# Asymptotic Behaviors of a Delayed Nonautonomous Predator-Prey System Governed by Difference Equations

Authors: Lili Liu; Zhijun Liu
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

Based on a predator-prey differential system with continuously distributed delays, we derive a corresponding difference version by using the method of a discretization technique. A good understanding of permanence of system and global attractivity of positive solutions of system is gained. An example and its numerical simulations are presented to substantiate our theoretical results.


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## Body

## 1. Introduction

It is well known that the classical Lotka-Volterra differential equations for predator-prey interaction can be written as(1.1)x′(t)=x(t)[a-bx(t)-cy(t)],y′(t)=y(t)[-d+px(t)-qy(t)],
where x(0)>0, y(0)>0, x(t), and y(t) are the population densities of prey and predator, respectively, at time t, and the coefficients a,b,c,d,p,q are positive constants. a is the intrinsic growth rate of prey species, and d represents the death rate of predator species; b,q and c,p, denote their self-inhibition and interaction rates, respectively. To account for time lag effects in their interaction, Cushing [1] suggested visualizing system (1.1) as a special case of the following more general system (1.2)x′(t)=x(t)[a-∫0+∞x(t-θ)dl1(θ)-∫0+∞y(t-θ)dl2(θ)],y′(t)=y(t)[-d+∫0+∞x(t-θ)dl3(θ)-∫0+∞y(t-θ)dl4(θ)],
where li(θ) are nondecreasing functions on [0,+∞) with ∫0+∞dli(θ)<+∞ for each i=1,…,4, and with limθ→0+[li(θ)-li(0)], respectively, equal to b,c,p,q for i=1,…,4.As Hale [2] pointed out, one expects that the lag effects tend to diminish gradually in ever-moderating pace as one moves backward in time after the initial instantaneous effect described by the assumption limθ→0+[li(θ)-li(0)]>0. The effects are negligible after a certain length of time τ>0. Mathematically, for θ>0, it is reasonable that dli(θ) can be assumed to be of the form dli(θ)=ki(θ)dθ, i=1,…,4, where(S1)
ki(s)≥0 and continuous for s∈[0,τ]; ki(s)=0 for s∈[τ,+∞];(S2)
ki′(s)≤0 and continuous for s∈(0,τ); lims→0+ki′(s) and lims→τ-ki′(s) both exist;(S3)
ki′′(s)≥0 and continuous for s∈(0,τ).The assumption(S2) represents gradual diminishing effect as one moves backward in time, and the assumption (S3) stands for the moderating pace of the decrease (see [2] or [3] for the assumptions (S1) to (S3)). Leung [4] revised system (1.2) and established the following predator-prey model with continuously distributed delays(1.3)x′(t)=x(t)[a-bx(t)-cy(t)-∫0τk1(s)x(t-s)ds-∫0τk2(s)y(t-s)ds],y′(t)=y(t)[-d+px(t)-qy(t)+∫0τk3(s)x(t-s)ds-∫0τk4(s)y(t-s)ds].
The parameters of system (1.3) are assumed to be constant; however, in the real world the parameters are not fixed constants owing to the variation of environment. The effect of a varying environment is significant for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, it is realistic to assume that the parameters of system (1.3) are continuous functions with respect to t, and then we obtain the following form:(1.4)x′(t)=x(t)[a(t)-b(t)x(t)-c(t)y(t)-∫0τk1(s)x(t-s)ds-∫0τk2(s)y(t-s)ds],y′(t)=y(t)[-d(t)+p(t)x(t)-q(t)y(t)+∫0τk3(s)x(t-s)ds-∫0τk4(s)y(t-s)ds],
