On a Two-Phase Size-Structured Population Model with Infinite States-At-Birth and Distributed Delay in Birth Process

In this paper we study the two-phase size-structured population model with infinite states-at-birth and distributed delay in birth process. The model distinguishes individuals by two different status: the ‘reproductive’ stage and the ‘nonreproductive’ stage. We establish the well-posedness for this model and show that the solution of this model exhibits asynchronous exponential growth by means of semigroups. We also consider a special case in which the individuals in the ‘reproductive’ stage and the ‘nonreproductive’ stage have the same growth rates and give a comparison between this two-phase model with the classical one-phase model.


Introduction
Among those size-structured population models with infinite states-at-birth, the fundamental one is as follows: β(x, y)n(t, y) dy, 0 ≤ x ≤ā, t ≥ 0, Here, the unknown function n(t, x) denotes the density of individuals of size x ∈ [0,ā] at time t ∈ [0, ∞), whereā > 0 represents the (finite) maximum size of any individual in the population. The vital rates μ(x) and γ (x) denote the death and growth rates, respectively. It is assumed that individuals may have different sizes at birth and therefore β(x, y) denotes the probability which individuals of size y give birth to the individuals of size x. The asymptotic behaviour of its solution can be obtained by using the generalized relative entropy method [12]. Recently the similar nonlinear model was studied in [10,11]. An interesting situation to consider is when the birth process undergoes an observable period so that there is a time lag between conception and birth [6,15,16,18]. In such considerations, the model (1) should be improved by the following model with the distributed delay: Here β(σ , x, y) denotes the rate at which individuals of size y give birth to the individuals of size x after a time lag −σ starting from conception and τ is a constant denoting the maximal delay. For example, in a host-parasite system, the distributed delay in model (2) may be given by the time lag between laying and hatching of the parasite eggs [2]. Moreover, unlike the non-distributed delay case, the time lag considered here can change from 0 to τ , i.e. it is distributed in the interval [0, τ ]. The idea of considering the distributed delay is inspired by the work of Piazzer and Tonetto [14], where a different age-structured population model with distributed delayed birth process was studied. Recently, the model (2) has been proved in [4] that the problem is globally wellposed, and the solutions exhibit so-called asynchronous exponential growth (we refer the readers to [1,4,9,14] for definition).
In this paper we study a model which distinguishes individuals by two different status: the 'reproductive' stage and the 'nonreproductive' stage. In this model, only individuals in the 'reproductive' stage reproduce. In fact the 'reproductive' stage and the 'nonreproductive' stage occur in the evolution of many populations generally. We denote by p(t, x) and n(t, x) the densities of individuals in the 'reproductive' stage and the 'nonreproductive' stage of size x ∈ [0,ā] at time t ∈ [0, ∞), respectively. Then the model reads as follows: Here γ 1 (x) and γ 2 (x) represent the growth rates of the individuals in the 'reproductive' stage and in the 'nonreproductive' stage, ρ 1 (x) and ρ 2 (x) represent the transferring rates between the 'reproductive' stage and the 'nonreproductive' stage, respectively, and μ 1 (x) and μ 2 (x) represent the death rates of the individuals in the 'reproductive' stage and the 'nonreproductive' stage, respectively. Also β(σ , x, y) represent the rate at which the individuals in the 'reproductive' stage of size y give birth to the individuals in the 'reproductive' stage or the 'non-reproductive' stage of size x after a time lag −σ starting from conception, and ν is a constant, 0 ≤ ν ≤ 1. In addition, p and n 0 are given functions defined in [−τ , 0] × [0,ā] and [0,ā], respectively. Later on we shall denotep The similar model without distributed delay has been proved in [9] that the problem is globally well-posed, and the solutions exhibit asynchronous exponential growth. The purpose of this work is to extend the results in [4,9] to the model. We shall prove that under suitable assumptions on (3) is globally well-posed, and its solution possesses the properties of asynchronous exponential growth. Throughout this paper, μ 1 (x), μ 2 (x), ρ 1 (x), ρ 2 (x), γ 1 (x), γ 2 (x) and β(σ , x, y) are supposed to satisfy the following conditions: (H.1) μ 1 , μ 2 , ρ 1 and ρ 2 are nonnegative and continuous functions defined on [0,ā].
In order to prove the property of asynchronous exponential growth, we make the additional assumptions: We introduce the subspace A of Our first main result establishes the global well-posedness of the problem (1) and reads as follows: The proof of this result will be given in Section 2. Actually, from the proof of Theorem 1.1 we shall see that for any F ∈ E, where and it satisfies an integral equation which is equivalent to Equation (3) in a suitable sense. This defines, Our second main result of this article studies the asymptotic behaviour of this semigroup and reads as follows: where · denotes the operator norm on E.
The proof of this result is given in Section 3. The parameter λ 0 is called intrinsic rate of natural increase or Malthusian parameter [17]. Theorem 1.2 shows that the solutions of the model (3) exhibit asynchronous exponential growth.
Next we consider the special case of the model (3), where γ 1 (x) = γ 2 (x) = γ (x) and give a comparison between this two-phase model with the one-phase model. We want to give a comparison between the asymptotic behaviours of the sum of the densities of individuals in the 'reproductive' stage and the 'nonreproductive' stage and the solution of the one-phase model after modifying the death rates and β(σ , c, y) properly. Note that the above result says that there exists a positive vector function (w,v) ∈ E, such that for any (p, n 0 ) ∈ A × B, the mild solution (p(t + ·, ·), n(t, ·)) of the model has the following asymptotic expression: whereÛ We can see that θ(x) is the asymptotic proportion of the individuals in the 'reproductive' stage in the population. Since the model (6) describes the evolution of the sum of the densities of individuals in the 'reproductive' stage and the 'nonreproductive' stage in the asymptotic sense, one might expect that N(t, x) −N(t, x) → 0 as t → ∞. But to our surprise, this is actually not the case. In fact, we have the following result: Theorem 1.3 Let the notation be as above. We have the following relation: where c is a constant which is generally non-vanishing.

