Global stability of positive periodic solutions and almost
periodic solutions for a discrete competitive system
Heping Ma∗ , Jianguo Gao† , Lingling Xie‡
School of Mathematics and Information Science, Beifang University of Nationalities,
Yinchuan 750021, China
Abstract: In this paper, a discrete two-species competitive model is investigated.
By using some preliminary lemmas and constructing a Lyapunov function, the existence and uniformly asymptotic stability of positive almost periodic solutions of the
system is derived. In addition, an example and numerical simulations are presented
to illustrate and substantiate the results of this paper.
Keywords: Discrete two-species competitive system; Lyapunov function; Almost
periodic solutions; Uniformly asymptotic stability.
∗
E-mail: [email protected] (H.Ma)
Corresponding author. E-mail:[email protected] (J.Gao)
‡
E-mail:[email protected] (L.Xie)
†
1
1. Introduction
K.Gopalsamy has presented the following Lotka-Volterra competitive system with continuous
time version in 1992(see[1])

 x′ (t) = x (t)[r (t) − b (t)x (t) −
1
1
1
1
1
 x′2 (t) = x2 (t)[r2 (t) − b2 (t)x2 (t) −
a2 (t)x2 (t)
1+d1 (t)x1 (t) ],
a1 (t)x1 (t)
1+d2 (t)x2 (t) ],
n = 0, 1, 2, ...
(1.1)
where x1 (t), x2 (t) represent the population densities of two competing species; r1 (t), r2 (t) are
the intrinsic growth rates of species; b1 (t), b2 (t) stand for the rates of intraspeciﬁc competition
of the ﬁrst and second species, respectively;
a1 (t)x1 (t)
a2 (t)x2 (t)
1+d1 (t)x1 (t) , 1+d2 (t)x2 (t)
denote the competitive
response function, respectively. All the coeﬃcients above are continuous and bounded above
and below by positive constants.
The discrete-time systems governed by diﬀerence equations recently have been won widespread attention and applied in studying population growth, the transmission of tuberculosis
and HIV/AIDS and inﬂuenza prevention and control(see[4, 5]), just because discrete-time
models conform better to the reality than the continuous ones, especially for the populations
with a short life expectancy or non-overlapping generations. In addition, some works about
the bifurcation, chaos, and complex dynamical behaviors of the discrete specie systems have
been done(see[2, 3]). In practice, according to the discrete data measured, the discrete-time
models commonly provide eﬃcient computational models of continuous models for numerical
simulations (see [4–11]). Therefore, we derive the discrete analogue of system (1.1) by using
the same discretization method (see [11]):

 x (n + 1) = x (n) exp[r (n) − b (n)x (n) −
1
1
1
1
1
 x2 (n + 1) = x2 (n) exp[r2 (n) − b2 (n)x2 (n) −
a2 (n)x2 (n)
1+d1 (n)x1 (n) ],
a1 (n)x1 (n)
1+d2 (n)x2 (n) ],
n = 0, 1, 2, ...
(1.2)
where xi (n) (i = 1, 2; i ̸= j) denote the densities of species xi at the nth generation, ri (n)
stand for the natural growth rates of species xi at the nth generation, bi (n) represent the selfinhibition rate respectively, aj (n) and di (n) are the inter-speciﬁc eﬀects of the nth generation
of species xi on species xj .
From an evolutionary perspective, as the selective forces on species, the periodically varying environments are of vital importance for survival of the ﬁttest. For instance, any periodic change of climate tends to impose its period upon oscillations of internal origin or to
cause such oscillations to have a harmonic relation to periodic climatic changes (see[11–16]).
Therefore, the coeﬃcients of many a system constructed in ecology are usually considered as
periodic functions (see [12, 13]). Not long ago, Wang(see[12]) studied a delayed predator-prey
model with Hassell-Varley type functional responses and obtained the suﬃcient conditions
for the existence of positive periodic solutions by applying the coincidence degree theorem.
Many excellent results concerned with the discrete periodic systems are obtained (see[14–16]).
2
In nature, however, there hardly exists necessarily commensurate periods in the various
environment components like seasonal weather change, food supplies, mating habits and
harvesting, etc. Compared with the periodic systems, we can thus incorporate the assumption
of almost periodicity of the coeﬃcients of (1.1) to reﬂect the time-dependent variability
of the environment(see[6, 8–10, 21]). Recently, Li et al.(see[17])have proposed an almost
periodic discrete predator-prey models with time delays and investigated permanence of the
system and the existence of a unique uniformly asymptotically stable positive almost periodic
sequence solution. Afterwards, by using Mawhins continuation theorem of the coincidence
degree theory, [18] achieved some suﬃcient conditions for the existence of positive almost
periodic solutions for a class of delay discrete models with Allee-eﬀect.
Notice that the investigation of periodic solutions and almost periodic solutions is one of
the most important topics in the qualitative theory of the diﬀerence equations. In this paper,
based on the ideas mentioned above, for system (1.2), one carries out two main works:
• Assume that all the coeﬃcients {ri (n)}, {bi (n)}, {ai (n)}, {di (n)} for are bounded
nonnegative periodic sequences. We explore the existence and global stability of positive
periodic solutions of system (1.2) with positive periodic coeﬃcients.
• Furtherly, one discusses the almost periodic solutions of system (1.2) with positive
almost periodic coeﬃcients.
The organization of this paper is as follows. In the next section, we present some notations
and preliminary lemmas. In Section 3, we seek suﬃcient conditions which ensure the existence
and global stability of positive periodic solutions of system (1.2). In Section 4, we further
investigate the existence, uniqueness, and uniformly asymptotic stability of positive almost
periodic solutions for the above system (1.2). In Section 5, we present an example and its
numerical simulations are carried out to illustrate the feasibility of our main results. In the
last section, a conclusion is given to conclude this work.
2. Notations and Preliminaries Lemmas
Throughout this paper, the notations below will be used:
hu = sup {h(n)},
hl = inf {h(n)},
n∈Z +
n∈Z +
(2.1)
where {h(n)} is a bounded sequence and Z + = {0, 1, 2, 3, ...}.
Denote by R, R+ , Z and Z + the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. R2 and Rk the cone of 2-dimensional and
k-dimensional real Euclidean apace, respectively.
