Permanence for a Discrete Model with Feedback Control and Delay

Abstract

The permanence of a single-species population discrete model with feedback control is considered. We found that if we use the method of comparison theorem, then the additional condition, to some extent, is necessary. But for the system itself, this condition may not be necessary. Here, we use new methods instead of the comparison theorem to get the permanence of the system in consideration; the additional condition in Chen's paper (2007) is deleted.

1. Introduction

In [1], Li and Zhu investigated the single-species population discrete model with feedback control:𝑁(𝑛+1)=𝑁(𝑛)exp𝑟(𝑛)1−𝑁(𝑛−𝑚),𝑘(𝑛)−𝑐(𝑛)𝜇(𝑛)Δ𝜇(𝑛)=−𝑎(𝑛)𝜇(𝑛)+𝑏(𝑛)𝑁(𝑛−𝑚), (1.1)which is a difference form of the single model with feedback control:𝑑𝑁(𝑡)𝑁𝑑𝑡=𝑟(𝑡)𝑁(𝑡)1−𝑡−𝜏(𝑡),𝑘(𝑡)−𝑐(𝑡)𝜇(𝑡)𝑑𝜇(𝑛).𝑑𝑡=−𝑎(𝑡)𝜇(𝑡)+𝑏(𝑡)𝑁𝑡−𝜏(𝑡) (1.2)

In [1, Theorem 2.1], under the assumptions that 𝑎∶𝑍→(0,1),𝑐,𝑘,𝑟,𝑏∶𝑍→𝑅⁺∶={𝑥∣𝑥>0} are all 𝜔-periodic sequences, 𝑚 is a nonnegative integer, which may be zero, and Δ is the first forward difference operator: Δ𝜇(𝑛)=𝜇(𝑛+1)−𝜇(𝑛). They obtained what follows.

Lemma 1.1. System (1.1) has at least one positive 𝜔-periodic solution.

The following notations and definition will be useful to our discussion.

Let 𝐶 denote the set of all bounded sequences 𝑓∶ℤ→ℝ, and let 𝐶₊ be the set of all 𝑓∈𝐶 such that 𝑓>0. Given 𝑓∈𝐶₊, denote𝑓^𝑢=sup_{𝑘∈[0,+∞]}𝑓(𝑘),𝑓^𝑙=inf_{𝑘∈[0,+∞]}𝑓(𝑘). (1.3)Also for 𝑓∈𝐶_𝜔∶={𝑓∈𝐶₊∣𝑓(𝑘+𝜔)=𝑓(𝑘)}, set1𝑓=𝜔^𝜔−1_𝑖=0𝑓(𝑖). (1.4)

Definition 1.2. System (1.1) is said to be permanent if there exist two positive constants 𝜆₁, 𝜆₂ such that𝜆₁≤liminf_𝑘→∞𝑁_𝑖(𝑘)≤limsup_𝑘→∞𝑁_𝑖(𝑘)≤𝜆₂,𝑖=1,2, (1.5)for any solution (𝑁₁(𝑘),𝑁₂(𝑘)) of (1.1).

Recently, by using the comparison theorem, Chen [2] investigated the permanence of system (1.1) under the basic assumptions that

(H)𝑎∶𝑍→(0,1),𝑐,𝑘,𝑟,𝑏∶𝑍→𝑅⁺∶={𝑥∣𝑥>0} are all bounded sequences.

He obtained what follows.

Lemma 1.3. Assume that (H) and𝑐^𝑢𝑀₂<1 (1.6)hold, then system (1.1) is permanent, where𝑀₂=𝑏^𝑢𝑀₁𝑎^𝑙,𝑀₁=𝑘^𝑢exp𝑟^𝑢(𝑚+1)−1𝑟^𝑢. (1.7)

In [3, 4], we studied the discrete predator-prey system which takes the form𝑁₁(𝑘+1)=𝑁₁(𝑘)exp𝑏₁(𝑘)−𝑎₁(𝑘)𝑁₁𝑘−𝜏₁−𝛼₁(𝑘)𝑁₁(𝑘)𝑁₂(𝑘)𝑁²₁(𝑘)+𝑚²𝑁²₂,𝑁(𝑘)₂(𝑘+1)=𝑁₂(𝑘)exp−𝑏₂𝛼(𝑘)+²(𝑘)𝑁²₁𝑘−𝜏₂𝑁²₁𝑘−𝜏₂+𝑚²𝑁²₂𝑘−𝜏₂. (1.8)Using the method of comparison theorem, we obtained the following lemma.

Lemma 1.4. Assume that the following conditions hold:
(H1)2𝑚𝑏₁>𝛼₁,(H2)𝛼₂>𝑏₂.

