such boundary conditions. Here u(s, t) denotes the density of individuals of size s ∈ [0, m] at time t ≥ 0. The function d stands for the size-specific diffusion coefficient while µ, γ denote respectively the mortality and growth rate of the individuals. Furthermore the non-local integral term in (1) represents the recruitment of individuals into the population. More precisely, β(s, y) is the rate at which individuals of size y produce individuals of size s.
The object of this work is to improve and extend [11] in various directions.
Actually, to deal with well-posedness of the Cauchy problem, the term can be ignored since it can be treated by elementary (bounded) perturbation arguments. In [11], the authors define first A on smooth functions where D(A s ) = {(u, u 0 , u m ) ∈ C 2 [0, m] × R 2 : u(0) = u 0 , u(m) = u m } and show the dissipativity of A s . Then they refer to C α -theory of elliptic equations ( [14] Theorem 6.31) for the proof that the closure of A s denoted by A, is a generator. A priori such an argument gives no information on the domain of A apart from the fact that The authors claim that the generator A is resolvent compact because the embedding of Thus, there is a priori a gap in their proof that A is resolvent compact.
Here we define A on an explicit domain is the usual Sobolev space of functions in L 1 (0, m) having the first two distributional derivatives in L 1 (0, m). Indeed, besides dissipativity arguments following [11], we show here directly that the operator is closed, densely defined and satisfies the rank condition. Thus, a self-contained generation theory with an explicit generator is given. (In particular, the knowledge of D(A) allows to assert that A is resolvent compact.) This is the first contribution of this work.
We show here that this strict positivity assumption is unnecessary. Indeed, e tA ≥ e tA and we show that e tA t≥0 is irreducible by using Hopf's maximum principle. In particular, e tA t≥0 is irreducible even if β = 0. This is our second contribution. We show the existence of an algebraically simple leading real eigenvalue of A. This is our third contribution.
We deal also with a much more important issue. Indeed, in [11] the authors "deduce" from the fact that A is resolvent compact and e tA t≥0 is irreducible that e tA t≥0 converges (in operator norm) exponentially to the spectral projection P associated to the leading eigenvalue λ of A e −t λ e tA → P (t → ∞).
A priori, such a proof is not complete. Indeed, such a conclusion can be reached only if we know that the semigroup e tA t≥0 has a spectral gap (i.e. its essential type is strictly less than its type) which is not at all a consequence of the resolvent compactness of A and the irreducibility of e tA . In fact, we need to study the spectrum of the semigroup e tA t≥0 itself. We can show this property by using tools developped in the context of Transport theory [20,22]. Indeed, by using weak compactness arguments (we assume that K is weakly compact), we show first that the semigroups e tA t≥0 and e tA t≥0 have the same essential type ω ess ( e tA t≥0 ) = ω ess ( e tA t≥0 ); (the weak compactness of K is insured e.g. if there existsβ ∈ L 1 (0, m) such that β(s, y) ≤β(s); in particular, it is trivially satisfied if β is continuous on [0, m] 2 ). It follows that the essential type of e tA t≥0 is less than or equal to the spectral bound of A ω ess ( e tA t≥0 ) ≤ s(A) := sup { (λ); λ ∈ σ(A)} . Secondly, by exploiting the fact that A is resolvent compact and Marek's results [18], we show that the spectral bound of A is strictly less than that of A i.e. once β(., .) is not equal to zero almost everywhere. This implies that e tA t≥0 exhibits a spectral gap ω ess ( e tA t≥0 ) < ω( e tA t≥0 ) where ω( e tA t≥0 ) is the type of e tA t≥0 or equivalently the spectral bound of its generator A, i.e. ω ess ( e tA t≥0 ) < s(A) (the type of a positive semigroup in L p spaces coincides with the spectral bound of its generator [10] and is an element of the spectrum [20]). The fact that e −ts(A) e tA → P (t → ∞) exponentially is then just a consequence of standard functional analytic results (see e.g. [35] Proposition 2.3). This is our fourth (key) contribution.
A fifth contribution is the generalization of this theory to the case m = ∞ allowing arbitrary sizes, i.e. we study also the model To our knowledge, the spectral analysis of this model appears here for the first time.