where the coefficients a(t),b(t),c(t),d(t),p(t),q(t) are positive continuous functions and ki(s)ds satisfy ∫0τki(s)ds=1 for i=1,…,4.Although much progress on the predator-prey models with discrete or distributed delays has been made, such models are not sufficiently researched in the sense that most results are continuous time versions related (see [5–13]). Many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous time ones when populations have a short life expectancy, nonoverlapping generations in the real world (see [14–18]). Meanwhile, discrete time models can provide efficient computational models of continuous time models for numerical simulations. To the best of our knowledge, no such work has been done for the corresponding discrete version of system (1.4).In the following, we employ the discretization technique to derive the discrete version of system (1.4). Throughout this paper, let ℤ, ℤ+ and ℝ2 denote the sets of all integers, nonnegative integers, and two-dimensional vector space, respectively. We begin to approximate the continuous time system by replacing the integral term with discrete sums of the form(1.5)∫0τki(s)x(t-s)ds≈∑[s/h]=0Mki([sh]h)x([th]h-[sh]h)w(h)
for t∈[nh,(n+1)h), s∈[rh,(r+1)h), n,r∈ℤ+, i=1,…,4, where [·] denotes the greatest integer function. M=[τ/h], w(h)=h+O(h) for h>0, where the fixed number h denotes an uniform discretization step size. We approximate system (1.4) by differential equations with piecewise constant arguments of the form(1.6)x′(t)=x(t)[∑[s/h]=0Ma([th]h)-b([th]h)x([th]h)-c([th]h)y([th]h)-∑[s/h]=0Mk1([sh]h)x([th]h-[sh]h)h-∑[s/h]=0Mk2([sh]h)y([th]h-[sh]h)h],y′(t)=y(t)[∑[s/h]=0M-d([th]h)+p([th]h)x([th]h)-q([th]h)y([th]h)+∑[s/h]=0Mk3([sh]h)x([th]h-[sh]h)h-∑[s/h]=0Mk4([sh]h)y([th]h-[sh]h)h]
for t∈[nh,(n+1)h), s∈[rh,(r+1)h), n,r∈ℤ+. Noting that [t/h]=n, [s/h]=r, we integrate (1.6) over [nh,t), where t<(n+1)h, then (1.6) can be reformulated as(1.7)x′(t)=x(t)[a(nh)-b(nh)x(nh)-c(nh)y(nh)-∑r=0Mk1(rh)x(nh-rh)h-∑r=0Mk2(rh)y(nh-rh)h]y′(t)=y(t)[-d(nh)+p(nh)x(nh)-q(nh)y(nh)+∑r=0Mk3(rh)x(nh-rh)h-∑r=0Mk4(rh)y(nh-rh)h].
Denoting x(n)=x(nh), y(n)=y(nh), a(n)=a(nh), b(n)=b(nh), c(n)=c(nh), d(n)=d(nh), p(n)=p(nh), q(n)=q(nh), ki(r)=ki(rh)h, i=1,…,4, then we have(1.8)x(t)=x(n)exp⁡{[a(n)-b(n)x(n)-c(n)y(n)-∑r=0Mk1(r)x(n-r)-∑r=0Mk2(r)y(n-r)](t-nh)},y(t)=y(n)exp⁡{[-d(n)+p(n)x(n)-q(n)y(n)+∑r=0Mk3(r)x(n-r)-∑r=0Mk4(r)y(n-r)](t-nh)}.
Setting t→(n+1)h in (1.8) and simplifying, we get a discrete time analogue of continuous time system (1.4) with the form(1.9)x(n+1)=x(n)exp⁡{a(n)-b(n)x(n)-c(n)y(n)-∑r=0Mk1(r)x(n-r)-∑r=0Mk2(r)y(n-r)}y(n+1)=y(n)exp⁡{-d(n)+p(n)x(n)-q(n)y(n)+∑r=0Mk3(r)x(n-r)-∑r=0Mk4(r)y(n-r)}.
Our main purpose of this paper is to derive a set of easily verifiable sufficient conditions concerning the permanence and global attractivity of system (1.9). For convenience, we introduce the following definitions and notations.Definition 1.1.
It is said that system (1.9) is permanent if there exist positive constants mi,Mi, i=1,2 and any positive solution {(x(n),y(n))⊤} of system (1.9) satisfies
(1.10)m1≤lim⁡inf⁡n→+∞x(n)≤lim⁡sup⁡n→+∞x(n)≤M1,m2≤lim⁡inf⁡n→+∞y(n)≤lim⁡sup⁡n→+∞y(n)≤M2.Definition 1.2.
A positive solution{(x(n),y(n))⊤} of system (1.9) is global attractive if each other positive solution {(x*(n),y*(n))⊤} of system (1.9) satisfies
(1.11)lim⁡n→+∞|x(n)-x*(n)|=0,lim⁡n→+∞|y(n)-y*(n)|=0.
Let f(n) be a bounded sequence; we denote
(1.12)fu=sup⁡n∈Z+{f(n)},fl=inf⁡n∈Z+{f(n)}.
Meanwhile, we make a convention that ∏n=mn=kf(n)=1 for all m>k.This paper proceeds as follows. System (1.9) is analyzed to study the permanence and global attractivity of system (1.9) in the next two sections, respectively. In the final section, we give an example, and its numerical simulations are presented to substantiate our theoretical results.


