Permanence of a Discrete Periodic Volterra Model with Mutual Interference

Abstract

This paper discusses a discrete periodic Volterra model with mutual interference and Holling II type functional response. Firstly, sufficient conditions are obtained for the permanence of the system. After that, we give an example to show the feasibility of our main results.

1. Introduction

In 1971, Hassell introduced the concept of mutual interference between the predators and preys. Hassell [1] established a Volterra model with mutual interference as follows:𝑥̇(𝑡)=𝑥𝑔(𝑥)−𝜑(𝑥)𝑦^𝑚,𝑦(̇𝑡)=𝑦(−𝑑+𝑘𝜑(𝑥)𝑦^𝑚−1−𝑞(𝑦)), (1.1) where 𝑚 denote mutual interference constant and 0<𝑚≤1.

Motivated by the works of Hassell [1], Wang and Zhu [2] considered the following Volterra model with mutual interference and Holling II type functional response: 𝑥̇(𝑡)=𝑥(𝑡)(𝑟₁(𝑡)−𝑏₁𝑐(𝑡)𝑥(𝑡))−¹(𝑡)𝑥(𝑡)𝑦𝑘+𝑥(𝑡)^𝑚̇(𝑡),𝑦(𝑡)=𝑦(𝑡)(−𝑟₂(𝑡)−𝑏₂𝑐(𝑡)𝑦(𝑡))+²(𝑡)𝑥(𝑡)𝑦𝑘+𝑥(𝑡)^𝑚(𝑡). (1.2)Sufficient conditions which guarantee the existence, uniqueness, and global attractivity of positive periodic solution are obtained by employing Mawhin's continuation theorem and constructing suitable Lyapunov function.

On the other hand, it has been found that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations (see [3–15]). However, to the best of the author's knowledge, until today, there are still no scholars propose and study a discrete-time analogue of system (1.2). Therefore, the main purpose of this paper is to study the following discrete periodic Volterra model with mutual interference and Holling II type functional response:𝑥(𝑛+1)=𝑥(𝑛)exp{𝑟₁(𝑛)−𝑏₁𝑐(𝑛)𝑥(𝑛)−¹(𝑛)𝑦𝑘+𝑥(𝑛)^𝑚(𝑛)},𝑦(𝑛+1)=𝑦(𝑛)exp{−𝑟₂(𝑛)−𝑏₂𝑐(𝑛)𝑦(𝑛)+²(𝑛)𝑥(𝑛)𝑦𝑘+𝑥(𝑛)^𝑚−1(𝑛)}, (1.3) where 𝑥(𝑛) is the density of prey species at 𝑛th generation and 𝑦(𝑛) is the density of predator species at 𝑛th generation. Also, 𝑟₁(𝑛),𝑏₁(𝑛) denote the intrinsic growth rate and density-dependent coefficient of the prey, respectively, 𝑟₂(𝑛),𝑏₂(𝑛) denote the death rate and density-dependent coefficient of the predator, respectively, 𝑐₁(𝑛) denote the capturing rate of the predator and 𝑐₂(𝑛) represent the transformation from preys to predators. Further, 𝑚 is mutual interference constant and 𝑘 is a positive constant. In this paper, we always assume that {𝑟_𝑖(𝑛)}, {𝑏_𝑖(𝑛)}, {𝑐_𝑖(𝑛)}, 𝑖=1,2, are positive 𝑇-periodic sequences and 0<𝑚<1. Here, for convenience, we denote 𝑓=(1/𝑇)∑^𝑇−1_𝑛=0𝑓(𝑛), 𝑓^𝑀=sup_{𝑛∈𝐼𝑇}{𝑓(𝑛)}, and 𝑓^𝐿=inf_{𝑛∈𝐼𝑇}{𝑓(𝑛)}, where 𝐼_𝑇={0,1,2,…,𝑇−1}.

This paper is organized as follows. In Section 2, we will introduce a definition and establish several useful lemmas. The permanence of system (1.3) is then studied in Section 3. In Section 4, we give an example to show the feasibility of our main results.

From the view point of biology, we only need to focus our discussion on the positive solution of system (1.3). So it is assumed that the initial conditions of (1.3) are of the form𝑥(0)>0,𝑦(0)>0. (1.4)One can easily show that the solution of (1.3) with the initial condition (1.4) are defined and remain positive for all 𝑛∈𝑁 where 𝑁={0,1,2,…}.