3
Definition 2.1. (see [10]) A sequence y : Z → Rk is called an almost periodic sequence
provided that the following ε-translation set of y
I{ε, y} := {τ ∈ Z : |y(n + τ ) − y(n)| < ε, ∀n ∈ Z}
is a relatively dense set in Z for all ε > 0; i.e., for any given ε > 0, there exists an integer
l(ε) > 0 such that each discrete interval of length l(ε) contains a τ = τ (ε) ∈ I{ε, y} such that
|y(n + τ ) − y(n)| < ε, ∀n ∈ Z,
τ is referred to as the ε-translation number of y(n).
Definition 2.2. (see [10]) Suppose h : Z × A → Rk , where A is an open set in Rk . h(n, x) is
said to be almost periodic in n uniformly for x ∈ A, or uniformly almost periodic for short,
if for any ε > 0 and any compact set S in A there exists a positive integer l(ε, S) such that
any interval of length l(ε, S) contains an integer τ for which
|h(n + τ, x) − h(n, x)| < ε, ∀n ∈ Z, x ∈ S,
(2.2)
τ is called the ε-translation number of h(n, x).
Lemma 2.1. (see [10]) {x(n)} is an almost periodic sequence if and only if for any sequence
{p′t } ⊂ Z there exists a subsequence {pt } ⊂ {p′t } such that x(n + pt ) converges uniformly on
n ∈ Z as t → ∞. Thus, the limit sequence is also an almost periodic sequence.
Furthermore, we consider the following almost periodic diﬀerence system:
x(n + 1) = h(n, x(n)), n ∈ Z + ,
(2.3)
where h : Z + × CB → Rk , CB = {x ∈ C : ||x|| < B}, and h(n, x) is almost periodic in n
uniformly for x ∈ CB and is continuous in x.
The product system of (2.3) is in the form of
x(n + 1) = h(n, xn ), y(n + 1) = h(n, yn ),
(2.4)
and [20] obtained the following lemma, where (x(n, ϕ), y(n, ψ)) is a solution of (2.4).
Lemma 2.2. (see [20]) Suppose there exist a Lyapunov function V (n, ϕ, ψ) deﬁned for n ∈
Z + , ∥ϕ∥ < B, ∥ψ∥ < B satisfying the following conditions:
(1) α(|ϕ − ψ|) ≤ V (n, ϕ, ψ) ≤ β(∥ϕ − ψ∥), where α, β ∈ p with p = {η : [0, ∞) →
[0, ∞)|η(0) = 0 and η(ν) is continuous, increasing in ν};
(2) |V (n, ϕ1 , ψ1 ) − V (n, ϕ2 , ψ2 )| ≤ L(||ϕ1 − ϕ2 || + ||ψ1 − ψ2 ||), where L > 0 is a constant;
(3) △V(2.2) (n, ϕ, ψ) ≤ −γV (n, ϕ, ψ), where 0 < γ < 1 is a constant and
△V(2.1) (n, ϕ, ψ) = V (n + 1, xn+1 (n, ϕ), yn+1 (n, ψ)) − V (n, ϕ, ψ).
4
Moreover, suppose that there exists a solution x(n) of system (2.3) such that ||xn || ≤ B ∗ < B
for all n ∈ Z + , then there exists a unique uniformly asymptotically stable almost periodic
solution q(n) of system (2.3) which satisﬁes |q(n)| ≤ B ∗ . In particular, if h(n, ϕ) is periodic
of period ω, then system (2.3) has a unique uniformly asymptotically stable periodic solution
with period ω.
Lemma 2.3.(see[19])Any positive solution (x1 (n), x2 (n)) of system (1.2) satisﬁes
lim sup xi (n) ≤
n→+∞
exp(riu − 1)
≡ Mi ,
bli
i = 1, 2.
(2.5)
Lemma 2.4.(see[19])Let system (1.2) satisfy the following assumptions:
{
}
min r1l − au2 M2 , r2l − au1 M1 > 0.
(2.6)
Then, any positive solution (x1 (n), x2 (n)) of system (1.2) satisﬁes
xi (n) ≥
ril − auj Mj
exp(ril − auj Mj − bui Mi ) ≡ mi ,
bui
i ̸= j; i, j = 1, 2.
(2.7)
3 Existence and stability of positive periodic solutions
Apparently, the permanence of system (1.2) can be obtained according to Lemma 2.3 and
Lemma 2.4. In the following, we shall show the existence and stability of positive periodic
solutions of system (1.2). To this end, let us assume that all the coeﬃcients of system (1.2)
are δ-periodic, namely,
ri (n + δ) = ri (n), ai (n + δ) = ai (n), bi (n + δ) = bi (n), di (n + δ) = di (n), i = 1, 2. (3.1)
Lemma 3.1.(see[16]) If the assumption (2.6) hold, then system (1.2) has at least one strictly
positive δ-periodic solution and denoted by (x∗1 (n), x∗2 (n)).
Definition 3.1. A positive periodic solution (x∗1 (n), x∗2 (n)) of system (1.2) is globally stable
if each other solution (x1 (n), x2 (n)) with positive initial value deﬁned for all n > 0 satisﬁes
lim |x1 (n) − x∗1 (n)| = 0,
lim |x2 (n) − x∗2 (n)| = 0.
n→+∞
n→+∞
(3.2)
Now, we present the main results.
Theorem 3.1. Let the following assumption
{
}
ξ1 = max |1 + au2 du1 M1 M2 − bl1 m1 |, |1 + al2 dl1 m1 m2 − bu1 M1 | + au2 M2 + au2 du1 M1 M2 < 1,
{
ξ2 = max |1 + au1 du2 M1 M2 − bl2 m2 |, |1 + al1 dl2 m1 m2 − bu2 M2 |} + au1 M1 + au1 du2 M1 M2 < 1
(3.3)
5
and (2.6) hold, then the positive periodic solution of system (1.2) is globally stable.
Proof. Let (x∗1 (n), x∗2 (n)) be a positive periodic solution of system (1.2).
