 Research
 Open Access
 Published:
Global attractivity in a nonmonotone agestructured model with agedependent diffusion and death rates
Advances in Difference Equations volume 2018, Article number: 419 (2018)
Abstract
In this paper, the global attractivity of the homogeneous equilibrium solution for the diffusive agestructured model
is established when the diffusion and death rates, \(D(a)\) and \(d(a)\), respectively, are age dependent during the whole life of the species, and when the birth function \(f(w)\) is nonmonotone. In the paper, we also present some demonstrative examples.
Introduction
Temporal maturation and spatial movement play a major role in many biological systems. Therefore, many mathematical models and studies have appeared recently dealing with the interaction between them; e.g., see [2,3,4,5,6, 8, 9, 11, 13,14,15,16,17, 20, 24, 26, 27, 29]. One of the most popular techniques that has been used to study the interaction between the temporal maturation and the spatial movement is the Smith–Thieme agestructure technique [17]. In this technique, the species population is divided into two categories: the immature population and the mature population. At different ages, the diffusive agestructured model is given by
where \(u(t,a,x)\) is the size of the species population at any time \(t\geq 0\), age \(a\geq 0\), and a spatial location \(x\in \varOmega \subseteq \mathbb {R}\); the functions \(D(a)\) and \(d(a)\) are the agedependent diffusion and death rates of the species; see [12]. Let \(\tau \geq 0\) be the maturation time of the species and \(A_{l}>0\) be its life span. Then the size of the mature population at any time \(t\geq 0\) and a spatial location \(x\in \varOmega \) is given by
Since only the mature individuals can reproduce, we assume that
where \(f(w)\) is a given birth function.
In [20], So et al. assumed that the diffusion and death rates of the mature population are age independent. Particularly, \(D_{m}(a)=D\) and \(d_{m}(a)=d\), where D and d are positive constants. By using this assumption and by applying the method of characteristics, So et al. derived the following reaction–diffusion equation:
The function \(u(t,\tau ,x)\) appears above is called the maturation rate of the species, and is formulated by
where
and
The agedependent functions \(D_{I}(a)\) and \(d_{I}(a)\) are the immature population diffusion and death rates, respectively. In conclusion, So et al. deduced the nonlocal timedelayed reaction–diffusion equation:
and they investigated the existence of monotone traveling wave solutions of this equation for the specific birth function \(f(w)=pw\exp \{aw ^{\frac{1}{q}}\}\). In [11], Mei and So showed the stability of these traveling wave solutions. In [10], Liang and Wu investigated the existence of monotone traveling wave solutions of (1.5) for different birth functions. In [9], by using a numerical simulation, Liang et al. investigated the long time behavior of the solution of the agestructured model (1.1)–(1.3) when the spatial domain Ω is the closed interval \([0,l]\). In their study, Liang et al. assumed the diffusion and death rates of the mature population to be age independent, so that they can easily derive nonlocal timedelayed reaction–diffusion equations similar to equation (1.5).
In [24], Thieme and Zhao showed the existence of monotone traveling waves for the following stagestructured model:
where \(u(t,a,x)\) is the density of the species population at any time \(t\geq \tau \), age \(a\geq 0\), and a location \(x\in \mathbb {R}^{n}\), \(u_{m}(t,x)\) is the density of the mature population at any time \(t\geq \tau \) and a location \(x\in \mathbb {R}^{n}\), the functions \(f(u_{m})\) and \(g(u_{m})\) are the birth and mortality rates for the mature individuals, respectively, the agedependent functions \(D_{I}(a)\) and \(d_{I}(a)\) are the diffusion and mortality rates for the immature population, respectively, and the positive constants \(D_{m}\) and \(d_{m}\) are the ageindependent diffusion and death rates for the mature population, respectively. When the spatial domain is a closed and bounded set to \(\mathbb {R}^{n}\), the above agestructured model has been investigated by Xu and Zhao [26] and by Jin and Zhao [29]. In these two studies, the authors transformed Eq. (1.6) to the following nonlocal timedelayed reaction–diffusion equation:
where \(\varGamma (\chi (\tau ),x,y)\) is the Green’s function associated with the Laplacian operator \(\Delta_{x}\), \(Bu_{m}:=\frac{\partial u _{m}}{\partial n}+\beta u\), \(\chi (\tau ):=\int_{0}^{\tau }D_{I}(s)\,ds\), \(\mathcal{F}(\tau ):=e^{\int_{0}^{\tau }d_{I}(s)\,ds}\), and \(\psi (t,x)\) is a positive initial data. Particularly, Xu and Zhao in [26] investigated the global dynamics of (1.7) when the birth function \(f(u_{m})\) is monotonic; Jin and Zhao in [29] investigated the existence of asymptotic spreading speeds and the dynamics of periodic solutions.