2. Preliminaries

In this section, we will introduce the definition of permanence and several useful lemmas.

Definition 2.1. System (1.3) is said to be permanent if there exist positive constants 𝑥^∗,𝑦^∗,𝑥_∗,𝑦_∗, which are independent of the solution of the system, such that for any positive solution (𝑥(𝑛),𝑦(𝑛)) of system (1.3) satisfies𝑥_∗≤liminf_𝑛→∞𝑥(𝑛)≤limsup_𝑛→∞𝑥(𝑛)≤𝑥^∗,𝑦_∗≤liminf_𝑛→∞𝑦(𝑛)≤limsup_𝑛→∞𝑦(𝑛)≤𝑦^∗. (2.1)

Lemma 2.2. Assume that 𝑥(𝑛) satisfies𝑥(𝑛+1)≤𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)}∀𝑛≥𝑛₀, (2.2)where {𝑎(𝑛)} and {𝑏(𝑛)} are positive sequences, 𝑥(𝑛₀)>0 and 𝑛₀∈𝑁. Then, one haslimsup_𝑛→∞𝑥(𝑛)≤𝐵, (2.3)where 𝐵=exp(𝑎^𝑀−1)/𝑏^𝐿.

Lemma 2.3. Assume that 𝑥(𝑛) satisfies𝑥(𝑛+1)≥𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)}∀𝑛≥𝑛₀, (2.4)where {𝑎(𝑛)} and {𝑏(𝑛)} are positive sequences, 𝑥(𝑛₀)>0 and 𝑛₀∈𝑁. Also, limsup_𝑛→∞𝑥(𝑛)≤𝐵 and 𝑏^𝑀𝐵/𝑎^𝐿>1. Then, one hasliminf_𝑛→∞𝑥(𝑛)≥𝐴, (2.5)where 𝐴=(𝑎^𝐿/𝑏^𝑀)exp(𝑎^𝐿−𝑏^𝑀𝐵).

Proof. The proofs of Lemmas 2.2 and 2.3 are very similar to that of [8, Lemmas 1 and 2], respectively. So, we omit the detail here.

The following Lemma 2.4 is Lemma 2.2 of Fan and Li [12].

Lemma 2.4. The problem𝑥(𝑛+1)=𝑥(𝑛)exp{𝑎(𝑛)−𝑏(𝑛)𝑥(𝑛)}, (2.6)with 𝑥(0)=𝑥₀>0 has at least one periodic positive solution 𝑥^∗(𝑛) if both 𝑏∶𝑍→𝑅⁺ and 𝑎∶𝑍→𝑅 are 𝑇-periodic sequences with 𝑎>0. Moreover, if 𝑏(𝑛)=𝑏 is a constant and 𝑎^𝑀<1, then 𝑏𝑥(𝑛)≤1 for 𝑛 sufficiently large, where 𝑥(𝑛) is any solution of (2.6).

The following comparison theorem for the difference equation is Theorem 2.1 of L. Wang and M. Q. Wang [15, page 241].

Lemma 2.5. Suppose that 𝑓∶𝑍⁺×[0,+∞) and 𝑔∶𝑍⁺×[0,+∞) with 𝑓(𝑛,𝑥)≤𝑔(𝑛,𝑥) (𝑓(𝑛,𝑥)≥𝑔(𝑛,𝑥)) for 𝑛∈𝑍⁺ and 𝑥∈[0,+∞). Assume that 𝑔(𝑛,𝑥) is nondecreasing with respect to the argument 𝑥. If 𝑥(𝑛) and 𝑢(𝑛) are solutions of𝑥(𝑛+1)=𝑓(𝑛,𝑥(𝑛)),𝑢(𝑛+1)=𝑔(𝑛,𝑢(𝑛)), (2.7)respectively, and 𝑥(0)≤𝑢(0) (𝑥(0)≥𝑢(0)), then𝑥(𝑛)≤𝑢(𝑛),(𝑥(𝑛)≥𝑢(𝑛)) (2.8)for all 𝑛≥0.

3. Permanence

In this section, we establish a permanent result for system (1.3).