TIME ASYMPTOTICS OF STRUCTURED POPULATIONS 4091
A priori the domain of the generator is larger than the space but we show that this space is a core of the domain generator. As previously, the irreducibility of the semigroup is shown by using Hopf's maximum principle. Similarly, if is weakly compact, (e.g. if there existsβ ∈ L 1 (0, ∞) such that β(s, y) ≤β(s) ), then the semigroups e tA t≥0 and e tA t≥0 have the same essential type. On the other hand, we cannot appeal to Marek's arguments [18] to infer the existence of a spectral gap because A is not a priori resolvent compact. In this case, we show that the spectral gap property where r σ refers to a spectral radius. We do not know whether (9) is always satisfied.
In particular, we do not know whether lim λ→s(A) always holds. Note that if then the semigroup generated by A + cK has a spectral gap once c > η −1 . If β is bounded below by a separable kernel then we show that where (U ) 1 refers to the first component of U ∈ X . In particular (9) is satisfied if lim λ→s(A) Note that (11) holds e.g. if β is continuous at some point (x, y) with β(x, y) > 0. Whether (10) is a general property of such biological models is an open problem.
The authors are indebted to the referees for their constructive remarks and suggestions.
2.1. Framework and hypotheses. In order to analyze the problem described by (1)-(2)-(3), following [11] we rewrite the boundary conditions (2)-(3). We substitute the diffusion term in (2)-(3), by the remainder of (1) evaluated in 0 and m respectively. We thus get the following dynamic equations where Following [11], the Banach space is endowed with the norm .
We denote by X + the nonnegative cone of X . We introduce some hypotheses on the different parameters: 1. γ, d ∈ W 1,∞ (0, m) and µ, β 0 , β m ∈ L ∞ (0, m), 2. the functions µ, γ and s → β(s, y) are continuous at s = 0 and s = m for every y ∈ [0, m], 3. the operator Using (1)-(12)-(13), we define the operator A by: where the domain of A is given by We are then concerned with the following Cauchy problem