## 2. Permanence of System (1.9)

In this section, we devote to investigating the permanence of system (1.9). To do so, we introduce the following lemmas.Lemma 2.1 (see [19]).
Assume that{x(n)} satisfies  x(n)>0 and
(2.1)x(n+1)≤x(n)exp⁡[r(n)(1-ax(n))]
for n∈[n1,+∞), where a is a positive constant and n1∈ℤ+. Then
(2.2)lim⁡n→+∞sup⁡x(n)≤1aruexp⁡(ru-1).Lemma 2.2 (see [19]).
Assume that{x(n)} satisfies  x(n)>0 and
(2.3)x(n+1)≥x(n)exp⁡[r(n)(1-ax(n))]
for n∈[n2,+∞), limsupn→+∞x(n)≤H and x(n2)>0, where a is a positive constant such that aH>1 and n2∈ℤ+. Then
(2.4)lim⁡n→+∞inf⁡x(n)≥1aexp⁡[ru(1-aH)].For the simplicity of description, we denote(2.5)Δ1ɛ=-dl+[pu+(M+1)k3u](M1+ɛ),Δ2ɛ=al-cu(M2+ɛ)-(M+1)[k1u(M1+ɛ)+k2u(M2+ɛ)],Δ3ɛ=-du+pl(m1-ɛ)+(M+1)[k3l(m1-ɛ)-k4u(M2+ɛ)],Δi=lim⁡ɛ→0Δiɛ,i=1,2,3,
where M1,M2,m1 are, respectively, defined in (2.8) and (2.11)–(2.14) and ɛ is a sufficient small positive constant. Now, we begin to search the conditions for the permanence of (1.9).Theorem 2.3.
If the conditions(2.6)min⁡{Δi,i=1,2,3}>0
hold, then (1.9) is permanent.Proof.
It is easy to verify thatΩ={(x(n),y(n))⊤∣x(n)>0,y(n)>0} is a positive invariant set of (1.9) with x(0)>0, y(0)>0. Suppose that {(x(n),y(n))⊤} is any positive solution of (1.9), we prove Theorem 2.3 by the following two steps.
Step 1.
We show that{(x(n),y(n))⊤} is uniformly ultimately upper bounded.
From the first equation of (1.9), we have
(2.7)x(n+1)≤x(n)exp⁡[a(n)-b(n)x(n)]≤x(n)exp⁡{a(n)[1-blaux(n)]}.
Applying Lemma 2.1, we can obtain that
(2.8)lim⁡n→+∞sup⁡x(n)≤exp⁡(au-1)bl=defM1.
From (2.8), there exists a sufficient large n1∈ℤ+ and any constant ɛ>0 such that
(2.9)x(n)≤M1+ɛ,n≥n1.
It follows from the second equation of (1.9) that
(2.10)y(n+1)≤y(n)exp⁡{-dl+[pu+(M+1)k3u](M1+ɛ)-qly(n)}=y(n)exp⁡{Δ1ɛ[1-qlΔ1ɛy(n)]}.
So by Lemma 2.1 and letting ɛ→0 in (2.10), we have
(2.11)lim⁡n→+∞sup⁡y(n)≤exp⁡(Δ1-1)ql=defM2.Step 2.
We prove that{(x(n),y(n))⊤} is uniformly ultimately lower bounded.
For any sufficient smallɛ>0, it follows from (2.6) that Δ2ɛ>0. According to (2.11), there exists an n0>n1 such that y(n)≤M2+ɛ for n≥n0 and the above constant ɛ. By the first equation of (1.9), it gives that
(2.12)x(n+1)≥x(n)exp⁡{al-cu(M2+ɛ)-(M+1)[k1u(M1+ɛ)+k2u(M2+ɛ)]-bux(n)}=x(n)exp⁡{Δ2ɛ[1-buΔ2ɛx(n)]}.
Note that
(2.13)buΔ2ɛM1=buΔ2ɛexp⁡(au-1)bl≥bualexp⁡(au-1)bl>exp⁡(au-1)au>1,
where we employ the inequality result that exp(x-1)>x for x>0. Therefore, applying Lemma 2.2 and setting ɛ→0, one has
(2.14)lim⁡n→+∞inf⁡x(n)≥Δ2buexp⁡[Δ2(1-buΔ2M1)]=defm1.
For the above constant ɛ>0, it follows from (2.14) that there exists a sufficient large n1*∈ℤ+ such that
(2.15)x(n)≥m1-ɛ,n≥n1*.
From the second equation of (1.9), we have
(2.16)y(n+1)≥y(n)exp⁡{-du+pl(m1-ɛ)+(M+1)[k3l(m1-ɛ)-k4u(M2+ɛ)]-quy(n)}=y(n)exp⁡{Δ3ɛ[1-quΔ3ɛy(n)]}.
Clearly,
(2.17)quΔ3ɛM2=quΔ3ɛexp⁡(Δ1-1)ql≥qlexp⁡{-du+[pl+(M+1)k3l]m1-1}[-du+(pl+(M+1)k3l)m1]ql>1.
Thus, applying Lemma 2.2 and letting ɛ→0, one has
(2.18)lim⁡n→+∞inf⁡y(n)≥Δ3quexp⁡[Δ3(1-quΔ3M2)]=defm2.
Based on the above Step 1 with Step 2, we can see that system (1.9) is permanent. This completes the proof of Theorem 2.3.


