386 

© Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This 

 is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License 

 

 
  A Mathematical Approach to Oscillation of a Discrete Hematopoiesis Model 

with Positive and Negative Coefficients 

 

Iman Sabeeh Hadeed1* , Hussain Ali Mohamad2   
1Department of Mathematics, College of Sciences, University of Baghdad, Baghdad, Iraq. 

2Department of Mathematics, College of Sciences for women, University of Baghdad, Baghdad, Iraq. 

*Corresponding Author.  

 

 Received:19 April 2023                            Accepted:12 June 2023                        Published:20 July 2024  

 doi.org/10.30526/37.3.3432 

 
 Abstract   

     In this paper, we investigate a mathematical model of hematopoiesis, a process responsible for 

the regular replacement of circulating blood cells. Since differential delay equations are difficult to 

control analytically, numerous studies have considered the models as difference equations. The 

main objective of this work is to provide the necessary and sufficient conditions for oscillation. We 

address our problem mainly on the basis of oscillatory behavior. Moreover, the latest findings on 

the qualitative behavior of the biological mathematical model of discrete hematopoiesis are taken 

into account. More specifically, we explain the mathematical differential equation of discrete 

hematopoiesis. Moreover, certain significant, necessary and sufficient criteria for the solution of 

this discrete problem are found, which guarantee either the convergence of the non-oscillating 

solutions towards zero or the oscillation of all solutions of the discrete hematopoiesis model to the 

nonlinear lag difference with positive and negative coefficients. Some numerical examples are also 

given to illustrate the most important results. 

Keywords: oscillation, delay differential equations, difference equations, delay differential 

equations, hematopoiesis model. 

 
 

1. Introduction 

     The oscillatory behavior of difference equations' and dynamic equations' solutions has recently 

been the subject of a lot of research. Numerous recent works have been focused on oscillations of 

delay differential equation (DDE) solutions among these investigations [1-12]. This discovery has 

garnered a lot of interest since it has numerous practical applications in mathematical models of 

biology, ecology, and the transmission of various infectious diseases in people, and other areas. The 

reader can turn to [13-22] and the references therein for more details on this study. The behavior of 

the solutions for DDE has received a lot of attention in relation to the study of the oscillations of 

the analytical solutions. To the best of our knowledge, there have been very few studies that address 

the oscillations of nonlinear DDE solutions. In our work, we focus on this subject. An equation or 

a system of equations used to describe a natural occurrence is known as a mathematical model. 

Numerous scholars investigate how nonlinear delay mathematical models behave qualitatively in 

single species as well as in species that interact. 

https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0002-1059-7790
mailto:%20iyman.sobaih1103a@sc.uobaghdad.edu.iq
https://orcid.org/0000-0002-4684-786x
mailto:hussainam_math@scw.uobaghdad.edu.iq.


IHJPAS. 2024, 37( 3 ) 

387 
 

There has been a great deal of research done on the qualitative analysis of delay models with 

constant coefficients (autonomous models). We are aware that a variety of biological and 

ecological dynamical systems heavily depend on the environment. For instance, consider the 

physical environment's elements, like temperature and humidity, as well as the accessibility of 

resources like food, water, and other essentials, typically change with time on a seasonal or daily 

basis. Therefore, nonautonomous systems would have more accurate representations [2,20]. 

Studying the oscillation and non- oscillation of a certain kind of nonautonomous hematopoiesis 

delay model in biology is one of the goals of our work. Many authors looked for sufficient 

conditions to ensure oscillatory properties for different differential equations. Hematopoiesis refers 

to the process of producing blood cells. In 1978, Mackey and Glass proposed the first mathematical 

models of hematopoiesis dynamics [23]. In Saker, S. H. [24], consider the following equation of 

the hematopoiesis model with positive and negative coefficients: 

ℋ′(𝑡) =
𝛽(𝑡)