Semigroup generation.
We show here that A is the generator of a C 0semigroup. The dissipativity arguments are essentially those in [11] but we prove directly that A is closed, densely defined and satisfies the rank condition.
Proof. We may restrict ourselves to the operator A; straightforward (bounded) perturbation arguments will end the proof.
1. Let us show that D(A) = X . Let u, u 0 , u m T ∈ X . Let u j j be C ∞ functions with compact supports such that u j → u in L 1 (0, m) and We look for a parabola This amounts to c = u 0 and We find Let us show that v j → u in L 1 (0, m). It suffices to show that We have We have to prove that By definition of H, we have where µ(s) := ω + µ(s), ρ 0 := ω + ρ 0 , ρ m := ω + ρ m .
We multiply (14) by sign(u(s)), integrate between 0 and m and then multiply (15) and (16) respectively by sign(u 0 ) and sign(u m ). We get which is equivalent to and λ U X ≤ H X for ω large enough..This ends the proof of the dissipativity of A − ω.
3. Let us prove that (A, D(A)) is a closed operator.
Let (U n ) n∈N := (u n , u n 0 , u n m ) n∈N ⊂ D(A) and let U := (u, u 0 , u m ) ∈ X and G := (g, g 0 , g m ) ∈ X such that lim n→∞ U n − U X = 0 and lim n→∞ AU n − G X = 0. Note that implies that (u n ) converges in L 1 (0, m) so that u ∈ W 2,1 (0, m) and u n → u in W 2,1 (0, m). Finally U ∈ D(A), G = AU. This ends the proof of the closedness of A. 4. Let us prove that (λI − A) : D(A) → X is a surjective operator for λ > 0 large enough. We consider first a particular case We look for U := u, u 0 , u m T ∈ D(A) such that (λI − A)U = H, i.e.
We multiply (21) by v ∈ H 1 (0, m) and integrate between 0 and m to get hv.
We define the bilinear form by the left hand side and a linear form L : H 1 (0, m) → R by the right hand side of (24), to get a(u, v) = L(v).
We deal now with the surjectivity of (λI − A) . Let There exists a sequence (H n ) n≥0 = (h n , h 0 , h m ) ∈ L 2 (0, m) × R 2 such that lim n→∞ H n − H X = 0. We know that ∀n ≥ 0, ∃! U n ∈ D(A) : (λI − A)U n = H n . In particular ∀n, m ≥ 0, (λI − A)(U n − U m ) = H n − H m . Using the dissipativity result shown before, we get It follows that (U n ) n≥0 is a Cauchy sequence in X . Let U be its limit. Since AU n = −H n + λU n then AU n converges to −H + λU. The closedness of A implies that U ∈ D(A) and (λI − A)U = H and this ends the proof of the surjectivity. Thus A generates a C 0 -semigroup {T (t)} t≥0 by Lumer-Phillips Theorem (see [26] Theorem 4.3, p. 14). Finally, as a bounded perturbation of A, A generates also a quasi-contraction C 0 -semigroup {U (t)} t≥0 .
2.3. On irreducibility. To understand time asymptotics of {U (t)} t≥0 , we need to prove a key result related to positivity. We remind first some definitions and results about positive and irreducible operators. We denote by ., . the duality pairing between X and X .
1. For x ∈ X , the notation x > 0 means x ∈ X + and x = 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.
is positive. 4. A positive operator O ∈ L(X ) is said to be positivity improving if for any x > 0 and x > 0, we have Ox, x > 0.

A positive operator O ∈ L(X ) is said to be irreducible if for any x > 0 and
x > 0 there exists an integer n such that O n x, x > 0. 6. A C 0 -semigroup {Z(t), t ≥ 0} on X is said to be irreducible if for any x > 0 and x > 0 there exists t > 0 such that Z(t)x, x > 0.
We recall that a C 0 -semigroup {Z(t), t ≥ 0} on X with generator B is positive if and only if, for λ large enough, the resolvent operator (λI − B) −1 is positive. We recall also that a C 0 -semigroup {Z(t), t ≥ 0} on X with generator B is irreducible if, for λ large enough, the resolvent operator (λI − B) −1 is positivity improving, (see e.g. [8] p. 165). The main result of this subsection is: Proof. We have to show that the resolvent (λI − A) −1 is positivity improving for large λ. It is easy to see that for large λ because K is a positive operator. Hence it suffices to prove that (λI − A) −1 is positivity improving.
Let us show first that , we may assume without loss of generality that h ∈ C + ([0, m]) .

The first equation is
Hence, without loss of generality, we may assume that H = (h, h 0 , h m ) ∈ X + is such that If u reaches its minimum in (0, m) then v reaches its maximum in (0, m) . By the maximum principle (see [27]

2.5.
On asynchronous exponential growth. Let us remind some definitions and results about asynchronous exponential growth (see [10], [24] and [35] for the details).
Definition 2.6. 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 {Z(t); t ≥ 0} be a C 0 -semigroup on X with generator B : D(B) ⊂ X → X . The growth bound (or type) of {Z(t); t ≥ 0} is given by and the essential growth bound (or essential type) of {Z(t); t ≥ 0} is given by