M. Bai and S. Xu
The proof of this result will be given in Section 3. Theorem 1.3 shows that the asymptotic behaviours of the sum of the densities of individuals in the 'reproductive' stage and the 'nonreproductive' stage and the solution of the one-phase model are different and the research of the model with two stages is meaningful.
The layout of the rest of the paper is as follows. In Section 2 we reduce the model (3) into an abstract Cauchy problem and establish the well-posedness of it by means of strongly continuous semigroups. In Section 3, we prove that the solution of the model (3) has asynchronous exponential growth. In Section 4, we give the proof of Theorem 1.3.

Reduction and well-posedness
In this section we reduce the problem (1) into an abstract Cauchy problem and establish the wellposedness of it by means of strongly continuous semigroups. We refer the reader to see [5,14] for similar reductions.
First, we introduce the following operators on the Banach spaces X : We note that A ∈ L(D(A), X), B ∈ L(X) and C ∈ L(E, X). Using this notation, we rewrite the model (1) into the following abstract initial value problem for a retarded differential equation in the Banach space X: where u :  (7) in usual sense, we say that functions (u, v) is a classical solution of the problem (7). It is evident that a necessary condition for the problem (7) to have a classical solution is thatp ∈ W 1,1 ([−τ , 0], X) and the functionsp 0 defined by Equation (4) and n 0 belong to Y 0 .
We note that G ∈ L(D(G), E) and Q ∈ L(D(G), X). We now let E := E × X, and introduce operators A 0 , B and A in E as follows: We note that A 0 ∈ L(D(A), E), B ∈ L(E, E) and A ∈ L(D(A), E). Using this notation, we see that the problem (7) can be equivalently rewritten into the following abstract initial value problem of an ordinary differential equation in the Banach space E: where