1 + ℋ𝑚(𝑡 − 𝜏)
− 𝛿(𝑡)ℋ(𝑡),     𝑡 ≥ 0 .                                                                                  (1) 

where 𝛽(𝑡), 𝛿(𝑡) ∈ ∁([0, ∞), 𝑅+), 𝜏 ∈  [0, ∞) , 𝑚 ∈ 𝑁. Equation (1) has a unique positive 

equilibrium point 𝐾, and satisfies the equation  

𝛽

1 + 𝐾𝑚
= 𝛿𝐾.                                                                                                                                               (2) 

Since differential delay equations are challenging to control analytically, numerous studies have 

looked at the models as difference equations. So, there are a number of analytical results 

concerning the oscillation, global attractivity, and periodicity of Equation (1) [20,23]. The 

corresponding nonlinear first order delay difference equation with positive and negative 

coefficients of Equation (1) in the discrete hematopoiesis model is:  

∆𝐻𝑛 + 𝛿𝑛𝐻𝑛 − 𝛽𝑛𝐺(𝐻𝑛−𝑙) = 𝑓𝑛 .                                                                                                             (3) 

Where𝛿𝑛,  𝛽𝑛 and 𝑓𝑛 are infinite sequences of real numbers and ∆ is the forward difference 

operator. And  𝐺(𝐻𝑛) =
1

1+𝐻𝑛
𝑚 ∈ (𝑅, 𝑅) is a flux function that depends on the size of cells 𝐻𝑛 and 

𝐻𝑛−𝑙at times n and n – l, respectively, and l is the time of maturation, such as that 𝐻𝑛𝐺(𝐻𝑛) > 0.  

There are some studies on the discrete hematopoiesis model, for instance. Wang et. al. (2013) [23]: 

With the assistance of two 𝜃-methods, it was possible to discuss the conditions under which the 

numerical solutions fluctuate for the nonlinear delay differential equations in the hematopoiesis 

model. Additionally, it has been established that every non-oscillatory numerical solution tends to 

an equilibrium point of (3). [25]: established sufficient conditions for the existence of at least three 

positive T-periodic solutions for a discrete delay hematopoiesis model. Wei Li and Xianyi Li 

(2018) [15]: They derived a semi-discrete system for a nonlinear model of blood cell production. 

In [15], the authors discovered a few prerequisites for the oscillation of all ∆𝑌(𝑛) +

𝑝(𝑛)𝑌(𝜏(𝑛)) = 0 solutions of the linear difference equation with varying delays. Every solution 

of first order linear difference equations with positive and negative coefficients of the form is given 

sufficient conditions to oscillate [26]. 

Ψ𝑛+1 − Ψ𝑛 + ℛΨ𝑛−𝑘 − 𝑄Ψ𝑛−𝑙 = 0,   𝑛 ∈ 𝑁.                                                                                         (4) 

Every solution to Equation (4) oscillates if  𝑄(𝑘 −  𝑙)  <  1 and  
(ℛ− 𝑄)(𝑘 + 1)𝑘+1

𝑘𝑘  >  1. The authors 

discovered sufficient conditions for the oscillation of each solution of first-order linear difference 

equations with multiple positive and negative coefficients in [27], extending the findings from 

[26]. 



IHJPAS. 2024, 37( 3 ) 

388 
 

Ψ𝑛+1 − Ψ𝑛 + ∑ ℛ𝑖Ψ𝑛−𝑘𝑖
− 𝑄𝑖Ψ𝑛−𝑙𝑖

𝑚

𝑖=1
= 0, 𝑛 ∈ 𝑁0 .                                                                        (5) 

It is proved that if  ∑ 𝑄𝑖(𝑘𝑖 −  𝑙𝑖)
𝑚
𝑖=0 < 1 and ∑

(ℛ𝑖− 𝑄𝑖)(𝑘𝑖 + 1)𝑘𝑖+1

𝑘𝑖
𝑘𝑖

𝑚
𝑖=0 > 1, then every solution of 

Equation (5) oscillates. According to Mohamad (2015) [4], all of the solutions to the first-order 

neutral difference equation with positive and negative coefficients will oscillate if the necessary 

and sufficient requirements are satisfied. The authors of [28] provided sufficient conditions to 

ensure that all solutions to Equation (4) are oscillatory or tend to zero. The oscillations and global 

attractivity of Equation (3) with periodic time coefficients were investigated by Agarwal and Saker 

[24]. A mathematical model of hematopoietic stem cell dynamics is also proposed and investigated 

in [29], which considers the dynamics of two cell populations: an immature population and a 

mature population.  