Proposition 3.1. If (𝐻₁)∶(1−𝑚)𝑟^𝑀₂<1 holds, then for any positive solution (𝑥(𝑛),𝑦(𝑛)) of system (1.3), there exist positive constants 𝑥^∗ and 𝑦^∗, which are independent of the solution of the system, such thatlimsup_𝑛→∞𝑥(𝑛)≤𝑥^∗,limsup_𝑛→∞𝑦(𝑛)≤𝑦^∗. (3.1)

Proof. Let (𝑥(𝑛),𝑦(𝑛)) be any positive solution of system (1.3), from the first equation of (1.3), it follows that𝑥(𝑛+1)≤𝑥(𝑛)exp{𝑟₁(𝑛)−𝑏₁(𝑛)𝑥(𝑛)}. (3.2)By applying Lemma 2.2, we obtainlimsup_𝑛→∞𝑥(𝑛)≤𝑥^∗, (3.3)where𝑥^∗=1𝑏^𝐿₁exp(𝑟^𝑀₁−1). (3.4)
Denote 𝑃(𝑛)=(1/𝑦(𝑛))^1−𝑚. Then, from the second equation of (1.3), it follows that𝑃(𝑛+1)=𝑃(𝑛)exp{(1−𝑚)𝑟₂(𝑛)+(1−𝑚)𝑏₂(𝑛)_1−𝑚−𝑃(𝑛)(1−𝑚)𝑐₂(𝑛)𝑥(𝑛)𝑘+𝑥(𝑛)𝑃(𝑛)}, (3.5)which leads to𝑃(𝑛+1)≥𝑃(𝑛)exp{(1−𝑚)𝑟₂(𝑛)−(1−𝑚)𝑐^𝑀₂𝑃(𝑛)}. (3.6)Consider the following auxiliary equation:𝑍(𝑛+1)=𝑍(𝑛)exp{(1−𝑚)𝑟₂(𝑛)−(1−𝑚)𝑐^𝑀₂𝑍(𝑛)}. (3.7)By Lemma 2.4, (3.7) has at least one positive 𝑇-periodic solution and we denote one of them as 𝑍^∗(𝑛). Now (𝐻₁) and Lemma 2.4 imply (1−𝑚)𝑐^𝑀₂𝑍(𝑛)≤1 for 𝑛 sufficiently large, where 𝑍(𝑛) is any solution of (3.7). Consider the following function:𝑔(𝑛,𝑍)=𝑍exp{(1−𝑚)𝑟₂(𝑛)−(1−𝑚)𝑐^𝑀₂𝑍}. (3.8)It is not difficult to see that 𝑔(𝑛,𝑍) is nondecreasing with respect to the argument 𝑍. Then, applying Lemma 2.5 to (3.6) and (3.7), we easily obtain that 𝑃(𝑛)≥𝑍^∗(𝑛). So liminf_𝑛→∞𝑃(𝑛)≥(𝑍^∗(𝑛))^𝐿, which together with that transformation 𝑃(𝑛)=(1/𝑦(𝑛))^1−𝑚, produceslimsup_𝑛→∞1𝑦(𝑛)≤_1−𝑚(𝑍^∗(𝑛))^𝐿≜𝑦^∗. (3.9)Thus, we complete the proof of Proposition 3.1.

Proposition 3.2. Assume that(𝐻₂)∶(𝑟₁𝑐(𝑛)−¹(𝑛)𝑘(𝑦^∗)^𝑚)^𝐿>0 (3.10)holds, then for any positive solution (𝑥(𝑛),𝑦(𝑛)) of system (1.3), there exist positive constants 𝑥_∗ and 𝑦_∗, which are independent of the solution of the system, such thatliminf_𝑛→∞𝑥(𝑛)≥𝑥_∗,liminf_𝑛→∞𝑦(𝑛)≥𝑦_∗, (3.11)where 𝑦^∗ can be seen in Proposition 3.1.

