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

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.

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: 𝑥^Δ(𝑡)=𝑥(𝑡)𝑏(𝑡)−𝑐(𝑡)𝑥𝛿₋𝜏₁−𝑑(𝑡)𝑦(𝑡),𝑡)(1−𝑚𝑎(𝑡)𝑦(𝑡)(𝑡)−ℎ(𝑡)𝑧(𝑡)+𝑥)(1−𝑚)(1−𝑚𝑘(𝑡)(𝑡)𝑥(𝑡)(𝑡)𝑧(𝑡),𝑦+𝑝)(1−𝑚+𝑛_Δ(𝑡)=𝑦(𝑡)−𝑑₁(𝑡)+𝑓(𝑡)𝑥𝛿₋𝜏₂,𝑡(1−𝑚)𝑎(𝑡)𝑦𝛿₋𝜏₂,𝑡+𝑥𝛿₋𝜏₂(),𝑧,𝑡1−𝑚_Δ(𝑡)=𝑧(𝑡)−𝑑₂(𝑡)+𝑔(𝑡)𝑥𝛿₋𝜏₃,𝑡(1−𝑚)𝑘(𝑡)+𝑝(𝑡)𝑥𝛿₋𝜏₃,𝑡(1−𝑚)+𝑛(𝑡)𝑧𝛿₋𝜏₃,,𝑡 (1.1) 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₋(𝜏_𝑖,𝑡), 𝑖=1,2,3, are delay functions with 𝑡∈𝕋 and 𝜏_𝑖∈[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 form𝑥(𝑡)=𝜑₁(𝑡),𝑦(𝑡)=𝜑₂(𝑡),𝑧(𝑡)=𝜑₃(𝑡),𝑡∈𝛿₋(𝜏,0),0_𝕋,𝜑_𝑖(0)>0,𝑖=1,2,3, (1.2) where 𝜏=max{𝜏_𝑖},𝜏_𝑖≥0,𝑖=1,2,3.

For convenience, we introduce the notation𝑓^𝑢=sup_𝑡∈𝕋𝑓(𝑡),𝑓^𝑙=inf_𝑡∈𝕋𝑓(𝑡), (1.3) where 𝑓 is a positive and bounded function.

2. Dynamic Inequalities on Time Scales

Let 𝕋 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, by𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡},𝜇(𝑡)=𝜎(𝑡)−𝑡. (2.1)

A point 𝑡∈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 𝕋^𝑘=𝕋⧵{𝑚}; otherwise 𝕋^𝑘=𝕋. If 𝕋 has a right scattered minimum 𝑚, then 𝕋_𝑘=𝕋⧵{𝑚}; otherwise 𝕋_𝑘=𝕋.

The basic theories of calculus on time scales, one can see [7].

A function 𝑝∶𝕋→ℝ is called regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋^𝑘. 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}.

If 𝑟 is a regressive function, then the generalized exponential function 𝑒_𝑟 is defined by 𝑒_𝑟(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)(𝑟(𝜏))Δ𝜏 (2.2) for all 𝑠,𝑡∈𝕋, with the cylinder transformation 𝜉_ℎ(𝑧)=Log(1+ℎ𝑧)ℎ,ifℎ≠0,𝑧,ifℎ=0. (2.3)

Let 𝑝,𝑞∶𝕋→ℝ be two regressive functions and define𝑝𝑝⊕𝑞=𝑝+𝑞+𝜇𝑝𝑞,⊖𝑝=−1+𝜇𝑝,𝑝⊖𝑞=𝑝⊕⊖𝑞. (2.4)

Lemma 2.1 (See [7]). Assume that 𝑝,𝑞∶𝕋→ℝ are two regressive functions, then (i)𝑒₀(𝑡,𝑠)≡1 and 𝑒_𝑝(𝑡,𝑡)≡1;(ii)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(iii)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊖𝑝(𝑠,𝑡);(iv)𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟);(v)(𝑒_⊖𝑝(𝑡,𝑠))^Δ=(⊖𝑝)(𝑡)𝑒_⊖𝑝(𝑡,𝑠).

