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Global attractivity in a non-monotone age-structured model with age-dependent diffusion and death rates
- M. Al-Jararha^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1884-4
© The Author(s) 2018
- Received: 4 June 2018
- Accepted: 9 November 2018
- Published: 16 November 2018
Abstract
Keywords
- Homogeneous equilibrium solution
- Global attractivity
- Age-structured model
- Non-monotone birth function
- Age-dependent death rate
- Age-dependent diffusion rate
MSC
- 34K10
- 35K55
- 35B35
- 92D25
1 Introduction
To study the dynamics of (1.9), particularly, the existence and the global stability of the homogeneous equilibrium solution of (1.9), Al-Jararha 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 non-monotone 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 non-local time-delayed reaction–diffusion predator–prey model, and later it has been used in many mathematical articles to deal with the non-monotone dynamics difficulties; e.g., see [7, 18, 21–24, 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.
2 Preliminaries
- (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)
By following the same argument in the proof of Lemma 6.1 [1], we have the following theorem.
Theorem 2.1
- (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
- (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).
3 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])
- (P0)
\(f(w)\) is a non-decreasing function on \([0,M]\).
- (P1)
\(wf(w)\) is a strictly increasing function on \((0,M]\).
- (P2)
\(f(w)\) is a non-increasing function for \(w\in [w^{*},M]\), and \(\frac{f(\pi^{*}f(w))}{w}\) is a strictly decreasing function for \(w\in (0,w^{*}]\).
- (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 \(\forall x \in [0,\pi ]\).
4 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
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 \(\pi^{*}f( \overline{w})\leq ( \frac{2}{a(q-2)} ) ^{\frac{1}{q}}\); 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 \(\overline{w}=(\frac{1}{a(q-1)})^{\frac{1}{q}}\) and \(f(\overline{w})=\frac{p(q-1)}{q} \overline{w}\). 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 \(1< p\pi^{*} \leq \frac{q}{q-1}\) (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 \([ 0, ( \frac{2}{a(q-2)} ) ^{\frac{1}{q}} ] \). Hence, if we consider \(M=\pi^{*}f(\overline{w})\), then (P1) holds provided that \(\pi^{*}f(\overline{w})\leq ( \frac{2}{a(q-2)} ) ^{\frac{1}{q}}\). 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
5 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 age-dependent 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.
- (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).
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 non-monotone function.
Declarations
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.
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Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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