On a Generalized Discrete Ratio-Dependent Predator-Prey System

Abstract

Verifiable criteria are established for the permanence and existence of positive periodic solutions of a delayed discrete predator-prey model with monotonic functional response. It is shown that the conditions that ensure the permanence of this system are similar to those of its corresponding continuous system. And the investigations generalize some well-known results. In particular, a more acceptant method is given to study the bounded discrete systems rather than the comparison theorem.

1. Introduction

Since the end of the 19th century, many biological models have been established to illustrate the evolutionary of species, among them, predator-prey models attracted more and more attention of biologists and mathematicians. There are many different kinds of predator-prey models in the literature. And since 1990s, the so-called ratio-dependent predator-prey models play an important role in the investigations on predator-prey models, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Under some simple assumptions, a general form of a ratio-dependent model is

𝑥^′𝑥=𝑟𝑥(1−𝐾𝑥)−𝜑(𝑦𝑦)𝑦,_′𝑥=𝑦(𝜇𝜑(𝑦)−𝐷). (1.1) Here the predator-prey interactions are described by 𝜑(𝑥/𝑦); this function replaces the functional response function 𝜑(𝑥) in the traditional prey-dependent model. For the study of ratio-dependent predator-prey models, most works have been done on the Michaelis-Menten type model

𝑥^′𝑥=𝑟𝑥(1−𝐾)−𝛼𝑥𝑦,𝑦𝑚𝑦+𝑥_′=𝑦(−𝑑+𝑓𝑥𝑚𝑦+𝑥), (1.2) or its periodic type

𝑥^′=𝑥𝑎(𝑡)−𝑏(𝑡)𝑥−𝛼(𝑡)𝑥𝑦,𝑦𝑚𝑦(𝑡)+𝑥_′=𝑦(−𝑑(𝑡)+𝑓(𝑡)𝑥𝑚𝑦+𝑥), (1.3) see [1–7] and references therein. It is easy to see that here the functional response function is 𝜑(𝑢)=𝑐𝑢/(𝑚+𝑢), 𝑢=𝑥/𝑦, as we know, this functional response function was first used by Holling [8], and later biologists call it Holling type II functional response function, it usually describes the uptake of substrate by the microorganisms in microbial dynamics or chemical kinetics [9]. And in the present paper, we will concentrate on the general form of the ratio-dependent predator-prey model.

For the sake of convenience, we introduce some notations and definitions. Denote ℤ,ℝ, and ℝ⁺ as the sets of all integers, real numbers, and nonnegative real numbers, respectively. Let 𝐶 denote the set of all bounded sequences 𝑓∶ℤ→ℝ, 𝐶₊ is the set of all 𝑓∈𝐶 such that 𝑓>0, and 𝐶_𝜔={𝑓∈𝐶₊∣𝑓(𝑘+𝜔)=𝑓(𝑘),𝑘∈ℤ},𝐼_𝜔={0,1,…,𝜔−1}. We define

𝑓^𝑀=sup_𝑘∈ℤ𝑓(𝑘),𝑓^𝐿=inf_𝑘∈ℤ𝑓(𝑘), (1.4) for any 𝑓∈𝐶. Obviously, if 𝑓 is an 𝜔-periodic sequence, then

𝑓^𝑀=max_{𝑘∈𝐼𝜔}𝑓(𝑘),𝑓^𝐿=min_{𝑘∈𝐼𝜔}𝑓(𝑘). (1.5) We also define

1𝑓=𝜔^𝜔−1_𝑘=0𝑓(𝑘), (1.6) if 𝑓 is an 𝜔-periodic sequence. And denote

1𝑓=𝜔^𝜔₀𝑓(𝑡)𝑑𝑡,𝑓^𝐿=min_{𝑡∈[0,𝜔]}|𝑓(𝑡)|,𝑓^𝑀=max_{𝑡∈[0,𝜔]}|𝑓(𝑡)|, (1.7) when 𝑓 is a periodic continuous function with period 𝜔.

In view of the periodicity of the actual environment, we begin with the following periodic continuous ratio-dependent predator-prey system:

𝑥^′(𝑡)=𝑥(𝑡)[𝑏(𝑡)−𝑎(𝑡)𝑥(𝑡−𝜏₁)]−𝑐(𝑡)𝑔(𝑥(𝑡)𝑦𝑦(𝑡))𝑦(𝑡),_′(𝑡)=𝑦(𝑡)[𝑒(𝑡)𝑔(𝑥(𝑡−𝜏₂)𝑦(𝑡−𝜏₂))−𝑑(𝑡)], (1.8) where 𝑥(𝑡) and 𝑦(𝑡) represent the densities of the prey population and predator population at time 𝑡 respectively; 𝜏₁≥0,𝜏₂≥0 are real constants; 𝑏,𝑑∶ℝ→ℝ and 𝑎,𝑐,𝑒∶ℝ→ℝ⁺ are continuous periodic functions with period 𝜔>0 and ∫^𝜔₀𝑏(𝑡)𝑑𝑡>0, ∫^𝜔₀𝑑(𝑡)𝑑𝑡>0,𝑎,𝑐,𝑒 is not always zero; and 𝑔(𝑢(𝑡)) (here 𝑢(𝑡)=𝑥(𝑡)/𝑦(𝑡)) denotes the ratio-dependent response function, which reflects the capture ability of the predator. Here we assume that 𝑔(𝑢) satisfies the following monotonic condition, for short, we call it (𝑀):

(i)𝑔∈𝐶¹[0,+∞),𝑔(0)≥0;(ii)𝑔^′(𝑢)≥0 for 𝑢∈[0,+∞);(iii)lim_𝑢→∞𝑔(𝑢)=𝛼≠0.