Denote exp z1 (n) =
x1 (n)
x∗1 (n) ,
exp z2 (n) =
x2 (n)
x∗2 (n) ,
then we have
z1 (n + 1)
= z1 (n) + b1 (n)x∗1 (n)(1 − exp z1 (n)) +
≤ z1 (n) + b1 (n)x∗1 (n)(1 − exp z1 (n)) +
a2 (n)(x∗2 (n)−x2 (n))+a2 (n)d1 (n)(x1 (n)x∗2 (n)−x∗1 (n)x2 (n))
(1+d1 (n)x∗1 (n))(1+d1 (n)x1 (n))
∗
a2 (n)x2 (n)(1 − exp z2 (n))
+a2 (n)d1 (n)x∗1 (n)x∗2 (n)[(1 − exp z2 (n)) − (1 − exp z1 (n))],
z2 (n + 1)
= z2 (n) + b2 (n)x∗2 (n)(1 − exp z2 (n)) +
≤ z2 (n) + b2 (n)x∗2 (n)(1 − exp z2 (n)) +
a1 (n)(x∗1 (n)−x1 (n))+a1 (n)d2 (n)(x∗1 (n)x2 (n)−x1 (n)x∗2 (n))
(1+d2 (n)x∗2 (n))(1+d2 (n)x2 (n))
a1 (n)x∗1 (n)(1 − exp z1 (n))
+a1 (n)d2 (n)x∗1 (n)x∗2 (n)[(1 − exp z1 (n)) − (1 − exp z2 (n))],
(3.4)
which, according to the mean value, yields
z1 (n + 1)
≤ z1 (n)[1 − b1 (n)x∗1 (n) exp(ϑ1 z1 (n)) + a2 (n)d1 (n)x∗1 (n)x∗2 (n) exp(ϑ1 z1 (n))]
−z2 (n)[a2 (n)x∗2 (n) exp(ϑ2 z2 (n)) + a2 (n)d1 (n)x∗1 (n)x∗2 (n) exp(ϑ2 z2 (n))],
z2 (n + 1)
(3.5)
≤ z2 (n)[1 − b2 (n)x∗2 (n) exp(ϑ3 z2 (n)) + a1 (n)d2 (n)x∗1 (n)x∗2 (n) exp(ϑ3 z2 (n))]
−z1 (n)[a1 (n)x∗1 (n) exp(ϑ4 z1 (n)) + a1 (n)d2 (n)x∗1 (n)x∗2 (n) exp(ϑ4 z1 (n))],
where all the constants ϑ1 , ϑ2 , ϑ3 , ϑ4 ∈ (0, 1). Obviously, together with (3.3) we can ﬁnd a
suﬃcient small ε such that
}
{
ξ1∗ = max 1 + au2 du1 (M1 + ε)(M2 + ε) − bl1 (m1 − ε)|, |1 + al2 dl1 (m1 − ε)(m2 − ε) − bu1 (M1 + ε)
+au2 (M2 + ε) + au2 du1 (M1 + ε)(M2 + ε) < 1,
}
{
ξ2∗ = max 1 + au1 du2 (M1 + ε)(M2 + ε) − bl2 (m2 − ε)|, |1 + al1 dl2 (m1 − ε)(m2 − ε) − bu2 (M2 + ε)
+au1 (M1 + ε) + au1 du2 (M1 + ε)(M2 + ε) < 1.
(3.6)
It follows from Lemma 2.3 and 2.4 that there exists an N0 such that n > N0 , we have
0 < m1 − ε ≤ x∗1 (n) ≤ M1 + ε, 0 < m2 − ε ≤ x∗2 (n) ≤ M2 + ε,
0 < m1 − ε ≤ x1 (n) ≤ M1 + ε, 0 < m2 − ε ≤ x2 (n) ≤ M2 + ε.
(3.7)
Then one obtains that both x∗1 (n) exp(ϑ1 z1 (n)) and x∗1 (n) exp(ϑ4 z1 (n)) are between x∗1 (n)
and x1 (n). Similarly, both x∗2 (n) exp(ϑ2 z2 (n)) and x∗2 (n) exp(ϑ3 z2 (n)) are between x∗2 (n) and
x2 (n). From the ﬁrst equation of (1.2), one has
6
|z1 (n + 1)|
≤ |z1 (n)|[1 − b1 (n)x∗1 (n) exp(ϑ1 z1 (n)) + a2 (n)d1 (n)x∗1 (n)x∗2 (n) exp(ϑ1 z1 (n))]
+|z2 (n)|[a2 (n)x∗2 (n) exp(ϑ2 z2 (n)) + a2 (n)d1 (n)x∗1 (n)x∗2 (n) exp(ϑ2 z2 (n))]
{
≤ max |1 + au2 du1 (M1 + ε)(M2 + ε) − bl1 (m1 − ε)|,
}
|1 + al2 dl1 (m1 − ε)(m2 − ε) − bu1 (M1 + ε)| |z1 (n)|
(3.8)
+[au2 (M2 + ε) + au2 du1 (M1 + ε)(M2 + ε)]|z2 (n)|
≤ ξ1∗ max{z1 (n), z2 (n)}.
Similar to the arguments as above, we must have
|z2 (n + 1)| ≤ ξ2∗ max{z1 (n), z2 (n)}.
(3.9)
We denote ξ ∗ = max{x∗1 , x∗2 }, then ξ ∗ < 1. Therefore, provided that n > N0 ,
max{|z1 (n + 1)|, |z2 (n + 1)|} ≤ ξ ∗ max{|z1 (n)|, |z2 (n)|} ≤ (ξ ∗ )n−N0 .
(3.10)
Consequently, limn→+∞ |xi (n) − x∗i (n)| = 0, where i = 1, 2. By Deﬁnition 3.1, it follows
that the positive periodic solution {x∗1 (n), x∗2 (n)} of sysytem (1.2) is globally stable. This
completes the proof.
4 Existence and stability of positive almost periodic solutions
In this section, we discuss the existence of positive almost periodic solutions of system (1.2).
Lemma 4.1. If the assumptions (2.6) is true, then Ω ̸= Ø.
Proof. According to an inductive argument, system (1.2) is actually described as follows
a2 (t)x2 (t) ]
,
1 + d1 (t)x1 (t)
t=0
n−1
∑[
a1 (t)x1 (t) ]
x2 (n) = x2 (0) exp
r2 (t) − b2 (t)x2 (t) −
.
