# Permanence for a Discrete Model with Feedback Control and Delay

Authors: Yong-Hong Fan; Lin-Lin Wang
Journal: Discrete Dynamics in Nature and Society
 (2008)

---

## 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.


---

## Body

## 1. Introduction

In [1], Li and Zhu investigated the single-species population
discrete model with feedback control:(1.1)N(n+1)=N(n)exp⁡[r(n)(1−N(n−m)k(n)−c(n)μ(n))],Δμ(n)=−a(n)μ(n)+b(n)N(n−m),which is a difference form of
the single model with feedback control:(1.2)dN(t)dt=r(t)N(t)(1−N(t−τ(t))k(t)−c(t)μ(t)),dμ(n)dt=−a(t)μ(t)+b(t)N(t−τ(t)).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.LetC 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+, denote(1.3)fu=sup⁡k∈[0,+∞]f(k),fl=inf⁡k∈[0,+∞]f(k).Also for f∈Cω:={f∈C+∣f(k+ω)=f(k)},
set(1.4)f¯=1ω∑i=0ω−1f(i).Definition 1.2.
System (1.1) is said to be permanent if there exist two positive
constants λ1, λ2 such that(1.5)λ1≤liminf⁡k→∞⁢Ni(k)≤limsup⁡k→∞⁢Ni(k)≤λ2,i=1,2,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) and(1.6)cuM2<1hold, then system (1.1) is
permanent, where(1.7)M2=buM1al,M1=kuexp⁡{ru(m+1)−1}ru.In [3, 4], we studied the discrete predator-prey system which
takes the form(1.8)N1(k+1)=N1(k)exp⁡{b1(k)−a1(k)N1(k−[τ1])−α1(k)N1(k)N2(k)N12(k)+m2N22(k)},N2(k+1)=N2(k)exp⁡{−b2(k)+α2(k)N12(k−[τ2])N12(k−[τ2])+m2N22(k−[τ2])}.Using the method of comparison theorem, we obtained
the following lemma.Lemma 1.4.
Assume
that the following conditions hold:
(H1)
2mb¯1>α¯1,(H2)
α¯2>b¯2.Furthermore, assume that(1.9)(b2(k))u<1holds. 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
condition(2.1)x(−m),x(−m+1),…,x(−1)≥0,x(0),μ(0)>0.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 thatA>0 and y(0)>0, and further suppose that
(1)
(2.2)y(n+1)≤Ay(n)+B(n),n=1,2,….Then for any integerk≤n,(2.3)y(n)≤Aky(n−k)+∑i=0k−1AiB(n−i−1).Especially, ifA<1andBis bounded above with respect toM,
then(2.4)lim⁡sup⁡n→∞⁢y(n)≤M1−A,(2)
(2.5)y(n+1)≥Ay(n)+B(n),n=1,2,….Then for any integerk≤n,(2.6)y(n)≥Aky(n−k)+∑i=0k−1AiB(n−i−1).Especially, ifA<1andBis bounded below with respect tom*,
then(2.7)lim⁡inf⁡n→∞⁢y(n)≥m*1−A.Proof.
Since the proof of
(2) is similar to that of (1), we only need to prove (1). For any integerk≤n,(2.8)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),then(2.9)Ay(n−1)≤A2y(n−2)+AB(n−2),⋯Ak−1y(n−k+1)≤Aky(n−k)+Ak−1B(n−k),which implies that(2.10)y(n)≤Aky(n−k)+B(n−1)+AB(n−2)+Ak−1B(n−k)=Aky(n−k)+∑i=0k−1AiB(n−i−1).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)
(2.11)x(n+1)≤x(n)exp⁡{r(n)(1−ax(n))}
for n∈[n1,+∞),whereais a positive constant andn1is a positive integer. Then,
(2.12)lim⁡sup⁡n→∞⁢x(n)≤1aruexp⁡(ru−1),(2)
(2.13)x(n+1)≥x(n)exp⁡{r(n)(1−ax(n))}
forn∈[n1,+∞),whereais a positive constant andn1is a positive integer. Further assume thatlim⁡sup⁡n→∞x(n)≤x*andax*>1.Then,
