A Global Attractivity in a Nonmonotone Age-Structured Model with Age Dependent Diffusion and Death Rates

In this paper, we investigated the global attractivity of the positive constant steady state solution of the mature population $w(t,x)$ governed by the age-structured model: \begin{equation*} \left\{\begin{array}{ll} \frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}=D(a)\frac{\partial ^2 u}{\partial x^2} - d(a)u,&t\geq t_0\geq A_l,\;a\geq 0,\; 0<x<\pi,\\ w(t,x)=\int_r^{A_l}u(t,a,x)da,&t\geq t_0\geq A_l,\; 0<x<\pi,\\ u(t,0,x)=f(w(t,x)),&t\geq t_0\geq A_l,\; 0<x<\pi,\\ u_x(t,a,0)=u_x(t,a,\pi)=0,\;&t\geq t_0\geq A_l,\; a \geq 0, \end{array} \right. \end{equation*} when the diffusion rate $D(a)$ and the death rate $d(a)$ are age dependent, and when the birth function $f(w)$ is nonmonotone. We also presented some illustrative examples.


Introduction
Spatial movement and temporal maturation are two important characters in most of biological systems; modeling the interaction between them has attracted considerable attention recently [3,4,5,6,8,9,11,13,14,15,16,18,19,21,25,26]. One of the most important methods applied is the Smith-Thieme age-structure technique [18]. In this approach, species population is divided into two groups: mature and immature. At different ages, the standard model with age-structured and diffusion is incorporated ( see [12]): Here u := u(t, a, x) denotes to the density of the population of the species at time t ≥ 0, age a ≥ 0, and location x ∈ [0, π]. The age functions D(a) and d(a) are the diffusion and death rates, respectively. Let r ≥ 0 be the maturation time for the species and A l > 0 be the life span of the species. Then the density of the mature population at time t ≥ 0 and location x ∈ [0, π] is given by Since only the mature individuals can reproduce, one can assume where f (.) is a birth function.
In [19], So, Wu, and Zou assume that the diffusion and death rates, D(a) and d(a), of the mature population are age independent. i.e., D(a) = D m and d(a) = d m .
By this assumption and by substituting (1.1) into (1.2), they derived the following reaction diffusion equation: where u(t, r, x) is called the maturation rate and it can be obtained by the Fourier transforms from (1.1) and the boundary condition (1.3), with a formula given by The functions D I (a) and d I (a) given above are the diffusion and death rates of the immature population, respectively. As such, a non-local time-delayed reaction diffusion equation for the mature population can be obtained: During the past decade, there have been some further studies on this model. In [11], Mei and So investigated the stability of traveling wave solution in the case of Nicholson's blowflies birth function. Liang and Wu in [10] investigated the existence of traveling wave solutions for different birth functions. In [9], Liang, So, Zhang, and Zou considered the above model on a bounded domain where they assumed the diffusion and death rates of the mature population to be constants. In fact, they investigated the long time behavior of the solution by using a numerical simulation. Thieme and Zhao in [21] considered the following general stage-structured model: where u m and u are the population density of the mature and the immature populations, f (u m ) and g(u m ) are the birth and death functions, D I (a) and µ I (a) are the diffusion and death rates of the immature population, and D m and d m are age independent diffusion and death rates of the mature population. In fact, they investigated the existence of traveling wave solutions of this model, when the spatial domain is R n . When the spatial domain is a bounded region Ω ⊂ R n , the model given in Eq. (1.5) was investigated by Xu and Zhao in [25], and by Jin and Zhao in [26]. In fact, the authors investigated the existence and the global attractivity of the steady-state solutions when the function f (u m ) is monotone. In these articles the authors assumed the diffusion and death rates of the mature population to be age-independent. i.e., D(a) = D m and d(a) = d m , respectively. Under these assumptions they transformed (1.5) into the following non-local time-delayed reaction diffusion equation: where Ω is a bounded region in R n , Γ(η(τ ), x, y) is the Green's function associated with the Laplacian operator ∆ x , Bu m = ∂um ∂n + αu, η(a) = a 0 D I (s)ds, F(a) = e − a 0 µ i (s)ds , and φ(t, x) is positive initial function. As a special case of the this model, Zhao in [28] considered the following time-delayed reaction diffusion equation: where Ω is bounded region in R n , D, α > 0, τ ≥ 0, ∆ is the Laplacian operator, and ∂u ∂n is the normal derivative of u in the direction of the outer normal n to the ∂Ω. In fact, the author proved the global attractivity of the positive steady state when f (u m ) is a nonmonotone function in u m . Conclusively, all these studies assumed the diffusion and death rates of the mature populations to be age-independent.
Biologically, it is more realistic to include the age effects in the mathematical models during the whole life of the species. For example, women in the age between 15-40 years have higher birth rate and lower death rate. This causes a variation in the diffusion and death rates among the different ages of the mature individuals. Therefore, the authors of [1] investigated the agestructured model (1.1)-(1.3) when the diffusion and death rates are age-dependent. For the spatial domain they considered two cases. The first case is when the spatial domain is whole r4eal line R. In this case they investigated the existence of monotone traveling waves solutions. The second case is when the spatial domain is closed and bounded interval in R. For this case they considered the model (1.1)-(1.3) with different types of boundary conditions. Particularly, for the Neumann boundary conditions, u x (t, a, 0) = u x (t, a, π) = 0, the authors derived the following integral equation: (1.8) The kernel function K(a, x, y) is given by Moreover, they investigated the global attractivity of the positive steady state solution w * when the birth function f (w) is monotone. In this paper, we investigate the global attractivity of the positive and steady state solution w * when the birth function f (w) is nonmonotone. The paper is organized as follows. In section 2, we present some preliminary results. In section 3, we prove our main result. In section 4, we present some illustrative examples.
By applying the same argument used in the proof of Lemma 6.1 in [1], we have the following theorem: Theorem 2.1. Assume that (F1) and (F2) hold. Then for any φ ∈ Y + , a unique solution w(t, x, φ) of (2.1) exists, and lim sup t→∞ w(t, x, φ) ≤ M uniformly ∀x ∈ [0, π]. Furthermore, the semiflow Φ(t) : Y + → Y + admits a connected global attractor on Y + which attracts every bounded set in Y + .
Also, by applying the same argument used in the proof of Lemma 6.2 and Theorem 6.3 in [1], we have the following theorem: Theorem 2.2. Let F1 and F2 hold, and let w(t, x, φ) be a solution of (2.1) for φ ∈ Y. Then the following statements are valid: I) If pk * < 1 and φ ∈ Y, then lim t→∞ w(t, x, φ) = 0. II) If pk * > 1 , then (2.1) admits at least one homogeneous steady state solution w * ∈ [0, M ], and there exists a positive constant δ such that lim inf t→∞ w(t, x, φ) ≥ δ uniformly, for all φ ∈ Y + and x ∈ [0, π].
Since f (w) w is assumed to be a strictly decreasing function on (0, M ]. Then u ≤ w * ≤ v. Also, by (3.3) and (3.4), we have Since f (w) satisfies th property (P), then we get w * = u = v. Moreover, we have Finally, we show that lim t→∞ w(t, x) = w * uniformly ∀x ∈ [0, π]. In fact, it is enough to show that . Thus, we get ω(φ) = w * , which implies that w(t, ., φ) converges to w * in X as t → ∞.

Examples
In this section, we present three examples of the birth function f (w) to demonstrate the application of our main result. First, we begin with the Ricker type function f (w) = pwe −aw q , a, p > 0, and q > 0. Then, we have the following theorem: Then the unique positive steady state solution w * = 1 a ln(pk * ) 1 q attracts all positive solutions of (2.1).