Conclusively, all these studies assumed that the diffusion and death rates of the mature population to be age independent. In fact, many biological aspects could cause a variation in the diffusion and death rates among different ages of the mature individuals. For example, the reproductivity of the mature individuals varies among different ages. A particular example is the human population where women at the ages between 15 to 40 years have a lower death rate and a higher birth rate. Another example is the predation of animals where the predation of mature animals could be heavier on some certain age groups. Therefore, the death rate of the mature individuals varies dependently on the age of mature individuals. In epidemiology, the disease infection rate could be higher in some age groups of the mature populations. An example is the sexually transmitted diseases (STDs) that spread among the mature individuals, and it implies to a change in the death rate of the mature individuals depending on their age. Thus, it is more logistic to include the age effects of the mature population in the mathematical models. Therefore, AlJararha and Ou in [1] investigated the agestructured model (1.1)–(1.3) with the assumption that \(D(a)\) and \(d(a)\) are agedependent functions during the whole life of the species. In their study, the authors considered two cases: the unbounded domains case and the bounded domains case. For the unbounded domains case, AlJararha and Ou derived from (1.1)–(1.3) the following integral equation:
where
and
and then they proved the existence of monotone traveling wave solutions for it. In the bounded domains case, particularly, \(\varOmega =[0,\pi ]\), AlJararha and Ou derived from (1.1)–(1.3) equipped with the Neumann boundary conditions the following integral equation:
where the kernel function \(\varPi (a,x,y)\) is given by
To study the dynamics of (1.9), particularly, the existence and the global stability of the homogeneous equilibrium solution of (1.9), AlJararha and Ou assumed that \(f(w)\) is a monotone function. So, they can apply the theory of monotone dynamics and the comparison concepts to prove the desired result.
In this paper, we prove the global attractivity of such homogeneous equilibrium solution with the assumption that \(f(w)\) is a nonmonotone function. In this case, the theory of monotone dynamics and the comparison arguments cannot be applied. Therefore, to prove our main result, we apply the fluctuation method. The fluctuation method has been improved in [23] to study the global dynamics of the nonlocal timedelayed reaction–diffusion predator–prey model, and later it has been used in many mathematical articles to deal with the nonmonotone dynamics difficulties; e.g., see [7, 18, 21,22,23,24, 31].
The paper is organized as follows. In Sect. 2, we present some preliminary results and concepts. In Sect. 3, we prove the main result. In Sect. 4, we present some demonstrative examples. Section 5 is devoted to the concluding remarks and discussions.
Preliminaries
In this section, we present some preliminary concepts and some previous results. Let \(\mathbb{X}=C([0,\pi ],\mathbb {R})\) be the space of all continuous real valued functions defined on the closed interval \([0,\pi ]\) equipped with the usual supremum norm:
