# Dynamic Inequalities on Time Scales with Applications in Permanence of Predator-Prey System

Authors: Meng Hu; Lili Wang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

By using the theory of calculus on time scales and some mathematical methods, several dynamic inequalities on time scales are established. Based on these results, we derive some sufficient conditions for permanence of predator-prey system incorporating a prey refuge on time scales. Finally, examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.


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## Body

## 1. Introduction

An important and ubiquitous problem in predator-prey theory and related topics in mathematical ecology concerns the long-term coexistence of species. In the past few years, permanence of different classes of continuous or discrete ecosystem has been studied wildly both in theories and applications; we refer the readers to [1–6] and the references therein.However, in the natural world, there are many species whose developing processes are both continuous and discrete. Hence, using the only differential equation or difference equation cannot accurately describe the law of their developments. Therefore, there is a need to establish correspondent dynamic models on new time scales.The theory of calculus on time scales (see [7] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [8] in order to unify continuous and discrete analysis, and it has a tremendous potential for applications and has recently received much attention since his foundational work; one may see [9–15]. Therefore, it is practicable to study that on time scales which can unify the continuous and discrete situations. However, to the best of the authors’ knowledge, there are few papers considered permanence of predator-prey system on time scales.Motivated by the previous, in this paper, we first establish some dynamic inequalities on time scales by using the theory of calculus on time scales and some mathematical methods, then, based on these results, as an application, we will study the permanence of the following delayed predator-prey system incorporating a prey refuge with Michaelis-Menten and Beddington-DeAngelis functional response on time scales:(1.1)xΔ(t)=x(t)[b(t)-c(t)x(δ-(τ1,t))-d(t)y(t)(1-m)a(t)y(t)+x(t)(1-m)-h(t)z(t)(1-m)k(t)+p(t)x(t)(1-m)+n(t)z(t)],yΔ(t)=y(t)[-d1(t)+f(t)x(δ-(τ2,t))(1-m)a(t)y(δ-(τ2,t))+x(δ-(τ2,t))(1-m)],zΔ(t)=z(t)[-d2(t)+g(t)x(δ-(τ3,t))(1-m)k(t)+p(t)x(δ-(τ3,t))(1-m)+n(t)z(δ-(τ3,t))],
where t∈𝕋, 𝕋 is a time scale. x(t) denotes the density of prey specie and y(t) and z(t) denote the density of two predators species. a(t), b(t), c(t), d(t), f(t), g(t), h(t), k(t), p(t), n(t), d1(t), d2(t) are continuous, positive, and bounded functions, m∈[0,1] is a constant, and m denotes the prey refuge parameter. δ-(τi,t), i=1,2,3, are delay functions with t∈𝕋 and τi∈[0,∞)𝕋, i=1,2,3, where δ- be a backward shift operator on the set 𝕋* and 𝕋* is a nonempty subset of the time scale 𝕋. For the ecological justification of (1.1), one can refer to [2, 12].The initial conditions of (1.1) are of the form(1.2)x(t)=φ1(t),y(t)=φ2(t),z(t)=φ3(t),t∈[δ-(τ,0),0]T,φi(0)>0,i=1,2,3,
where τ=max{τi},τi≥0,i=1,2,3.For convenience, we introduce the notation(1.3)fu=sup⁡t∈T{f(t)},fl=inf⁡t∈T{f(t)},
where f is a positive and bounded function.


