Permanence of a Discrete Predator-Prey Systems with Beddington-DeAngelis Functional Response and Feedback Controls

Abstract

We propose a discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system.

1. Introduction

Zhang and Wang [1] considered the following nonautonomous discrete predator-prey systems with the Beddington-DeAngelis functional response 𝑥(𝑘+1)=𝑥(𝑘)𝑒𝑥𝑝{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑐(𝑘)𝑦(𝑘)𝛼(𝑘)+𝛽(𝑘)𝑥(𝑘)+𝛾(𝑘)𝑦(𝑘)},𝑦(𝑘+1)=𝑦(𝑘)𝑒𝑥𝑝{−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝛼(𝑘)+𝛽(𝑘)𝑥(𝑘)+𝛾(𝑘)𝑦(𝑘)}. (1.1) By using a continuation theorem, sufficient criteria are established for the existence of positive periodic solutions of the system (1.1).

As we know, permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. Biologically, when a system of interacting species is persistent in a suitable sense, it means that all the species survive in the long term. It is reasonable to ask for conditions under which the system is permanent. However, Zhang and Wang [1] did not investigate this property of the system (1.1).

As we know, ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. Already, Gopalsamy and Weng [2] have studied the Logistic growth model with feedback control. To the author knowledge, there is few works dealt with system (1.1) with feedback control.

Therefore, one objective of this paper is to study the following discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls 𝑥(𝑘+1)=𝑥(𝑘)𝑒𝑥𝑝{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑐(𝑘)𝑦(𝑘)𝛼(𝑘)+𝛽(𝑘)𝑥(𝑘)+𝛾(𝑘)𝑦(𝑘)−𝑒₁(𝑘)𝑢₁(𝑘)},𝑦(𝑘+1)=𝑦(𝑘)𝑒𝑥𝑝{−𝑑(𝑘)+𝑓(𝑘)𝑥(𝑘)𝛼(𝑘)+𝛽(𝑘)𝑥(𝑘)+𝛾(𝑘)𝑦(𝑘)−𝑒₂(𝑘)𝑢₂(𝑘)},Δ𝑢₁(𝑘)=−𝜂₁(𝑘)𝑢₁(𝑘)+𝑞₁(𝑘)𝑥(𝑘),Δ𝑢₂(𝑘)=−𝜂₂(𝑘)𝑢₂(𝑘)+𝑞₂(𝑘)𝑦(𝑘), (1.2) where 𝑎(𝑘),𝑏(𝑘),𝑐(𝑘),𝑑(𝑘),𝑓(𝑘),𝛼(𝑘),𝛽(𝑘),𝛾(𝑘),𝑒₁(𝑘),𝑒₂(𝑘), 𝜂₁(𝑘),𝜂₂(𝑘), 𝑞₁(𝑘),and𝑞₂(𝑘) are all bounded nonnegative sequence. For more biological background of system (1.2), one could refer to [1] and the references cited therein.

Throughout this paper, we use the following notations for any bounded sequence {𝑎(𝑘)}: 𝑎^𝑢=𝑠𝑢𝑝_𝑘∈𝑁𝑎(𝑘),𝑎^𝑙=𝑖𝑛𝑓_𝑘∈𝑁𝑎(𝑘), (1.3) and assume that 0<𝜂^𝑙₁≤𝜂^𝑢₁<1,0<𝜂^𝑙₂≤𝜂^𝑢₂<1.

The aim of this paper is, by further developing the analysis technique of Chen [3], to obtain a set of sufficient conditions which ensure the permanence of the system (1.2).