Our study includes recent findings on the qualitative behavior of a biological mathematical model 

of discrete hematopoiesis. Let's introduce an invariant oscillation transformation based on the 

analogous procedure in [20]. 

 𝐻𝑛 =  𝑈𝑛 − 𝐾,                                                                                                                                              (6)  

where 𝐾 is the unique positive equilibrium point of Equation (1). Then Equation (3) can be reduced 

to  

∆𝑈𝑛 + 𝛿𝑛𝑈𝑛 − 𝛽𝑛𝑉(𝑈𝑛−𝑙) = 𝜉𝑛 ,                                                                                                            (7) 

Where  𝑉(𝑈𝑛) =
1

1+(𝑈𝑛+𝐾)𝑚 and 𝜉𝑛 = 𝑓𝑛 − 𝛿𝑛𝐾. Then 𝐻𝑛oscillates about K if and only if 𝑈𝑛 

oscillates about zero.  The following arguments are established in this article: 

𝐴1 : ∑ 𝛽𝑛

∞

𝑛=0
< ∞ .  

𝐴2 :  
 𝑉(휁𝑛)

휁𝑛
≤ 𝜆2. 

𝐴3: There exists a sequence 𝐹𝑛 such that  ∆𝐹𝑛 = 𝜉𝑛 and lim
𝑛→∞

𝐹𝑛 = 0 .  

𝐴4 : ∑ 𝛿𝑛

∞

𝑛=𝑛∗

< ∞, 𝑛∗ ≥ 𝑛0.  

For the conditions produced 𝐴1, 𝐴2, 𝐴3 is the assertion that there is no non-oscillating solution that 

achieves the equation of hematopoiesis or the resulting inequality from it. In return, the biological 

interpretation of these conditions is like the doses that are given to blood cells in order to control 

the number of cells produced. A sequence, 𝑈𝑛 satisfying Equation (7) for 𝑛 > 0 is referred to as a 

solution of Equation(7). If there is 𝑛 > 𝑛𝑗 such that, 𝑈𝑛. 𝑈𝑛+1 > 0, then a nontrivial solution, 𝑈𝑛. 

is said to be oscillatory (oscillating about zero). Otherwise, it is argued that the solution is 

nonoscillatory [30]. In this work, several new necessary conditions are found to cause oscillations 

or convergence to equilibrium 𝐾 in the solutions and bounded solutions of Equation (3). The 

theoretical results are supported by a few examples. 

 

 

 

 

 

 



IHJPAS. 2024, 37( 3 ) 

389 
 

2.  Background 

Definition 1 [30] A point 𝐾 in the domain of 𝐻𝑛 is said to be an equilibrium point of (3) if it is a 

fixed point of 𝐻𝑛, that is, 𝐻𝑛(𝐾)  =  𝐾 .  

Definition 2 [20] If a solution to Equation (3), 𝐻𝑛 oscillates around equilibrium 𝐾, then 𝐻𝑛 − 𝑘 

𝐾 oscillate about zeros. If not, 𝐻𝑛 is regarded as non-oscillatory. When 𝐾 = 0, we say that 𝐻𝑛 

simply oscillates or oscillates about zero. 

Based on Theorem 1in [4], the result follows. 

 

3.  Results   and Discussion 

      In this section, some sufficient conditions are established for the oscillation of all solutions to   

Equation (3). It is simple to demonstrate that 𝐻𝑛 oscillates about K if and only if 𝑈𝑛 .We just need 

to take into account the oscillations of Equation (3) in order to investigate the oscillations of 

Equation (7).  