Let \(\mathbb{X}^{+}= \{ \psi (x)\in \mathbb{X} \mid \psi (x) \geq 0, x\in [0,\pi ] \} \) be its positive cone. It is well known that \(\operatorname{Int}( X^{+})\neq \phi \). Hence, a strongly positive relation on \(\mathbb{X}^{+}\) can be defined. This relation is defined as follows: for any two functions η and ψ in \(\mathbb{X}\), we have \(\eta (x)\preceq \psi (x)\) if and only if \(\eta (x)\leq \psi (x)\), \(\forall x \in [0,\pi ]\). Thus, the pair \((\mathbb{X}, \mathbb{X}^{+})\) forms a strongly ordered Banach space. Also, we consider the functions space \(\mathbb{Y}=C([t_{0}A_{l}, t_{0}], \mathbb{X})\), where \(t_{0}\geq A_{l}\) is fixed, with its ordered positive cone \(\mathbb{Y}^{+}=C([t_{0}A_{l},t_{0}],\mathbb{X}^{+})\). For convenience, we identify each \(\psi \in \mathbb{Y}^{+}\) as a function from \([t_{0}A_{l}, t_{0}]\times [0,\pi ]\) to \(\mathbb{R}\) as follows: \(\psi (s,x)=\psi (s)(x)\). For any function \(\xi (\cdot): [ t_{0}A_{l},c ) \rightarrow \mathbb{X}\), where \(c>t_{0}\), define \(\xi_{t} \in \mathbb{Y}\) by \(\xi_{t}(s)=\xi (t+s)\), \(\forall s\in [t_{0}A_{l},t _{0})\). Let \(M >0\). Then define the positive cone \(\varSigma_{M}:= \{ \psi (x)\in \mathbb{X}^{+} \mid \psi (x)\leq M, x\in [0,\pi ] \} \) and the function space \(\mathbb{Z}_{M}=C([t_{0}A_{l},t_{0}],\varSigma_{M})\). Assume that \(f(w)\) is a Lipschitz continuous function. Then by applying the method of steps (for example, see [25]) the nonlinear integral equation
where
has a unique solution \(w(t,x,\psi )\) for any \(t\geq t_{0}\) and \(\psi \in \mathbb{Y}^{+}\). Therefore, we can define the semiflow \(\varPhi (t):\mathbb{Y}^{+}\rightarrow \mathbb{Y}^{+}\), by \((\varPhi (t) \psi )(s,x)=w(t+s,x,\psi ), \forall s\in [t_{0}A_{l},t_{0}]\) and \(x\in [0,\pi ]\). In addition, the semiflow \(\varPhi (t):\mathbb{Y}^{+} \rightarrow \mathbb{Y}^{+}\) is compact for \(\forall t> t_{0}\); see [25]. The concept of the semiflow can be found in [30, p. 8] (also, one can see [19, p. 2]). We note that the kernel function \(\varPi (a,x,y)\) is continuous, positive, and uniformly bounded on \([\tau ,A_{l}]\times [0,\pi ]\times [0,\pi ]\); see [1]. Moreover, \(\int_{0}^{\pi }\varPi (a,x,y)\,dy=1\). So, if we set \(\pi^{*}:=\int_{\tau }^{A_{l}}\int_{0}^{\pi }\varPi (a,x,y)\,dy\,da\), then \(\pi^{*}=\int_{\tau }^{A_{l}}\beta (a)\,da\).
To prove our main result, we need the following assumptions on the birth function \(f(w)\):

(F)
Assume that:

(F1)
\(f:\mathbb {R}^{+}\rightarrow \mathbb {R}\) is a Lipschitz continuous function \(\forall w\geq 0\), \(f(0)=0\), f is a differentiable function at 0 with \(f^{\prime }(0)=p>0\), and \(f(w)\leq p w\), \(\forall w\geq 0 \).

(F2)
There exists a positive constant M, such that \(\forall w>M\) we have \(\pi^{*}\bar{f}(w)\leq w\), where \(\bar{f}(w):= \max_{v\in [0,w]}f(v)\).

(F1)
Consider the following linearized equation:
Let \(w(t,x)=e^{\lambda t}w(x)\) in the above equation. Then we have the following eigenvalue problem:
Set \(w(x)=1\) in (2.4). Then the characteristic equation of (2.4) is given by \(p\varGamma_{0}(\lambda )=1\), where \(\varGamma_{0}(\lambda )=\int_{\tau }^{A_{l}}\exp \{  ( \lambda a+\gamma (a) ) \}\,da\) and \(\gamma (a):=\int_{0}^{a}d( \xi )\,d\xi \). By solving this characteristic equation, we can uniquely determine the principal eigenvalue of (2.4). Clearly, \(\varGamma_{0}(\lambda )\) is a decreasing function in λ and it satisfies the following inequality:
Thus, \(\lim_{\lambda \rightarrow \infty }\varGamma_{0}(\lambda )=0\), \(\lim_{\lambda \rightarrow \infty }\varGamma_{0}(\lambda )=\infty \), and \(\varGamma_{0}(0)=\pi^{*}:=\int_{\tau }^{A_{l}}e^{\gamma (a)}\,da>0\). Therefore, there exists a unique \(\lambda_{0}\) that solves \(p\varGamma _{0}(\lambda )=1\). So, \(\lambda_{0}\) is the required principal eigenvalue of (2.4); see [1, Theorem 5.1]. Moreover, \(\lambda_{0}>0\) if \(p\pi^{*}>1\) and \(\lambda_{0}<0\) if \(p\pi^{*}<1\).
By following the same argument in the proof of Lemma 6.1 [1], we have the following theorem.
Theorem 2.1
Assume that (F1) and (F2) hold. Then the following statements are valid:

(I)
for any \(\psi \in \mathbb{Y}^{+}\), a unique solution \(w(t,x,\psi )\) of (2.1) globally exists and \(\lim \sup_{t\rightarrow \infty }w(t,x,\psi )\leq M\) uniformly for all \(x\in [0,\pi ]\).