We say that system (1.2) is permanent if there are positive constants 𝑀 and 𝑚 such that for each positive solution (𝑥(𝑘),𝑦(𝑘),𝑢₁(𝑘),𝑢₂(𝑘)) of system (1.2) satisfies 𝑚≤𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑥(𝑘)≤𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑥(𝑘)≤𝑀,𝑚≤𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑦(𝑘)≤𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑦(𝑘)≤𝑀,𝑚≤𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑢_𝑖(𝑘)≤𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑢_𝑖(𝑘)≤𝑀,𝑖=1,2. (1.4)

For biological reasons, we only consider solution (𝑥(𝑘),𝑦(𝑘),𝑢₁(𝑘),𝑢₂(𝑘)) with 𝑥(0)>0;𝑦(0)>0;𝑢_𝑖(0)>0,𝑖=1,2. Then system (1.2) has a positive solution (𝑥(𝑘),𝑦(𝑘),𝑢₁(𝑘),𝑢₂(𝑘)) passing through (𝑥(0),𝑦(0),𝑢₁(0),𝑢₂(0)).

2. Permanence

In this section, we establish a permanence result for system (1.2).

First, let us consider the first order difference equation 𝑦(𝑛+1)=𝐴𝑦(𝑛)+𝐵,𝑛=1,2,…, (2.1) where 𝐴,𝐵 are positive constants. Following Lemma 2.1 is a direct corollary of Theorem 6.2 of L. Wang and M. Q. Wang [4, page 125]. Lemma 2.1. Assume that |𝐴|<1, for any initial value 𝑦(0), there exists a unique solution y(n) of (2.1) which can be expressed as follows: 𝑦(𝑛)=𝐴^𝑛(𝑦(0)−𝑦^∗)+𝑦^∗, (2.2) where 𝑦^∗=𝐵/(1−𝐴). Thus, for any solution {𝑦(𝑛)} of system (2.1), 𝑙𝑖𝑚_𝑛→+∞𝑦(𝑛)=𝑦^∗. (2.3)

Following Comparison Theorem of difference equation is Theorem 2.1 of [4, page 241]. Lemma 2.2. Let 𝑘∈𝑁⁺_𝑘0={𝑘₀,𝑘₀+1,…,𝑘₀+𝑙,…},𝑟≥0. For any fixed 𝑘,𝑔(𝑘,𝑟) is a nondecreasing function with respect to 𝑟, and for 𝑘≥𝑘₀, the following inequalities hold: 𝑦(𝑘+1)≤𝑔(𝑘,𝑦(𝑘)),𝑢(𝑘+1)≥𝑔(𝑘,𝑢(𝑘)). (2.4) If 𝑦(𝑘₀)≤𝑢(𝑘₀), then 𝑦(𝑘)≤𝑢(𝑘) for all 𝑘≥𝑘₀.