Let the sequence 𝑍𝑛 be defined as 

𝑍𝑛 = 𝑈𝑛 + ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
𝑉(𝑈𝑖−𝑙) − 𝐹𝑛  .                                                                                                     (8) 

The following lemma helps to demonstrate the main findings. 

Lemma 1. suppose,  𝛿𝑛 − 𝛽𝑛+𝑙𝜆2 ≥ 0 and assume that  (𝐴1) − (𝐴3) hold. Let  𝐻𝑛 be a 

nonoscillatory solution to 𝐾 of Equation (3). Then , 𝑍𝑛 ≥ 0 and a nonincreasing sequence. 

Proof. Let  𝐻𝑛 be a nonoscillatory solution to 𝐾 of Equation (3), that is  𝐻𝑛 > 𝐾, 𝑛 ≥ 𝑛0, (the 

proof of the case 0 < 𝐻𝑛 < 𝐾 is similar and will be omitted), hence 𝑈𝑛 > 0 eventually. From 

Equation (8), we obtain 

 ∆𝑍𝑛 = −𝛿𝑛𝑈𝑛 + 𝛽𝑛+𝑙𝑉(𝑈𝑛) ≤ −(𝛿𝑛 − 𝛽𝑛+𝑙𝜆2)𝑈𝑛 ≤ 0  .                                                                (9) 

Hence, 𝑍𝑛 is a nonincreasing sequence, and lim
𝑛→∞

𝑍𝑛  =  𝐿, where −∞ ≤ 𝐿 < ∞. We claim that 

𝐿 ≥ 0. Otherwise 𝐿 < 0, so there exists 𝑛1 ≥ 𝑛0 and 𝛼 < 0, such that 𝑍𝑛 ≤ 𝛼 < 0  for  𝑛 ≥ 𝑛1. 

From Equation (8), we get 

𝑈𝑛 = 𝑍𝑛 − ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
𝑉(𝑈𝑖−𝑙) + 𝐹𝑛 , 

𝑈𝑛 ≤ 𝛼 − ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
𝑉(𝑈𝑖−𝑙) + 𝐹𝑛 , 

𝑈𝑛 ≤ 𝛼 + 𝐹𝑛 < 𝛼 + 휀,   휀 > 0,   𝑛 ≥ 𝑛2 ≥ 𝑛1.                                                                                     (10) 

Since 휀 is arbitrary, then Equation (10) leads to  𝑈𝑛 ≤ 𝛼 which is a contradiction since 𝑈𝑛 is 

positive. So, since our claim has been proven,  𝐿 ≥ 0, or  𝑍𝑛 ≥ 0, follows. 

 

Theorem 1. Assume  𝛿𝑛 − 𝛽𝑛+𝑙𝜆2 ≥ 0 and let (𝐴1) − (𝐴3)  hold, and 

limsup
𝑛→∞

∑ 𝛿𝑖

𝑛−1

𝑖=𝑛∗

= ∞    , 𝑛∗ ≥ 𝑛0  .                                                                                                        (11) 

Then every solution of Equation (3) either oscillates about a unique equilibrium 𝐾 or 

nonoscillatory tends to 𝐾 as 𝑡 → ∞. 

Proof. Assume that Equation (3) posse nonoscillatory solution 𝐻𝑛 about 𝐾, Let 𝐻𝑛 > 𝐾, 𝑛 ≥ 𝑛0, 

hence 𝑈𝑛 > 0  eventually. Then by Lemma 1:  

𝑍𝑛 ≥ 0 and ∆𝑍𝑛 ≤ 0, hence  lim
𝑛→∞

𝑍𝑛 = 𝐿, where −∞ ≤  𝐿 <  ∞. We look at two situations: 