## 3. Global Attractivity of Positive Solutions of System (1.9)

In this section, we investigate the global attractivity of positive solutions of system (1.9).Theorem 3.1.
Assume that there exists a positive constantη such that
(3.1)min⁡{bl,2M1-bu}-(M+1)(k1u+k3u)-pu≥η,min⁡{ql,2M2-qu}-(M+1)(k2u+k4u)-cu≥η
hold. Then any positive solution {(x(n),y(n))⊤} of system (1.9) is global attractive.Proof.
Denote{(x*(n),y*(n))⊤} be any other positive solution of system (1.9). Let V11(n)=|lnx(n)-lnx*(n)|, then it follows from the first equation of (1.9) that
(3.2)V11(n+1)=|ln⁡x(n+1)-ln⁡x*(n+1)|≤|ln⁡x(n)-ln⁡x*(n)-b(n)(x(n)-x*(n))|+c(n)|y(n)-y*(n)|+∑r=0Mk1(r)|x(n-r)-x*(n-r)|+∑r=0Mk2(r)|y(n-r)-y*(n-r)|.
By the Mean Value theorem, we get
(3.3)ln⁡x(n)-ln⁡x*(n)=1θ1(n)(x(n)-x*(n)),
where θ1(n) lies between x(n) and x*(n). Then we have
(3.4)|ln⁡x(n)-ln⁡x*(n)-b(n)(x(n)-x*(n))|=|ln⁡x(n)-ln⁡x*(n)|-|ln⁡x(n)-ln⁡x*(n)|+|ln⁡x(n)-ln⁡x*(n)-b(n)(x(n)-x*(n))|=|ln⁡x(n)-ln⁡x*(n)|-1θ1(n)|x(n)-x*(n)|+|1θ1(n)(x(n)-x*(n))-b(n)(x(n)-x*(n))|=|ln⁡x(n)-ln⁡x*(n)|-(1θ1(n)-|1θ1(n)-b(n)|)|x(n)-x*(n)|.
Combining (3.2) with (3.4), we have
(3.5)ΔV11(n)=V11(n+1)-V11(n)≤-(1θ1(n)-|1θ1(n)-b(n)|)|x(n)-x*(n)|+c(n)|y(n)-y*(n)|+∑r=0Mk1(r)|x(n-r)-x*(n-r)|+∑r=0Mk2(r)|y(n-r)-y*(n-r)|.
Next, we let
(3.6)V12(n)=∑r=0M∑s=n-rn-1k1(r)|x(s)-x*(s)|+∑r=0M∑s=n-rn-1k2(r)|y(s)-y*(s)|.
Then we have
(3.7)ΔV12(n)=V12(n+1)-V12(n)=∑r=0M∑s=n+1-rnk1(r)|x(s)-x*(s)|-∑r=0M∑s=n-rn-1k1(r)|x(s)-x*(s)|+∑r=0M∑s=n+1-rnk2(r)|y(s)-y*(s)|-∑r=0M∑s=n-rn-1k2(r)|y(s)-y*(s)|=∑r=0M∑s=n+1-rn-1k1(r)|x(s)-x*(s)|+∑r=0Mk1(r)|x(n)-x*(n)|-∑r=0M∑s=n+1-rn-1k1(r)|x(s)-x*(s)|-∑r=0Mk1(r)|x(n-r)-x*(n-r)|+∑r=0M∑s=n+1-rn-1k2(r)|y(s)-y*(s)|+∑r=0Mk2(r)|y(n)-y*(n)|-∑r=0M∑s=n+1-rn-1k2(r)|y(s)-y*(s)|-∑r=0Mk2(r)|y(n-r)-y*(n-r)|=∑r=0Mk1(r)|x(n)-x*(n)|-∑r=0Mk1(r)|x(n-r)-x*(n-r)|+∑r=0Mk2(r)|y(n)-y*(n)|-∑r=0Mk2(r)|y(n-r)-y*(n-r)|.
We set
(3.8)V1(n)=V11(n)+V12(n),
then it follows from (3.5) and (3.7) that
(3.9)ΔV1(n)=ΔV11(n)+ΔV12(n)≤-(1θ1(n)-|1θ1(n)-b(n)|-∑r=0Mk1(r))|x(n)-x*(n)|+(c(n)+∑r=0Mk2(r))|y(n)-y*(n)|.
Similarly, we define
(3.10)V2(n)=V21(n)+V22(n),
where
(3.11)V21(n)=∑r=0M∑s=n-rn-1k3(r)|x(s)-x*(s)|+∑r=0M∑s=n-rn-1k4(r)|y(s)-y*(s)|,V22(n)=|ln⁡y(n)-ln⁡y*(n)|.
