# On multi-periodicity in a delayed model of hematopoiesis

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## Abstract

In this paper, we study a periodic model of hematopoiesis with a time-varying delay. Some new criteria are established to ensure that there are at least two positive periodic solutions by applying Krasnoselskii’s fixed point theorem, which are essentially new and complement some existing ones. Moreover, numerical simulations are performed to substantiate the effectiveness of the theoretical analysis.

## Keywords

Hematopoiesis Multi-periodicity Krasnoselskii’s fixed point theorem## MSC

34G20 34K13## 1 Introduction

*a*is the rate at which cells are lost from the circulation, the flux \(g(x(t-\tau))=\frac{bx^{m}(t-\tau)}{1+x^{n}(t-\tau)}\) of the cells into the circulation from the stem cell compartment depends on \(x(t-\tau)\) at time \(t-\tau\), and

*τ*is the time delay between the production of immature cells in the bone marrow and their maturation for release in circulating bloodstreams.

However, it is noteworthy that when \(0< m< n\), the flux function in model (1.2) has stronger nonlinearity than the cases of \(m=0\) or \(m=1\), and thus it may show more complex and rich dynamic behaviors. On the other hand, the aforementioned periodic solution (can be regarded as a special case of almost periodic solution) is unique, as mentioned by May in [13] that a large number of empirical observations shows that many natural communities have a multiplicity of stable states. The multiplicity of periodic solutions is an interesting problem in the qualitative study of delay differential equations, and such an issue of Eq. (1.2) has been seldom considered up to now. Motivated by the above discussions, in this paper we aim to establish some sufficient conditions ensuring that Eq. (1.2) has at least two positive *T*-periodic solutions. Our approach is based on Krasnoselskii’s fixed point theorem.

The structure of the remaining part of this paper is as follows. In Sect. 2, we present some necessary lemmas. In Sect. 3, some sufficient conditions are established to guarantee that Eq. (1.2) has at least two positive periodic solutions. In Sect. 4, we demonstrate the validity of these theoretical results with numerical simulations. Finally, some conclusions are made and future directions are pointed out in Sect. 5.

## 2 Preliminaries

In this section, we first introduce some notations and recall well-known Krasnoselskii’s fixed point theorem.

*T*-periodic function, we denote

On the other hand, let \(g(x)=\frac{x^{m}}{1+x^{n}}\), if \(0< m< n\), we can easily verify that \(g(x)\) increases strictly on \([0, \sqrt[n]{\frac{m}{n-m}} ]\) and decreases on \([ \sqrt[n]{\frac{m}{n-m}}, \infty )\). Thus, there exists a unique \(c_{0}\in (\sqrt[n]{\frac{m}{n-m}}, \infty )\) such that \(g(c_{0})=g (\rho\sqrt[n]{\frac{m}{n-m}} )\), where \(0<\rho<1\).

### Definition 2.1

*X*be a Banach space, and let

*P*be a closed, nonempty subset of

*X*.

*P*is a cone if

- (i)
\(\alpha x+\beta y \in P\) for all \(x, y\in P\) and all \(\alpha , \beta\geq 0\);

- (ii)
\(x, -x\in P\) imply \(x=0\).

### Lemma 2.1

*Let*

*X*

*be a Banach space*,

*and let*\(P\subset X\)

*be a cone in*

*X*.

*Assume that*\(\varOmega_{1}\), \(\varOmega_{2}\)

*are open bounded subsets of*

*X*

*with*\(0\in\varOmega_{1}\subset\overline{\varOmega}_{1}\subset\varOmega_{2}\),

*and let*

*be a completely continuous operator such that either*

- (i)
\(\|\varPhi x\|\leq\|x\|\), \(\forall x\in P\cap\partial \varOmega_{1}\)

*and*\(\|\varPhi x\|\geq\|x\|\), \(\forall x\in P\cap\partial\varOmega_{2}\);

*or*

- (ii)
\(\|\varPhi x\|\geq\|x\|\), \(\forall x\in P\cap\partial \varOmega_{1}\)

*and*\(\|\varPhi x\|\leq\|x\|\), \(\forall x\in P\cap\partial\varOmega_{2}\).

*Then*

*Φ*

*has a fixed point in*\(P\cap(\overline{\varOmega}_{2}\setminus\varOmega_{1})\).

*X*is a Banach space equipped with the above norm \(\|\cdot\|\). If \(x(t)\in X\) is a solution of Eq. (1.2), then

*T*-periodic solution, it suffices to prove that

*Φ*has a fixed point on

*X*. To establish the main results, we also make the following assumptions:

- (H1)
\(a, b, \tau\in C(\mathbb{R}, (0, \infty)) \) are all

*T*-periodic functions; - (H2)
\(Nb^{-}Tg(c_{0}) >c_{0}\).

### Lemma 2.2

*The mapping**Φ**maps**P**into**P*, *that is*, \(\varPhi P\subset P\).