In [10], we gave a sufficient condition for the permanence of the continuous model

𝑥^′(𝑡)=𝑥(𝑡)[𝑎(𝑡)−𝑏(𝑡)𝑥(𝑡)]−𝑐(𝑡)𝑔(𝑥(𝑡)𝑦𝑦(𝑡))𝑦(𝑡),_′(𝑡)=𝑦(𝑡)[𝑒(𝑡)𝑔(𝑥(𝑡−𝜏)𝑦(𝑡−𝜏))−𝑑(𝑡)]. (1.9) In which 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡),𝑑(𝑡), and 𝑒(𝑡) are all positive periodic continuous functions with period 𝜔>0; 𝜏 is a positive constant. In addition to condition (𝑀), the functional response function 𝑔 also satisfies the following.

(iv)There exists a positive constant ℎ such that 𝑢²𝑔^′(𝑢)≤ℎ.

Without loss of generality, in this paper, we always assume that 𝛼=1 (if 𝛼≠1, let 𝑐^∗(𝑡)=𝛼𝑐(𝑡) and still denote 𝑐^∗(𝑡) as 𝑐(𝑡)).

Some special cases of system (1.8) have been studied, see [11, 12] and so forth. In those papers, the authors mainly concentrated on the existence of periodic solutions and permanence for systems they considered.

Set

𝑚=sup_{𝑢∈[0,+∞)}𝑔(𝑢)𝑢, (1.10) from (ii) and (iii), we can easily obtain 0<𝑚<+∞.

Through the above assumptions, we can see that, one of the main results in [10] can be given as follows.

Theorem 1.1. If(H1)𝑏>𝑚𝑐and(H2)𝑒>𝑑hold, then system (1.9) is permanent.

Remark 1.2. By similar methods proposed in [10], we can show that under conditions (H1) and (H2), system (1.8) is also permanent.

We also need to mention that conditions (H1) and (H2) are sufficient to assure the existence of positive periodic solutions of (1.8); this problem has been solved in [13].

However, when the size of the population is rarely small or the population has nonoverlapping generations [14, 15], a more realistic model should be considered, that is, the discrete time model. Just as pointed out in [16], even if the coefficients are constants, the asymptotic behavior of the discrete system is rather complex and “chaotic” than the continuous one, see [16] for more details. Similar to the arguments of [17], we can obtain a discrete time analogue of (1.8):

𝑁₁(𝑘+1)=𝑁₁(𝑘)exp{𝑏(𝑘)−𝑎(𝑘)𝑁₁(𝑘−[𝜏₁𝑁])−𝑐(𝑘)ℎ(¹(𝑘)𝑁₂𝑁(𝑘))},₂(𝑘+1)=𝑁₂𝑁(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(¹(𝑘−[𝜏₂])𝑁₂(𝑘−[𝜏₂]))}, (1.11) where [𝑡] denotes the integer part of 𝑡 (𝑡>0) and ℎ(𝑁₁(𝑘)/𝑁₂(𝑘))=𝑁₂(𝑘)𝑔(𝑁₁(𝑘)/𝑁₂(𝑘))/𝑁₁(𝑘)=𝑔(𝑢(𝑘))/𝑢(𝑘),𝑢(𝑘)=𝑁₁(𝑘)/𝑁₂(𝑘). Correspondingly, the basic assumptions of (1.11) is the same as that in (1.8), of cause, here 𝑏,𝑑∶ℤ→ℝ and 𝑎,𝑐,𝑒∶ℤ→ℝ⁺ are periodic sequences with period 𝜔>0 and ∑^𝜔−1_𝑘=0𝑏(𝑘)>0, ∑^𝜔−1_𝑘=0𝑑(𝑘)>0, and 𝑔 satisfies (𝑀). To the best of our knowledge, a few investigations have been carried out for the permanence on delayed discrete ecological systems, since the dynamics of these systems are usually more complicated than the continuous ones.

The exponential form of system (1.11) assures that, for any initial condition 𝑁(0)>0, 𝑁(𝑘) remains positive. In the remainder of this paper, for biological reasons, we only consider solutions 𝑁(𝑘) with

𝑁_𝑖(−𝑘)≥0,𝑘=1,2,…,max{[𝜏₁],[𝜏₂]},𝑁_𝑖(0)>0,𝑖=1,2. (1.12)

System (1.11) includes many biological models as its special cases, which have been studied by many authors; see [17–20] and so forth. Among them, Fan and Wang (see [17]) considered the existence of positive periodic solutions for delayed periodic Michaelis-Menten type ratio-dependent predator-prey system

𝑁₁(𝑘+1)=𝑁₁(𝑘)exp{𝑏₁(𝑘)−𝑎₁(𝑘)𝑁₁𝛼(𝑘)−¹(𝑘)𝑁₂(𝑘)𝑁₁(𝑘)+𝑚(𝑘)𝑁₂𝑁(𝑘)},₂(𝑘+1)=𝑁₂(𝑘)exp{−𝑏₂𝛼(𝑘)+²(𝑘)𝑁₁(𝑘)𝑁₁(𝑘)+𝑚(𝑘)𝑁₂(𝑘)}, (1.13) and obtained the following theorem.