## 2. Dynamic Inequalities on Time Scales

Let𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators σ,ρ:𝕋→𝕋, and the graininess μ: 𝕋→ℝ+ are defined, respectively, by(2.1)σ(t)=inf⁡{s∈T:s>t},ρ(t)=sup⁡{s∈T:s<t},μ(t)=σ(t)-t.A pointt∈𝕋 is called left dense if t>inf𝕋 and ρ(t)=t, left scattered if ρ(t)<t, right dense if t<sup𝕋 and σ(t)=t, and right scattered if σ(t)>t. If 𝕋 has a left scattered maximum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋. If 𝕋 has a right scattered minimum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋.The basic theories of calculus on time scales, one can see [7].A functionp:𝕋→ℝ is called regressive provided 1+μ(t)p(t)≠0 for all t∈𝕋k. The set of all regressive and rd-continuous functions p:𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ). We define the set ℛ+=ℛ+(𝕋,ℝ)={p∈ℛ:1+μ(t)p(t)>0,forallt∈𝕋}.Ifr is a regressive function, then the generalized exponential function er is defined by
(2.2)er(t,s)=exp⁡{∫stξμ(τ)(r(τ))Δτ}
for all s,t∈𝕋, with the cylinder transformation
(2.3)ξh(z)={Log(1+hz)h,ifh≠0,z,ifh=0.Letp,q:𝕋→ℝ be two regressive functions and define(2.4)p⊕q=p+q+μpq,⊖p=-p1+μp,p⊖q=p⊕(⊖q).Lemma 2.1 (See [7]).
Assume thatp,q:𝕋→ℝ are two regressive functions, then (i)
e0(t,s)≡1 and ep(t,t)≡1;(ii)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(iii)
ep(t,s)=1/ep(s,t)=e⊖p(s,t);(iv)
ep(t,s)ep(s,r)=ep(t,r);(v)
(e⊖p(t,s))Δ=(⊖p)(t)e⊖p(t,s).Lemma 2.2.
Assume thata>0, b>0 and -a∈ℛ+. Then
(2.5)yΔ(t)≥(≤)b-ay(t),y(t)>0,t∈[t0,∞)T
implies
(2.6)y(t)≥(≤)ba[1+(ay(t0)b-1)e(-a)(t,t0)],t∈[t0,∞)T.Proof.
We only prove the “≥” case; the proof of the “≤” case is similar. For(2.7)[ye⊖(-a)(⋅,t0)]Δ(t)=yΔ(t)e⊖(-a)(σ(t),t0)+y(t)(⊖(-a))e⊖(-a)(t,t0)=yΔ(t)e⊖(-a)(σ(t),t0)+y(t)(⊖(-a))1+μ(t)(⊖(-a))e⊖(-a)(σ(t),t0)=[yΔ(t)-(⊖(⊖(-a)))y(t)]e⊖(-a)(σ(t),t0)=[yΔ(t)-(-a)y(t)]e⊖(-a)(σ(t),t0),
then, integrate both side from t0 to t to conclude
(2.8)y(t)e⊖(-a)(t,t0)-y(t0)=∫t0t[yΔ(s)-(-a)y(s)]e⊖(-a)(σ(s),t0)Δs≥∫t0tbe⊖(-a)(σ(s),t0)Δs=b∫t0te(-a)(t0,σ(s))Δs,
that is, y(t)≥y(t0)e(-a)(t,t0)+b∫t0te(-a)(t,σ(s))Δs. So
(2.9)y(t)≥y(t0)e(-a)(t,t0)+b∫t0te(-a)(t,σ(s))Δs=y(t0)e(-a)(t,t0)-ba∫t0te(-a)(t,σ(s))(-a)Δs=y(t0)e(-a)(t,t0)-ba[e(-a)(t,t0)-1]=e(-a)(t,t0)[y(t0)-ba]+ba,
that is, y(t)≥(b/a)[1+(ay(t0)/b-1)e(-a)(t,t0)]. This completes the proof.Lemma 2.3.
Assume thata>0, b>0. Then
(2.10)yΔ(t)≥(≤)b-ay(σ(t)),y(t)>0,t∈[t0,∞)T
implies
(2.11)y(t)≥(≤)ba[1+(ay(t0)b-1)e⊖a(t,t0)],t∈[t0,∞)T.Proof.
