Asymptotic behavior of a hierarchical size-structured population model

We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by \begin{document} $C_0-{\rm{semigroup}}$ \end{document} framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.

1. Introduction. In this paper we study the following system of a hierarchical size structured population model ) should be understood as the probability that an individual of size s reproduces after a time lag −τ starting from conception, here τ > 0 is a constant denoting the maximal time delay the birth process takes. Such a form of the delayed birth process was initialed by Pizzera [30] by taking into account the time which the birth process may take, and it has been adopted in many papers, such as in papers [18,19,37,38], to discuss the long-time behavior of the corresponding nonlinear systems and explore how the time lag influences the asymptotic behaviors of the considered systems.
framework to study analytical questions for structured populations (see [9,10,11]), including those pertaining to linear/nonlinear stability of population equilibria. In this context it was proven for large classes of structured population models, formulated as integral (or delay) equations, that the nonlinear dynamics of a population equilibrium, such as stability and bifurcations, is completely determined by its linearized equation, which is commonly referred to as the "Principle of Linearization", see Theorem 2.6 and Section 4 in [10] and Theorem 1.3 in [23]. By virtue of this principle, we investigate in this paper the dynamics of the linearized equation of the system (1.1). Precisely, we shall explore the linearized stability and instability of stationary solutions of the system (1.1) by using semigroup techniques and spectral methods based on the characteristic equation, and meanwhile, the asynchronous exponential growth property (AEG, for short, it will be interpreted in details in Section 6 below) of solutions will be studied based on the spectral analysis as well.
We first present the necessary conditions of the existence of positive equilibrium and linearize the considered system (1.1) around the equilibriums. We then employ the Perron-Frobenius techniques as in [20,30,31] to carry on our discussion and some results on linearized stability, instability and AEG for the linearized equations are obtained under proper conditions. Particularly, we find interestingly that the AEG phenomena may occur around both the equilibriums under the given conditions. The examples and the simulation illustrate well the applications of the obtained results. Clearly, the system discussed here is biologically and mathematically more general than that in [30,31] because the discussed equations there are merely linear systems. As described above, the nonlinearity of the boundary condition in (1.1) requires us study the problems by using the linearization arguments, and correspondingly, the obtained results in this article are richer than those in [30,31] (note that there is only the trivial stationary) and improve their main theorems directly. On the other hand, it can be seen that, the stability/instability results obtained here also extend and develop the corresponding ones in [1,2,14,15,16] as the effects of time lag in birth process on asymptotic behavior are studied in this paper. Moreover, the techniques employed here differs greatly from these papers due to the extra time delay. We also would like to point out that, it should be more reasonable or practical if the rates γ(·) and µ(·) in (1.1) depend on the environment Q like in [1,2,6,7]. However it is very difficult to obtain the explicit expression of the characteristic equation and so we only discuss here the simpler case for which the rates γ(·) and µ(·) are independent of Q.
The organization of this paper is as follows. In Section 2, we collect some notations and results on theory of C 0 -semigroup which will be used in the later sections, then we linearize the system (1.1). In Section 3, we set the linear system in the framework of semigroup theory, and prove existence and uniqueness of the solutions for the simplified system by showing that the related abstract Cauchy problem gives rise to a strongly continuous semigroup via Hille-Yosida operator theory. In Section 4, some regularity properties are derived for the linearized system, following that we also deduce the characteristic equation in this section. Then we discuss in Section 5 the linearized stability and instability of the stationary solution. And in Section 6 we devote to discussing the AEG property for the linearized system under some conditions. Finally, in Section 7, we provide some simple examples to illustrate the obtained results through numerical simulations.