(II)
the semiflow \(\varPhi (t):\mathbb{Y}^{+}\rightarrow \mathbb{Y}^{+}\) admits a connected global attractor on \(\mathbb{Y} ^{+}\) which attracts every bounded set in \(\mathbb{Y}^{+}\).
Also, by applying the same argument as in the proof of Lemma 6.2 and Theorem 6.3 in [1], we have the following theorem.
Theorem 2.2
Assume that (F1) and (F2) hold. Let \(w(t,x,\psi )\) be a solution of (2.1) for \(\psi \in \mathbb{Y}^{+}\). Then the following statements hold:

(I)
If \(p\pi^{*}<1\) and \(\psi \in \mathbb{Y}^{+}\), then \(\lim_{t\rightarrow \infty } w(t,x,\psi )=0\).

(II)
If \(p\pi^{*}>1\), then (2.1) admits at least one positive homogeneous equilibrium solution \(w^{*}\in [0,M]\), and there exists a positive constant σ such that \(\liminf_{t\rightarrow \infty }w(t,x,\psi )\geq \sigma \) uniformly, for all \(\psi \in \mathbb{Y}^{+}\) and \(x\in [0,\pi ]\).
Remark 2.1
Assume that \(p\pi^{*}>1\) and assume that (F1) and (F2) hold. Let \(F(w)= \pi^{*}f(w)w\). Since \(f(w)\) satisfies (F1), \(F(0)=0\) and \(F^{\prime }(0)=p\pi^{*}1>0\). Moreover, since \(f(w)\) satisfies (F2), \(F(M)\leq 0\). Therefore, there exists some \(w^{*}\in (0,M]\) such that \(F(w^{*})=0\). Hence, \(w^{*}\) is a positive homogeneous equilibrium solution of (2.1).
The main result
In this section, we prove the global attractivity of the homogeneous equilibrium solution \(w^{*}\). To prove this result we apply the fluctuation method. First, we start with the following definition.
Definition 3.1
The function \(f(w):(0,M]\rightarrow \mathbb {R}\) satisfies the property (P); if for any \(u,v\in (0,M]\) with \(u\leq w^{*} \leq v\), \(u\geq \pi^{*} f(v)\), and \(v\leq \pi^{*} f(u)\), then we have \(v=u\).
Lemma 3.1
(Lemma 2.2 [7])
A function \(f(w)\) satisfies the property (P) if one of the following statements hold:

(P0)
\(f(w)\) is a nondecreasing function on \([0,M]\).

(P1)
\(wf(w)\) is a strictly increasing function on \((0,M]\).

(P2)
\(f(w)\) is a nonincreasing function for \(w\in [w^{*},M]\), and \(\frac{f(\pi^{*}f(w))}{w}\) is a strictly decreasing function for \(w\in (0,w^{*}]\).
To prove our main result, we need more assumptions on the birth function \(f(w)\); therefore we assume that

(F3)
\(f^{\prime }(0)>1\), \(\frac{f(w)}{w}\) is a strictly decreasing function \(\forall w \in (0,M]\), and \(f(w)\) satisfies the property (P).
Lemma 3.2
Let \(\psi \in \mathbb{Y}^{+}\) with \(\psi (t_{0},\cdot)\not \equiv 0\). Moreover, Let \(\omega (\psi )\) be the omega limit set of the positive orbits through ψ for the solution semiflow \(\varPsi (t)\). Then \(\mathbb{Z}_{M}\) is positively invariant, i.e., \(\varPhi (t)\mathbb{Z} _{M}\subset \mathbb{Z}_{M}\). In addition, \(\omega (\psi )\subset \mathbb{Z}_{M}\).
Proof
Let \(\psi \in \mathbb{Y}^{+}\) with \(\psi (t_{0},\cdot) \not \equiv 0\), and let \(\omega (\psi )\) be the omega limit set of the positive orbits through ψ for the solution semiflow \(\varPhi (t)\). Then the conclusion of Theorem 2.1 implies that \(\lim \sup_{t\rightarrow \infty }w(t,x,\psi )\leq M\), \(\forall x \in [0,\pi ]\). Hence, \(\varPhi (t)\mathbb{Z}_{M}\subset \mathbb{Z}_{M}\), and so, \(\omega (\psi )\subset \mathbb{Z}_{M}\). □
Theorem 3.1
Assume that \(p\pi^{*}>1\). Moreover, assume that (F1)–(F3) hold. Then, for any \(\psi \in \mathbb{Y}^{+}\) with \(\psi (t_{0},\cdot)\not \equiv 0\), we have \(\lim_{t\rightarrow \infty } w(t,x,\psi )=w^{*}\) uniformly \(\forall x \in [0,\pi ]\).