Case 1. If 𝑈𝑛 is unbounded. So there exists a subsequence  𝑛𝑗  of 𝑛 such that 

lim
𝑗→∞

𝑛𝑗 = ∞, lim
𝑗→∞

𝑈𝑛𝑗
= ∞ and  𝑈𝑛𝑗

= max {𝑈𝑠: 𝑛0 < 𝑠 < 𝑛𝑗}. By using (𝐴4), we obtain from 

Equation (8); 



IHJPAS. 2024, 37( 3 ) 

390 
 

𝑍𝑛𝑗
= 𝑈𝑛𝑗

+ ∑ 𝛽𝑖

𝑛𝑗+𝑙−1

𝑖=𝑛𝑗

𝑉(𝑈𝑖−𝑙) − 𝐹𝑛𝑗
 . 

≥ 𝑈𝑛𝑗
− 𝐹𝑛𝑗

 

Hence lim
𝑗→∞

𝑍𝑛𝑗
= ∞ leads to a contradiction. 

Case 2. If  𝑈𝑛 is bounded. From Equation (9) and condition (𝐴2), we have ∆𝑍𝑛 ≤
−( 𝛿𝑛 − 𝛽𝑛+𝑙𝜆2)𝑈𝑛 .  

Let  liminf
𝑛→∞

𝑈𝑛 = 𝑙 ≥ 0. By taking summation to both sides of the above inequality, it yields:  

∑ ∆𝑍𝑖 ≤ − ∑ ( 𝛿𝑖 − 𝛽𝑖+𝑙𝜆2)𝑈𝑖
𝑛−1
𝑖=𝑛0

𝑛−1
𝑖=𝑛0

. 

Then  

𝑍𝑛 − 𝑍𝑛0
≤ − ∑ ( 𝛿𝑖 − 𝛽𝑖+𝑙𝜆2)𝑈𝑖

𝑛−1

𝑖=𝑛0

  .                                                                                              (12) 

We claim that 𝑙 = 0, otherwise if  𝑙 > 0, then there exists  𝑛1 ≥ 𝑛0 large enough such that 𝑈𝑛 ≥
𝑙, 𝑛 ≥ 𝑛1. 
From Equation (12), we have  

𝑍𝑛 − 𝑍𝑛0
≤ −𝑙 ∑ ( 𝛿𝑖 − 𝛽𝑖+𝑙𝜆2)

𝑛−1

𝑖=𝑛1

. 

Let 𝑛 → ∞, in virtue of Equation (11) and 𝐴1, the last inequality leads to lim
𝑛→∞

𝑍𝑛 = −∞ which is 

a contradiction. Hence, liminf
𝑛→∞

𝑈𝑛 = 0, then there exists a subsequence 𝑛𝑗  such that lim
𝑗→∞

𝑈𝑛𝑗
= 0. 

From equation (8), it can be easily got  𝑍𝑛 ≥ 𝑈𝑛, 𝑛 ≥ 𝑛2 ≥ 𝑛0. So from Equation (8), it follows   

𝑍𝑛 ≤ 𝑈𝑛 + ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
𝜆2𝑈𝑖−𝑙 − 𝐹𝑛 , 

 ≤ 𝑈𝑛 + 𝜆2 ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
𝑍𝑖−𝑙 − 𝐹𝑛 ,  

 Therefore  

𝑍𝑛𝑗
≤ 𝑈𝑛𝑗

+ 𝜆2𝑍𝑛−𝑙 ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
− 𝐹𝑛𝑗

≤ 𝑈𝑛𝑗
+ 𝜇2 ∑ 𝛽𝑖

𝑛+𝑙−1

𝑖=𝑛
− 𝐹𝑛𝑗

. 

Where  𝜆2𝑍𝑛−𝑙 ≤ 𝜇2.  Then by 𝐴1 and 𝐴3 , we obtain 

lim
𝑗→∞

𝑍𝑛𝑗
≤ lim

𝑗→∞
𝑈𝑛𝑗

= 0 

Thus, lim
𝑛→∞

𝑍𝑛 = 0   implies that lim
𝑛→∞

𝑈𝑛 = 0 that is  lim
𝑛→∞

𝐻𝑛 = 𝐾. The proof is finished. 