Lemma 2.2. Assume that 𝑎>0, 𝑏>0 and −𝑎∈ℛ⁺. Then 𝑦^Δ(𝑡)≥(≤)𝑏−𝑎𝑦(𝑡),𝑦(𝑡)>0,𝑡∈𝑡₀,∞_𝕋 (2.5) implies 𝑦(𝑡)≥(≤)𝑏𝑎𝑡1+𝑎𝑦₀𝑏−1𝑒_(−𝑎)𝑡,𝑡₀,𝑡∈𝑡₀,∞_𝕋. (2.6)

Proof. We only prove the “≥” case; the proof of the “≤” case is similar. For 𝑦𝑒_⊖(−𝑎)⋅,𝑡₀^Δ(𝑡)=𝑦^Δ(𝑡)𝑒_⊖(−𝑎)𝜎(𝑡),𝑡₀+𝑦(𝑡)(⊖(−𝑎))𝑒_⊖(−𝑎)𝑡,𝑡₀=𝑦^Δ(𝑡)𝑒_⊖(−𝑎)𝜎(𝑡),𝑡₀(𝑡)(⊖(+𝑦−𝑎))(𝑡⊖(𝑒1+𝜇)(−𝑎))^⊖(−𝑎)𝜎(𝑡),𝑡₀=𝑦^Δ(𝑡)−(⊖(⊖(−𝑎)))𝑦(𝑡)𝑒_⊖(−𝑎)𝜎(𝑡),𝑡₀=𝑦^Δ(𝑡)−(−𝑎)𝑦(𝑡)𝑒_⊖(−𝑎)𝜎(𝑡),𝑡₀, (2.7) then, integrate both side from 𝑡₀ to 𝑡 to conclude 𝑦(𝑡)𝑒_⊖(−𝑎)𝑡,𝑡₀−𝑦𝑡₀=^𝑡_𝑡0𝑦^Δ(𝑠)−(−𝑎)𝑦(𝑠)𝑒_⊖(−𝑎)𝜎(𝑠),𝑡₀≥Δ𝑠_𝑡𝑡0𝑏𝑒_⊖(−𝑎)𝜎(𝑠),𝑡₀Δ𝑠=𝑏^𝑡_𝑡0𝑒_(−𝑎)𝑡₀,𝜎(𝑠)Δ𝑠, (2.8) that is, 𝑦(𝑡)≥𝑦(𝑡₀)𝑒_(−𝑎)(𝑡,𝑡₀)+𝑏∫^𝑡_𝑡0𝑒_(−𝑎)(𝑡,𝜎(𝑠))Δ𝑠. So 𝑦(𝑡)≥𝑦𝑡₀𝑒_(−𝑎)𝑡,𝑡₀+𝑏^𝑡_𝑡0𝑒_(−𝑎)((𝑠𝑡𝑡,𝜎))Δ𝑠=𝑦₀𝑒_(−𝑎)𝑡,𝑡₀−𝑏𝑎^𝑡_𝑡0𝑒_(−𝑎)((𝑠)𝑡𝑡,𝜎))(−𝑎Δ𝑠=𝑦₀𝑒_(−𝑎)𝑡,𝑡₀−𝑏𝑎𝑒_(−𝑎)𝑡,𝑡₀−1=𝑒_(−𝑎)𝑡,𝑡₀𝑦𝑡₀−𝑏𝑎+𝑏𝑎, (2.9) that is, 𝑦(𝑡)≥(𝑏/𝑎)[1+(𝑎𝑦(𝑡₀)/𝑏−1)𝑒_(−𝑎)(𝑡,𝑡₀)]. This completes the proof.

Lemma 2.3. Assume that 𝑎>0, 𝑏>0. Then 𝑦^Δ(𝑡)≥(≤)𝑏−𝑎𝑦(𝜎(𝑡)),𝑦(𝑡)>0,𝑡∈𝑡₀,∞_𝕋 (2.10) implies 𝑦(𝑡)≥(≤)𝑏𝑎𝑡1+𝑎𝑦₀𝑏−1𝑒_⊖𝑎𝑡,𝑡₀,𝑡∈𝑡₀,∞_𝕋. (2.11)