DONGXUE YAN AND XIANLONG FU
2. Preliminaries and the linearized system. In this section we recall some preliminaries on linear operators and C 0 -semigroup which will be used later. We will also linearize the system (1.1) around a stationary solution so that we may construct the C 0 -semigroup framework for our discussion.
Let L : D(L) ⊂ X → X be a linear operator on a Banach space X. Denote by ρ(L) the resolvent set of L. The spectrum of L is σ(L) = C \ ρ(L). The point spectrum of L is given by the set Definition 2.1. Let L : D(L) ⊆ X → X be a linear operator. Assume that there exist real constants M ≥ 1, and ω ∈ R, such that (ω, +∞) ⊆ ρ(L), and (λ − L) −n ≤ M (λ − ω) n , for n ∈ N + , and all λ > ω.
Then the linear operator (L, D(L)) will be called a Hille-Yosida operator.
If (L, D(L)) is a Hille-Yosida operator on the Banach space X and set Then the operator (L 0 , D(L 0 )) is called the part of L in X 0 and one has that.
If L is the generator of a strongly continuous semigroup (T (t)) t≥0 , we present briefly the concepts of the extrapolation space and Favard class here, see [27] for more details. Definition 2.3. On Banach space X, a new norm associated to linear operator L can be defined by Then the completion of the space (X, · −1 ) will be called the extrapolation space of X 0 associated to L and will be denoted by X L −1 (or simply X −1 ). The (unique) extension of T (t) to X −1 is denoted by T −1 (t), it is a semigroup on X −1 and its generator is denoted as A −1 .
Definition 2.4. The Favard class of the generator L is the space In addition, a crucial quantity associated to it is the growth bound ω 0 (L), defined as t .
The essential growth bound ω ess (L) is defined by where α is the measure of noncompactness defined as Then there holds ω 0 (L) = max{ω ess (L), s(L)}, where s(L) is the spectral bound of L, i.e. s(L) = sup{Reλ : λ ∈ σ(L)}. The essential spectral radius of T ∈ L (X) is given by With Definition 2.3, Definition 2.4 and these notations, we have the following perturbation result to be used to prove Theorem 6.4. Lemma 2.5. ( see [32], Corollary 2.2 or [35], Theorem 2.4 ) Let (L, D(L)) be a Hille-Yosida operator on a Banach space X and let C be a bounded operator from X 0 = D(L) to F L−1 , the Favard class of L (see Definition 2.4). Let (T 0 (t)) t≥0 and (T 0 (t)) t≥0 be the C 0 -semigroup on X 0 generated by the parts of L and L −1 + C in X 0 , respectively. If CT 0 (t) is a compact operator from X 0 to F L−1 for all t > 0, then r ess (T 0 (t)) = r ess (T 0 (t)).
Next we set about to linearize the system (1.1). Obviously Eqs. (1.1) admit the trivial solution u * ≡ 0. Realistically we also expect some additional positive (continuously differentiable) stationary solutions u * > 0 for (1.1). In the following proposition we formulate a necessary condition for the existence of a positive equilibrium solution of problem (1.1).
and the operator π is given for 0 ≤ s ≤ m by In this case, the unique positive stationary solution u * of problem (1.1) is given by where U * = m 0 u * (s)ds represents the positive population quantity.

DONGXUE YAN AND XIANLONG FU
Proof. Suppose that u * (s) is a positive stationary solution of the system (1.1). Then u * satisfies the equations The general solution of Eq. (2.7) is found as Observing that Taking (2.11) into (2.9), we get the form of (2.5).
Given any stationary solution u * in C 1 ([0, m]) of the system (1.1), we linearize the governing equations by introducing the infinitesimal perturbation v = v(s, t) and making the ansatz u = v + u * . Hence v has to satisfy the equations (2.12) Now we linearize the fertility rate β(s, σ, Q(s, t + σ)). To this end, we note that the functional dependence of the fertility rate on Q rather than on u requires the linearization about Q * . Thus by the approximation in system (2.12) and dropping all the nonlinear terms, we arrive at the linearized problem where we have set ρ(s) = γ (s) + µ(s), (2.14) 3. C 0 -semigroup for linear system. To analysis the asymptotic behavior for linearized system (2.13), we establish in this section the C 0 -semigroup framework for this system and through which rewrite it into an abstract evolution equation. Suppose u * is any positive stationary solution of problem (1.1). We denote the Banach space with the usual norm · and on this space we introduce the following operator Here the subscript m reminds that the operator is defined on its maximal domain and ρ(s) is given by (2.14). Moreover, we call the map the boundary operator, which is used to express the boundary condition (see [21,30]). Now look at the Banach space On this space we introduce the operator Φ ∈ L (E, C), called delay operator, by setting, for g ∈ E, Then with these operators the linearized system (2.13) can be cast in the form of an abstract boundary delay problem: In order to apply the C 0 -semigroup theory, we rewrite (3.1) as an abstract Cauchy problem. For this, as in [30,31], on the space E we consider the differential operator In addition, we introduce another boundary operator Q : D(G m ) → X defined as Qg := g(0).