1 + d2 (t)x2 (t)
x1 (n) = x1 (0) exp
n−1
∑[
r1 (t) − b1 (t)x1 (t) −
(4.1)
t=0
Combining Lemmas (2.3) and (2.4), for any solution (x1 (n), x2 (n)) of system (1.2) and an
arbitrarily small constant ε > 0, there must be n0 suﬃcient large such that
m1 − ε ≤ x1 (n) ≤ M1 + ε, m2 − ε ≤ x2 (n) ≤ M2 + ε, for all n ≥ n0 .
(4.2)
Assume that τk is any positive integer sequence such that {τk } → +∞ as k → +∞, we can
prove that there is a subsequence of {τk } still denoted by {τk }, such that xi (n + τk ) → x∗i (n),
i = 1, 2 uniformly in n on any ﬁnite subset L of Z + as k → +∞, where L = {l1 , l2 ...lm },
lj ∈ Z + (j = 1, 2, ...m), and m is a ﬁnite number.
7
In fact, for any ﬁnite subset L ⊂ Z + ,τk + lj > n0 , (j = 1, 2, ...m), when k is large enough.
Therefore, mi − ε ≤ xi (n + τk ) ≤ Mi − ε, i = 1, 2; that is, xi (n + τk ) are uniformly bounded
when k is suﬃcient large.
(1)
(1)
Next, for l1 ∈ L, we choose a subsequence {τk } of {τk } such that x1 (l1 + τk ) and
(1)
x2 (l1 + τk ) uniformly converge on Z + for k suﬃcient large.
(2)
(1)
Similar to the arguments of l1 , for l2 ∈ L, one can select a subsequence {τk } of {τk }
(2)
(2)
such that x1 (l2 + τk ) and x2 (l2 + τk ) uniformly converge on Z + for k large enough.
(m)
Repeating above-mentioned process, for lm ∈ L, one obtains a subsequence {τk
(m−1)
{τk
}
such that
(m)
x1 (lm + τk )
and
(m)
x2 (lm + τk )
} of
uniformly converge on Z + for k suﬃcient
large.
(m)
Based on the above, one selects the sequence {τk
} which is a subsequence of {τk } still
denoted by {τk }; then, for n ∈ L, one get that xi (n + τk ) → x∗i (n), i = 1, 2 uniformly in n ∈ L
as k → +∞. So the conclusion holds truely due to the arbitrariness of L.
Diﬀerent from the third section, we suppose that all the coeﬃcients {ri (n)},{ai (n)},{bi (n)}
and {di (n)}, i = 1, 2, are bounded nonnegative almost periodic sequences, for the above
sequence {τk }, τk → +∞ as k → +∞, there exists a subsequence denoted by {τk } such that
ri (n + τk ) → ri (n), bi (n + τk ) → bi (n), ai (n + τk ) → ai (n), di (n + τk ) → di (n), i = 1, 2,
(4.3)
as k → +∞ uniformly on
Z +.
For any ρ ∈ Z + , assume that τk + ρ ≥ N0 when k is large enough. By an inductive
argument of system (1.2) from τk + ρ to n + τk + ρ, where n ∈ Z + , one obtains
a2 (t)x2 (t) ]
,
1 + d1 (t)x1 (t)
t=τk +ρ
n+τ∑
k +ρ−1[
a1 (t)x1 (t) ]
.
x2 (n + τk + ρ) = x2 (τk + ρ) exp
r2 (t) − b2 (t)x2 (t) −
1 + d2 (t)x2 (t)
t=τ +ρ
x1 (n + τk + ρ) = x1 (τk + ρ) exp
n+τ∑
k +ρ−1[
r1 (t) − b1 (t)x1 (t) −
(4.4)
k
Hence, (4.4) yields
n+ρ−1[
a2 (t + τk )x2 (t + τk ) ]
,
1 + d1 (t + τk )x1 (t + τk )
t=ρ
n+ρ−1[
∑
a1 (t + τk )x1 (t + τk ) ]
r2 (t + τk )−b2 (t + τk )x2 (t + τk )−
x2 (n + τk + ρ) = x2 (τk + ρ) exp
.
1 + d2 (t + τk )x2 (t + τk )
t=ρ
x1 (n + τk + ρ) = x1 (τk + ρ) exp
∑
r1 (t + τk )−b1 (t + τk )x1 (t + τk )−
(4.5)
Let k → +∞, one has
a2 (t)x∗2 (t) ]
,
1 + d1 (t)x∗1 (t)
t=ρ
n+ρ−1
∑ [
a1 (t)x∗1 (t) ]
∗
∗
x2 (n + ρ) = x2 (ρ) exp
r2 (t) − b2 (t)x∗2 (t) −
.
1 + d2 (t)x∗2 (t)
t=ρ
x∗1 (n
+ ρ) =
x∗1 (ρ) exp
n+ρ−1
∑ [
r1 (t) − b1 (t)x∗1 (t) −
8
(4.6)
It is easy to see that (x∗1 (n), x∗2 (n)) is a solution of system (1.2) on Z + for arbitrary ρ, and
0 < m1 − ε ≤ x∗1 (n) ≤ M1 − ε, 0 < m2 − ε ≤ x∗2 (n) ≤ M2 − ε, n ∈ Z + .
(4.7)
Then we get the following (4.8) due to ε which is an arbitrarily small positive constant,
0 < m1 ≤ x∗1 (n) ≤ M1 , 0 < m2 ≤ x∗2 (n) ≤ M2 , n ∈ Z + .
(4.8)
This completes the proof.
Finally, we are ready to state our main result in this section.
Theorem 4.1 Let the assumption (2.6) be satisﬁed and the following statement which 0 <
γ < 1 holds true, where γ = min{β1 , β2 },
u u
u u
l l l
2
u u
au
au
u M )2 − 2(a2 d1 M1 M2 −a2 b1 d1 m1 m2 )
1 M1 +a1 b2 M1 M2
2 M2 +a2 b1 M1 M2
−
−
(b
1
l
l
l
2
1
1+d1 m1
1+d2 m2
(1+d1 m1 )
2
u
2
u
2
u
2
(au
au2
au2
(au
1 M1 )
2 d1 M1 M2
1 d2 M1 M2
2 d1 M1 M2 )
− (1+d
−
−
−
,
l m )2
(1+dl1 m1 )3
(1+dl2 m2 )3
(1+dl1 m1 )4
2 2
β1 = 2bl1 m1 −
u u
au
au M +au bu M M
2(au du M1 M2 −al2 bl2 dl2 m1 m22 )
2 M2 +a2 b1 M1 M2
− 1 11+d1l m2 1 2 − (bu2 M2 )2 − 1 2 (1+d
l m )2
1+dl1 m1
2
2
2 2
2
u2 du M M 2
u2 du M 2 M
u du M M )2
(au
M
)
a
a
(a
2
1 2
2
1 2
2
2
1
1
2
1
1 2
− (1+dl m )2 − (1+dl m )3 − (1+dl m )3 − (1+dl m )4 ,
1 1
1 1
2 2
2 2
β2 = 2bl2 m2 −
(4.9)
then system (1.2) admits a unique positive almost periodic solution, which is uniformly asymptotically stable.