Proof. We only prove the “≥” case; the proof of the “≤” case is similar. For 𝑦𝑒_𝑎⋅,𝑡₀^Δ(𝑡)=𝑦^Δ(𝑡)𝑒_𝑎𝑡,𝑡₀+𝑎𝑒_𝑎𝑡,𝑡₀𝑦(𝜎(𝑡))=𝑒_𝑎𝑡,𝑡₀𝑦^Δ(𝑡)+𝑎𝑦(𝜎(𝑡)), (2.12) then, integrate both side from 𝑡₀ to 𝑡 to conclude 𝑦(𝑡)𝑒_𝑎𝑡,𝑡₀−𝑦𝑡₀=^𝑡_𝑡0𝑒_𝑎𝑠,𝑡₀𝑦^Δ(𝑠)+𝑎𝑦(𝜎(𝑠))Δ𝑠≥𝑏^𝑡_𝑡0𝑒_𝑎𝑠,𝑡₀Δ𝑠, (2.13) then 𝑦(𝑡)≥𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀+𝑏^𝑡_𝑡0𝑒_⊖𝑎(𝑡,𝑠)Δ𝑠=𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀+𝑏^𝑡_𝑡01+𝜇(𝑠)𝑎𝑒_𝑎()1𝑠,𝑡1+𝜇(𝑠)𝑎Δ𝑠=𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀+𝑏^𝑡_𝑡0𝑒_𝑎(𝜎(𝑠))1,𝑡1+𝜇(𝑠)𝑎Δ𝑠=𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀+𝑏^𝑡_𝑡0𝑒_⊖𝑎((𝑠1𝑡,𝜎))1+𝜇(𝑠)𝑎Δ𝑠=𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀−𝑏𝑎^𝑡_𝑡0𝑒_⊖𝑎(𝑡,𝜎(𝑠))(⊖𝑎)Δ𝑠=𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀−𝑏𝑎𝑒_⊖𝑎𝑡,𝑡₀−1=𝑒_⊖𝑎𝑡,𝑡₀𝑦𝑡₀−𝑏𝑎+𝑏𝑎, (2.14) that is, 𝑦(𝑡)≥(𝑏/𝑎)[1+(𝑎𝑦(𝑡₀)/𝑏−1)𝑒_⊖𝑎(𝑡,𝑡₀)]. This completes the proof.

Lemma 2.4. Assume that 𝑎>0, 𝑏>0 and −𝑏∈ℛ⁺. Then 𝑦^Δ(𝑡)≤(≥)𝑦(𝜎(𝑡))𝑏−𝑎𝑦(𝑡),𝑦(𝑡)>0,𝑡∈𝑡₀,∞_𝕋 (2.15) implies 𝑦(𝑡)≤(≥)𝑏𝑎𝑏1+𝑎𝑦𝑡₀−1𝑒_(−𝑏)𝑡,𝑡₀⁻¹,𝑡∈𝑡₀,∞_𝕋. (2.16)

Proof. We only prove the “≤” case; the proof of the “≥” case is similar.
Let 𝑦(𝑡)=1/𝑥(𝑡), then (1/𝑥(𝑡))^Δ≤(1/𝑥(𝜎(𝑡)))(𝑏−𝑎/𝑥(𝑡)), that is, −𝑥^Δ(𝑡)/(𝑥(𝑡)𝑥(𝜎(𝑡)))≤(1/𝑥(𝜎(𝑡)))(𝑏−𝑎/𝑥(𝑡)), so, 𝑥^Δ(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 that 𝑎>0, 𝑏>0. Then 𝑦^Δ(𝑡)≤(≥)𝑦(𝑡)𝑏−𝑎𝑦(𝜎(𝑡)),𝑦(𝑡)>0,𝑡∈𝑡₀,∞_𝕋 (2.17) implies 𝑦(𝑡)≤(≥)𝑏𝑎𝑏1+𝑎𝑦𝑡₀−1𝑒_⊖𝑏𝑡,𝑡₀,𝑡∈𝑡₀,∞_𝕋. (2.18)