Furthermore, assume that𝑏₂(𝑘)^𝑢<1 (1.9)holds. Then, system (1.8) is permanent.

We should point out that conditions (H1) and (H2) are sufficient for the permanence of system (1.8) (the reader can refer to [5]); that is, condition (1.9) is not necessary for its permanence.

We found that if we use the method of comparison theorem, then the additional condition, to some extent, is necessary. But for the system itself, this condition may not be necessary. Motivated by the above problem, we discuss the permanence of system (1.1) again; our investigation shows that condition (1.6) is also not necessary.

2. Main Results

In the remainder of this paper, for biological reasons, we only consider the solution (𝑥(𝑘),𝜇(𝑘)) of system (1.1) with initial condition𝑥(−𝑚),𝑥(−𝑚+1),…,𝑥(−1)≥0,𝑥(0),𝜇(0)>0. (2.1)

One can easily show that any solution of system (1.1) with initial condition (2.1) remains positive.

First, we state our main result below.

Theorem 2.1. Under the basic assumptions (H), the system (1.1) is permanent.

In order to prove our main result, firstly we give some lemmas which will be useful for the following discussion.

Lemma 2.2. Assume that 𝐴>0 and 𝑦(0)>0, and further suppose that
(1)𝑦(𝑛+1)≤𝐴𝑦(𝑛)+𝐵(𝑛),𝑛=1,2,…. (2.2)Then for any integer𝑘≤𝑛,𝑦(𝑛)≤𝐴^𝑘𝑦(𝑛−𝑘)+^𝑘−1_𝑖=0𝐴^𝑖𝐵(𝑛−𝑖−1). (2.3)Especially, if 𝐴<1and𝐵is bounded above with respect to𝑀, thenlimsup_𝑛→∞𝑀𝑦(𝑛)≤,1−𝐴 (2.4)(2)𝑦(𝑛+1)≥𝐴𝑦(𝑛)+𝐵(𝑛),𝑛=1,2,…. (2.5)Then for any integer𝑘≤𝑛,𝑦(𝑛)≥𝐴^𝑘𝑦(𝑛−𝑘)+^𝑘−1_𝑖=0𝐴^𝑖𝐵(𝑛−𝑖−1). (2.6)Especially, if𝐴<1and 𝐵 is bounded below with respect to𝑚^∗, thenliminf_𝑛→∞𝑚𝑦(𝑛)≥^∗.1−𝐴 (2.7)

Proof. Since the proof of (2) is similar to that of (1), we only need to prove (1). For any integer 𝑘≤𝑛,⋯𝑦(𝑛)≤𝐴𝑦(𝑛−1)+𝐵(𝑛−1),𝑦(𝑛−1)≤𝐴𝑦(𝑛−2)+𝐵(𝑛−2),𝑦(𝑛−𝑘+1)≤𝐴𝑦(𝑛−𝑘)+𝐵(𝑛−𝑘), (2.8)then𝐴𝑦(𝑛−1)≤𝐴²⋯𝐴𝑦(𝑛−2)+𝐴𝐵(𝑛−2),_𝑘−1𝑦(𝑛−𝑘+1)≤𝐴^𝑘𝑦(𝑛−𝑘)+𝐴^𝑘−1𝐵(𝑛−𝑘), (2.9)which implies that𝑦(𝑛)≤𝐴^𝑘𝑦(𝑛−𝑘)+𝐵(𝑛−1)+𝐴𝐵(𝑛−2)+𝐴^𝑘−1𝐵(𝑛−𝑘)=𝐴^𝑘𝑦(𝑛−𝑘)+^𝑘−1_𝑖=0𝐴^𝑖𝐵(𝑛−𝑖−1). (2.10)From the above inequality, (2.4) is obvious and the proof is complete.

The following lemma can be found in [1, 6].

Lemma 2.3. Assume that {𝑥(𝑛)} satisfies 𝑥(𝑛₁)>0 and
(1)𝑥(𝑛+1)≤𝑥(𝑛)exp𝑟(𝑛)1−𝑎𝑥(𝑛) (2.11)for 𝑛∈[𝑛₁,+∞),where𝑎is a positive constant and 𝑛₁is a positive integer. Then,limsup_𝑛→∞1𝑥(𝑛)≤𝑎𝑟^𝑢exp𝑟^𝑢−1, (2.12)(2)𝑥(𝑛+1)≥𝑥(𝑛)exp𝑟(𝑛)1−𝑎𝑥(𝑛) (2.13)for𝑛∈[𝑛₁,+∞),where𝑎is a positive constant and𝑛₁ is a positive integer. Further assume that limsup_𝑛→∞𝑥(𝑛)≤𝑥^∗and 𝑎𝑥^∗>1.Then,liminf_𝑛→∞1𝑥(𝑛)≥𝑎exp𝑟^𝑢1−𝑎𝑥^∗. (2.14)

The following two lemmas are direct conclusions of [2].