Then we have
(3.12)ΔV21(n)=V21(n+1)-V21(n)=∑r=0M∑s=n+1-rnk3(r)|x(s)-x*(s)|-∑r=0M∑s=n-rn-1k3(r)|x(s)-x*(s)|+∑r=0M∑s=n+1-rnk4(r)|y(s)-y*(s)|-∑r=0M∑s=n-rn-1k4(r)|y(s)-y*(s)|=∑r=0M∑s=n+1-rn-1k3(r)|x(s)-x*(s)|+∑r=0Mk3(r)|x(n)-x*(n)|-∑r=0M∑s=n+1-rn-1k3(r)|x(s)-x*(s)|-∑r=0Mk3(r)|x(n-r)-x*(n-r)|+∑r=0M∑s=n+1-rn-1k4(r)|y(s)-y*(s)|+∑r=0Mk4(r)|y(n)-y*(n)|-∑r=0M∑s=n+1-rn-1k4(r)|y(s)-y*(s)|-∑r=0Mk4(r)|y(n-r)-y*(n-r)|=∑r=0Mk3(r)|x(n)-x*(n)|-∑r=0Mk3(r)|x(n-r)-x*(n-r)|+∑r=0Mk4(r)|y(n)-y*(n)|-∑r=0Mk4(r)|y(n-r)-y*(n-r)|,ΔV22(n)=V22(n+1)-V22(n)≤-(1θ2(n)-|1θ2(n)-q(n)|)|y(n)-y*(n)|+p(n)|x(n)-x*(n)|+∑r=0Mk3(r)|x(n-r)-x*(n-r)|+∑r=0Mk4(r)|y(n-r)-y*(n-r)|,
where θ2(n) is between y(n) and y*(n). Consequently, one has
(3.13)ΔV2(n)=ΔV21(n)+ΔV22(n)≤-(1θ2(n)-|1θ2(n)-q(n)|-∑r=0Mk4(r))|y(n)-y*(n)|+(p(n)+∑r=0Mk3(r))|x(n)-x*(n)|.
Now, we consider a Lyapunov-like discrete function V(n) defined by
(3.14)V(n)=V1(n)+V2(n).
According to (2.8) and (2.11), there exists an n*∈ℤ+ such that x(n)≤M1+ɛ, y(n)≤M2+ɛ for n≥n* and any positive constant ɛ. Obviously, V(n)≥0 for all n∈ℤ+ and V(n*+M)<+∞. In view of (3.1), we choose a sufficient small ɛ>0 such that
(3.15)min⁡{bl,2M1+ɛ-bu}-(M+1)(k1u+k3u)-pu≥η,min⁡{ql,2M2+ɛ-qu}-(M+1)(k2u+k4u)-cu≥η.
Therefore, combining (3.9) with (3.13) for all n≥n*+M, we obtain
(3.16)ΔV(n)=ΔV1(n)+ΔV2(n)≤-(1θ1(n)-|1θ1(n)-b(n)|-∑r=0Mk1(r)-∑r=0Mk3(r)-p(n))|x(n)-x*(n)|-(1θ2(n)-|1θ2(n)-q(n)|-∑r=0Mk4(r)-∑r=0Mk2(r)-c(n))|y(n)-y*(n)|≤-(min⁡{bl,2M1+ɛ-bu}-(M+1)(k1u+k3u)-pu)|x(n)-x*(n)|-(min⁡{ql,2M2+ɛ-qu}-(M+1)(k2u+k4u)-cu)|y(n)-y*(n)|≤-η(|x(n)-x*(n)|+|y(n)-y*(n)|).
Summing both sides of (3.16) from n*+M to n, it derives that
(3.17)∑s=n*+MnΔV(s)=∑s=n*+Mn(V(s+1)-V(s))≤-η∑s=n*+Mn(|x(s)-x*(s)|+|y(s)-y*(s)|),
which implies that
(3.18)V(n+1)+η∑s=n*+Mn(|x(s)-x*(s)|+|y(s)-y*(s)|)≤V(n*+M),foranyn≥n*+M,∑n=n*+M+∞(|x(n)-x*(n)|+|y(n)-y*(n)|)≤V(n*+M)η<+∞,
from which we conclude that
(3.19)lim⁡n→+∞(|x(n)-x*(n)|+|y(n)-y*(n)|)=0,
that is,
(3.20)lim⁡n→+∞|x(n)-x*(n)|=0,lim⁡n→+∞|y(n)-y*(n)|=0.
According to Definition 1.2, this result implies that {(x(n),y(n))⊤} is global attractive. The proof is complete.