Proof. We only prove the “≤” case; the proof of the “≥” case is similar.
Let 𝑦(𝑡)=1/𝑥(𝑡), then (1/𝑥(𝑡))^Δ≤(1/𝑥(𝑡))(𝑏−𝑎/𝑥(𝜎(𝑡))), that is, −𝑥^Δ(𝑡)/(𝑥(𝑡)𝑥(𝜎(𝑡)))≤(1/𝑥(𝑡))(𝑏−𝑎/𝑥(𝜎(𝑡))), so, 𝑥^Δ(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]). Let 𝕋^∗ 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,∞)_𝕋 in 𝛿₋(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: let 𝕋=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=ℕ^1/2 and 𝑡₀=0; then 𝛿₋(𝑠,𝑡)=√𝑡²−𝑠²; let 𝕋=2^ℕ and 𝑡₀=1; then 𝛿₋(𝑠,𝑡)=𝑡/𝑠; and so on.

Lemma 2.7. If 𝑎∈ℛ⁺ and 𝑦^Δ(𝑡)≤(≥)𝑎(𝑡)𝑦(𝑡),𝑡∈[𝑡₀,∞)_𝕋, then 𝑦𝛿₋(𝑠,𝑡)≥(≤)𝑒_⊖𝑎𝜎(𝑡),𝛿₋(𝑠,𝑡)𝑦(𝜎(𝑡)), (2.19) 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 𝑦𝑒_⊖𝑎⋅,𝑡₀^Δ(𝑡)=𝑦^Δ(𝑡)𝑒_⊖𝑎𝜎(𝑡),𝑡₀+𝑦(𝑡)(⊖𝑎)(𝑡)𝑒_⊖𝑎𝑡,𝑡₀=𝑦^Δ(𝑡)𝑒_⊖𝑎𝜎(𝑡),𝑡₀(𝑡)(𝑡)+𝑦⊖𝑎)((𝑡𝑡)𝑒1+𝜇)(⊖𝑎)(^⊖𝑎𝜎(𝑡),𝑡₀=𝑦^Δ(𝑡)−(⊖(⊖𝑎))(𝑡)𝑦(𝑡)𝑒_⊖𝑎𝜎(𝑡),𝑡₀=𝑦^Δ(𝑡)−𝑎(𝑡)𝑦(𝑡)𝑒_⊖𝑎𝜎(𝑡),𝑡₀≤0, (2.20) then, integrate both side from 𝛿₋(𝑠,𝑡) to 𝜎(𝑡) to conclude 𝑦(𝜎(𝑡))𝑒_⊖𝑎𝜎(𝑡),𝑡₀≤𝑦𝛿₋(𝑠,𝑡)𝑒_⊖𝑎𝛿₋(𝑠,𝑡),𝑡₀, (2.21) then 𝑦𝛿₋(𝑠,𝑡)≥𝑒_⊖𝑎𝜎(𝑡),𝛿₋(𝑠,𝑡)𝑦(𝜎(𝑡)). (2.22) This completes the proof.

3. Permanence

As an application, based on the results obtained in Section 2, 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⊆Intℝ³₊, 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 scale 𝕋 that satisfies 𝑒_𝑎(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 𝑒_𝑎(𝜎(𝑡),𝛿₋(𝜏,𝑡))=𝑒^𝑎𝜏; when 𝕋=ℤ, then 𝑒_𝑎(𝜎(𝑡),𝛿₋(𝜏,𝑡))=(1+𝑎)^1+𝜏; when 𝕋=ℎℤ, then 𝑒_𝑎(𝜎(𝑡),𝛿₋(𝜏,𝑡))=(1+𝑎ℎ)^1+𝜏/ℎ, and so on.