 

In the next result, the sequence 𝑊𝑛 will be used:  

𝑊𝑛 = 𝑈𝑛 + ∑ 𝛿𝑖

𝑛−1

𝑖=𝑛−𝑙
𝑈𝑖 − 𝐹𝑛 .                                                                                                              (13) 

Lemma 2: Assume, 𝛽𝑛𝜆2 − 𝛿𝑛−𝑙 ≤ 0 and suppose that (𝐴1) − (𝐴3) hold. Let  𝐻𝑛 be a 

nonoscillatory solution to 𝐾 of Equation (3). Then, 𝑊𝑛 ≥ 0  and a nonincreasing sequence. 

Proof.  Let  𝐻𝑛 be a nonoscillatory solution to 𝐾 of Equation (3). Let  𝐻𝑛 > 𝐾, 𝑛 ≥ 𝑛0. From 

Equation (13) and Equation (3), we have 

∆𝑊𝑛 = 𝛽𝑛𝑉(𝑈𝑛−𝑙) − 𝛿𝑛−𝑙𝑈𝑛−𝑙 ≤ (𝛽𝑛𝜆2 − 𝛿𝑛−𝑙)𝑈𝑛−𝑙 ≤ 0   .                                                         (14) 

Hence, 𝑊𝑛 is a nonincreasing sequence and lim
𝑛→∞

𝑊𝑛  = 𝐿, where −∞ ≤ 𝐿 < ∞. We claim that 

 𝐿 ≥ 0. Otherwise, 𝐿 < 0, and then there exists 𝑛1 ≥ 𝑛0 and 𝛼 < 0,  such that  𝑊𝑛 ≤ 𝛼 < 0  for 

 𝑛 ≥ 𝑛1. From Equation (13), we get 

𝑈𝑛 = 𝑊𝑛 − ∑ 𝛿𝑖
𝑛−1
𝑖=𝑛−𝑙 𝑈𝑖 + 𝐹𝑛 ≤ 𝛼 + 𝐹𝑛 ,  

𝑈𝑛 < 𝛼 + 휀,   𝑛 ≥ 𝑛2 ≥ 𝑛1,   휀 > 0. 

Since 휀 is arbitrary, the last inequality leads to 𝑈𝑛 ≤ 𝛼 . This is a contradiction in terms of the fact 



IHJPAS. 2024, 37( 3 ) 

391 
 

that 𝑈𝑛 is positive. So, since our claim has been proven,𝐿 ≥ 0 or 𝑊𝑛 ≥ 0.  

Theorem 2 Assume 𝛽𝑛𝜆2 − 𝛿𝑛−𝑙 ≤ 𝛾 < 0 and let (𝐴1) − (𝐴4) hold. Then, every solution of 

Equation (3) either oscillates about a positive equilibrium  𝐾 or nonoscillatory tends to 𝐾 as  𝑡 →
∞. 

Proof. Let 𝐻𝑛 be a nonoscillatory solution to 𝐾 of Equation (3); assume 𝐻𝑛 > 𝐾  eventually, hence 

 𝑈𝑛 > 0, 𝑛 ≥ 𝑛0. From Lemma 2, it yields  𝑊𝑛 ≥ 0 and decreasing sequence. This means 𝑊𝑛 is a 

bounded sequence. Hence, from Equation (13), it follows that there exists 𝑛1 ≥ 𝑛0  such that 

𝑊𝑛 ≥ 𝑈𝑛,   𝑛 ≥ 𝑛1 .                                                                                                                                    (15) 

which means  𝑈𝑛 is bounded. Let liminf
𝑛→∞

𝑈𝑛 = 𝑙 ≥ 0. We claim that 𝑙 = 0 otherwise 𝑙 > 0. Then, 

there exists 𝑛1 ≥ 𝑛0 large enough, such that 𝑈𝑛 ≥ 𝑙, 𝑛 ≥ 𝑛1. Taking summation to both sides of 