(2.14)lim⁡inf⁡n→∞x(n)≥1aexp⁡(ru(1−ax*)).The following two lemmas are direct conclusions of
[2].Lemma 2.4.
There exists a positive constantK1 such that(2.15)lim⁡sup⁡n→∞N(n)≤K1.In fact, one can
choose(2.16)K1=kuexp⁡{ru(m+1)−1}ru.Lemma 2.5.
There exists a positive constantK2 such that(2.17)lim⁡sup⁡n→∞μ(n)≤K2.Similarly, one can
choose(2.18)K2=buK1al.Lemma 2.6.
There exists a positive constantk1 such that(2.19)lim⁡inf⁡n→∞N(n)≥k1.Proof.
By Lemmas2.4 and 2.5
and by the first equation of system (1.1), we have(2.20)N(n+1)=N(n)exp⁡[r(n)(1−N(n−m)k(n)−c(n)μ(n))]≥N(n)exp⁡{r(n)(1−K1k(n)−c(n)K2)}for n sufficiently large; then(2.21)∏i=n−mn−1N(i+1)N(i)≥exp⁡{∑i=n−mn−1r(i)(1−K1k(i)−c(i)K2)}.Notice that the sequence ∑i=n−mn−1r(i)(1−K1/k(i)−c(i)K2) is bounded below; thus(2.22)N(n−m)≤N(n)exp⁡{−∑i=n−mn−1r(i)(1−K1k(i)−c(i)K2)}.For simplicity, we set −r(i)(1−K1/k(i)−c(i)K2)=D(i); then(2.23)N(n−m)≤N(n)exp⁡{∑i=n−mn−1D(i)}.From the second equation of
system (1.1), we have(2.24)μ(n+1)=(1−a(n))μ(n)+b(n)N(n−m)≤(1−al)μ(n)+b(n)N(n−m)≤(1−al)μ(n)+b(n)N(n)exp⁡{∑i=n−mn−1D(i)}:=Aμ(n)+B(n).Then, Lemma 2.2 implies that for
any k≤n−m,(2.25)μ(n)≤Akμ(n−k)+∑i=0k−1AiB(n−i−1)=Akμ(n−k)+∑i=0k−1Aib(n−i+1)N(n−i−1−m)≤Akμ(n−k)+∑i=0k−1AibuN(n)exp⁡{∑j=n−(i+1+m)n−1D(j)}.Note that(2.26)0≤Akμ(n−k)≤K2Ak⟶0,ask⟶∞.Hence, there exists a
positive integer K such that for any solution (N(n),μ(n)) of system (1.1), cuAkμ(n−k)<1/2, as k>K. In fact, we can choose K=max⁡{1,−log⁡A2cuK2}, then we get(2.27)μ(n)≤AKμ(n−K)+∑i=0K−1AibuN(n)exp⁡{∑j=n−(i+1+m)n−1D(j)}≤AKK2+[∑i=0K−1Aibuexp⁡{(i+m+1)Du}]N(n)for n>K.
Since∑i=0K−1Aibuexp⁡{(i+m+1)Du} is bounded above, set M1=[∑i=0K−1Aibuexp⁡{(i+m+1)Du}]u; then(2.28)μ(n)≤AKK2+M1N(n).Considering the first equation
of system (1.1), we have(2.29)N(n+1)≥N(n)exp⁡{r(n)[(1−cuAKK2)−(cuM1+mexp⁡{Du}kl)N(n)]}≥N(n)exp⁡{12r(n)[1−2(cuM1+mexp⁡{Du}kl)N(n)]}.
In order to prove this lemma, by Lemma2.2(2), for the
rest we only need to prove(2.30)2(cuM1+mexp⁡{Du}kl)K1>1.Notice that(2.31)2(cuM1+mexp⁡{Du}kl)K1>1klK1=kuklexp⁡{ru(m+1)−1}ru>m+1>1.Here, we use the Bernoulli
inequality ex>1+x for x>−1.
Thus,(2.32)lim⁡inf⁡n→∞N(n)≥k1,where k1 can be chosen as(2.33)k1=exp⁡{(1/2)ru(1−M2K1)}M2,M2=2(cuM1+mDukl).
This completes the proof.Lemma 2.7.
There exists a positive constantk2 such that(2.34)lim⁡inf⁡n→∞⁢μ(n)≥k2.Proof.
Without loss of generality, for any0<ε<(1/2)k1, there exists a positive integer N2>K such that(2.35)N(n)≥m1−ε∀n>N2.Then, the second equation
implies that(2.36)Δμ(n)≥−auμ(n)+bl(m1−ε)∀n>N2.
By Lemma2.2(2), we have(2.37)lim⁡inf⁡n→∞⁢μ(n)≥k2,where k2 can be chosen as(2.38)k2=blk1au.
The proof is complete.Proof  of Theorem2.1.
From Lemmas2.4–2.7 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.



---

*Source: 945109-2008-06-18.xml*