For convenience, we introduce the following notations:𝑀₁=𝑏^𝑢𝑒_𝑏𝑢𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑐^𝑙𝑀+𝜀,₂=𝑓^𝑢−𝑑^𝑙₁(1−𝑚)𝑀₁𝑒_{(𝑓𝑢−𝑑𝑙1)}𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑎^𝑙𝑑^𝑙₁𝑀+𝜀,₃=𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/𝑝^𝑢(1−𝑚)𝑀₁𝑝^𝑢𝑀₁(1−𝑚)𝑒_{(𝑔𝑢/𝑝𝑙−𝑑𝑙2)}𝜎(𝑡),𝛿₋𝜏₃,𝑡𝑑^𝑙₂𝑛^𝑙𝑚+𝜀,₁=𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙𝑒_𝛽𝑙𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑐^𝑢𝑚−𝜀,₂=𝑚₁/2𝑓^𝑙−𝑑^𝑢₁(1−𝑚)𝑒_(−𝑑^𝑢₁₎𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑎^𝑢𝑑^𝑢₁𝑚−𝜀,₃=𝑚₁/2𝑝^𝑙(1−𝑚)𝑔^𝑙−𝑑^𝑢₂−𝑘^𝑢𝑑^𝑢₂𝑒_(−𝑑^𝑢₂₎𝜎(𝑡),𝛿₋𝜏₃,𝑡𝑛^𝑢𝑑^𝑢₂−𝜀, (3.1) where 𝛽^𝑙=𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙−𝑐^𝑢𝑀₁.

Hereafter, we assume that(H₁)𝑓(𝑡)−𝑑₁(𝑡)∈ℛ⁺;(H₂)𝑓^𝑢−𝑑^𝑙₁>0;(H₃)𝑔(𝑡)/𝑝(𝑡)−𝑑₂(𝑡)∈ℛ⁺;(H₄)(𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂)/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/(𝑝^𝑢(1−𝑚)𝑀₁)>0;(H₅)𝑏(𝑡)−𝑑(𝑡)(1−𝑚)/𝑎(𝑡)−ℎ(𝑡)(1−𝑚)/𝑛(𝑡)−𝑐(𝑡)𝑀₁∈ℛ⁺;(H₆)𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙>0;(H₇)−𝑑₁(𝑡)∈ℛ⁺; (H₈)𝑓^𝑙−𝑑^𝑢₁>0;(H₉)−𝑑₂(𝑡)∈ℛ⁺;(H₁₀)(𝑚₁/2)(1−𝑚)(𝑔^𝑙−𝑑^𝑢₂𝑝^𝑢)−𝑘^𝑢𝑑^𝑢₂>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 𝑥(𝑡)≤𝑀₁,𝑦(𝑡)≤𝑀₂,𝑧(𝑡)≤𝑀₃. (3.2)