## 4. An Example and Its Numerical Simulations

In this paper, we have investigated the asymptotic behaviors of a delayed nonautonomous predator-prey difference system. Sufficient conditions which guarantee the permanence of system and global attractivity of positive solutions of system are obtained, respectively. The theoretical results are substantiated by the following example and numerical results.Example 4.1.
Consider the following system:(4.1)x(n+1)=x(n)exp⁡{0.5+0.01sin⁡n-(0.38+0.02sin⁡n)x(n)-(0.02+0.01sin⁡n)y(n)-0.024[x(n)+x(n-1)]-0.001[y(n)+y(n-1)]},y(n+1)=y(n)exp⁡{-0.015-0.005sin⁡n+(0.26+0.03sin⁡n)x(n)-(0.97+0.02sin⁡n)y(n)+0.01[x(n)+x(n-1)]-0.02[y(n)+y(n-1)]}.
Obviously,
(4.2)Δ1≈0.5175>0,Δ2≈0.3875>0,Δ3≈0.1347>0.
It is easy to verify that the assumptions of Theorem 2.3 are satisfied, so system (4.1) is permanent (see Figure 1). Furthermore, a calculation shows that
(4.3)min⁡{bl,2M1-bu}-(M+1)(k1u+k3u)-pu≈0.0020>0.001,min⁡{ql,2M2-qu}-(M+1)(k2u+k4u)-cu≈0.8780>0.001.
According to Theorem 3.1, any positive solution {(x(n),y(n))⊤} of system (4.1) is global attractive (see Figure 2). From Figures 2(a1) and 2(a2), we see that x with x(0)=1.03 tends to x* with x*(0)=1.0. Similarly, from Figures 2(b1) and 2(b2), we see that y with y(0)=0.31 tends to y* with y*(0)=0.315.Permanence of (1.9) with {(x(-1),y(-1))⊤}={(1.0,0.29)⊤} and {(x(0),y(0))⊤}={(0.95,0.3)⊤}. (a) Time series of x for n∈[0,40]. (b) Time series of y for n∈[0,40].
(a)(b)Dynamic behaviors of (1.9) with different initial values. (a1) and (a2) show x*,x with x*(-1)=1.06, x*(0)=1.0, x(-1)=1.04, x(0)=1.03 for k∈[0,800] and k∈[700,720], respectively. (b1) and (b2) show y*,y with y*(-1)=0.32, y*(0)=0.315, y(-1)=0.315, y(0)=0.31 for k∈[0,800] and k∈[700,720], respectively.



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*Source: 271928-2011-12-14.xml*