We only prove the “≥” case; the proof of the “≤” case is similar. For(2.12)[yea(⋅,t0)]Δ(t)=yΔ(t)ea(t,t0)+aea(t,t0)y(σ(t))=ea(t,t0)[yΔ(t)+ay(σ(t))],
then, integrate both side from t0 to t to conclude
(2.13)y(t)ea(t,t0)-y(t0)=∫t0tea(s,t0)[yΔ(s)+ay(σ(s))]Δs≥b∫t0tea(s,t0)Δs,
then
(2.14)y(t)≥e⊖a(t,t0)y(t0)+b∫t0te⊖a(t,s)Δs=e⊖a(t,t0)y(t0)+b∫t0t(1+μ(s)a)ea(s,t)11+μ(s)aΔs=e⊖a(t,t0)y(t0)+b∫t0tea(σ(s),t)11+μ(s)aΔs=e⊖a(t,t0)y(t0)+b∫t0te⊖a(t,σ(s))11+μ(s)aΔs=e⊖a(t,t0)y(t0)-ba∫t0te⊖a(t,σ(s))(⊖a)Δs=e⊖a(t,t0)y(t0)-ba(e⊖a(t,t0)-1)=e⊖a(t,t0)[y(t0)-ba]+ba,
that is, y(t)≥(b/a)[1+(ay(t0)/b-1)e⊖a(t,t0)]. This completes the proof.Lemma 2.4.
Assume thata>0, b>0 and -b∈ℛ+. Then
(2.15)yΔ(t)≤(≥)y(σ(t))(b-ay(t)),y(t)>0,t∈[t0,∞)T
implies
(2.16)y(t)≤(≥)ba[1+(bay(t0)-1)e(-b)(t,t0)]-1,t∈[t0,∞)T.Proof.
We only prove the “≤” case; the proof of the “≥” case is similar.
Lety(t)=1/x(t), then (1/x(t))Δ≤(1/x(σ(t)))(b-a/x(t)), that is, -xΔ(t)/(x(t)x(σ(t)))≤(1/x(σ(t)))(b-a/x(t)), so, xΔ(t)≥a-bx(t). By Lemma 2.2, we have x(t)≥(a/b)[1+(bx(t0)/a-1)e(-b)(t,t0)]. Therefore, y(t)≤(b/a)[1+(b/ay(t0)-1)e(-b)(t,t0)]-1. This completes the proof.Lemma 2.5.
Assume thata>0, b>0. Then
(2.17)yΔ(t)≤(≥)y(t)(b-ay(σ(t))),y(t)>0,t∈[t0,∞)T
implies
(2.18)y(t)≤(≥)ba[1+(bay(t0)-1)e⊖b(t,t0)],t∈[t0,∞)T.Proof.
We only prove the “≤” case; the proof of the “≥” case is similar.
Lety(t)=1/x(t), then (1/x(t))Δ≤(1/x(t))(b-a/x(σ(t))), that is, -xΔ(t)/(x(t)x(σ(t)))≤(1/x(t))(b-a/x(σ(t))), so, xΔ(t)≥a-bx(σ(t)). By Lemma 2.3, we have x(t)≥(a/b)[1+(bx(t0)/a-1)e⊖b(t,t0)]. Therefore, y(t)≤(b/a)[1+(b/ay(t0)-1)e⊖b(t,t0)]-1. This completes the proof.Definition 2.6 (see [16]).
Let𝕋* be a nonempty subset of the time scale 𝕋 and t0∈𝕋* a fixed number. The operator δ- associated with t0∈𝕋* (called the initial point) is said to be backward shift operator on the set 𝕋*. The variable s∈[t0,∞)𝕋inδ-(s,t) is called the shift size. The value δ-(s,t) in 𝕋* indicate s units translation of the term t∈𝕋* to the left.Now, we state some different time scales with their corresponding backward shift operators: let𝕋=ℝ and t0=0; then δ-(s,t)=t-s; let 𝕋=ℤ and t0=0; then δ-(s,t)=t-s; let 𝕋=hℤ and t0=0; then δ-(s,t)=t-s; let 𝕋=ℕ1/2 and t0=0; then δ-(s,t)=t2-s2; let 𝕋=2ℕ and t0=1; then δ-(s,t)=t/s; and so on.Lemma 2.7.