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 𝑥^Δ(𝑡)≤𝑥(𝑡)𝑏(𝑡)−𝑐(𝑡)𝑥𝛿₋𝜏₁,𝑡. (3.3) From (3.3), we can see 𝑥^Δ(𝑡)≤𝑏(𝑡)𝑥(𝑡); then by Lemma 2.7, we can get 𝑥𝛿₋𝜏₁,𝑡≥𝑒_⊖𝑏𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑥(𝜎(𝑡)). (3.4)
Together with (3.3) and (3.4), we have 𝑥^Δ(𝑡)≤𝑥(𝑡)𝑏(𝑡)−𝑐(𝑡)𝑒_⊖𝑏𝜎(𝑡),𝛿₋𝜏₁𝑥(𝜎(𝑡(𝑡)𝑏,𝑡))≤𝑥^𝑢−𝑐^𝑙𝑒_⊖𝑏𝑢𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑥(𝜎(𝑡)). (3.5) By Lemma 2.5, for arbitrary small positive constant 𝜀, there exists 𝑇₁>0 such that 𝑥(𝑡)≤𝑏^𝑢𝑐^𝑙𝑒_⊖𝑏𝑢𝜎(𝑡),𝛿₋𝜏₁=𝑏,𝑡+𝜀_𝑢𝑒_𝑏𝑢𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑐^𝑙+𝜀∶=𝑀₁,𝑡∈𝑇₁,∞_𝕋. (3.6)
Again, from the second equation of system (1.1) and (3.6), for 𝑡∈[𝑇₁+𝜏₂,∞)_𝕋, we have 𝑦^Δ(𝑡)≤𝑦(𝑡)−𝑑₁(𝑡)+𝑓(𝑡)(1−𝑚)𝑀₁𝑎(𝑡)𝑦𝛿₋𝜏₂,𝑡+(1−𝑚)𝑀₁. (3.7) From (3.7), we can see 𝑦^Δ(𝑡)≤(−𝑑₁(𝑡)+𝑓(𝑡))𝑦(𝑡); then by (𝐻₁) and Lemma 2.7, we can get 𝑦𝛿₋𝜏₂,𝑡≥𝑒_(⊖𝛼)𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡)), (3.8) where 𝛼(𝑡)=𝑓(𝑡)−𝑑₁(𝑡).
Together with (3.7) and (3.8), we have 𝑦^Δ(𝑡)≤𝑦(𝑡)−𝑑₁(𝑡)+𝑓(𝑡)(1−𝑚)𝑀₁𝑎(𝑡)𝑒_(⊖𝛼)𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡))+(1−𝑚)𝑀₁(𝑡)()𝑀≤𝑦1−𝑚₁𝑓(𝑡)−𝑑₁(𝑡)−𝑎(𝑡)𝑑₁(𝑡)𝑒_(⊖𝛼)𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡))𝑎(𝑡)𝑒_(⊖𝛼)𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡))+(1−𝑚)𝑀₁≤𝑦(𝑡)𝑓^𝑢−𝑑^𝑙₁−𝑎^𝑙𝑑^𝑙₁𝑒_{⊖(𝑓𝑢−𝑑𝑙1)}𝜎(𝑡),𝛿₋𝜏₂,𝑡(1−𝑚)𝑀₁𝑦(𝜎(𝑡)). (3.9)
By (H₂) and Lemma 2.5, for arbitrary small positive constant 𝜀, there exists 𝑇₂>𝑇₁+𝜏₂ such that 𝑦(𝑡)≤𝑓^𝑢−𝑑^𝑙₁(1−𝑚)𝑀₁𝑎^𝑙𝑑^𝑙₁𝑒_{⊖(𝑓𝑢−𝑑𝑙1)}𝜎(𝑡),𝛿₋𝜏₂=𝑓,𝑡+𝜀_𝑢−𝑑^𝑙₁(1−𝑚)𝑀₁𝑒_{(𝑓𝑢−𝑑𝑙1)}𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑎^𝑙𝑑^𝑙₁+𝜀∶=𝑀₂,𝑡∈𝑇₂,∞_𝕋. (3.10)
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 𝑧(𝑡)≤𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/𝑝^𝑢(1−𝑚)𝑀₁𝑝^𝑢𝑀₁(1−𝑚)𝑒_{(𝑔𝑢/𝑝𝑙−𝑑𝑙2)}𝜎(𝑡),𝛿₋𝜏₃,𝑡𝑑^𝑙₂𝑛^𝑙+𝜀∶=𝑀₃,𝑡∈𝑇₃,∞_𝕋. (3.11) 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 𝑥(𝑡)≥𝑚₁,𝑦(𝑡)≥𝑚₂,𝑧(𝑡)≥𝑚₃. (3.12)