Proof. To prove the global attractivity of \(w^{*}\), by Lemma 3.2, it is sufficient to prove the global attractivity of \(w^{*}\) on \(\mathbb{Z}_{M}\). Therefore, let \(\psi \in \mathbb{Z}_{M}\) be such that \(\psi (t_{0},\cdot)\not \equiv 0\). Then the solution of (2.1) through ψ satisfies
Let \(w^{\infty }(x)= \limsup_{t\rightarrow \infty }w(t,x)\) and \(w_{\infty }(x)= \liminf_{t\rightarrow \infty } w(t,x)\) for any \(x\in [0,\pi ]\). Then \(w^{\infty }(x)\geq w_{\infty }(x)\). Since \(p\pi^{*}>1\), by Theorem 2.2, we have
Moreover, if we let \(w^{\infty }= \sup_{x\in [0,\pi ]}w^{\infty }(x)\) and \(w_{\infty }= \inf_{x\in [0,\pi ]}w_{\infty }(x)\), then \(0<\sigma \leq w_{\infty } \leq w^{\infty }\leq M\). Now, we define the diagonal function
Then \(F(u,v):[0,M]\times [0,M]\rightarrow \mathbb {R}\) is a continuous function, nondecreasing in \(u\in [0,M]\), nonincreasing in \(v\in [0,M]\), and \(f(w)=F(w,w)\); see, e.g., [22, Sect. 3.6]. Since the kernel function \(\varPi (a,x,y)\) is uniformly bounded for all \((a,x,y)\in [ \tau ,A_{l}]\times [0,\pi ]\times [0,\pi ]\), by Fatou’s lemma, we have
Thus,
Using the same argument, we have the following inequality:
Obviously, by the definition of \(F(u,v)\), there exist \(u,v\in [w_{ \infty },w^{\infty }]\subset [0,M]\) such that \(f(u)=F(w^{\infty },w _{\infty })\) and \(f(v)=F(w_{\infty },w^{\infty })\). Hence,
and
Consequently,
\(\frac{f(w)}{w}\) is assumed to be a strictly decreasing function on \((0,M]\). Then \(u\leq w^{*} \leq v\). Also, from (3.3) and (3.4), we get
and
That is,
Since \(f(w)\) satisfies the property (P), \(w^{*}=u=v\). Moreover, we have
So, \(w^{*}=w_{\infty }=w^{\infty }\). Recall that
Thus, \(w^{\infty }(x)= w_{\infty }(x)=w^{*}\), \(\forall x \in [0,\pi ]\). Hence,
To complete the proof, we need to show that \(\lim_{t\rightarrow \infty } w(t,x)=w^{*}\) uniformly \(\forall x \in [0,\pi ]\). In fact, it is enough to show that \(\omega (\psi )=\{w ^{*}\},\forall \psi \in \mathbb{Y}^{+}\). Let \(\eta \in \omega (\psi )\). By the definition of the omega limit set, there exists a positive time sequence \(t_{n}\rightarrow \infty \) such that \(\varPhi (t_{n})\psi \rightarrow \eta \) in \(\mathbb{Y}\) as \(n\rightarrow \infty \). This implies that
uniformly for \((s,x)\in [t_{0}A_{l},t_{0}]\times [0,\pi ]\). Hence, from (3.7), we have \(\eta (s,x)=w^{*}\), \(\forall (s,x)\in [t_{0}A _{l},t_{0}]\times [0,\pi ]\). It follows that \(\omega (\psi )=w^{*}\). Thus \(w(t,\cdot,\psi )\) converges to \(w^{*}\) in \(\mathbb{X}\) as \(t\rightarrow \infty \).
Examples
In this section, we present some examples to demonstrate the applicability of the main result. First, we begin with the Nicholson blowflies birth function \(f(w)=pwe^{aw^{q}}\) where a, p, and q are positive constants. Then we have the following theorem.
Theorem 4.1
Let \(f(w)=pwe^{aw^{q}}\), where \(a>0\), \(p>0\), and \(q>0\). Assume that \(1< \pi^{*} p\leq e^{\frac{2}{q}}\). Then the unique positive steady state solution \(w^{*}= [ \frac{1}{a}\ln (p\pi^{*}) ] ^{ \frac{1}{q}}\) attracts all positive solutions of (2.1).