DONGXUE YAN AND XIANLONG FU
Finally, we consider the product space on which we define the operator matrix With these notations, we obtain the following abstract Cauchy problem associated to the operator (A, D(A)) on the space X . Here the function V : [0, +∞) → X is given by To obtain the well-posedness of solutions for the abstract Cauchy problem (3.3), in the sequel we will verify that (A, D(A)) generates a C 0 -semigroup on X .
In the first step, we consider the Banach space and the matrix operator . We shall use the following well-known perturbation result to show the operator (A , D(A )) to be a Hille-Yosida operator: [13,29]). Let (A, D(A)) be a Hille-Yosida operator on a Banach space X and B be a bounded linear operator on X, then the sum C = A + B is also a Hille-Yosida operator.
By this lemma we have that Proof. The operator A can be written as the sum of two operators on X as The restriction (G 0 , D(G 0 )) of G m to the kernel of Q generates the nilpotent left shift semigroup (S 0 (t)) t≥0 on E given by the formula Similarly, the restriction (A 0 , D(A 0 )) of A m to the kernel of P generates a strongly continuous positive semigroup (T 0 (t)) t≥0 on X given by where We claim that A 1 is a Hille-Yosida operator. In fact, for any λ ∈ C and which shows σ(A 0 ) = ∅ as γ(s) > 0. And similarly, σ(G 0 ) = ∅. So for every λ ∈ C, its resolvent is given by Therefore, we have λR(λ, A 1 ) ≤ 1, and A 1 is a Hille-Yosida operator.
Since the perturbation operator A 2 is clearly bounded, A is a Hille-Yosida operator as well by Lemma 3.1.
According to Lemma 2.2, we have actually obtained by Proposition 2 that the operator (A 0 , D(A 0 )), the part of the operator (A , D(A )) in the closure of its domain, generates a C 0 -semigroup on the space E × {0} × X × {0}. Now we show that the operator (A, D(A)) generates a strongly continuous semigroup on X by the following theorem. In particular, Proof. From Lemma 2.2, we know that the part (A 0 , D(A 0 )) of (A , D(A )) in the closure of its domain generates a strongly continuous semigroup.
Observe that , Therefore, the operator (A, D(A)) is isomorphic to (A 0 , D(A 0 )) and thus generates a strongly continuous semigroup on the space X .
The following well-posedness result for (3.1) is then a direct consequence of Theorem 3.2.
where Π 2 is the projection operator of T (t) on the space X.