Theorem 1.3. Assume that the following conditions hold:(A1)𝑏₁>(𝛼₁(𝑘)/𝑚(𝑘)),(A2)𝛼₂>𝑏₂.Then (1.13) has at least one positive 𝜔-periodic solution.

Later in [20], we proved that under conditions (A1) and (A2), system (1.13) is also permanent, so by the main result in [21], we can also obtain Theorem 1.3, which gives another method to prove the existence of periodic solutions.

From the works above, it is not difficult to find: that for the continuous time model (1.3) and the discrete time model (1.13), conditions that assure the existence of positive periodic solutions are exactly the same. In addition, when we comparing the work in [11] with that in [20], we found amazedly that conditions that assure the permanence of the discrete models are also the same as those of the continuous models. This motivated us to consider the permanence of system (1.11) only under conditions (H1) and (H2), since we have already obtained the permanence of system (1.9).

Until very recently, Yang [22] studied the permanence of system (1.11) when 𝜏₁=0 and obtained the following conclusion.

Theorem 1.4. Assume that(B1)𝑒^𝑀>𝑑^𝐿, (1.14)(B2)𝑏^𝐿>𝑚𝑐^𝑀, (1.15)(B3)𝑒^𝐿𝑚𝑔(¹𝑀₂)=𝑑^𝑀, (1.16) hold, where𝑚₁=𝑏^𝐿−𝑚𝑐^𝑀𝑎^𝑀exp{𝑏^𝐿−𝑚𝑐^𝑀−𝑎^𝑀𝑀₁𝑀},₁=1𝑎^𝐿exp{𝑏^𝑀−1},𝑀₂=exp{2(𝑒^𝑀−𝑑^𝐿)}. (1.17) Then (1.11) is permanent.

Remark 1.5. In Theorem 1.4, condition (B3) implies condition (B1); and (B3) is an equality, it is too strong to satisfy.

As pointed out in [23], if we use the method of comparison theorem, then the additional condition (like (B3)), to some extent, is necessary. But for system itself, this condition may be not necessary. In this paper, our aim is to improve the above results. One of the main results in this paper is given below, furthermore, we can conclude Corollary 3.5 similarly, from which we could show that condition (B3) can be deleted. Now we list the main result in the following.

Theorem 1.6. Assume that (H1) and (H2)hold. Then system (1.11) is permanent.

Remark 1.7. Condition (H2) is a necessary condition.

Corollary 1.8. Assume that (H1) and (H2) hold, then system (1.11) has at least one positive 𝜔-periodic solution.

Clearly, Theorem 1.6 extends and improves [19, Theorem  3.1], [20, Theorem  1.4]; Theorem 1.6 also extends and improves Theorem 1.4 by weaker conditions (H1) and (H2) instead of (B1–B3) when the coefficients are all periodic. In particular, our investigation gives a more acceptant method to study the bounded discrete systems, which is better than the comparison theorem.

For the permanence of biology systems, one can refer to [24–33] and the references cited therein.

The tree of this paper is arranged as follows. In the next section, we give some useful lemmas which are essential to prove our conclusions. And in the third section, we give a proof to the main result.

2. Preliminary

In this section, we list the definition of permanence and establish some useful lemmas.

Definition 2.1. System (1.11) is said to be permanent if there exist two positive constants 𝜆₁,𝜆₂ such that 𝜆₁≤liminf_𝑘→∞𝑁_𝑖(𝑘)≤limsup_𝑘→∞𝑁_𝑖(𝑘)≤𝜆₂,𝑖=1,2, (2.1) for any solution (𝑁₁(𝑘),𝑁₂(𝑘)) of (1.11).

Lemma 2.2 ([20]). The problem 𝑥(𝑘+1)=𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)},𝑥(0)=𝑥₀>0, (2.2) has at least one periodic solution 𝑈 if 𝑏∈𝐶_𝜔,𝑎∈𝐶and 𝑎 is an 𝜔-periodic sequence with 𝑎>0, moreover, the following properties hold.(a)𝑈 is positive 𝜔-periodic.(b)𝑈 has the following estimations for it's boundary: 𝑎𝑏exp{−(|𝑎|+𝑎)𝜔}≤𝑈(𝑘)≤𝑎𝑏exp{(|𝑎|+𝑎)𝜔}, (2.3) especially, 𝑎𝑏exp{−𝑎𝜔}≤𝑈(𝑘)≤𝑎𝑏exp{𝑎𝜔}, (2.4) if 𝑎∈𝐶_𝜔.

Lemma 2.3. For any positive constant 𝐾, the problem 𝐾𝑥(𝑘+1)=𝑥(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(𝑥(𝑘))}, (2.5) has at least one periodic solution 𝑈 if 𝑑∈𝐶and 𝑑 is an 𝜔-periodic sequence provided that (H2) holds. Moreover, the following properties hold.(a)𝑈 is positive 𝜔-periodic.(b)𝑈 has the following estimations for its boundary:𝐾𝑔⁻¹(𝑑/𝑒)exp{−(|𝑑|+𝐾𝑑)𝜔}≤𝑈(𝑘)≤𝑔⁻¹(𝑑/𝑒)exp{(|𝑑|+𝑑)𝜔}, (2.6) especially, 𝐾𝑔⁻¹(𝑑/𝑒)exp{−𝐾𝑑𝜔}≤𝑈(𝑘)≤𝑔⁻¹(𝑑/𝑒)exp{𝑑𝜔}, (2.7) if 𝑑∈𝐶_𝜔, where 𝑔⁻¹ represents the inverse of 𝑔.