Now let us consider the following single species discrete model: 𝑁(𝑘+1)=𝑁(𝑘)𝑒𝑥𝑝{𝑎(𝑘)−𝑏(𝑘)𝑁(𝑘)}, (2.5) where {𝑎(𝑘)} and {𝑏(𝑘)} are strictly positive sequences of real numbers defined for 𝑘∈𝑁={0,1,2,…} and 0<𝑎^𝑙≤𝑎^𝑢,0<𝑏^𝑙≤𝑏^𝑢. Similarly to the proof of [5, Propositions 1 and 3 ], we can obtain the following. Lemma 2.3. Any solution of system (2.5) with initial condition 𝑁(0)>0 satisfies 𝑚≤𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑁(𝑘)≤𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑁(𝑘)≤𝑀, (2.6) where 1𝑀=𝑏^𝑙𝑒𝑥𝑝{𝑎^𝑢𝑎−1},𝑚=^𝑙𝑏^𝑢𝑒𝑥𝑝{𝑎^𝑙−𝑏^𝑢𝑀}. (2.7)Lemma 2.4 (see [6]). Let 𝑥(𝑛) and 𝑏(𝑛) be nonnegative sequences defined on 𝑁 and 𝑐≥0 is a constant. If 𝑥(𝑛)≤𝑐+^𝑛−1_𝑠=0𝑏(𝑠)𝑥(𝑠),for𝑛∈𝑁. (2.8) Then 𝑥(𝑛)≤𝑐^𝑛−1_𝑠=0[1+𝑏(𝑠)],for𝑛∈𝑁. (2.9)Proposition 2.5. Assume that −𝑑^𝑙+𝑓^𝑢𝛽^𝑙>0 (2.10) holds, then 𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑥(𝑘)≤𝑀₁,𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑦(𝑘)≤𝑀₂,𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑢_𝑖(𝑘)≤𝑊_𝑖,𝑖=1,2, (2.11) where 𝑀₁=1𝑏^𝑙𝑒𝑥𝑝{𝑎^𝑢𝑀−1},₂=𝑒𝑥𝑝{2(−𝑑^𝑙+𝑓^𝑢𝛽^𝑙𝑊)},_𝑖=𝑞^𝑢_𝑖𝑀_𝑖𝜂^𝑙_𝑖,𝑖=1,2. (2.12)Proof. Let 𝑠(𝑘)=(𝑥(𝑘),𝑦(𝑘),𝑢₁(𝑘),𝑢₂(𝑘)) be any positive solution of system (1.2); from (1.2), we have 𝑥(𝑘+1)≤𝑥(𝑘)𝑒𝑥𝑝{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)}. (2.13) By applying Lemmas 2.2 and 2.3, it immediately follows that 𝑙𝑖𝑚_𝑘→+∞1𝑠𝑢𝑝𝑥(𝑘)≤𝑏^𝑙𝑒𝑥𝑝{𝑎^𝑢−1}∶=𝑀₁. (2.14) From the second equation of the system (1.2), we can obtain 𝑦(𝑘+1)≤𝑦(𝑘)𝑒𝑥𝑝{−𝑑(𝑘)+𝑓(𝑘)}𝛽(𝑘)≤𝑦(𝑘)𝑒𝑥𝑝{−𝑑^𝑙+𝑓^𝑢𝛽^𝑙}. (2.15) Let 𝑦(𝑘)=𝑒𝑥𝑝{𝑢(𝑘)}, then 𝑢(𝑘+1)≤𝑢(𝑘)+(−𝑑^𝑙+𝑓^𝑢𝛽^𝑙)=^𝑘_𝑠=0𝑏(𝑠)𝑢(𝑠)+(−𝑑^𝑙+𝑓^𝑢𝛽^𝑙), (2.16) where 𝑏(𝑠)=0,0≤𝑠≤𝑘−1,1,𝑠=𝑘. (2.17) Condition (2.10) shows that Lemma 2.4 could be applied to (2.16), and so by applying Lemma 2.4, it immediately follows that 𝑢(𝑘+1)≤2(−𝑑^𝑙+𝑓^𝑢𝛽^𝑙). (2.18) This is 𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑦(𝑘)≤𝑒𝑥𝑝{2(−𝑑^𝑙+𝑓^𝑢𝛽^𝑙)}∶=𝑀₂. (2.19) For any positive constant 𝜀 small enough, it follows from (2.14) and (2.19) that there exists enough large 𝐾₀ such that 𝑥(𝑘)≤𝑀₁+𝜀,𝑦(𝑘)≤𝑀₂+𝜀,∀𝑘≥𝐾₀. (2.20) From the third and fourth equations of the system (1.2) and (2.20), we can obtain Δ𝑢₁(𝑘)≤−𝜂₁(𝑘)𝑢₁(𝑘)+𝑞₁(𝑘)(𝑀₁+𝜀),Δ𝑢₂(𝑘)≤−𝜂₂(𝑘)𝑢₂(𝑘)+𝑞₂(𝑘)(𝑀₂+𝜀). (2.21) So 𝑢₁(𝑘+1)≤(1−𝜂^𝑙₁)𝑢₁(𝑘)+𝑞^𝑢₁(𝑀₁𝑢+𝜀),₂(𝑘+1)≤(1−𝜂^𝑙₂)𝑢₂(𝑘)+𝑞^𝑢₂(𝑀₂+𝜀). (2.22) By applying Lemmas 2.1 and 2.2, it immediately follows that 𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑢₁𝑞(𝑘)≤^𝑢1(𝑀₁+𝜀)𝜂^𝑙₁,𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑢₂𝑞(𝑘)≤^𝑢2(𝑀₂+𝜀)𝜂^𝑙₂. (2.23) Setting 𝜀→0 in the above inequality leads to 𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑢₁𝑞(𝑘)≤^𝑢1𝑀₁𝜂^𝑙₁,𝑙𝑖𝑚_𝑘→+∞𝑠𝑢𝑝𝑢₂𝑞(𝑘)≤^𝑢2𝑀₂𝜂^𝑙₂. (2.24) This completes the proof of Proposition 2.5.