Proof. Denote u1 (n) = ln x1 (n), u2 (n) = ln x2 (n). It follows from (1.2) that

 u1 (n + 1) = u1 (n) + r1 (n) − b1 (n)eu1 (n) − a2 (n)eu2u(n)(n) ,
1+d (n)e 1
 u2 (n + 1) = u2 (n) + r2 (n) − b2 (n)eu2 (n) −
1
a1 (n)eu1 (n)
.
1+d2 (n)eu2 (n)
(4.10)
By Lemma (4.1), it is easy to see that the system (4.10) exists a bounded solution (u1 (n), u2 (n))
satisfying
ln m1 ≤ u1 (n) ≤ ln M1 , ln m2 ≤ u2 (n) ≤ ln M2 , n ∈ Z + .
(4.11)
Thus |u1 (n)| ≤ s1 , |u2 (n)| ≤ s2 , where s1 = max{| ln M1 |, | ln m1 |}, s2 = max{| ln M2 |, | ln m2 |}.
Deﬁne the norm
∥(u1 (n), u2 (n))∥ = |u1 (n)| + |u2 (n)|,
(4.12)
where (u1 (n), u2 (n)) ∈ R2 .
Consider the product system of system (4.10) as follow:
a2 (n)eu2 (n)
,
1+d1 (n)eu1 (n)
u
(n)
a1 (n)e 1
u2 (n + 1) = u2 (n) + r2 (n) − b2 (n)eu2 (n) − 1+d
u (n) ,
2 (n)e 2
w
a2 (n)e 2 (n)
w1 (n + 1) = w1 (n) + r1 (n) − b1 (n)ew1 (n) − 1+d
w (n) ,
1 (n)e 1
w
a1 (n)e 1 (n)
w2 (n + 1) = w2 (n) + r2 (n) − b2 (n)ew2 (n) − 1+d
w (n) .
2 (n)e 2
u1 (n + 1) = u1 (n) + r1 (n) − b1 (n)eu1 (n) −
9
(4.13)
Assume that H = (u1 (n), u2 (n)), I = (w1 (n), w2 (n)) are any two solution of system (3.7) deﬁned on Γ. And then ∥H∥ ≤ S, ∥I∥ ≤ S, where S = s1 + s2 , and Γ = {(u1 (n), u2 (n))| ln mi ≤
ui (n) ≤ ln Mi , i = 1, 2, n ∈ Z + }.
In the following, constructing a Lyapunov function which is deﬁned on Z + × Γ × Γ:
V (n, H, I) = (u1 (n) − w1 (n))2 + (u2 (n) − w2 (n))2 .
(4.14)
Notice that the form ∥H − I∥ = |u1 (n) − w1 (n)| + |u2 (n) − w2 (n)| is equivalent to ∥H − I∥△ =
1
[(u1 (n) − w1 (n))2 + (u2 (n) − w2 (n))2 ] 2 , which implies that there exist two positive constants
K1 , K2 such that K1 ∥H − I∥ ≤ ∥H − I∥ ≤ K2 ∥H − I∥. Obviously, K12 (∥H − I∥)2 ≤
V (n, H, I) ≤ K22 (∥H − I∥)2 .
Let α(x) = K12 x2 , β(x) = K22 x2 . Then the condition (1) of Lemma 2.2 is satisﬁed. Furˆ I)
ˆ ∈ Z + ×Γ×Γ, we replace (ui (n)−wi (n)), (ˆ
thermore, for any (n, H, I), (n, H,
ui (n)− w
ˆi (n)),
ˆ i (n), respectively, we have
i = 1, 2 with Di (n), D
ˆ I)|
ˆ
|V (n, H, I) − V (n, H,
ˆ 2 (n) − D
ˆ 2 (n)|
= |D2 (n) + D2 (n) − D
1
|D21 (n)
2
1
2
ˆ 2 (n)| + |D2 (n) − D
ˆ 2 (n)|
≤
−D
1
2
2
ˆ 1 (n)||D1 (n) − D
ˆ 1 (n)|
= |D1 (n) + D
ˆ 2 (n)||D2 (n) − D
ˆ 2 (n)|
+|D2 (n) + D
(4.15)
≤ (|u1 (n)| + |w1 (n)| + |ˆ
u1 (n)| + |w
ˆ1 (n)|)(|u1 (n) − u
ˆ1 (n)| + |w1 − w
ˆ1 (n)|)
+(|u2 (n)| + |w2 (n)| + |ˆ
u2 (n)| + |w
ˆ2 (n)|)(|u2 (n) − u
ˆ2 (n)| + |w2 − w
ˆ2 (n)|)
≤ G{|u1 (n) − u
ˆ1 (n)| + |u2 (n) − u
ˆ2 (n)| + |w1 − w
ˆ1 (n)| + |w2 − w
ˆ2 (n)|}
ˆ + ∥I − I∥},
ˆ
= G{∥H − H∥
ˆ = (ˆ
where H
u1 (n), u
ˆ2 (n)), Iˆ = (w
ˆ1 (n), w
ˆ2 (n)), and G = 4 max{s1 , s2 }. Consequently, the
condition (2) of Lemma 2.2 is satisﬁed. At last, calculating the △ V (n) of V (n) along the
solutions of system (4.13) yields
△V4.13 (n)
= V (n + 1) − V (n)
= D21 (n + 1) + D22 (n + 1) − D21 (n) − D22 (n)
[
] [
]
= D21 (n + 1) − D1 (n)2 + D22 (n + 1) − D2 (n)2
[
(
)]2
a2 (n)eu2 (n)