Proof. We only prove that (2.6) holds, for the rest of the proof, one can refer to [17]. Let 𝑥(𝑛) be any possible 𝜔-periodic positive solution of (2.5), then 𝐾ln𝑥(𝑛+1)−ln𝑥(𝑛)=−𝑑(𝑛)+𝑒(𝑛)𝑔(𝑥(𝑛)), (2.8) therefore ^𝜔−1_𝑛=0[ln𝑥(𝑛+1)−ln𝑥(𝑛)]=^𝜔−1_𝑛=0𝐾[−𝑑(𝑛)+𝑒(𝑛)𝑔(𝑥(𝑛))]=0, (2.9) this leads to ^𝜔−1_𝑛=0𝐾𝑒(𝑛)𝑔(𝑥(𝑛))=𝑑𝜔. (2.10) We claim that there exist some 𝑛₁ and 𝑛₂ such that 𝑛₁,𝑛₂∈𝐼_𝜔,𝑥(𝑛₁𝐾)≤𝑔⁻¹(𝑑/𝑒),𝑥(𝑛₂𝐾)≥𝑔⁻¹(𝑑/.𝑒) (2.11) If this is not true, then either 𝐾𝑥(𝑛)<𝑔⁻¹(𝑑/𝑒), (2.12) or 𝑥(𝑛₂𝐾)>𝑔⁻¹(𝑑/𝑒), (2.13) for any 𝑛∈𝐼_𝜔, in any case, we can obtain ∑^𝜔−1_𝑛=0𝑒(𝑛)𝑔(𝐾/𝑥(𝑛))≠𝑑𝜔, this contradiction shows that our claim is true.
Note that for any 𝑛∈𝐼_𝜔,
ln𝑥(𝑛)−ln𝑥(𝑛₁)≤^𝜔−1_𝑛=0𝐾|−𝑑(𝑛)+𝑒(𝑛)𝑔(𝑥(𝑛))|,ln𝑥(𝑛)−ln𝑥(𝑛₂)≥−^𝜔−1_𝑛=0𝐾|−𝑑(𝑛)+𝑒(𝑛)𝑔(𝑥(𝑛))|, (2.14) then by virtue of equality (2.10), we complete the proof.

Lemma 2.4. Consider the inequality problem 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)}. (2.15) If 𝑏∈𝐶_𝜔,𝑎∈𝐶,and 𝑎 is an 𝜔-periodic sequence with 𝑎>0, then any positive solutions 𝑥(𝑘) of (2.15) satisfy limsup_𝑘→∞𝑥(𝑘)≤𝐻₁exp{𝑏^𝑀𝐻₁}, (2.16) where 𝐻₁=𝑎𝑏exp{(𝑎+|𝑎|)𝜔}. (2.17) Moreover, if 𝑎∈𝐶_𝜔, then limsup_𝑘→∞𝑥(𝑘)≤𝐻₂exp{𝑏^𝑀𝐻₂}, (2.18) where 𝐻₂=𝑎𝑏exp{𝑎𝜔}. (2.19)