Now we are in the position of stating the permanence of the system (1.2). Theorem 2.6. In addition to (2.10), assume further that 𝑎^𝑙−𝑐^𝑢𝛾^𝑙−𝑒^𝑢₁𝑊₁>0,−𝑑^𝑢+𝑓^𝑙𝑚₁−𝑒^𝑢₂𝑊₂>0, (2.25) then system (1.2) is permanent, where 𝑚₁=𝑎^𝑙−𝑐^𝑢/𝛾^𝑙−𝑒^𝑢₁𝑊₁𝑏^𝑢𝑒𝑥𝑝{𝑎^𝑙−𝑐^𝑢𝛾^𝑙−𝑒^𝑢₁𝑊₁−𝑏^𝑢𝑀₁}. (2.26)Proof. By applying Proposition 2.5, we see that to end the proof of Theorem 2.6, it is enough to show that under the conditions of Theorem 2.6, 𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑥(𝑘)≥𝑚₁,𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑦(𝑘)≥𝑚₂,𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑢_𝑖(𝑘)≥𝑤_𝑖,𝑖=1,2. (2.27) From Proposition 2.5, for all 𝜀>0, there exists a 𝐾₁>0,𝐾₁∈𝑁, for all 𝑘>𝐾₁, 𝑥(𝑘)≤𝑀₁+𝜀,𝑦(𝑘)≤𝑀₂+𝜀;𝑢_𝑖(𝑘)≤𝑊_𝑖+𝜀,𝑖=1,2. (2.28) From the first equation of systems (1.2) and (2.28), we have 𝑥(𝑘+1)≥𝑥(𝑘)𝑒𝑥𝑝{𝑎(𝑘)−𝑏(𝑘)𝑥(𝑘)−𝑐(𝑘)𝛾(𝑘)−𝑒₁(𝑘)(𝑊₁+𝜀)},=𝑥(𝑘)𝑒𝑥𝑝{𝑎(𝑘)−𝑐(𝑘)𝛾(𝑘)−𝑒₁(𝑘)(𝑊₁+𝜀)−𝑏(𝑘)𝑥(𝑘)} (2.29) for all 𝑘>𝐾₁.
Condition (2.25) shows that Lemmas 2.2 and 2.3 could be applied to (2.29), and so by applying Lemmas 2.2 and 2.3 to (2.29), it immediately follows that 𝑙𝑖𝑚_𝑘→+∞𝑎𝑖𝑛𝑓𝑥(𝑘)≥^𝑙−𝑐^𝑢/𝛾^𝑙−𝑒^𝑢₁(𝑊₁+𝜀)𝑏^𝑢𝑒𝑥𝑝{𝑎^𝑙−𝑐^𝑢𝛾^𝑙−𝑒^𝑢₁(𝑊₁+𝜀)−𝑏^𝑢𝑀₁}. (2.30) Setting 𝜀→0 in (2.30) leads to 𝑙𝑖𝑚_𝑘→+∞𝑎𝑖𝑛𝑓𝑥(𝑘)≥^𝑙−𝑐^𝑢/𝛾^𝑙−𝑒^𝑢₁𝑊₁𝑏^𝑢exp{𝑎^𝑙−𝑐^𝑢𝛾^𝑙−𝑒^𝑢₁𝑊₁−𝑏^𝑢𝑀₁}∶=𝑚₁. (2.31)