a2 (n)ew2 (n)
= D1 (n) − b1 (n)(eu1 (n) − ew1 (n) ) − 1+d
−
− D1 (n)2
u (n)
1+d1 (n)ew1 (n)
1 (n)e 1
[
(
)]2
a1 (n)eu1 (n)
a1 (n)ew1 (n)
+ D2 (n) − b2 (n)(eu2 (n) − ew2 (n) ) − 1+d
−
− D2 (n)2
u
(n)
w
(n)
2
1+d2 (n)e 2
2 (n)e (
)
u2 (n)
ew2 (n)
= −2b1 (n)D1 (n)(eu1 (n) − ew1 (n) ) − 2a2 (n)D1 (n) 1+d e(n)e
u1 (n) − 1+d (n)ew1 (n)
1
1
(
)
eu2 (n)
ew2 (n)
u
(n)
w
(n)
1
1
+2a2 (n)b1 (n)(e
−e
) 1+d (n)eu1 (n) − 1+d (n)ew1 (n)
1
( 1 u (n)
)2
w2 (n)
e 2
2
u
(n)
w
(n)
2
2
1
1
+b1 (n)(e
−e
) + a2 (n) 1+d (n)eu1 (n) − 1+d e(n)e
w1 (n)
1
10
1
(
)
u1 (n)
ew1 (n)
−2b2 (n)D2 (n)(eu2 (n) − ew2 (n) ) − 2a1 (n)D2 (n) 1+d e(n)e
u2 (n) − 1+d (n)ew2 (n)
2
2
(
)
eu1 (n)
ew1 (n)
u
(n)
w
(n)
2
2
+2a1 (n)b2 (n)(e
−e
) 1+d (n)eu2 (n) − 1+d (n)ew2 (n)
2
)2
( 2 u (n)
w1 (n)
e 1
2
u
(n)
w
(n)
2
2
2
2
.
+b2 (n)(e
−e
) + a1 (n) 1+d (n)eu2 (n) − 1+d e(n)e
w2 (n)
2
(4.16)
2
Applying the one-dimensional and two-dimensional mean value theorem, we arrive at a simple
result as follows
eui (n) − ewi (n) = ζi (n)(ui (n) − wi (n)),
eui (n)
1+dj (n)euj (n)
ewi (n)
1+dj (n)ewj (n)
i (n)
= 1+djθ(n)θ
(ui (n)
j (n)
−
(4.17)
− wi (n)) −
dj (n)θi (n)θj (n)
(uj (n)
(1+dj (n)θj (n))2
− wj (n)),
where i, j = 1, 2, i ̸= j and ζi (n), θi (n) lie between eui (n) and ewi (n) respectively. Substituting
(4.17) into (4.16), one obtains
△v(n)
(n)d1 (n)θ1 (n)θ2 (n) 2
2a2 (n)θ2 (n)
D1 (n) − 1+d
D1 (n)D2 (n)
= −2b1 (n)ζ1 (n)D21 (n) + 2a2(1+d
2
1 (n)θ1 (n))
1 (n)θ1 (n)
]
[
2
2
θ2 (n)
2d1 (n)θ1 (n)θ22 (n)
(d1 (n)θ1 (n)θ2 (n))
2
2
2
+a2 (n) (1+d1 (n)θ1 (n))4 D1 (n) + (1+d1 (n)θ1 (n))2 D2 (n) − (1+d1 (n)θ1 (n))3 D1 (n)D2 (n)
(n)d1 (n)ζ1 (n)θ1 (n)θ2 (n) 2
− 2a2 (n)b1(1+d
D1 (n) +
2
1 (n)θ1 (n))
2a2 (n)b1 (n)ζ1 (n)θ2 (n)
D1 (n)D2 (n) + b21 (n)ζ12 (n)D21 (n)
1+d1 (n)θ1 (n)
2a1 (n)θ1 (n)
2a1 (n)d2 (n)θ1 (n)θ2 (n) 2
D2 (n) − 1+d
D1 (n)D2 (n)
(1+d2 (n)θ2 (n))2
2 (n)θ2 (n)
−2b2 (n)ζ2 (n)D22 (n) +
[
2
2 (n)θ1 (n)θ2 (n))
2
2
+a1 (n) (d(1+d
4 D2 (n) +
2 (n)θ2 (n))
]
θ12 (n)
(n)θ1 (n)2 (n)θ2 (n)
D2 (n)− 2d2(1+d
D1 (n)D2 (n)
3
(1+d2 (n)θ2 (n))2 1
2 (n)θ2 (n))
(n)d2 (n)ζ2 (n)θ1 (n)θ2 (n) 2
2 (n)ζ2 (n)θ1 (n)
− 2a1 (n)b2(1+d
D2 (n) + 2a1 (n)b
D1 (n)D2 (n) + b22 (n)ζ22 (n)D22 (n)
2
1+d2 (n)θ2 (n)
2 (n)θ2 (n))
[
2
(a1 (n)θ1 (n))2
(n)d1 (n)θ1 (n)θ2 (n)
1 (n)θ1 (n)θ2 (n))
+ b21 (n)ζ12 (n) + (a2 (n)d
+ (1+d
= − 2b1 (n)ζ1 (n) + 2a2(1+d
2
2
(1+d1 (n)θ1 (n))4
1 (n)θ1 (n))
2 (n)θ2 (n))
]
[
(n)d1 (n)ζ1 (n)θ1 (n)θ2 (n)
(n)d2 (n)θ1 (n)θ2 (n)
D21 (n) + − 2b2 (n)ζ2 (n) + 2a1(1+d
− 2a2 (n)b1(1+d
2
2
1 (n)θ1 (n))
2 (n)θ2 (n))
]
2
2
2a
(n)b
(n)d
(n)ζ
(n)θ
(n)θ
(n)
(a
(n)θ
(a
(n)d
(n)θ
(n)θ
(n))
1
2
2
2
1
2
2
2 (n))
1
2
1
2
2
2
−
+ (1+d1 (n)θ1 (n))2 D22 (n)
+b2 (n)ζ2 (n) +
(1+d2 (n)θ2 (n))4
(1+d2 (n)θ2 (n))2
[
2a2 (n)d (n)θ (n)θ 2 (n)
2a2 (n)θ2 (n)
1 (n)ζ1 (n)θ2 (n)
+ − 1+d
− 2(1+d11 (n)θ11 (n))23 + 2a2 (n)b
1+d1 (n)θ1 (n)
1 (n)θ1 (n)
]
2a21 (n)d2 (n)θ2 (n)θ12 (n)
2a1 (n)θ1 (n)
2a1 (n)b2 (n)ζ2 (n)θ1 (n)