Proof
First, we remark that \(f(w)\) satisfies the conditions (F1)–(F3), \(f(w)/w\) is a strictly decreasing function on \([0,\infty )\). Moreover, \(f^{\prime }(0)=p>0\), and \(f(w)\) takes its maximum at the point \(\overline{w}=(\frac{1}{aq})^{\frac{1}{q}}\) and \(f(\overline{w})=p( \frac{1}{aqe})^{\frac{1}{q}}\). Assume that \(1<\pi^{*}p\leq e^{ \frac{1}{q}}\), then \(f(w)\) is increasing function on \([0,w^{*}]\). Therefore, (P0) holds with \(M=w^{*}\). Now, assume that \(\pi^{*}p>e ^{\frac{1}{q}}\). In this case, we consider \(M=f(\overline{w})\). Hence, \(f(w)\) is decreasing function on \([w^{*},M]\). Moreover, the function
is a strictly decreasing function on \([0,u^{*}]\) if \(e^{\frac{1}{q}}< p \pi^{*} \leq e^{\frac{2}{q}}\). Thus, property (P2) holds, so the conditions of Theorem 3.1 are satisfied. As a result, \(w^{*}\) attracts every positive solution of (2.1). □
Next, we consider the Beverton–Holt function \(f(w)=\frac{pw}{1+aw ^{q}}\), \(a>0\), \(p>0\), and \(q>0\). Then we have the following theorem.
Theorem 4.2
Let \(f(w)=\frac{pw}{1+aw^{q}}\), where \(a>0\), \(p>0\), and \(q>0\). Assume that \(q\in ( 0,\max ( 2, \frac{p\pi^{*}}{p\pi^{*}1} ) ] \), or \(q>\max ( 2,\frac{p\pi^{*}}{p\pi^{*}1} ) \) and \(\pi^{*}f( \overline{w})\leq ( \frac{2}{a(q2)} ) ^{\frac{1}{q}}\); where \(p\pi^{*}>1\) and w̅ is the value where \(f(w)\) takes its maximum. Then the unique positive steady state solution \(w^{*}= ( \frac{p \pi^{*}1}{a} ) ^{1/q}\) attracts every positive solutions of (2.1).
Proof
First, we remark that \(f(w)\) satisfies the conditions (F1)–(F3), and \(f(w)/w\) is a strictly decreasing function on \([0,\infty )\). Moreover, \(f^{\prime }(0)=p>0\), and \(f(w)\) takes its maximum at \(\overline{w}=(\frac{1}{a(q1)})^{\frac{1}{q}}\) and \(f(\overline{w})=\frac{p(q1)}{q} \overline{w}\). Assume that \(q \in (0,1]\), then \(f(w)\) is monotone increasing on \([0,\infty )\), and hence, (P0) holds with \(M=w^{*}\). Now, if we assume \(1< q\leq 2\), then \(wf(w)\) is increasing function on \([0,\infty )\). Hence (P1) holds with \(M=w^{*}\). Moreover, if \(1< p\pi^{*} \leq \frac{q}{q1}\) (i.e., \(q \in (1,\frac{p\pi^{*}}{p\pi^{*}1})\)), then \(w^{*}\leq \overline{w}\). Hence, if we let \(M=w^{*}\), then (P0) holds. Conclusively, if \(q\in ( 0,\max ( 2, \frac{p\pi^{*}}{p\pi^{*}1} ) ] \), then either (P0) or (P1) holds. If \(q>\max ( 2,\frac{p\pi^{*}}{p\pi^{*}1} ) \), then \(h(w):=wf(w)=\frac{p w^{2}}{1+a w^{q}}\) is a monotone increasing function on \([ 0, ( \frac{2}{a(q2)} ) ^{\frac{1}{q}} ] \). Hence, if we consider \(M=\pi^{*}f(\overline{w})\), then (P1) holds provided that \(\pi^{*}f(\overline{w})\leq ( \frac{2}{a(q2)} ) ^{\frac{1}{q}}\). Thus, the conditions of Theorem 3.1 hold, and so, \(w^{*}\) attracts every positive solution of (2.1). □
Finally, we consider the logistic function \(f(w)=pw(1\frac{w}{K})\), where p and K are positive constants. Then we have the following theorem.
Theorem 4.3
Let \(f(w)=pw(1\frac{w}{K})\), \(p>0\), and \(K>0\) in (2.1). Moreover, assume that \(1< p\pi^{*}\leq 3\). Then the unique positive steady state solution \(w^{*}=K ( 1\frac{1}{p\pi ^{*}} ) \) attracts every positive solution of (2.1).