Proof. Let (𝑥(𝑛),𝑦(𝑛)) be any positive solution of system (1.3). From (𝐻₂), there exists a small enough positive constant 𝜀 such that(𝑟₁𝑐(𝑛)−¹(𝑛)𝑘(𝑦^∗+𝜀)^𝑚)^𝐿>0. (3.12)Also, according to Proposition 3.1, for above 𝜀, there exists 𝑁₁>0 such that for 𝑛≥𝑁₁,𝑦(𝑛)≤𝑦^∗+𝜀. (3.13)Then, from the first equation of (1.3), for 𝑛≥𝑁₁, we have𝑥(𝑛+1)≥𝑥(𝑛)exp{𝑟₁𝑐(𝑛)−¹(𝑛)𝑘(𝑦^∗+𝜀)^𝑚−𝑏₁(𝑛)𝑥(𝑛)}. (3.14)Let 𝑎₁(𝑛,𝜀)=𝑟₁(𝑛)−(𝑐₁(𝑛)/𝑘)(𝑦^∗+𝜀)^𝑚, so the above inequality follows that𝑥(𝑛+1)≥𝑥(𝑛)exp{𝑎₁(𝑛,𝜀)−𝑏₁(𝑛)𝑥(𝑛)}. (3.15)Because (𝑎₁(𝑛,𝜀))^𝐿<𝑟^𝐿₁ and 𝑏^𝐿₁<𝑏^𝑀₁, we have𝑏^𝑀₁(𝑎₁(𝑛,𝜀))^𝐿𝑥^∗>𝑏^𝑀₁𝑟^𝐿₁exp(𝑟^𝑀₁−1)𝑏^𝐿₁>1. (3.16)Here, we use the fact exp(𝑟^𝑀₁−1)>𝑟^𝑀₁. From (3.12) and (3.15), by Lemma 2.3, we haveliminf_𝑛→∞𝑥(𝑛)≥(𝑎₁(𝑛,𝜀))^𝐿𝑏^𝑀₁exp{(𝑎₁(𝑛,𝜀))^𝐿−𝑏^𝑀₁𝑥^∗}. (3.17)Setting 𝜀→0 in the above inequality leads toliminf_𝑛→∞𝑎𝑥(𝑛)≥^𝐿1𝑏^𝑀₁exp{𝑎^𝐿₁−𝑏^𝑀₁𝑥^∗}≜𝑥_∗, (3.18)where𝑎₁(𝑛)=𝑟₁𝑐(𝑛)−¹(𝑛)𝑘(𝑦^∗)^𝑚. (3.19)For above 𝜀, there exists 𝑁₂>𝑁₁ such that for 𝑛≥𝑁₂,𝑥(𝑛)≥𝑥_∗−𝜀. So from (3.5), we obtain that𝑃(𝑛+1)≤𝑃(𝑛)exp{(1−𝑚)(𝑟₂(𝑛)+𝑏₂(𝑛)(𝑦^∗+𝜀))−(1−𝑚)𝑐₂(𝑛)(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀𝑃(𝑛)}. (3.20)Consider the following auxiliary equation:𝑊(𝑛+1)=𝑊(𝑛)exp{(1−𝑚)(𝑟₂(𝑛)+𝑏₂(𝑛)(𝑦^∗+𝜀))−(1−𝑚)𝑐₂(𝑛)(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀𝑊(𝑛)}. (3.21)By Lemma 2.4, (3.21) has at least one positive 𝑇-periodic solution and we denote one of them as 𝑊^∗(𝑛).
Let𝑅(𝑛)=ln(𝑃(𝑛)),𝑌(𝑛)=ln(𝑊^∗(𝑛)). (3.22)
Then,𝑅(𝑛+1)−𝑅(𝑛)≤(1−𝑚)(𝑟₂(𝑛)+𝑏₂(𝑛)(𝑦^∗+𝜀))−(1−𝑚)𝑐₂(𝑛)(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀exp{𝑅(𝑛)},𝑌(𝑛+1)−𝑌(𝑛)=(1−𝑚)(𝑟₂(𝑛)+𝑏₂(𝑛)(𝑦^∗+𝜀))−(1−𝑚)𝑐₂(𝑛)(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀exp{𝑌(𝑛)}. (3.23)
Set𝑈(𝑛)=𝑅(𝑛)−𝑌(𝑛). (3.24)Then,𝑈(𝑛+1)−𝑈(𝑛)≤−(1−𝑚)𝑐₂(𝑛)(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀exp{𝑌(𝑛)}[exp{𝑈(𝑛)}−1]. (3.25)In the following we distinguish three cases.
Case 1. {𝑈(𝑛)} is eventually positive. Then, from (3.25), we see that 𝑈(𝑛+1)<𝑈(𝑛) for any sufficiently large 𝑛. Hence, lim_𝑛→∞𝑈(𝑛)=0, which implies thatlimsup_𝑛→∞𝑃(𝑛)≤(𝑊^∗(𝑛))^𝑀. (3.26)
Case 2. {𝑈(𝑛)} is eventually negative. Then, from (3.24), we can also obtain (3.26).
Case 3. {𝑈(𝑛)} oscillates about zero. In this case, we let {𝑈(𝑛_𝑠𝑡)}(𝑠,𝑡∈𝑁) be the positive semicycle of {𝑈(𝑛)}, where 𝑈(𝑛_𝑠1) denotes the first element of the 𝑠th positive semicycle of {𝑈(𝑛)}. From (3.25), we know that 𝑈(𝑛+1)<𝑈(𝑛) if 𝑈(𝑛)>0. Hence, limsup_𝑛→∞𝑈(𝑛)=limsup_𝑠→∞𝑈(𝑛_𝑠1). From (3.25), and 𝑈(𝑛_𝑠1−1)<0, we can obtain𝑈(𝑛_𝑠1)≤(1−𝑚)𝑐₂(𝑛_𝑠1)(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀exp{𝑌(𝑛_𝑠1)}[1−exp{𝑈(𝑛_𝑠1≤)}],(1−𝑚)𝑐^𝑀₂(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀(𝑊^∗(𝑛))^𝑀. (3.27)
From (3.22) and (3.24), we easily obtainlimsup_𝑛→∞𝑃(𝑛)≤(𝑊^∗(𝑛))^𝑀exp{(1−𝑚)𝑐^𝑀₂(𝑥_∗−𝜀)𝑘+𝑥_∗−𝜀(𝑊^∗(𝑛))^𝑀}. (3.28)Setting 𝜀→0 in the above inequality leads tolimsup_𝑛→∞𝑃(𝑛)≤(𝑊^∗(𝑛))^𝑀exp{(1−𝑚)𝑐^𝑀₂𝑥_∗𝑘+𝑥_∗(𝑊^∗(𝑛))^𝑀}≜𝑃^∗, (3.29)which together with that transformation 𝑃(𝑛)=(1/𝑦(𝑛))^1−𝑚, we haveliminf_𝑛→∞1𝑦(𝑛)≥_1−𝑚√𝑃^∗≜𝑦_∗. (3.30)Thus, we complete the proof of Proposition 3.2.