Remark 2.2 As usual, we say that a function
is a classical solution of the problem (7), then is a classical solution of the problem (8). Conversely, if U is a classical solution of the problem (8), then U has the form U(t) = Proof See Lemma 2.1 of Bai and Xu [4] and Theorem 2.2 of Piazzer and Tonetto [14].
In the sequel, we consider the semigroup generated by the operator A. We first consider the one generated by the principle part A 0 of A. We have the following results: Proof We note that A generates a nilpotent semigroup on X(see Theorem 2.2 of Farkas and Hinow [9]). Since B ∈ L(X), by using the perturbation theorem for generators of strongly continuous semigroups in Banach spaces (see Theorem III.1.3 of Engel and Nagel [8]), we get this lemma.
The operator A 0 generates a strongly continuous semigroup (T 0 (t)) t≥0 on E, given by where (S(t)) t≥0 is a nilpotent left shift semigroup on E, given by and T t : X → E are linear operators defined as where π 1 is the projection onto the first coordinate.
Since B ∈ L(E), by using the perturbation theorem for generators of strongly continuous semigroups in Banach spaces (see [8,Theorem III.1.3]), we get the following lemma: The operator A generates a strongly continuous semigroup (T(t)) t≥0 on E.
By using the theory of strongly continuous semigroups in Banach spaces, we get the following result: Theorem 2.5 For any given initial data U 0 = By Lemma 2.1 and Theorem 2.5, we see that Theorem 1.1 follows.

Asynchronous exponential growth
In this section we study the asymptotic behaviour of the solution of the problem (1). We shall prove that the semigroup (T(t)) t≥0 has the property of asynchronous exponential growth on E.
We denote by ω ess (A) the essential growth bound of the semigroup (T (t)) t≥0 with generator A, ω 0 (A) the growth bound, i.e.
and s(L) the spectral bound , i.e.
If we prove that the semigroup (T(t)) t≥0 is an irreducible positive strongly continuous semigroup (we refer the readers to [7,8] for definition) satisfying the inequality ω ess (A) < ω 0 (A), then by [7, Theorem 9.10 and Theorem 9.11], the semigroup (T(t)) t≥0 has the property of asynchronous exponential growth on E. Thus, in the sequel we step-by-step prove the above assertions about the semigroup (T(t)) t≥0 .
Lemma 3.1 The semigroup (T(t)) t≥0 generated by A is positive and eventually compact (we refer the readers to [7,8] for definition).
Proof Since B is a positive bounded linear operator in E, the positivity of (T(t)) t≥0 follows if we prove that the semigroup (T 0 (t)) t≥0 generated by A 0 is positive (see Corollary VI.1.11 of Engel and Nagel [8]). Since (S(t)) t≥0 is positive, by the expression (9) of (T 0 (t)) t≥0 , we only need to prove that the semigroup (T 0 (t)) t≥0 generated by  (11), we also see that T t (t) = 0 for t > + τ . Hence, from Equations (9) and (10), we have that T 0 (t) = 0 for t > + τ . This particularly implies that (T 0 (t)) t≥0 is compact for t > + τ . Thus, by Proposition III.1.14 of Engel and Nagel [8], the eventual compactness of (T(t)) t≥0 follows if we prove that B is compact. We note that the only nonzero component operator of operator matrix B is C : E → X. We use the method which is similar to Lemma 3.6 in [9] to prove that C is compact. Hence the desired assertion follows.
By the eventual compactness of the semigroup (T(t)) t≥0 and Corollary IV.3.12 of Engel and Nagel [8], the following result holds.
In the proof of Lemma 3.1, we have that the semigroup T 0 (t) = 0 for t > + τ . Then by the definition (14), we have that ω ess (A 0 ) = −∞. Since B is compact on E, by Proposition 2.12 of Clément et al. [7], we have the following result: Lemma 3.4 The semigroup (T(t)) t≥0 generated by A is irreducible.