Proof
First, we remark that \(f(w)\) satisfies the conditions (F1)–(F3), and \(f(w)/w\) is a strictly decreasing function on \((0,K]\). Moreover, \(f^{\prime }(0)=p>0\), and \(f(w)\) takes its maximum at \(\overline{w}=\frac{K}{2}\) with \(f(\overline{w})=\frac{pK}{4}\). Assume that \(1< p\pi^{*}\leq 2\), then \(f(w)\) is a monotone increasing function on \([0,\frac{2}{K}]\), and hence, (P0) holds with \(M=u^{*}\). Assume that \(2< p\pi^{*}<4\), and let \(M=\frac{p\pi^{*}}{4}K\). Then the function
is a strictly decreasing function on \([0,w^{*}]\) provided that \(2< p\pi^{*}\leq 3\). Hence, the property (P2) holds. Therefore, the assumptions of Theorem 3.1 hold. Thus \(w^{*}\) attracts every positive solution of (2.1). □
Results and discussions
Since many biological aspects could cause a variation in the diffusion and death rates among different ages of the mature individuals, it is important to investigate the dynamics of the ecological model (1.1)–(1.3) when the diffusion and death rates are agedependent functions along the whole life of the species. For this purpose, the authors of [1] investigated (1.1)–(1.3) under this crucial assumption. In their paper they showed the existence of a unique positive and homogeneous equilibrium solution \(w^{*}\), and they proved its global stability when the birth function \(f(w)\) is monotone. If we assume that (F1)–(F2) hold, then (1.9) has a positive homogeneous equilibrium solution \(w^{*}\). Moreover, if we assume that (F3) holds and the inequality \(p\pi^{*}>1\) is satisfied, then \(w^{*}\) is attracting every positive solution of (2.1). To show the implication of this result, we applied it to three types of birth functions.
We address a particular case; let \(D(a)=D\) and \(d(a)=d\), where D and d are positive constants, and let \(f(w)=pwe^{aw}\). Assume the life span of the species is large (i.e., \(A_{l}\gg 1\)) and suppose that \(1< \pi^{*} p\leq e^{2}\). Since \(A_{l}\gg 1\), \(\pi^{*}\sim \frac{e ^{d\tau }}{d}\). Thus, by Theorem 4.1, the positive equilibrium solution \(w^{*}=\frac{1}{a} \ln ( \frac{pe^{d\tau }}{d} ) \) is globally stable. Hence, by Theorem 2.2 and Theorem 4.1, the following two statements are valid:

(I)
if \(pe^{d\tau }< d\), then the trivial solution is attracting every positive solution of (2.1);

(II)
if \(1<\frac{pe^{d\tau }}{d}\leq e^{2}\), then the equilibrium solution \(w^{*}=\frac{1}{a} \ln ( \frac{pe^{d\tau }}{d} ) \) is attracting every positive solution of (2.1).
Here, we remark that a similar result to above threshold dynamics can be found in [28] when the diffusion and death rates are age independent.
In the paper, we employed the method of fluctuation to prove the global attractivity of a positive and homogeneous equilibrium solution \(w^{*}\) of (1.9) in the case that \(f(w)\) is a nonmonotone function.
References
 1.
AlJararha, M., Ou, C.: Populations dynamics with agedependent diffusion and death rates. Eur. J. Appl. Math. 24, 471–500 (2013)
 2.
Diekmann, O.: Thresholds and traveling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1978)
 3.
Gourley, S.A., Kuang, Y.: Wavefronts and global stability in a timedelayed population with stage structure. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 459(2034), 1563–1579 (2003)
 4.
Gourley, S.A., So, J.W.H.: Extinction and wavefront propagation in a reaction–diffusion model of a structured population with distributed maturation delay. Proc. R. Soc. Edinb. 133A, 527–548 (2003)
 5.
Gourley, S.A., So, J.W.H., Wu, J.: Nonlocality of reaction–diffusion equations induced by delay: biological modeling and nonlinear dynamics. J. Math. Sci. 124(4), 5119–5153 (2004)
 6.
Gourley, S.A., Wu, J.: Delayed nonlocal diffusive systems in biological invasion and disease spread. In: Nonlinear Dynamics and Evolution Equations. Fields Inst. Commun., vol. 48, pp. 137–200. Amer. Math. Soc., Providence, RI (2006). https://doi.org/10.1090/fic/048/06
 7.
Hsu, S.B., Zhao, X.Q.: Spreading speeds and traveling waves for nonmonotone integrodifference equations. SIAM J. Math. Anal. 40, 776–789 (2008)
 8.
Huddleston, J.V.: Population dynamics with age and timedependent birth and death rates. Bull. Math. Biol. 45(5), 827–836 (1983)
 9.