Theorem 3.3. Assume that (𝐻₁) and (𝐻₂) hold, then system (1.3) is permanent.

It should be noticed that, from the proofs of Propositions 3.1 and 3.2, one knows that under the conditions of Theorem 3.3, the set Ω={(𝑥,𝑦)∣𝑥_∗≤𝑥≤𝑥^∗,𝑦_∗≤𝑦≤𝑦^∗} is an invariant set of system (1.3).

4. Example

In this section, we give an example to show the feasibility of our main result.

Example 4.1. Consider the following system 𝑥(𝑛+1)=𝑥(𝑛)exp{0.7+0.1sin(𝑛)−0.7𝑥(𝑛)−0.4𝑦^0.6(𝑛)2.4+𝑥(𝑛)},𝑦(𝑛+1)=𝑦(𝑛)exp{−(0.8+0.1cos(𝑛))−(1.1+0.1sin(𝑛))𝑥(𝑛)−0.6𝑦^−0.4(𝑛)2.4+𝑥(𝑛)}, (4.1)where 𝑟₁(𝑛)=0.7+0.1sin(𝑛), 𝑏₁(𝑛)=0.7, 𝑐₁(𝑛)=0.4, 𝑟₂(𝑛)=0.8+0.1cos(𝑛), 𝑏₂(𝑛)=1.1+0.1sin(𝑛), 𝑐₂(𝑛)=0.6,𝑚=0.6,𝑘=2.4.

By simple computation, we have 𝑦^∗≈1.1405. Thus, one could easily see that(𝑟₁𝑐(𝑛)−¹(𝑛)𝑘(𝑦^∗)^𝑚)^𝐿≈0.4197>0,(1−𝑚)(𝑟₂(𝑛))^𝑀=0.36<1. (4.2)Clearly, conditions (𝐻₁) and (𝐻₂) are satisfied, then system (1.3) is permanent.

Figure 1 shows the dynamics behavior of system (1.3).

(a) x

(b) y

Figure 1 

Dynamics behavior of system (1.3) with initial condition (𝑥(0),𝑦(0))=(0.6,0.03).

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions which greatly improve the presentation of the paper. This work was supported by the Program for New Century Excellent Talents in Fujian Province University (NCETFJ) and the Foundation of Fujian Education Bureau (JB08028).