Lemma 2.4. There exists a positive constant 𝐾₁ such thatlimsup_𝑛→∞𝑁(𝑛)≤𝐾₁. (2.15)In fact, one can choose𝐾₁=𝑘^𝑢exp𝑟^𝑢(𝑚+1)−1𝑟^𝑢. (2.16)

Lemma 2.5. There exists a positive constant 𝐾₂ such thatlimsup_𝑛→∞𝜇(𝑛)≤𝐾₂. (2.17)Similarly, one can choose𝐾₂=𝑏^𝑢𝐾₁𝑎^𝑙. (2.18)

Lemma 2.6. There exists a positive constant 𝑘₁ such thatliminf_𝑛→∞𝑁(𝑛)≥𝑘₁. (2.19)

Proof. By Lemmas 2.4 and 2.5 and by the first equation of system (1.1), we have𝑁(𝑛+1)=𝑁(𝑛)exp𝑟(𝑛)1−𝑁(𝑛−𝑚)𝐾𝑘(𝑛)−𝑐(𝑛)𝜇(𝑛)≥𝑁(𝑛)exp𝑟(𝑛)1−¹𝑘(𝑛)−𝑐(𝑛)𝐾₂ (2.20)for 𝑛 sufficiently large; then^𝑛−1_{𝑖=𝑛−𝑚}𝑁(𝑖+1)𝑁(𝑖)≥exp^𝑛−1_{𝑖=𝑛−𝑚}𝐾𝑟(𝑖)1−¹𝑘(𝑖)−𝑐(𝑖)𝐾₂. (2.21)Notice that the sequence ∑^𝑛−1_{𝑖=𝑛−𝑚}𝑟(𝑖)(1−𝐾₁/𝑘(𝑖)−𝑐(𝑖)𝐾₂) is bounded below; thus𝑁(𝑛−𝑚)≤𝑁(𝑛)exp−^𝑛−1_{𝑖=𝑛−𝑚}𝐾𝑟(𝑖)1−¹𝑘(𝑖)−𝑐(𝑖)𝐾₂. (2.22)For simplicity, we set −𝑟(𝑖)(1−𝐾₁/𝑘(𝑖)−𝑐(𝑖)𝐾₂)=𝐷(𝑖); then𝑁(𝑛−𝑚)≤𝑁(𝑛)exp^𝑛−1_{𝑖=𝑛−𝑚}𝐷(𝑖). (2.23)From the second equation of system (1.1), we have≤𝜇(𝑛+1)=1−𝑎(𝑛)𝜇(𝑛)+𝑏(𝑛)𝑁(𝑛−𝑚)1−𝑎^𝑙≤𝜇(𝑛)+𝑏(𝑛)𝑁(𝑛−𝑚)1−𝑎^𝑙𝜇(𝑛)+𝑏(𝑛)𝑁(𝑛)exp^𝑛−1_{𝑖=𝑛−𝑚}𝐷(𝑖)∶=𝐴𝜇(𝑛)+𝐵(𝑛). (2.24)Then, Lemma 2.2 implies that for any 𝑘≤𝑛−𝑚,𝜇(𝑛)≤𝐴^𝑘𝜇(𝑛−𝑘)+^𝑘−1_𝑖=0𝐴^𝑖𝐵(𝑛−𝑖−1)=𝐴^𝑘𝜇(𝑛−𝑘)+^𝑘−1_𝑖=0𝐴^𝑖𝑏(𝑛−𝑖+1)𝑁(𝑛−𝑖−1−𝑚)≤𝐴^𝑘𝜇(𝑛−𝑘)+^𝑘−1_𝑖=0𝐴^𝑖𝑏^𝑢𝑁(𝑛)exp^𝑛−1_{𝑗=𝑛−(𝑖+1+𝑚)}𝐷(𝑗). (2.25)Note that0≤𝐴^𝑘𝜇(𝑛−𝑘)≤𝐾₂𝐴^𝑘⟶0,as𝑘⟶∞. (2.26)Hence, there exists a positive integer 𝐾 such that for any solution (𝑁(𝑛),𝜇(𝑛)) of system (1.1), 𝑐^𝑢𝐴^𝑘𝜇(𝑛−𝑘)<1/2, as 𝑘>𝐾. In fact, we can choose 𝐾=max{1,−log_𝐴2𝑐^𝑢𝐾₂}, then we get𝜇(𝑛)≤𝐴^𝐾𝜇(𝑛−𝐾)+^𝐾−1_𝑖=0𝐴^𝑖𝑏^𝑢𝑁(𝑛)exp^𝑛−1_{𝑗=𝑛−(𝑖+1+𝑚)}𝐷(𝑗)≤𝐴^𝐾𝐾₂+^𝐾−1_𝑖=0𝐴^𝑖𝑏^𝑢exp(𝑖+𝑚+1)𝐷^𝑢𝑁(𝑛) (2.27)for 𝑛>𝐾.
Since ∑^𝐾−1_𝑖=0𝐴^𝑖𝑏^𝑢exp{(𝑖+𝑚+1)𝐷^𝑢} is bounded above, set 𝑀₁=[∑^𝐾−1_𝑖=0𝐴^𝑖𝑏^𝑢exp{(𝑖+𝑚+1)𝐷^𝑢}]^𝑢; then𝜇(𝑛)≤𝐴^𝐾𝐾₂+𝑀₁𝑁(𝑛). (2.28)Considering the first equation of system (1.1), we have𝑁(𝑛+1)≥𝑁(𝑛)exp𝑟(𝑛)1−𝑐^𝑢𝐴^𝐾𝐾₂−𝑐^𝑢𝑀₁+𝑚exp𝐷^𝑢𝑘^𝑙1𝑁(𝑛)≥𝑁(𝑛)exp2𝑟(𝑛)1−2𝑐^𝑢𝑀₁+𝑚exp𝐷^𝑢𝑘^𝑙𝑁(𝑛). (2.29)
In order to prove this lemma, by Lemma 2.2(2), for the rest we only need to prove2𝑐^𝑢𝑀₁+𝑚exp𝐷^𝑢𝑘^𝑙𝐾₁>1. (2.30)Notice that2𝑐^𝑢𝑀₁+𝑚exp𝐷^𝑢𝑘^𝑙𝐾₁>1𝑘^𝑙𝐾₁=𝑘^𝑢𝑘^𝑙exp𝑟^𝑢(𝑚+1)−1𝑟^𝑢>𝑚+1>1. (2.31)Here, we use the Bernoulli inequality 𝑒^𝑥>1+𝑥 for 𝑥>−1.
Thus,liminf_𝑛→∞𝑁(𝑛)≥𝑘₁, (2.32)where 𝑘₁ can be chosen as𝑘₁=exp(1/2)𝑟^𝑢1−𝑀₂𝐾₁𝑀₂,𝑀₂=2𝑐^𝑢𝑀₁+𝑚𝐷^𝑢𝑘^𝑙. (2.33)
This completes the proof.