Equation (14), we get 

∑ ∆𝑊𝑛 ≤ ∑ ( 𝛽𝑖𝜆2 − 𝛿𝑖−𝑙)𝑈𝑖−𝑙

𝑛−1

𝑖=𝑛0

𝑛−1

𝑖=𝑛0

 , 

𝑊𝑛 − 𝑊𝑛0
≤ ∑ ( 𝛽𝑖𝜆2 − 𝛿𝑖−𝑙)𝑈𝑖−𝑙

𝑛−1

𝑖=𝑛0

 .                                                                                              (16) 

From Equation (16), we have  

𝑊𝑛 − 𝑊𝑛0
≤ 𝑙 ∑ ( 𝛽𝑖𝜆2 − 𝛿𝑖−𝑙)

𝑛−1

𝑖=𝑛0

≤ 𝑙𝛾(𝑛 − 1). 

As 𝑛 → ∞, the last inequality leads to lim
𝑛→∞

𝑊𝑛 = −∞ which is a contradiction. Hence, 

liminf
𝑛→∞

𝑈𝑛 = 0. Therefore, there exists a subsequence 𝑛𝑗  such that lim
𝑗→∞

𝑈𝑛𝑗
= 0. Let lim

𝑛→∞
𝑊𝑛 = 𝐿. 

From Equation (13), it follows  

𝑈𝑛 =  𝑊𝑛 − ∑ 𝛿𝑖

𝑛−1

𝑖=𝑛−𝑙
𝑈𝑖 + 𝐹𝑛 ,   

 ≥ 𝑊𝑛 − ∑ 𝛿𝑖

𝑛−1

𝑖=𝑛−𝑙
𝑊𝑖  − 𝐹𝑛 ,   

𝑈𝑛𝑗
≥ 𝑊𝑛𝑗

− 𝑊𝑛𝑗−𝑙 ∑ 𝛿𝑖

𝑛𝑗−1

𝑖=𝑛𝑗−𝑙
− 𝐹𝑛𝑗

 , 

lim
𝑗→∞

𝑊𝑛𝑗
≤ lim

𝑗→∞
𝑈𝑛𝑗

= 0 . 

 

Thus, lim
𝑛→∞

𝑊𝑛 = 0  implies that lim
𝑛→∞

𝑈𝑛 = 0,that is lim
𝑛→∞

𝐻𝑛 = 𝐾  and the proof is finished. 

We provide examples to discuss and illustrate the previous results. In the following, we discuss 

the accuracy of the numerical solution and the oscillatory behavior of Equation (17) and (18). 

Therefore, in comparison to the exponential θ-method in [25], our results have higher accuracy. 

 

Example 1 Consider the nonlinear first-order difference equation 

∆𝑈𝑛 +
𝑛

64
𝑈𝑛 − 6 (

1

𝑒
)

𝑛+4 1

1 + 𝑈𝑛−2
= 2(−1)𝑛3𝑛−2𝑒−2𝑛, 𝑛 ≥ 1  .                                                 (17) 

Where  

𝑙 = 2,    𝛿𝑛 =
𝑛

64
,   𝛽𝑛 = 6 (

1

𝑒
)

𝑛+4

, 𝑉(𝑈𝑛) =
1

1+𝑈𝑛
,   𝜆2 = 2, 𝑓𝑛 = 2(−1)𝑛3𝑛−2𝑒−2𝑛.  

To show that all conditions of theorem 1 are satisfying:  

(𝛿𝑛 − 𝛽𝑛+𝑙𝜆2) =  
𝑛

64
− 12 (

1

𝑒
)

𝑛+6

> 0, 𝑛 ≥ 1. 

limsup
𝑛→∞

∑ 𝛿𝑖

𝑛−1

𝑖=𝑛0

= lim
𝑛→∞

∑
𝑖

64

𝑛−1

𝑖=𝑛0

= ∞  . 