− 1+d2 (n)θ2 (n) − (1+d2 (n)θ2 (n))3 +
D1 (n)D2 (n)
1+d2 (n)θ2 (n)
[
2
(n)d1 (n)θ1 (n)θ2 (n)
1 (n)θ1 (n)θ2 (n))
≤ − 2b1 (n)ζ1 (n) + 2a2(1+d
+ b21 (n)ζ12 (n) + (a2 (n)d
2
(1+d1 (n)θ1 (n))4
1 (n)θ1 (n))
a2 (n)d (n)θ (n)θ2 (n)
(a1 (n)θ1 (n))2
a2 (n)θ2 (n)
+ 1+d
+ 2 (1+d11 (n)θ11 (n))23
(1+d2 (n)θ2 (n))2
1 (n)θ1 (n)
]
a21 (n)d2 (n)θ2 (n)θ12 (n)
a1 (n)θ1 (n)
a1 (n)b2 (n)ζ2 (n)θ1 (n)
1 (n)ζ1 (n)θ2 (n)
+
+
+
D21 (n)
+ a2 (n)b
1+d2 (n)θ2 (n)
(1+d2 (n)θ2 (n))3
1+d2 (n)θ2 (n)
[ 1+d1 (n)θ1 (n)
(a1 (n)d2 (n)θ1 (n)θ2 (n))2
(n)d2 (n)θ1 (n)θ2 (n)
2
2
(n)
+
+ − 2b2 (n)ζ2 (n) + 2a1(1+d
(n)ζ
+
b
2
2
2
(n)θ
(n))
(1+d2 (n)θ2 (n))4
2
2
a22 (n)d1 (n)θ1 (n)θ22 (n)
2a1 (n)b2 (n)d2 (n)ζ2 (n)θ1 (n)θ2 (n)
(a2 (n)θ2 (n))2
a2 (n)θ2 (n)
−
+
+
+
2
2
(1+d2 (n)θ2 (n))
(1+d1 (n)θ1 (n))
1+d1 (n)θ1 (n)
(1+d1 (n)θ
]1 (n))3
a21 (n)d2 (n)θ2 (n)θ12 (n)
a1 (n)θ1 (n)
a1 (n)b2 (n)ζ2 (n)θ1 (n)
a2 (n)b1 (n)ζ1 (n)θ2 (n)
D22 (n)
+ 1+d1 (n)θ1 (n) + 1+d2 (n)θ2 (n) + (1+d2 (n)θ2 (n))3 + 1+d2 (n)θ2 (n)
[
au M +au bu M M
2(au du M1 M2 −al2 bl1 dl1 m21 m2 )
au M +au bu M M
− 2bl1 m1 + 2 21+d2l m1 1 1 2 + 1 11+d1l m2 2 1 2 + (bu1 M1 )2 + 2 1 (1+d
l m )2
1
2
1 1
]
2
2
2
2
u
u
u
u
u2
u2
u
(a1 M1 )
a2 d1 M1 M2
a1 d2 M1 M2
(a2 d1 M1 M2 )
2
+ (1+dl m2 )2 + (1+dl m1 )3 + (1+dl m2 )3 + (1+dl m1 )4 D1 (n)
(n)d1 (n)ζ1 (n)θ1 (n)θ2 (n)
− 2a2 (n)b1(1+d
+
2
1 (n)θ1 (n))
≤
2
1
2
1
11
[
+ − 2bl2 m2 +
=
u u
au
au M +au bu M M
2(au du M1 M2 −al2 bl2 dl2 m1 m22 )
2 M2 +a2 b1 M1 M2
+ 1 11+d1l m2 2 1 2 + (bu2 M2 )2 + 1 2 (1+d
l m )2
1+dl1 m1
2
2 2
]
2
u2 du M M 2
u2 du M 2 M
u du M M ) 2
(au
M
)
a
a
(a
2
1
2
1
2
2
2
2
1
2
1
2
1
1 2
(n)
+ (1+d
+
+
+
D
l
l
3
2
(1+dl2 m2 )3
(1+dl2 m2 )4
1)
[ 1 m1 )2 u (1+d1 m
u bu M M
u M +au bu M M
a
M
+a
a
2(au du M1 M2 −al2 bl1 dl1 m21 m2 )
− 2bl1 m1 − 2 21+d2l m1 1 1 2 − 1 11+d1l m2 2 1 2 − (bu1 M1 )2 − 2 1 (1+d
l m )2
1
1
2
1
]
2
u
u
u
2
2
2
(au
au2
au2
(au
2
1 M1 )
2 d1 M1 M2
1 d2 M1 M2
2 d1 M1 M2 )
− (1+d
D
(n)
−
−
−
l
l
1
(1+dl2 m2 )3
(1+dl1 m1 )4
[ 2 m2 )2 u (1+d1um1u)3
u M +au bu M M
a
M
+a
a
2(au du M1 M2 −al2 bl2 dl2 m1 m22 )
b
M
M
2
1
2
1
1
2
− 2bl2 m2 − 2 1+d2l m1 1
− 1 1+d1l m2 2
− (bu2 M2 )2 − 1 2 (1+d
l m )2
1
2
2 2
]
u2 du M M 2
u2 du M 2 M
u du M M )2
2
a
a
(a
(au
M
)
2
1 2
2
1 2
2
2
2
1
1
2
1
1 2
− (1+dl m1 )2 − (1+dl m1 )3 − (1+dl m2 )3 − (1+dl m2 )4 D2 (n)
1
1
2
2
= −[β1 D21 (n) + β2 D22 (n)]
= −γ[D21 (n) + D22 (n)]
= −γVn ,
(4.18)
where γ = min{β1 , β2 }, and 0 < γ < 1 which has been pointed out in the Theorem 4.1. In
addition, condition (3) of Lemma 2.2 is also satisﬁed. According to Lemma 2.2, there exists
a uniformly asymptotically stable almost periodic solution (u∗1 (n), u∗2 (n)) of system (4.10)
which is bounded by Γ for all n ∈ Z + . Namely, there exists a uniformly asymptotically
stable almost periodic solution (x∗1 (n), x∗2 (n)) of system (1.2) which is bounded by Ω for all
n ∈ Z + . This completed the proof.