Proof. Consider the following auxiliary equation: 𝑧(𝑘+1)=𝑧(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑧(𝑘)}, (2.20) by Lemma 2.2, (2.20) has at least one positive 𝜔-periodic solution, denote it as 𝑧^∗(𝑘), then 𝑧^∗(𝑘)≤𝐻₁. (2.21) Let 𝑥(𝑘)=exp{𝑢₁(𝑘)},𝑧^∗(𝑘)=exp{𝑢₂(𝑘)}, (2.22) then 𝑢₁(𝑘+1)−𝑢₁(𝑘)≤𝑎(𝑘)−𝑏(𝑘)exp{𝑢₁𝑢(𝑘)},₂(𝑘+1)−𝑢₂(𝑘)=𝑎(𝑘)−𝑏(𝑘)exp{𝑢₂(𝑘)}. (2.23) Make the transformation 𝑢(𝑘)=𝑢₁(𝑘)−𝑢₂(𝑘), we can obtain 𝑢(𝑘+1)−𝑢(𝑘)≤−𝑏(𝑘)𝑧^∗(𝑘)[exp{𝑢(𝑘)}−1]. (2.24) Now we divide the proof into two cases according to the oscillating property of 𝑢(𝑘). First we assume that 𝑢(𝑘) does not oscillate about zero, then 𝑢(𝑘) will be either eventually positive or eventually negative. If the latter holds, that is, 𝑢₁(𝑘)<𝑢₂(𝑘), we have 𝑥(𝑘)<𝑧^∗(𝑘)≤𝐻₁(if𝑎∈𝐶_𝜔,then𝑥(𝑘)≤𝐻₂). (2.25) Either if the former holds, then by (2.24), we know 𝑢(𝑘+1)<𝑢(𝑘), which means that 𝑢(𝑘) is eventually decreasing, also in terms of its positivity, we know that lim_𝑘→∞𝑢(𝑘) exists. Then (2.24) implies lim_𝑘→∞𝑢(𝑘)=0, which leads to limsup_𝑘→∞𝑥(𝑘)≤𝐻₁(if𝑎∈𝐶_𝜔,thenlimsup_𝑘→∞𝑥(𝑘)≤𝐻₂). (2.26)
Now we assume that 𝑢(𝑘) oscillates about zero, by (2.24), we know that 𝑢(𝑘)>0implies 𝑢(𝑘+1)≤𝑢(𝑘). Thus, if we let {𝑢(𝑘_𝑙)} be a subsequence of {𝑢(𝑘)}, where 𝑢(𝑘_𝑙) is the first element of the 𝑙th positive semicycle of {𝑢(𝑘)}, then limsup_𝑘→∞𝑢(𝑘)=limsup_𝑙→∞𝑢(𝑘_𝑙). For the definition of semicycle and other related concepts, we refer to [34]. Notice that
𝑢(𝑘_𝑙)≤𝑢(𝑘_𝑙−1)−𝑏(𝑘_𝑙−1)𝑧^∗(𝑘_𝑙−1)[exp{𝑢(𝑘_l−1)}−1] (2.27) and 𝑢(𝑘_𝑙−1)<0, then we know 𝑢(𝑘_𝑙)≤𝑏(𝑘_𝑙−1)𝑧^∗(𝑘_𝑙−1)[1−exp{𝑢(𝑘_𝑙−1)}]≤𝑏(𝑘_𝑙−1)𝑧^∗(𝑘_𝑙−1)≤(𝑏(𝑘_𝑙−1)𝑧^∗(𝑘_𝑙−1))^𝑀. (2.28) Therefore limsup_𝑙→∞𝑢(𝑘_𝑙)≤(𝑏(𝑘_𝑙−1)𝑧^∗(𝑘_𝑙−1))^𝑀. (2.29) By the medium of (2.22), (2.25), and (2.26), we have limsup_𝑘→∞𝑥(𝑘)≤𝐻₁exp{𝑏^𝑀𝐻₁}(if𝑎∈𝐶_𝜔,thenlimsup_𝑘→∞𝑥(𝑘)≤𝐻₂exp{𝑏^𝑀𝐻₂}). (2.30)

Corollary 2.5. Any positive solution of the inequality problem (2.15) satisfies limsup_𝑘→∞1𝑥(𝑘)≤min{𝑏^𝐿exp{𝑎^𝑀𝑎−1},^𝑀𝑏^𝐿exp{𝑎^𝑀}}, (2.31) where 𝑎∈𝐶, 𝑎^𝑀>0 and 𝑏∈𝐶₊.

Proof. Define the function 𝑓(𝑥)=𝑥exp{𝑎−𝑏𝑥},𝑥>0,𝑎∈ℝ,𝑏>0, (2.32) it is easy to see 1𝑓(𝑥)≤𝑏exp{𝑎−1}, (2.33) which immediately leads to limsup_𝑘→∞𝑥(𝑘)≤limsup_𝑘→∞11𝑏(𝑘)exp{𝑎(𝑘)−1}≤𝑏^𝐿exp{𝑎^𝑀−1}. (2.34) From (2.15), we have 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)}≤𝑥(𝑘)exp{𝑎^𝑀−𝑏^𝐿𝑥(𝑘)}. (2.35) By Lemma 2.4, for any 𝜔>0,limsup_𝑘→∞𝑎𝑥(𝑘)≤^𝑀𝑏^𝐿exp{𝑎^𝑀𝜔}exp{𝑎^𝑀exp{𝑎^𝑀𝜔}}, (2.36) let 𝜔→0, we can obtain limsup_𝑘→∞𝑎𝑥(𝑘)≤^𝑀𝑏^𝐿exp{𝑎^𝑀}, (2.37) by (2.34) and (2.37), we complete the proof.

Remark 2.6. Note that when 𝑎^𝑀≤1/𝑒,(𝑎^𝑀/𝑏^𝐿)exp{𝑎^𝑀}≤(1/𝑏^𝐿)exp{𝑎^𝑀−1}.
Similarly, we can obtain the following result.

Lemma 2.7. If any positive solution 𝑥(𝑘) of the inequality problem 𝑥(𝑘+1)≥𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)x(𝑘)} (2.38) satisfies limsup_𝑘→∞𝑥(𝑘)≤𝐻, (2.39) here 𝐻 is a positive constant. Then if 𝑏∈𝐶_𝜔,𝑎∈𝐶,and 𝑎 is an 𝜔-periodic sequence with 𝑎>0, one has liminf_𝑘→∞𝑥(𝑘)≥𝐻₃exp{−𝐻𝑏^𝑀𝐻},₃=𝑎𝑏exp{−(|𝑎|+𝑎)𝜔}. (2.40) Moreover, if 𝑎∈𝐶_𝜔, then liminf_𝑘→∞𝑥(𝑘)≥𝐻₄exp{−𝐻𝑏^𝑀}, (2.41) where 𝐻₄=𝑎𝑏exp{−𝑎𝜔}. (2.42)

The proof is similar to that of Lemma 2.4.

Corollary 2.8. If any positive solution 𝑥(𝑘) of the inequality problem 𝑥(𝑘+1)≤𝑥(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)} (2.43) satisfies limsup_𝑘→∞𝑥(𝑘)≤𝐻, (2.44) here 𝐻 is a positive constant, then liminf_𝑘→∞𝑎𝑥(𝑘)≥^𝐿𝑏^𝑀exp{𝑎^𝐿−𝑏^𝑀𝐻}, (2.45) where 𝑎∈𝐶 and 𝑏∈𝐶₊.