IHJPAS. 2024, 37( 3 ) 

392 
 

That is, the analytic solutions of Equation (17) are oscillatory about 0 as 𝑡 → ∞ which is illustrated 

in Figure 1 since all conditions of theorem 1 holds. So, all the solutions of the corresponding 

hematopoiesis model of equation (17) are oscillatory around equilibrium 𝐾 as 𝑡 → ∞. The 

MATLAB solver ode23 that allows to numerically solve delay differential Equation (17) is used 

to perform the numerical result. 

 

 
Figure 1. The solution of Equation (17) is oscillatory about 0 as 𝑡 → ∞. 

 

Example 2 Consider the nonlinear Hematopoiesis difference equation 

∆𝐻𝑛 + 3 (
1

𝑒
)

𝑛+4

𝐻𝑛 − 𝑛(1 + 3𝑒 + 𝑒3)
1

1 + 𝐻𝑛−2
=

2

9
(

1

2
)

𝑛

. (
−3

2
)

𝑛

 .                                          (18) 

Where  

L=2, 𝜆2 = 1, 𝛿𝑛 = 3 (
1

𝑒
)

𝑛+4

, 𝛽𝑛 = 𝑛(1 + 3𝑒 + 𝑒3), 𝑓𝑛 = −𝑒 (
1

𝑒
)

𝑛

. (
−1

𝑒
)

𝑛

, 𝐺(𝐻𝑛) =
1

1+𝐻𝑛−2
. 

It's easy to show that all condition of theorem 2 are satisfies: 

(𝛽𝑛𝜆2 − 𝛿𝑛−𝑙) =  3 (
1

𝑒
)

𝑛+2

− 𝑛(1 + 3𝑒 + 𝑒3) < 0. 

limsup
𝑛→∞

(𝛽𝑛) = limsup
𝑛→∞

(𝑛(1 + 3𝑒 + 𝑒3)) = ∞.      

Since all conditions of theorem 2 hold, the solution is non-oscillatory and tends to equilibrium 𝐾 =

10,  as  𝑡 → ∞,  as illustrated in Figure 2.  The MATLAB solver ode23 that allows to numerically 

solve delay differential Equation (18) is used to perform the numerical result. 

 

 

 



IHJPAS. 2024, 37( 3 ) 

393 
 

 
Figure 2. Non-oscillatory solution for example2 of (18) tends to equilibrium K=10 as  𝑡 → ∞. 

 

The Matlap program is used to illustrate the numerical oscillation solutions of Equations (17) and 

(18).  

 

5. Conclusions 

     Homeostasis is a reasonably stable internal state of physical and chemical conditions that is 

regulated by living systems through a self-regulating mechanism, despite the changes necessary 

for existence. The blood maintains homeostasis. Part of this process that allows us to adapt to 

change and maintain life are negative feedback loops. Mathematically, homeostasis is the stability 

of a state of equilibrium or oscillation. The discrete hematopoiesis model (1), which has positive 

and negative coefficients oscillating around the equilibrium K, has the primary purpose of 

identifying appropriate conditions for this oscillation. The hematopoiesis model is therefore 

analyzed as a difference equation (3). Its oscillation is guaranteed by sufficient conditions, which 

is a new discovery for the oscillatory behavior of the presented model. It is also necessary to find 

suitable conditions to guarantee the convergence of non-oscillating solutions to the equilibrium K. 

We also show that non-oscillatory numerical solutions, as shown in Example 2, can retain the 

associated properties of the analytical solutions. It is clear that the technique used here can be 

applied to models that are periodic or nearly periodic  as long as a positive nearly periodic solution 

exists. 

 

Acknowledgment 

     The authors would like to thank Prof. Dinesh Verma for his guidance. 

 

Conflict of Interest  

      “Conflict of Interest: The authors declare that they have no conflicts of interest.” 

Funding 

     There is no financial support in preparation for the publication. 

 

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