5 Example and numerical simulations
In this section, we only give the following example about almost periodic to check the feasibility of the assumptions of Theorem 4.1 considering that the simulation about periodic
model is similar.
Example 5.1 Consider the following discrete system:

[
√
√

x
(n
+
1)
=
x
(n)
exp
1.09 − 0.03 sin( 2nπ) − (1.15 − 0.01 cos( 2nπ))x1 (n)

1
1


]
√


(0.035+0.005 cos( 2nπ))x2 (n)

√
− 1+(2.10+0.02
,
[ cos( 5nπ))x1 (n) √
√

x2 (n + 1) = x2 (n) exp 1.06 + 0.03 cos( 2nπ) − (1.11 + 0.01 sin( 2nπ))x2 (n)



]
√


(0.025+0.005 cos( 2nπ))x1 (n)

√
− 1+(2.07+0.03
,
sin( 5nπ))x (n)
2
with the following intial conditions:
x1 (n)∗ (0) = 0.98, x2 (n)∗ (0) = 1.01.
By a computation, we get
M1 ≈ 0.9890,
M2 ≈ 0.9947,
m2 ≈ 0.7971, β1 ≈ 0.3781,
(r1l
−
au2 M2 )
≈ 1.0202 > 0,
β2 ≈ 0.4438
(r2l
12
m1 ≈ 0.7745,
− au1 M1 ) ≈ 1.0003 > 0
(5.1)
bu
1 M1
r1l −au
2 M2
bu
2 M2
r2l −au
2 M2
≈ 1.1245 > 1,
≈ 1.1251 > 1.
(5.2)
Clearly, the assumption of Theorem 4.1 are satisﬁed and all the coeﬃcients are appropriate.
Hence, system (5.1) admits a unique uniformly asymptotically stable positive almost periodic solution. From Fig.1, we easily see that there exists a positive almost periodic solution
(x∗1 (n), x∗2 (n)), and the 2-dimensional and 3-dimensional phase portraits of almost periodic system (5.1) are revealed in Fig.2, respectively. Fig.3 shows that any positive solution
(x1 (n), x2 (n)) tends to the almost periodic solution (x∗1 (n), x∗2 (n)).
6 Conclusions
In this paper, we consider a discrete two-species competitive model whose periodic solutions
and almost periodic solutions are discussed, respectively. By the scale law and mean-value
theorem, a good understanding of the existence and stability of positive periodic solutions is
gained. Furthermore, by constructing Lyapunov functions, the conditions on the asymptotic
stability of the positive almost periodic solution are established. The assumption in (2.6)
implies that the r(t) should be suitably large.
Acknowledgements
The work is supported by the National Natural Science Foundation of China (no.61261044).
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15
1.02
1.02
(b)
1
1
0.98
0.98
x*1(n),x1(n)
x*1(n)
(a)
0.96
0.96
0.94
0.94
0.92
0.92
0.9
0
20
40
60
80
0.9
900
100
920
940
n
1000
960
980
1000
(d)
(c)
1
1
0.98
0.98
x*2(n)
x*2(n)
980
1.02
1.02
0.96
0.96
0.94
0.94
0.92
0.92
0.9
960
n
0
20
40
60
80
0.9
900
100
n
920
940
n
Figure 1: Positive almost periodic solution of system (4.1). (a), (c) Time-series x∗1 (n) and x∗2 (n) with initial
value x∗1 (0) = 0.98, x∗2 (0) = 1.01, for n ∈ [0, 100], (b) and (d) show that x∗1 (n) and x∗2 (n) have the same initial
data for n ∈ [900, 1000], respectively.
16
1.02
1.02
(b)
1
1
0.98
0.98
x*2(n)
x*2(n)
(a)
0.96
0.96
0.94
0.94
0.92
0.92
0.9
0.9
0.92
0.94
0.96
0.98
1
0.9
0.9
1.02
0.92
0.94
x*1(n)
1
0.98
1
1.02
1
(c)
(d)
0.95
x*2(n)
0.95
x*2(n)
0.96
x*1(n)
0.9
0.9
0.85
0.85
0.8
1.1
0.8
1.1
100
1
1000
1
80
980
60
40
0.9
x*1(n)
960
940
0.9
20
0
x*1(n)
n
920
900
n
Figure 2: Phase portrait, 2-dimensional and 3-dimensional phase portrait of almost periodic solution of
system (4.1). Time-series x∗1 (n) and x∗2 (n) with initial value x∗1 (0) = 0.98, x∗2 (0) = 1.01, (a), (c) indicate
n ∈ [0, 100] and (b), (d) indicate n ∈ [900, 1000], respectively.
17
1.02
1.02
x1 (n) ∗;
(b)
x∗1 (n) ◦
1
1
0.98
0.98
x*1(n),x1(n)
x*1(n),x1(n)
(a)
0.96
0.94
0.92
0.92
0
20
40
60
80
0.9
900
100
920
940
n
980
1000
1.02
(c)
x2 (n) ∗;
(d)
x∗2 (n) ◦
1
1
0.98
0.98
x*2(n),x2(n)
x*2(n),x2(n)
960
n
1.02
0.96
0.94
0.92
0.92
0
20
40
60
80
0.9
900
100
n
x2 (n) ∗;
x∗2 (n) ◦
0.96
0.94
0.9
x∗1 (n) ◦
0.96
0.94
0.9
x1 (n) ∗;
920
940
960
980
1000
n
Figure 3: Uniformly asymptotic stability. Time-series x∗1 (n) and x∗2 (n) with initial value x∗1 (0) = 0.98, x∗2 (0) =
1.01, x1 (n) and x2 (n) with initial value x1 (0) = 0.92, x2 (0) = 0.91, (a), (c) indicate n ∈ [0, 100] and (b), (d)
indicate n ∈ [900, 1000], respectively.
18