3. Proof of the Main Result

For the rest of this paper, we only consider the solution of (1.11) with initial conditions (1.12). To prove Theorem 1.6, we need the following several propositions.

Proposition 3.1. There exists a positive constant 𝐾₁ such that limsup_𝑘→+∞𝑁₁(𝑘)≤𝐾₁.

Proof. Given any positive solution (𝑁₁(𝑘),𝑁₂(𝑘)) of (1.11), from the first equation of (1.11), we have 𝑁₁(𝑘+1)≤𝑁₁(𝑘)exp{𝑏(𝑘)−𝑎(𝑘)𝑁₁(𝑘−[𝜏₁])}. (3.1) Set 𝑁₁(𝑘)=exp{𝑢₁(𝑘)}, (3.2) then 𝑢₁(𝑘+1)−𝑢₁(𝑘)≤𝑏(𝑘)−𝑎(𝑘)exp{𝑢₁(𝑘−[𝜏₁])}, (3.3) thus ^𝑘−1_{𝑖=k−[𝜏1]}(𝑢₁(𝑖+1)−𝑢₁(𝑖))≤^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖), (3.4) which is equivalent to 𝑢₁(𝑘)−^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖)≤𝑢₁(𝑘−[𝜏₁]), (3.5) hence 𝑁₁(𝑘−[𝜏₁])=exp{𝑢₁(𝑘−[𝜏₁])}≥exp{𝑢₁(𝑘)−^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖)}=𝑁₁(𝑘)exp{−^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖)}. (3.6) Therefore 𝑁₁(𝑘+1)≤𝑁₁(𝑘)exp{𝑏(𝑘)−𝑎(𝑘)exp{−^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖)}𝑁₁(𝑘)}. (3.7) By Lemma 2.4, we have limsup_𝑘→+∞𝑁₁(𝑘)≤𝐾₁, (3.8) where 𝐾₁=𝐺₁exp{(𝑎(𝑘)exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖)})^𝑀𝐺₁𝐺},₁=𝑏𝑎(𝑘)exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏1]}𝑏(𝑖)}exp{(𝑏+|𝑏|)𝜔}. (3.9)

Proposition 3.2. Under condition (H1), there exists a positive constant 𝑘₁ such that liminf_𝑘→∞𝑁₁(𝑘)≥𝑘₁.

Proof. Given any positive solution (𝑁₁(𝑘),𝑁₂(𝑘)) of (1.11), from the first equation of (1.11), we have 𝑁₁(𝑘+1)≥𝑁₁(𝑘)exp{𝑏(𝑘)−𝑚𝑐(𝑘)−𝑎(𝑘)𝑁₁(𝑘−[𝜏₁])}. (3.10) Set 𝑁₁(𝑘)=exp{𝑢₁(𝑘)}, then 𝑢₁(𝑘+1)−𝑢₁(𝑘)≥𝑏(𝑘)−𝑚𝑐(𝑘)−𝑎(𝑘)exp{𝑢₁(𝑘−[𝜏₁])}, (3.11) which yields ^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑢₁(𝑖+1)−𝑢₁(𝑖))≥^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚𝑐(𝑖)−𝑎(𝑖)𝐾₁), (3.12) that is, 𝑢₁(𝑘−[𝜏₁])≤𝑢₁(𝑘)−^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚𝑐(𝑖)−𝑎(𝑖)𝐾₁), (3.13) thus 𝑁₁(𝑘−[𝜏₁])=exp{𝑢₁(𝑘−[𝜏₁])}≤exp{𝑢₁(𝑘)−^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚𝑐(𝑖)−𝑎(𝑖)𝐾₁)}=𝑁₁(𝑘)exp{−^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚𝑐(𝑖)−𝑎(𝑖)𝐾₁)}. (3.14) Therefore 𝑁₁(𝑘+1)≥𝑁₁(𝑘)×exp{𝑏(𝑘)−𝑚𝑐(𝑘)−𝑎(𝑘)exp{−^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚𝑐(𝑖)−𝑎(𝑖)𝐾₁)}𝑁₁(𝑘)}. (3.15) Since (H1) holds, then by Lemma 2.7 and Proposition 3.1, we have liminf_𝑘→∞𝑥(𝑘)≥𝐺₂exp{−𝐾₁(𝑎(𝑘)exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚𝑐(𝑖)−𝑎(𝑖)𝐾₁)})^𝑀𝐺},₂=𝑏−𝑚𝑐𝑎(𝑘)exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏1]}(𝑏(𝑖)−𝑚(𝑖)−𝑎(𝑖)𝐾₁)}exp{−(|𝑏−𝑚𝑐|+𝑏−𝑚𝑐)𝜔}. (3.16)

Proposition 3.3. If (H2) holds, then there exists a positive constant 𝐾₂ such that limsup_𝑘→∞𝑁₂(𝑘)≤𝐾₂. (3.17)