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 𝑥^Δ(𝑡)(𝑡)𝑏(𝑡)−𝑑(𝑡)≥𝑥)(1−𝑚𝑎(𝑡)−ℎ(𝑡)(1−𝑚)𝑛(𝑡)−𝑐(𝑡)𝑥𝛿₋𝜏₁,𝑡, (3.13) then 𝑥^Δ(𝑡)(𝑡)𝑏(𝑡)−𝑑(𝑡)≥𝑥)(1−𝑚𝑎(𝑡)−ℎ(𝑡)(1−𝑚)𝑛(𝑡)−𝑐(𝑡)𝑀₁. (3.14) By (3.14), (H₅), and Lemma 2.7, we can get 𝑥𝛿₋𝜏₁,𝑡≤𝑒_⊖𝛽𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑥(𝜎(𝑡)), (3.15) 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 𝑥^Δ(𝑡)(𝑡)𝑏(𝑡)−𝑑(𝑡)≥𝑥)(1−𝑚𝑎(𝑡)−ℎ(𝑡)(1−𝑚)𝑛(𝑡)−𝑐(𝑡)𝑒_⊖𝛽𝜎(𝑡),𝛿₋𝜏₁𝑥(𝜎(𝑡(𝑡)𝑏,𝑡))≥𝑥^𝑙−𝑑^𝑢(1−𝑚)𝑎^𝑙−ℎ^𝑢(1−𝑚)𝑛^𝑙−𝑐^𝑢𝑒_⊖𝛽𝑙𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑥(𝜎(𝑡)), (3.16) where 𝛽^𝑙=𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙−𝑐^𝑢𝑀₁.
By (H₆) and Lemma 2.5, for arbitrary small positive constant 𝜀, there exists 𝑇₄>𝑇₁+𝜏₁ such that 𝑥(𝑡)≥𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙𝑐^𝑢𝑒_⊖𝛽𝑙𝜎(𝑡),𝛿₋𝜏₁=𝑏,𝑡−𝜀_𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙𝑒_𝛽𝑙𝜎(𝑡),𝛿₋𝜏₁,𝑡𝑐^𝑢−𝜀∶=𝑚₁,𝑡∈𝑇₄,∞_𝕋. (3.17)
Again, from the second equation of system (1.1) and (3.17), for 𝑡∈[𝑇₄+𝜏₂,∞)_𝕋, we have 𝑦^Δ(𝑡)≥𝑦(𝑡)−𝑑₁(𝑡)+𝑚₁/2𝑓(𝑡)(1−𝑚)𝑎(𝑡)𝑦𝛿₋𝜏₂,𝑡+𝑚₁/2(1−𝑚). (3.18) From (3.18), we can see 𝑦^Δ(𝑡)≥−𝑑₁(𝑡)𝑦(𝑡); then by (H₇) and Lemma 2.7, we can get 𝑦𝛿₋𝜏₂,𝑡≤𝑒_{⊖(−𝑑1)}𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡)). (3.19)
Together with (3.18) and (3.19), we have 𝑦^Δ(𝑡)≥𝑦(𝑡)−𝑑₁(𝑡)+𝑚₁/2𝑓(𝑡)(1−𝑚)𝑎(𝑡)𝑒_{⊖(−𝑑1)}𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡))+𝑚₁()(𝑡)𝑚/21−𝑚≥𝑦¹/2𝑓^𝑙−𝑑^𝑢₁(1−𝑚)−𝑎^𝑢𝑑^𝑢₁𝑒_⊖(−𝑑^𝑢₁₎𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑦(𝜎(𝑡))𝑎^𝑢𝑒_⊖(−𝑑^𝑢₁₎𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑀₂+𝑚₁()./21−𝑚 (3.20)
By (H₈) and Lemma 2.5, for arbitrary small positive constant 𝜀, there exists 𝑇₅>𝑇₄+𝜏₂ such that 𝑦(𝑡)≥𝑚₁/2𝑓^𝑙−𝑑^𝑢₁(1−𝑚)𝑎^𝑢𝑑^𝑢₁𝑒_⊖(−𝑑^𝑢₁₎𝜎(𝑡),𝛿₋𝜏₂=𝑚,𝑡−𝜀₁/2𝑓^𝑙−𝑑^𝑢₁(1−𝑚)𝑒_(−𝑑^𝑢₁₎𝜎(𝑡),𝛿₋𝜏₂,𝑡𝑎^𝑢𝑑^𝑢₁−𝜀∶=𝑚₂,𝑡∈𝑇₅,∞_𝕋. (3.21)
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 𝑧(𝑡)≥𝑚₁/2𝑝^𝑙(1−𝑚)𝑔^𝑙−𝑑^𝑢₂−𝑘^𝑢𝑑^𝑢₂𝑒_(−𝑑^𝑢₂₎𝜎(𝑡),𝛿₋𝜏₃,𝑡𝑛^𝑢𝑑^𝑢₂−𝜀∶=𝑚₃,𝑡∈𝑇₆,∞_𝕋. (3.22) The conclusion of Proposition 3.3 follows. This completes the proof.

Together with Propositions 3.2 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 with 𝑚=0.6:𝑥^Δ(𝑡)(𝑡𝜋=𝑥)0.08+0.05sin2𝑡−0.2𝑥𝛿₋𝜏₁−(𝑡),𝑡0.05𝑦)(1−𝑚(𝑡)(𝑡)−(𝑡)2𝑦+𝑥)(1−𝑚0.05𝑧)(1−𝑚(𝑡)(𝑡),𝑦0.005+𝑥)(1−𝑚+2𝑧_Δ(𝑡)=𝑦(𝑡)−0.7−0.25cos5𝜋2𝑡+2𝑥𝛿₋𝜏₂,𝑡(1−𝑚)2𝑦𝛿₋𝜏₂,𝑡+𝑥𝛿₋𝜏₂),𝑧,𝑡(1−𝑚_Δ(𝑡)=𝑧(𝑡)−0.75+0.2sin3𝜋2𝑡+4𝑥𝛿₋𝜏₃,𝑡(1−𝑚)0.005+𝑥𝛿₋𝜏₃,𝑡(1−𝑚)+2𝑧𝛿₋𝜏₃.,𝑡 (4.1)