Ifa∈ℛ+ and  yΔ(t)≤(≥)a(t)y(t),t∈[t0,∞)𝕋, then
(2.19)y(δ-(s,t))≥(≤)e⊖a(σ(t),δ-(s,t))y(σ(t)),
where δ-(s,t) be defined in Definition 2.6 and (s,t)∈[t0,∞)𝕋×[s,∞)𝕋.Proof.
We only prove the “≤” case; the proof of the “≥” case is similar. For(2.20)[ye⊖a(⋅,t0)]Δ(t)=yΔ(t)e⊖a(σ(t),t0)+y(t)(⊖a)(t)e⊖a(t,t0)=yΔ(t)e⊖a(σ(t),t0)+y(t)(⊖a)(t)1+μ(t)(⊖a)(t)e⊖a(σ(t),t0)=[yΔ(t)-(⊖(⊖a))(t)y(t)]e⊖a(σ(t),t0)=[yΔ(t)-a(t)y(t)]e⊖a(σ(t),t0)≤0,
then, integrate both side from δ-(s,t) to σ(t) to conclude
(2.21)y(σ(t))e⊖a(σ(t),t0)≤y(δ-(s,t))e⊖a(δ-(s,t),t0),
then
(2.22)y(δ-(s,t))≥e⊖a(σ(t),δ-(s,t))y(σ(t)).
This completes the proof.


## 3. Permanence

As an application, based on the results obtained in Section2, we will establish a permanent result for system (1.1).Definition 3.1.
System (1.1) is said to be permanent if there exists a compact region D⊆Intℝ+3, such that for any positive solution (x(t),y(t),z(t)) of system (1.1) with initial condition (1.2) eventually enters and remains in region D.In this section, we consider the time scale𝕋 that satisfies ea(σ(t),δ-(τ,t)) to be a constant on [0,∞)𝕋, where a∈ℛ+, τ∈[0,∞)𝕋 are constants. For example, let t0=0 (the initial point), when 𝕋=ℝ, then ea(σ(t),δ-(τ,t))=eaτ; when 𝕋=ℤ, then ea(σ(t),δ-(τ,t))=(1+a)1+τ; when 𝕋=hℤ, then ea(σ(t),δ-(τ,t))=(1+ah)1+τ/h, and so on.For convenience, we introduce the following notations:(3.1)M1=buebu(σ(t),δ-(τ1,t))cl+ε,M2=(fu-d1l)(1-m)M1e(fu-d1l)(σ(t),δ-(τ2,t))ald1l+ε,M3=((gu-pld2l)/pl-kld2l/(pu(1-m)M1))puM1(1-m)e(gu/pl-d2l)(σ(t),δ-(τ3,t))d2lnl+ε,m1=[bl-du(1-m)/al-hu(1-m)/nl]eβl(σ(t),δ-(τ1,t))cu-ε,m2=(m1/2)(fl-d1u)(1-m)e(-d1u)(σ(t),δ-(τ2,t))aud1u-ε,m3=[(m1/2)pl(1-m)(gl-d2u)-kud2u]e(-d2u)(σ(t),δ-(τ3,t))nud2u-ε,
where βl=bl-du(1-m)/al-hu(1-m)/nl-cuM1.Hereafter, we assume that(H1)
f(t)-d1(t)∈ℛ+;(H2)
fu-d1l>0;(H3)
g(t)/p(t)-d2(t)∈ℛ+;(H4)
(gu-pld2l)/pl-kld2l/(pu(1-m)M1)>0;(H5)
b(t)-d(t)(1-m)/a(t)-h(t)(1-m)/n(t)-c(t)M1∈ℛ+;(H6)
bl-du(1-m)/al-hu(1-m)/nl>0;(H7)
-d1(t)∈ℛ+;(H8)
fl-d1u>0;(H9)
-d2(t)∈ℛ+;(H10)
(m1/2)(1-m)(gl-d2upu)-kud2u>0.Proposition 3.2.