Liang, D., So, J.W.H., Zhang, F., Zou, X.: Population dynamics models with nonlocal delay on bounded domains and their numerical computation. Differ. Equ. Dyn. Syst. 11(1–2), 117–139 (2003)
 10.
Liang, D., Wu, J.: Traveling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects. J. Nonlinear Sci. 13, 289–310 (2003)
 11.
Mei, M., So, J.W.H.: Stability of strong traveling waves for nonlocal timedelayed reaction diffusion equation. Proc. R. Soc. Edinb. 138A, 551–568 (2008)
 12.
Metz, J.A.J., Diekmann, O. (eds.): The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, vol. 68. Springer, Berlin (1986). https://doi.org/10.1007/9783662131596
 13.
Ou, C., Wu, J.: Existence and uniqueness of a wavefront in a delayed hyperbolicparabolic model. Nonlinear Anal. 63(3), 364–387 (2005)
 14.
Ou, C., Wu, J.: Persistence of wavefronts in delayed nonlocal reaction–diffusion equation. J. Differ. Equ. 235, 219–261 (2007)
 15.
Ou, C., Wu, J.: Traveling wavefronts in a delayed foodlimited population model. SIAM J. Math. Anal. 39(1), 103–125 (2007)
 16.
Ou, G., Wu, J.: Spatial spread of rabies revisited: influence of agedependent diffusion on nonlinear dynamics. SIAM J. Appl. Math. 67(1), 138–163 (2006)
 17.
Smith, H., Thieme, H.: Strongly order preserving semiflows generated by functional differential equations. J. Differ. Equ. 93, 332–363 (1991)
 18.
Smith, H., Thieme, H.: Dynamical Systems and Population Persistence. Graduate Studies in Mathematics, vol. 118. Amer. Math. Soc., New York (2011)
 19.
Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Math Surveys and Monographs, vol. 41. American Mathematical Society, Providence, RI (1995)
 20.
So, J.W.H., Wu, J., Zou, X.: A reaction–diffusion model for a single species with agestructured. I. Traveling wavefronts on unbounded domains. Proc. R. Soc. Lond. A 457, 1841–1853 (2001). https://doi.org/10.1098/rspa.2001.0789
 21.
Thieme, H.R.: Persistence under relaxed pointdissipativity with applications to an endemic model. SIAM J. Math. Anal. 24(2), 407–435 (1993)
 22.
Thieme, H.R.: Densitydependent regulation of spatially distributed populations and their asymptotics speed of spread. J. Math. Biol. 195, 173–187 (2003)
 23.
Thieme, H.R., Zhao, X.Q.: A nonlocal delayed and diffusive predator–prey model. Nonlinear Anal., Real World Appl. 2, 145–160 (2001)
 24.
Thieme, H.R., Zhao, X.Q.: Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion model. J. Differ. Equ. 195, 430–470 (2003)
 25.
Wu, J.: Theory and Applications of Partial Functional Differential Equations. Applied Math. Sci., vol. 119. Springer, New York (1996)
 26.
Xu, D., Zhao, X.Q.: A nonlocal reaction–diffusion population model with stage structure. Can. Appl. Math. Q. 11(3), 303–319 (2003)
 27.
Yi, T., Zou, X.: Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary conditions: a nonmonotone case. J. Differ. Equ. 245, 3376–3388 (2008)
 28.
Yi, T., Zou, X.: Global dynamics of a delay equation with spatial nonlocality in an unbounded domain. J. Differ. Equ. 251, 2598–2611 (2011)
 29.
Yu, J., Zhao, X.Q.: Spatial dynamics of a nonlocal periodic reaction–diffusion model with stage structure. SIAM J. Math. Anal. 40(6), 2496–2516 (2009)
 30.
Zhao, X.Q.: Dynamical Systems in Population Biology. Springer, New York (2003)
 31.
Zhao, X.Q.: Global attractivity in a class of nonmonotone reaction–diffusion equations with time delay. Can. Appl. Math. Q. 17(1), 271–281 (2009)
Acknowledgements
The author would like to thank the reviewers for their valuable comments on the paper. Also, the author would like to thank the deanship of graduate studies and scientific research at Yarmouk University for the partial support of this research.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Author information
Affiliations
Contributions
The author read and approved the final manuscript.
Corresponding author
Correspondence to M. AlJararha.
Ethics declarations
Competing interests
The author declares that he has no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
MSC
 34K10
 35K55
 35B35
 92D25
Keywords
 Homogeneous equilibrium solution
 Global attractivity
 Agestructured model
 Nonmonotone birth function
 Agedependent death rate
 Agedependent diffusion rate