Spectral analysis, regularities and characteristic equation.
In this section, we prove two regularity results about the C 0 -semigroup generated by (A, D(A)) which imply that the spectrally determined growth property holds true and that the linearized stability of the steady-state solution is governed by the location of the leading eigenvalue. Then we drive the characteristic equation for the linearized system. The first main result of this section is that Proof. We observe the abstract Cauchy problem (3.3) with A becomes With the definition of A, we have subject to the boundary condition in the system (1.1). From the equation (4.1), we get the solution To obtain the solution of v(s, t), let us introduce, for t 0 > 0, where t(s) = t 0 + Γ(s) with Γ defined in (3.4). Then With the equation Therefore, if t > Γ(m) + τ , v is continuous in s and t. Consequently, Eq. (4.2) implies that v is continuous differentiable if t > 2(Γ(m) + τ ). Hence the semigroup generated by A is differentiable for t > 2(Γ(m) + τ ). Since W 1,1 (0, m) is compactly imbedded in X, the claim follows. Theorem 4.1 has the following immediate and noteworthy consequence (see [13,29]). Moreover, the semigroup is spectrally determined, i.e. the growth rate ω(T (t)) of the C 0 -semigroup (T (t)) t≥0 and the spectral bound s(A) of its generator coincide.
Because of Corollary 2 the linear stability of the steady-state solution is spectrally determined (see [13,29], Theorem VI.1.15). Hence in the sequel it suffices to investigate the location of the leading eigenvalue of the generator of the C 0 -semigroup (T (t)) t≥0 .
In order to state and prove the second main theorem of this section, we need state several existing lemmas and theorems, for this we introduce two operators as follows. For λ ∈ ρ(G 0 ) ∩ ρ(A 0 ), we define L λ (g) = Φ(g)ϕ λ , f or g ∈ E, where λ and ϕ λ are from (3.5). The next result was formulated in [26].
The decomposition of the operator λ − A below has also been proved in ( [26], Lemma 2.6).
for the operator L λ K λ ∈ L (X). Then one has the implications (a) ⇐ (b) ⇔ (c).
If, in particular, K λ and L λ are compact operators, then the statements (a), (b) and (c) are equivalent.

8). A strongly continuous semigroup (T (t)) t≥0 on a Banach lattice X is positive if and only if the resolvent R(λ, A) of its generator
A is positive for all sufficiently large λ.
Now, using this lemma, we can conclude this section by formulating conditions for the positivity of the obtained semigroup (T (t)) t≥0 . Proof. We consider the operator K λ L λ . By the definitions of K λ and L λ in the above of Lemma 4.2, it is easy to see that lim Reλ→+∞ K λ L λ = 0.
Therefore K λ L λ < 1 for Reλ sufficiently large. Thus the operator (1 − K λ L λ ) is invertible and its inverse (1 − K λ L λ ) −1 is given by the Neumann series. Obviously K λ L λ is a positive operator by the condition (4.5), and hence (1 − K λ L λ ) −1 is positive as well for Reλ big enough. With the resolvent representation of A in (4.4), R(λ, A) is nonnegative for such λ. Thus, using Lemma 4.6 above, we infer that the operator (A, D(A)) generates a positive semigroup on the Banach lattice E × X. Then we get the assertion.
The positivity and eventual compactness of the C 0 -semigroup (T (t)) t≥0 enable us to draw the following important conclusion. Now we turn to address the characteristic equation. From Corollary 2, we know that the linearized stability of stationary solutions of the system (1.1) is entirely determined by the eigenvalues of the semigroup generator (A, D(A)). Hence, it is essential to determine the eigenvalues of A. The eigenvalue equation for λ ∈ C and nontrivial function v, is equivalent to the system For the remainder of this section let us assume that α ∈ [0, 1). From (4.10) we obtain Using the relations (4.11) we can rewrite system (4.7)-(4.9) in terms of V and its derivatives as follows Then the general solution of (4.12) can be formulated as For the case α = 1, by the definition (4.10) of V , we have that V is a constant. When we solve the problem (4.8)-(4.9) directly, we obtain again the condition ∆(λ) = 0, (4.20) where ∆(λ) is given by (4.19) with α = 1. Hence (4.19) is the characteristic equation for all 0 ≤ α ≤ 1.