Assume that(x(t),y(t),z(t)) is any positive solution of system (1.1) with initial condition (1.2). If (H1)–(H4) hold, then
(3.2)x(t)≤M1,y(t)≤M2,z(t)≤M3.Proof.
Assume that(x(t),y(t),z(t)) is any positive solution of system (1.1) with initial condition (1.2). From the first equation of system (1.1), for t∈[τ1,∞)𝕋, we have
(3.3)xΔ(t)≤x(t)(b(t)-c(t)x(δ-(τ1,t))).
From (3.3), we can see xΔ(t)≤b(t)x(t); then by Lemma 2.7, we can get
(3.4)x(δ-(τ1,t))≥e⊖b(σ(t),δ-(τ1,t))x(σ(t)).
Together with (3.3) and (3.4), we have
(3.5)xΔ(t)≤x(t)(b(t)-c(t)e⊖b(σ(t),δ-(τ1,t))x(σ(t)))≤x(t)(bu-cle⊖bu(σ(t),δ-(τ1,t))x(σ(t))).
By Lemma 2.5, for arbitrary small positive constant ε, there exists T1>0 such that
(3.6)x(t)≤bucle⊖bu(σ(t),δ-(τ1,t))+ε=buebu(σ(t),δ-(τ1,t))cl+ε:=M1,t∈[T1,∞)T.
Again, from the second equation of system (1.1) and (3.6), for t∈[T1+τ2,∞)𝕋, we have
(3.7)yΔ(t)≤y(t)(-d1(t)+f(t)(1-m)M1a(t)y(δ-(τ2,t))+(1-m)M1).
From (3.7), we can see yΔ(t)≤(-d1(t)+f(t))y(t); then by (H1) and Lemma 2.7, we can get
(3.8)y(δ-(τ2,t))≥e(⊖α)(σ(t),δ-(τ2,t))y(σ(t)),
where α(t)=f(t)-d1(t).
Together with (3.7) and (3.8), we have
(3.9)yΔ(t)≤y(t)(-d1(t)+f(t)(1-m)M1a(t)e(⊖α)(σ(t),δ-(τ2,t))y(σ(t))+(1-m)M1)≤y(t)((1-m)M1(f(t)-d1(t))-a(t)d1(t)e(⊖α)(σ(t),δ-(τ2,t))y(σ(t))a(t)e(⊖α)(σ(t),δ-(τ2,t))y(σ(t))+(1-m)M1)≤y(t)(fu-d1l-ald1le⊖(fu-d1l)(σ(t),δ-(τ2,t))(1-m)M1y(σ(t))).
By (H2) and Lemma 2.5, for arbitrary small positive constant ε, there exists T2>T1+τ2 such that
(3.10)y(t)≤(fu-d1l)(1-m)M1ald1le⊖(fu-d1l)(σ(t),δ-(τ2,t))+ε=(fu-d1l)(1-m)M1e(fu-d1l)(σ(t),δ-(τ2,t))ald1l+ε:=M2,t∈[T2,∞)T.
Similarly, under conditions (H3)-(H4), by Lemmas 2.5 and 2.7, we can get that, for arbitrary small positive constant ε, there exists T3>T2+τ3 such that
(3.11)z(t)≤((gu-pld2l)/pl-kld2l/(pu(1-m)M1))puM1(1-m)e(gu/pl-d2l)(σ(t),δ-(τ3,t))d2lnl+ε:=M3,t∈[T3,∞)T.
The conclusion of Proposition 3.2 follows. This completes the proof.Proposition 3.3.
Assume that(x(t),y(t),z(t)) is any positive solution of system (1.1) with initial condition (1.2). If (H1)–(H10) hold, then
(3.12)x(t)≥m1,y(t)≥m2,z(t)≥m3.Proof.