Definition 2.7 (Asynchronous Exponential Growth). [35, Definition 2.2]
Let {Z(t)} t≥0 be a C 0 -semigroup with infinitesimal generator B in the Banach space X. We say that {Z(t)} t≥0 has asynchronous exponential growth with intrinsic growth constant λ 0 ∈ R if there exists a nonzero finite rank operator P 0 in X such that lim t→∞ e −λ0t Z(t) = P 0 .
We recall the following standard result (see e.g. [8] Theorem 9.11, p. 224). We are ready to give the main result of this subsection.
Theorem 2.9. If K = 0 then the semigroup {U (t)} t≥0 generated by A has asynchronous exponential growth.
Proof. The semigroups {U (t)} t≥0 and {T (t)} t≥0 are related by the Duhamel equation Since K is a weakly compact operator then so is T (t − s)KU (s) for all s ≥ 0. It follows that the strong integral In addition, according to ([24] Proposition 2.5, p. 67), Remark 4. Note that in Theorem 2.9, the requirement K = 0 amounts to the fact that the function β is not identically zero.
3. Models with unbounded sizes. From now on, we consider the general model, described by (7)-(8).
As previously, we are concerned with the Cauchy problem

Semigroup generation.
The main result of this subsection is: Proof. As previously, we restrict ourselves to A ∞ since K ∞ is bounded.
As in the finite case, we introduce the functions where f j 0 (s) = j 2 u 0 s 2 − 2ju 0 s + u 0 = u 0 (js − 1) 2 and we verify that 2. Let us prove that (A ∞ , D(A ∞ )) is a closed operator. We argue as previously.
Note that (28) shows that Let H = h, h 0 T . We have to prove that By definition of H, we have Since u ∈ W 2,1 loc (R + ) ⊂ C 1 (0, ∞), we get, for every finite m > 0 Since for ω large enough. Finally A ∞ − ωI is dissipative. 4. Let us prove that (λI − A ∞ ) : D(A ∞ ) → X ∞ is a surjective operator for λ > 0 large enough. We consider first a particular case Multiply (29) by v ∈ H 1 (0, ∞) and integrate to get hv.

An integration by parts and (30) lead to
One can show that the bilinear form defined by the left hand of (31) is coercive. By Lax-Milgram's Theorem, there exists a unique u ∈ H 1 (R + ) satisfying (31) for all v ∈ H 1 (R + ). It follows easily that u ∈ H 2 (R + ). One sees that U = (u, u(0)) satisfies (29)- (30). Since u ∈ H 2 (R + ) then u ∈ W 2,1 (0, c) for every c > 0 and Let us prove that u ∈ L 1 (R + ). Consider λ :=λ + ω, withλ, ω > 0. Since and so, using the above estimates, . Equation (29) shows that (du ) − (γu) ∈ L 1 (0, ∞). As for the previous finite case, by exploiting the closedness of A ∞ , we get the surjectivity of (λI − A ∞ ) : Note that a priori the domain of the generator is not but this subspace turns out to be a core of D(A ∞ ). Indeed, we have: Then B is closable with closure A ∞ .
Proof. Note first that A ∞ is closed and (in the sense of graphs) so B ⊂ A ∞ and B is a graph, i.e. B is closable.
3.3. On irreducibility. The main result of this subsection is: Proof. As for the previous finite case, it suffices to prove that (λI − A ∞ ) −1 is positivity improving. Let us show first that Let us show now that (λI −A ∞ ) −1 is positivity improving. As for the previous finite case, by using the resolvent identity, we may assume, without loss of generality, that H ∈ D(A ∞ ) ∩ X + .
Let us show by contradiction that u > 0 everywhere. If the absolute minimum of u is not achieved, then u > 0 since u ≥ 0. Consequently we only need to deal with the case where it is achieved at some s ∈ [0, ∞). If u reaches its minimum in (0, c) then v reaches its maximum in (0, c). By the maximum principle (see [27] Theorem 3, p. 6), v must be constant and then u is equal to the constant u(s) = 0. It follows that which is contradictory.
If lim λ→s(A∞) then the semigroup {U ∞ (t)} t≥0 generated by A ∞ has asynchronous exponential growth.