Proof. Given any positive solution (𝑁₁(𝑘),𝑁₂(𝑘)) of (1.11). Set 𝑁₂(𝑘)=exp{𝑢₁(𝑘)}, from the second equation of (1.11) and notice that conditions (ii) and (iii) on 𝑔 imply that 𝑔(𝑘,𝑢(𝑘))≤1, then 𝑢₁(𝑘+1)−𝑢₁(𝑘)≤𝑒(𝑘)−𝑑(𝑘), (3.18) thus ^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑢₁(𝑖+1)−𝑢₁(𝑖))≤^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖)), (3.19) which is equivalent to 𝑢₁(𝑘)−^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖))≤𝑢₁(𝑘−[𝜏₂]), (3.20) hence 𝑁₂(𝑘−[𝜏₂])=exp{𝑢₁(𝑘−[𝜏₂])}≥exp{𝑢₁(𝑘)−^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖))}=𝑁₂(𝑘)(exp{−^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖))})^𝐿. (3.21) Therefore for any given 𝜀>0, we have 𝑁₂(𝑘+1)=𝑁₂𝑁(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(¹(𝑘−[𝜏₂])𝑁₂(𝑘−[𝜏₂]))}≤𝑁₂𝐾(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(¹+𝜀𝑁₂(𝑘)(exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖))})^𝐿)}, (3.22) for sufficiently large 𝑘. Here we use the monotonicity of the function 𝑔(𝑢).
Consider the following auxiliary equation:
𝐾𝑧(𝑘+1)=𝑧(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(¹+𝜀𝑧(𝑘)(exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖))})^𝐿)}. (3.23) By Lemma 2.3 and condition (H2), we can obtain that (3.23) has at least one positive 𝜔-periodic solution, denote it as 𝑧^∗₁(𝑘) and 𝑧^∗₁𝐾(𝑘)≤𝑔⁻¹(𝑑/𝑒)exp{(|𝑑|+𝑑)𝜔}∶=𝐺₃, (3.24) where 𝐾𝐾=¹(exp{−∑^𝑘−1_{𝑖=𝑘−[𝜏2]}(𝑒(𝑖)−𝑑(𝑖))})^𝐿. (3.25) Let 𝑧^∗₁(𝑘)=exp{𝑢₂(𝑘)}, (3.26) then 𝑢₁(𝑘+1)−𝑢₁𝐾(𝑘)≤−𝑑(𝑘)+𝑒(𝑘)𝑔(exp{𝑢₁𝑢(𝑘)}),₂(𝑘+1)−𝑢₂𝐾(𝑘)=−𝑑(𝑘)+𝑒(𝑘)𝑔(exp{𝑢₂(𝑘)}). (3.27) Denote 𝑢(𝑘)=𝑢₁(𝑘)−𝑢₂(𝑘), we have 𝐾𝑢(𝑘+1)−𝑢(𝑘)≤𝑒(𝑘)[𝑔(exp{𝑢₁𝐾(𝑘)})−𝑔(exp{𝑢₂(𝑘)})]. (3.28) First we assume that 𝑢(𝑘) does not oscillate about zero, then 𝑢(𝑘) will be either eventually positive or eventually negative. If the latter holds, that is, 𝑢₁(𝑘)<𝑢₂(𝑘), we have 𝑁₂(𝑘)<𝑧^∗₁(𝑘)≤𝐺₃. (3.29) Either if the former holds, then by (3.28), we have 𝑢(𝑘+1)<𝑢(𝑘), which means that 𝑢(𝑘) is eventually decreasing, also in terms of its positivity, we obtain that lim_𝑘→∞𝑢(𝑘) exists. Then (3.28) leads to lim_𝑘→∞𝑢(𝑘)=0, this implies limsup_𝑘→∞𝑁₂(𝑘)≤𝐺₃. (3.30)
Now we assume that 𝑢(𝑘) oscillates about zero, in view of (3.28), we know that 𝑢(𝑘)>0implies 𝑢(𝑘+1)≤𝑢(𝑘). Thus, if we let {𝑢(𝑘_𝑙)} be a subsequence of {𝑢(𝑘)} where 𝑢(𝑘_𝑙) is the first element of the 𝑙th positive semicycle of {𝑢(𝑘)}, then limsup_𝑘→∞𝑢(𝑘)=limsup_𝑙→∞𝑢(𝑘_𝑙). Also, from
𝑢(𝑘_𝑙)≤𝑢(𝑘_𝑙−1)+𝑒(𝑘_𝑙𝐾−1)[𝑔(exp{𝑢₁(𝑘_𝑙𝐾−1)})−𝑔(exp{𝑢₂(𝑘_𝑙−1)})] (3.31) and 𝑢(𝑘_𝑙−1)<0, we know 𝑢(𝑘_𝑙)≤𝑒(𝑘_𝑙−1). (3.32) Therefore limsup_𝑙→∞𝑢(𝑘_𝑙)≤(𝑒(𝑘_𝑙−1))^𝑀. (3.33) Thus we have limsup_𝑘→∞𝑁₂(𝑘)≤𝐾₂, (3.34) where 𝐾₂=𝐺₃exp{𝑒^𝑀}. (3.35)

Proposition 3.4. Under conditions (H1) and (H2), there exists a positive constant 𝑘₂ such that liminf_𝑘→+∞𝑁₂(𝑘)≥𝑘₂.