Let 𝕋=ℝ; then 𝜇(𝑡)=0. Obviously, (H₁), (H₃), (H₅), (H₇), and (H₉) hold. Taking 𝜏_𝑖=1, 𝑖=1,2,3, by a direct calculation, we can get(H₂)𝑓^𝑢−𝑑^𝑙₁=1.5500>0; (H₄)(𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂)/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/(𝑝^𝑢(1−𝑚)𝑀₁)=3.4407>0;(H₆)𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙=0.0100>0;(H₈)𝑓^𝑙−𝑑^𝑢₁=1.0500>0;(H₁₀)(𝑚₁/2)(1−𝑚)(𝑔^𝑙−𝑑^𝑢₂𝑝^𝑢)−𝑘^𝑢𝑑^𝑢₂=0.0218>0.

From the above results, we can see that all conditions of Theorem 3.4 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.

Let 𝕋=ℤ; then 𝜇(𝑡)=1. It is easy to check (H₁), (H₃), (H₅), (H₇), and (H₉) hold. Taking 𝜏_𝑖=1, 𝑖=1,2,3, by a direct calculation, we can get(H₂)𝑓^𝑢−𝑑^𝑙₁=1.5500>0; (H₄)(𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂)/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/(𝑝^𝑢(1−𝑚)𝑀₁)=3.4086>0;(H₆)𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙=0.0100>0;(H₈)𝑓^𝑙−𝑑^𝑢₁=1.0500>0;(H₁₀)(𝑚₁/2)(1−𝑚)(𝑔^𝑙−𝑑^𝑢₂𝑝^𝑢)−𝑘^𝑢𝑑^𝑢₂=0.0244>0.

From the above results, we can see that all conditions of Theorem 3.4 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.

Let 𝕋=⋃^∞_𝑘=0[2𝑘,2𝑘+1]; then 𝜇(𝑡)=^{0,for⋃𝑡∈∞𝑘=0[2𝑘,2𝑘+1),}_{1,for⋃𝑡∈∞𝑘=0{2𝑘+1}.} It is easy to check (H₁), (H₃), (H₅), (H₇), and (H₉) hold. Taking 𝜏_𝑖=2,𝑖=1,2,3, by a direct calculation, we can get(H₂)𝑓^𝑢−𝑑^𝑙₁=1.5500>0; (H₆)𝑏^𝑙−𝑑^𝑢(1−𝑚)/𝑎^𝑙−ℎ^𝑢(1−𝑚)/𝑛^𝑙=0.0100>0;(H₈)𝑓^𝑙−𝑑^𝑢₁=1.0500>0.

Furthermore, if 𝑡∈⋃^∞_𝑘=0[2𝑘,2𝑘+1), then(H₄)(𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂)/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/(𝑝^𝑢(1−𝑚)𝑀₁)=3.4418>0;(H₁₀)(𝑚₁/2)(1−𝑚)(𝑔^𝑙−𝑑^𝑢₂𝑝^𝑢)−𝑘^𝑢𝑑^𝑢₂=0.0175>0;

if 𝑡∈⋃^∞_𝑘=0{2𝑘+1}, then(H₄)(𝑔^𝑢−𝑝^𝑙𝑑^𝑙₂)/𝑝^𝑙−𝑘^𝑙𝑑^𝑙₂/(𝑝^𝑢(1−𝑚)𝑀₁)=3.4133>0;(H₁₀)(𝑚₁/2)(1−𝑚)(𝑔^𝑙−𝑑^𝑢₂𝑝^𝑢)−𝑘^𝑢𝑑^𝑢₂=0.0233>0.

From the above results, we can see that all conditions of Theorem 3.4 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.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant 61073065.