Assume that(x(t),y(t),z(t)) is any positive solution of system (1.1) with initial condition (1.2). From Theorem 3.4, there exists a T1>0, such that x(t)≤M1,t∈[T1,∞)𝕋. By the first equation of system (1.1), for t∈[T1+τ1,∞)𝕋, we have
(3.13)xΔ(t)≥x(t)(b(t)-d(t)(1-m)a(t)-h(t)(1-m)n(t)-c(t)x(δ-(τ1,t))),
then
(3.14)xΔ(t)≥x(t)(b(t)-d(t)(1-m)a(t)-h(t)(1-m)n(t)-c(t)M1).
By (3.14), (H5), and Lemma 2.7, we can get
(3.15)x(δ-(τ1,t))≤e⊖β(σ(t),δ-(τ1,t))x(σ(t)),
where β=b(t)-d(t)(1-m)/a(t)-h(t)(1-m)/n(t)-c(t)M1.
Together with (3.13) and (3.15), we have
(3.16)xΔ(t)≥x(t)(b(t)-d(t)(1-m)a(t)-h(t)(1-m)n(t)-c(t)e⊖β(σ(t),δ-(τ1,t))x(σ(t)))≥x(t)(bl-du(1-m)al-hu(1-m)nl-cue⊖βl(σ(t),δ-(τ1,t))x(σ(t))),
where βl=bl-du(1-m)/al-hu(1-m)/nl-cuM1.
By (H6) and Lemma 2.5, for arbitrary small positive constant ε, there exists T4>T1+τ1 such that
(3.17)x(t)≥bl-du(1-m)/al-hu(1-m)/nlcue⊖βl(σ(t),δ-(τ1,t))-ε=[bl-du(1-m)/al-hu(1-m)/nl]eβl(σ(t),δ-(τ1,t))cu-ε:=m1,t∈[T4,∞)T.
Again, from the second equation of system (1.1) and (3.17), for t∈[T4+τ2,∞)𝕋, we have
(3.18)yΔ(t)≥y(t)(-d1(t)+(m1/2)f(t)(1-m)a(t)y(δ-(τ2,t))+(m1/2)(1-m)).
From (3.18), we can see yΔ(t)≥-d1(t)y(t); then by (H7) and Lemma 2.7, we can get
(3.19)y(δ-(τ2,t))≤e⊖(-d1)(σ(t),δ-(τ2,t))y(σ(t)).
Together with (3.18) and (3.19), we have
(3.20)yΔ(t)≥y(t)(-d1(t)+(m1/2)f(t)(1-m)a(t)e⊖(-d1)(σ(t),δ-(τ2,t))y(σ(t))+(m1/2)(1-m))≥y(t)((m1/2)(fl-d1u)(1-m)-aud1ue⊖(-d1u)(σ(t),δ-(τ2,t))y(σ(t))aue⊖(-d1u)(σ(t),δ-(τ2,t))M2+(m1/2)(1-m)).
By (H8) and Lemma 2.5, for arbitrary small positive constant ε, there exists T5>T4+τ2 such that
(3.21)y(t)≥(m1/2)(fl-d1u)(1-m)aud1ue⊖(-d1u)(σ(t),δ-(τ2,t))-ε=(m1/2)(fl-d1u)(1-m)e(-d1u)(σ(t),δ-(τ2,t))aud1u-ε:=m2,t∈[T5,∞)T.
Similarly, under conditions (H9) and (H10), by Lemmas 2.5 and 2.7, we can get that, for arbitrary small positive constant ε, there exists T6>T5+τ3 such that
(3.22)z(t)≥[(m1/2)pl(1-m)(gl-d2u)-kud2u]e(-d2u)(σ(t),δ-(τ3,t))nud2u-ε:=m3,t∈[T6,∞)T.
The conclusion of Proposition 3.3 follows. This completes the proof.Together with Propositions3.2 and 3.3, we can obtain the following theorem.Theorem 3.4.
Assume that (H1)–(H10) hold; then system (1.1) is permanent.