Lemma 2.7. There exists a positive constant 𝑘₂ such thatliminf_𝑛→∞𝜇(𝑛)≥𝑘₂. (2.34)

Proof. Without loss of generality, for any 0<𝜀<(1/2)𝑘₁, there exists a positive integer 𝑁₂>𝐾 such that𝑁(𝑛)≥𝑚₁−𝜀∀𝑛>𝑁₂. (2.35)Then, the second equation implies thatΔ𝜇(𝑛)≥−𝑎^𝑢𝜇(𝑛)+𝑏^𝑙𝑚₁−𝜀∀𝑛>𝑁₂. (2.36)
By Lemma 2.2(2), we haveliminf_𝑛→∞𝜇(𝑛)≥𝑘₂, (2.37)where 𝑘₂ can be chosen as𝑘₂=𝑏^𝑙𝑘₁𝑎^𝑢. (2.38)
The proof is complete.

Proof of Theorem 2.1. From Lemmas 2.4–2.7 and the definition of permanence, the conclusion is obvious.

3. Discussion

In many situations, as our investigations show, the permanence and the existence of periodic solutions of a system are closely related with each other. But sometimes when we want to get the permanence of the system under the precondition that the periodic solution exists, we need some additional conditions. This mainly charges upon the method we used, especially when we use the comparison theorem. To make the comparison theorem holds, some additional conditions must be given, while to the system itself, these conditions may not be necessary. Just as in [2, 3], the conditions for the permanence of the system include some additional conditions besides the conditions for the existence of periodic solutions, while in this text and in [5], the conditions for the permanence of the system are exactly the same as the conditions for the existence of periodic solutions. This is because in this text and in [5] we use new methods instead of the comparison theorem.

Acknowledgments

This work is supported by the NSF of Ludong University (24070301, 24070302, 24200301) and by the Program for Innovative Research Team at Ludong University.