Proof. Given any positive solution (𝑁₁(𝑘),𝑁₂(𝑘)) of (1.11), from the second equation of (1.11), we have 𝑁₂(𝑘+1)≥𝑁₂(𝑘)exp{−𝑑(𝑘)}, (3.36) then ^𝑘−1_{𝑖=𝑘−[𝜏2]}[ln𝑁₂(𝑖+1)−ln𝑁₂(𝑖)]≥−^𝑘−1_{𝑖=𝑘−[𝜏2]}𝑑(𝑖), (3.37) hence 𝑁₂(𝑘−[𝜏₂])≤𝑁₂(𝑘)exp{^𝑘−1_{𝑖=𝑘−[𝜏2]}𝑑(𝑖)}. (3.38) Therefore from the second equation of (1.11), we have 𝑁₂(𝑘+1)≥𝑁₂𝑘(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(¹𝑁₂(𝑘)(exp{∑^𝑘−1_{𝑖=𝑘−[𝜏2]}𝑑(𝑖)})^𝑀)}. (3.39) Consider the auxiliary equation 𝑘𝑧(𝑘+1)=𝑧(𝑘)exp{−𝑑(𝑘)+𝑒(𝑘)𝑔(¹𝑧(𝑘)(exp{∑^𝑘−1_{𝑖=𝑘−[𝜏2]}𝑑(𝑖)})^𝑀)}, (3.40) by Lemma 2.3 and (H2), (3.40) has at least one positive 𝜔-periodic solution, denoted it as 𝑧^∗₂(𝑘), then 𝑧^∗₂𝐺(𝑘)≥𝑔⁻¹(𝑑/𝑒)exp{−(|𝑑|+𝑑)𝜔}∶=𝐺₄. (3.41) Where 𝑘𝐺=¹(exp{∑^𝑘−1_{𝑖=𝑘−[𝜏2]}𝑑(𝑖)})^𝑀. (3.42) If we set 𝑁₂(𝑘)=exp{𝑢₁(𝑘)},𝑧^∗₂(𝑘)=exp{𝑢₂(𝑘)}, (3.43) then 𝑢₁(𝑘+1)−𝑢₁𝐺(𝑘)≥−𝑑(𝑘)+𝑒(𝑘)𝑔(exp{𝑢₁𝑢(𝑘)}),₂(𝑘+1)−𝑢₂𝐺(𝑘)=−𝑑(𝑘)+𝑒(𝑘)𝑔(exp{𝑢₂(𝑘)}). (3.44) And let 𝑢(𝑘)=𝑢₁(𝑘)−𝑢₂(𝑘), we have 𝐺𝑢(𝑘+1)−𝑢(𝑘)≥𝑒(𝑘)[𝑔(exp{𝑢₁𝐺(𝑘)})−𝑔(exp{𝑢₂(𝑘)})]. (3.45) If 𝑢(𝑘) does not oscillate about zero, then by a similar analysis as that in Proposition 3.1, we have liminf_𝑘→∞𝑁₂(𝑘)≥(𝑧^∗₂(𝑘))^𝐿≥𝐺₄. (3.46) Otherwise, if 𝑢(𝑘) oscillates about zero, by (3.45), we know that 𝑢(𝑘)<0implies 𝑢(𝑘+1)≥𝑢(𝑘). Thus, if we denote {𝑢(𝑘_𝑙)} as a subsequence of {𝑢(𝑘)} where 𝑢(𝑘_𝑙) is the first element of the 𝑙th negative semicycle of {𝑢(𝑘)}, then liminf_𝑘→∞𝑢(𝑘)=liminf_𝑙→∞𝑢(𝑘_𝑙). On the other hand, from 𝑢(𝑘_𝑙)≥𝑢(𝑘_𝑙−1)+𝑒(𝑘_𝑙𝐺−1)[𝑔(exp{𝑢₁(𝑘_𝑙𝐺−1)})−𝑔(exp{𝑢₂(𝑘_𝑙−1)})] (3.47) and 𝑢(𝑘_𝑙−1)>0 we can obtain 𝑢(𝑘_𝑙)≥−𝑒(𝑘_𝑙𝐺−1)𝑔(exp{𝑢₂(𝑘_𝑙)−1)}≥−𝑒(𝑘_𝑙−1). (3.48) Therefore liminf_𝑙→∞𝑢(𝑘_𝑙)≥(−𝑒(𝑘_𝑙−1))^𝐿. (3.49) By the medium of (3.43), we have liminf_𝑘→∞𝑁₁(𝑘)≥(𝑧^∗₂(𝑘))^𝐿exp{(−𝑒(𝑘_𝑙−1))^𝐿}. (3.50) Hence liminf_𝑘→∞𝑁₁(𝑘)≥𝑘₂, where 𝑘₂=𝐺₄exp{−𝑒^𝑀}. (3.51)

Proof of Theorem 1.6. From the Propositions 3.1–3.4, we can easily know that system (1.11) is permanent. The proof is complete.

By a similar process as above, we can obtain the following result.

Corollary 3.5. Assume that 𝑏,𝑑∈𝐶 and 𝑎,𝑐,𝑒∈𝐶₊. If(C1)(𝑏−𝑚𝑐)^𝐿>0, (3.52)(C2)(𝑒−𝑑)^𝐿>0, (3.53) then system (1.11) is permanent.

Obviously, (B2) includes (C2), (B1), and (B3) include (C1). Thus, Corollary 3.5 generalizes and improves Theorem 1.4.

Acknowledgments

This work was supported by NNSF of China (10771032), the Natural Science Foundation of Ludong University (24070301, 24070302, 24200301), Program for Innovative Research Team in Ludong University, and China Postdoctoral Science Foundation funded project.