## 4. Examples and Simulations

Consider the following system on time scales withm=0.6:(4.1)xΔ(t)=x(t)[(0.08+0.05sin⁡π2t)-0.2x(δ-(τ1,t))-0.05y(t)(1-m)2y(t)+x(t)(1-m)-0.05z(t)(1-m)0.005+x(t)(1-m)+2z(t)],yΔ(t)=y(t)[-(0.7-0.25cos⁡5π2t)+2x(δ-(τ2,t))(1-m)2y(δ-(τ2,t))+x(δ-(τ2,t))(1-m)],zΔ(t)=z(t)[-(0.75+0.2sin⁡3π2t)+4x(δ-(τ3,t))(1-m)0.005+x(δ-(τ3,t))(1-m)+2z(δ-(τ3,t))].Let𝕋=ℝ; then μ(t)=0. Obviously, (H1), (H3), (H5), (H7), and (H9) hold. Taking τi=1, i=1,2,3, by a direct calculation, we can get(H2)
fu-d1l=1.5500>0;(H4)
(gu-pld2l)/pl-kld2l/(pu(1-m)M1)=3.4407>0;(H6)
bl-du(1-m)/al-hu(1-m)/nl=0.0100>0;(H8)
fl-d1u=1.0500>0;(H10)
(m1/2)(1-m)(gl-d2upu)-kud2u=0.0218>0.From the above results, we can see that all conditions of Theorem3.4 hold. So, system (4.1) is permanent, see Figure 1.Figure 1
𝕋=ℝ, m=0.6. Dynamics behavior of system (4.1) with initial condition x(0)=0.3, y(0)=0.3, z(0)=0.3.Let𝕋=ℤ; then μ(t)=1. It is easy to check (H1), (H3), (H5), (H7), and (H9) hold. Taking τi=1, i=1,2,3, by a direct calculation, we can get(H2)
fu-d1l=1.5500>0;(H4)
(gu-pld2l)/pl-kld2l/(pu(1-m)M1)=3.4086>0;(H6)
bl-du(1-m)/al-hu(1-m)/nl=0.0100>0;(H8)
fl-d1u=1.0500>0;(H10)
(m1/2)(1-m)(gl-d2upu)-kud2u=0.0244>0.From the above results, we can see that all conditions of Theorem3.4 hold. So, system (4.1) is permanent, see Figure 2.Figure 2
𝕋=ℤ, m=0.6. Dynamics behavior of system (4.1) with initial condition x(0)=0.35, y(0)=0.2, z(0)=0.4.Let𝕋=⋃k=0∞[2k,2k+1]; then μ(t)={0,fort∈⋃k=0∞[2k,2k+1),1,fort∈⋃k=0∞{2k+1}. It is easy to check (H1), (H3), (H5), (H7), and (H9) hold. Taking τi=2,i=1,2,3, by a direct calculation, we can get(H2)
fu-d1l=1.5500>0;(H6)
bl-du(1-m)/al-hu(1-m)/nl=0.0100>0;(H8)
fl-d1u=1.0500>0.Furthermore, ift∈⋃k=0∞[2k,2k+1), then(H4)
(gu-pld2l)/pl-kld2l/(pu(1-m)M1)=3.4418>0;(H10)
(m1/2)(1-m)(gl-d2upu)-kud2u=0.0175>0;ift∈⋃k=0∞{2k+1}, then(H4)
(gu-pld2l)/pl-kld2l/(pu(1-m)M1)=3.4133>0;(H10)
(m1/2)(1-m)(gl-d2upu)-kud2u=0.0233>0.From the above results, we can see that all conditions of Theorem3.4 hold. So, system (4.1) is permanent, see Figure 3.Figure 3
𝕋=⋃k=0∞[2k,2k+1], m=0.6. Dynamics behavior of system (4.1) with initial condition x(0)=0.3, y(0)=0.2, z(0)=0.3.



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*Source: 281052-2012-05-30.xml*