5.
Linear stability results. Based on the discussion in the previous sections we can no explore the asymptotic behaviors of solutions for (1.1). Firstly, in this part, we will formulate sufficient conditions about the stability and instability of stationary solutions for the equation (1.1). First we can establish the stability result for the null solution as follows. Next we focus on the linearized stability/instability of the positive stationary solutions. For this purpose, in the light of the arguments in the above sections, we deduce the following theorem. where π(y) is defined as (2.4). By the above considerations we conclude easily that which gives (5.3).
From this theorem we get immediately the following results on the stability/ instability of stationary solutions for the system (1.1). If, however, β Q (s, σ, Q * ) ≥ 0 and β Q (s, σ, Q * ) ≡ 0, then u * is linearly unstable.
6. Asynchronous exponential growth. The purpose of this section is to gain a deeper insight into asymptotic properties of solutions of the linearized system (2.13). That is, we will use semigroup techniques and spectral analysis methods to obtain the property of asynchronous exponential growth (AEG for short) for (2.13) which is defined in the framework of semigroup theory as below.
Definition 6.1. A linear C 0 -semigroup (T (t)) t≥0 on Banach space X is said to exhibit an AEG if there exist a real number λ 0 ≥ 0 (called Malthusian parameter) and a rank one projection P on X such that lim t→+∞ e −λ0t T (t)x = P x, for all x ∈ X.
The existence of such a λ 0 is related to the existence of a nonnegative strictly dominant eigenvalue in the spectrum of the generator of the semigroup.
The phenomenon of AEG appears frequently in age/size-structured populations models (see [5,23,36]). It describes the situation when the population grows exponentially in time but the proportion of individuals within any range of age/size compared to the total population tends, as time tends to infinity, to a limit which just depends on the chosen range. This is an important characteristic of solutions of population equations both from the theoretical and application point of view. In the past years there is much work on this topic, see for instance [5,31,37,38].
For positive semigroups there exists the well-known characterization of AEG (see [28], Theorem C-IV.2.1). As in Ref. [31], our analytical approach will be guided toward the result as follows. where P denotes the spectral projection corresponding to the eigenvalue λ 0 having rgP = Ker(λ 0 − A).

Remark 1.
It is easy to see that if (6.1) holds and Ker(λ 0 − A) has dimension 1, then the semigroup (T (t)) t≥0 has AEG.
Before presenting the main result of this section, we need to do some preparation. Consider the following decomposition for the operator A (S 0 (t)) t≥0 and (T 0 (t)) t≥0 are defined as in Proposition 2. The generator of (T 0 (t)) t≥0 is Proof. The first part of the assertion is a consequence of Lemma 6.3 and Lemma 2.2. Moreover, Therefore, the operator ( A 0 , D( A 0 )) is isomorphic to the operator which generates the semigroup (T 0 (t)) t≥0 given in (6.3) (see [3], Proposition 3.1 for the proof ) on the Banach space E × X. Thus, the semigroup generated by ( A 0 , D( A 0 )) is isomorphic to (T 0 (t)) t≥0 .
Based on the C 0 -semigroup and the spectral analysis, we are on the position now to show the system (2.13) has AEG under some conditions, namely, In particular, (e tA ) t≥0 has AEG.
Proof. We apply Theorem 6.2 to show this result. From the positive condition (4.5) we obtain that the semigroup (e tA ) t≥0 is positive, so we just have to verify that λ 0 is a first order pole of R(λ, A) with one dimensional eigienspace, and ω ess (A) < ω 0 (A).
Finally, we show that λ 0 := ω 0 (A) has pole order 1 and that Ker(λ 0 − A 0 ) is one dimensional, so that the dimension of the range of the corresponding spectral projection P is also equal to 1. Similar to [31], we apply the technique developed by Greiner (see [20], Proof of Corollary 1.6). To achieve this assertion, first we note that Ker(λ 0 − A 0 ) has dimension 1 because all eigenvectors are multiples of where ε λ0 (·) and ϕ λ0 (·) are given by (3.5). Assume now that the pole λ 0 has order n > 1. Let then it is clearly a positive operator, and U h = 0 for every eigenvector h of A 0 , in particular, U ψ λ0 = 0. In addition, if |φ| ≤ ψ λ0 , then |U φ| ≤ U |φ| ≤ U ψ λ0 = 0. Therefore, U vanishes on the set {φ ∈ X 0 : |φ| ≤ ψ λ0 } which is total in X 0 (i.e. the set of all linear combinations lin{φ ∈ X 0 : |φ| ≤ ψ λ0 } is dense in X 0 ) because ψ λ0 is strictly positive. Hence U = 0, i.e. λ 0 is a pole of order less than n which is a contradiction.
Then, from Theorem 6.2, the claim follows.