### Proof

### Lemma 2.3

\(\varPhi: P\to P\)*is completely continuous*.

### Proof

*Φ*is continuous. For any \(L > 0\) and \(\varepsilon> 0\), there exists \(\delta> 0\) such that, for \(\varphi, \psi\in X\), \(\|\varphi\|\leq L\), \(\|\psi\|\leq L\), and \(\|\varphi-\psi\|<\delta\) imply

*Φ*is continuous.

*Φ*is compact. Let \(B > 0\) be any constant, and let \(\mathscr{T}=\{x\in X: \|x\|\leq B\}\) be a bounded set . For any \(x\in\mathscr{T}\), it follows from (2.2) and (2.3) that

*Φ*is compact, and so it is completely continuous. The proof is completed. □

## 3 Main results

We are now in a position to state and prove our main results of this paper.

### Theorem 3.1

*Let*\(0< m< n\)*and* (H1)*–*(H2) *hold*. *Then Eq*. (1.2) *has at least two positive**T*-*periodic solutions*.

### Proof

Obviously, \(\varOmega_{i}\) (\(i=1,2,3,4\)) are open bounded subsets of *X* with \(0\in\varOmega_{1}\subset\overline{\varOmega}_{1}\subset\varOmega_{2} \subset\overline{\varOmega}_{2}\subset\varOmega_{3}\subset \overline{\varOmega}_{3}\subset\overline{\varOmega}_{4}\). Since \(\varPhi(P)\subset P\) and *Φ* is a completely continuous operator on *X*, we conclude from Lemma 2.1 that *Φ* has one fixed point \(x_{1}\in P\cap(\overline{\varOmega}_{2}\setminus\varOmega_{1})\) and another fixed point \(x_{2}\in P\cap(\overline{\varOmega}_{4}\setminus\varOmega_{3})\), that is, \(x_{i}(t)=(\varPhi x_{i})(t)\), \(i=1,2\), and \(x_{1}(t)\geq\rho c_{1}>0\) and \(x_{2}(t)\geq\rho c_{0}>0\), i.e., \(x_{1}(t)\) and \(x_{2}(t)\) are two positive *T*-periodic solutions of Eq. (1.2). The proof is completed. □

### Remark 3.1

### Theorem 3.2

*Let*\(0< m< n\)*and* (H2) *hold*. *Then Eq*. (3.3) *has at least two positive**T*-*periodic solutions*.

## 4 A numerical example

In this section, we give a numerical example with simulations to illustrate the feasibility of our main results.

### Example 4.1

*π*-periodic model of hematopoiesis with a time-varying delay:

*π*-periodic solutions, see Fig. 1.

### Remark 4.1

In recent years, by using the continuation theorem, the existence of multiple periodic solutions of delayed population models has widely been studied (see [16, 17] and the references therein), and the multiplicity is heavily dependent on the harvesting term. It is readily seen that our methods are quite different from the previous works and the considered model is without the harvesting term. On the other hand, to the best of authors’ knowledge, there is no research work concerning the multiplicity of periodic solutions of Eq. (1.2). Therefore, the results established in this paper are essentially new and complement some existing ones.

## 5 Conclusion

In this paper, we have studied the multiplicity of periodic solutions for a delayed model of hematopoiesis, a new set of criteria ensuring the existence of at least two periodic solutions have been derived. The effectiveness of the theoretical results has been demonstrated by a numerical example.

It is known that almost periodic problem is a hot research topic in science [18, 19] and engineering [20, 21]. However, it would be more difficult to find the sufficient condition for the multiplicity of almost periodic solutions than the periodic case since the compact condition fails the almost periodic function family, and then Krasnoselskii’s fixed point theorem controlled by compact conditions cannot be used to solve the existence of almost periodic solutions. Therefore, interesting problems of the existence and stability of multiple almost periodic solutions for the kinds of models described by delay differential equations are still open, and we leave them as our future research work.

## Notes

### Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their constructive comments. This revised form was finished when the first author was visiting Prof. Xianhua Tang at Central South University, and he would like to thank the staff in the School of Mathematics and Statistics for their help and the university for its excellent facilities and support during his stay.

### Authors’ contributions

All authors contributed equally to this manuscript. All authors read and approved the final manuscript.

### Funding

This work was jointly supported by the National Natural Science Foundation of China (11701007, 61572035), Natural Science Foundation of Anhui Province (1808085QA01), Key Program of University Natural Science Research Fund of Anhui Province (KJ2017A088), China Postdoctoral Science Foundation (2018M640579), Key Program of Scientific Research Fund for Young Teachers of AUST (QN201605), Open Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (2018MMAEZD17), and the Student’s Platform for Innovation and Entrepreneurship Training Program (201810361096).

### Competing interests

The authors declare that they do not have any competing interests in this manuscript.

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