Then, for any positive constant 𝜀 small enough, from (2.31) we know that there exists an enough large 𝐾₂>𝐾₁ such that 𝑥(𝑘)≥𝑚₁−𝜀,∀𝑘≥𝑘₂. (2.32) From the second equation of systems (1.2), (2.28), and (2.32), we have 𝑦(𝑘+1)≥𝑦(𝑘)𝑒𝑥𝑝{−𝑑(𝑘)+𝑓(𝑘)−𝛽(𝑘)𝑓(𝑘)(𝛽(𝑘)𝛼(𝑘)+𝛾(𝑘)𝑦(𝑘)𝛼(𝑘)+𝛽(𝑘)𝑥(𝑘)+𝛾(𝑘)𝑦(𝑘))−𝑒₂(𝑘)𝑢₂(𝑘)}≥𝑦(𝑘)𝑒𝑥𝑝{−𝑑(𝑘)+𝑓(𝑘)−𝛽(𝑘)𝑓(𝑘)(𝛽(𝑘)𝛼(𝑘)𝛼(𝑘)+𝛽(𝑘)(𝑚₁)−−𝜀)𝑓(𝑘)(𝛽(𝑘)𝛾(𝑘)𝑦(𝑘)𝛼(𝑘)+𝛽(𝑘)(𝑚₁−𝜀))−𝑒₂(𝑘)(𝑊₂+𝜀)}≥𝑦(𝑘)𝑒𝑥𝑝{−𝑑(𝑘)+𝑓(𝑘)(𝑚₁−𝜀)−𝑒₂(𝑘)(𝑊₂+𝜀)−𝑓(𝑘)𝛾(𝑘)𝛽(𝑘)[𝛼(𝑘)+𝛽(𝑘)(𝑚₁−𝜀)]𝑦(𝑘)} (2.33) for all 𝑘>𝐾₂.
Condition (2.25) shows that Lemmas 2.2 and 2.3 could be applied to (2.33), and so by applying Lemmas 2.2 and 2.3 to (2.33), it immediately follows that 𝑙𝑖𝑚_𝑘→+∞𝛽𝑖𝑛𝑓𝑦(𝑘)≥^𝑙[𝛼^𝑙+𝛽^𝑙(𝑚₁−𝜀)][−𝑑^𝑢+𝑓^𝑙(𝑚₁−𝜀)−𝑒^𝑢₂(𝑊₂+𝜀)]𝑓^𝑢𝛾^𝑢×𝑒𝑥𝑝{−𝑑^𝑢+𝑓^𝑙(𝑚₁−𝜀)−𝑒^𝑢₂(𝑊₂𝑓+𝜀)−^𝑢𝛾^𝑢𝛽^𝑙[𝛼^𝑙+𝛽^𝑙(𝑚₁𝑀−𝜀)]²}. (2.34) Setting 𝜀→0 in (2.34) leads to 𝑙𝑖𝑚_𝑘→+∞𝛽𝑖𝑛𝑓𝑦(𝑘)≥^𝑙(𝛼^𝑙+𝛽^𝑙𝑚₁)(−𝑑^𝑢+𝑓^𝑙𝑚₁−𝑒^𝑢₂𝑊₂)𝑓^𝑢𝛾^𝑢×𝑒𝑥𝑝{−𝑑^𝑢+𝑓^𝑙𝑚₁−𝑒^𝑢₂𝑊₂−𝑓^𝑢𝛾^𝑢𝛽^𝑙(𝛼^𝑙+𝛽^𝑙𝑚₁)𝑀₂}∶=𝑚₂. (2.35) Without loss of generality, we may assume that 𝜀<(1/2)min{𝑚₁,𝑚₂}. For any positive constant 𝜀 small enough, it follows from (2.31) and (2.35) that there exists enough large 𝐾₃>𝐾₂ such that 𝑥(𝑘)≥𝑚₁−𝜀,𝑦(𝑘)≥𝑚₂−𝜀,∀𝑘≥𝐾₃. (2.36) From the third and fourth equations of the system, (1.2) and (2.36), we can obtain that Δ𝑢₁(𝑘)≥−𝜂₁(𝑘)𝑢₁(𝑘)+𝑞₁(𝑘)(𝑚₁−𝜀),Δ𝑢₂(𝑘)≥−𝜂₂(𝑘)𝑢₂(𝑘)+𝑞₂(𝑘)(𝑚₂−𝜀). (2.37) So 𝑢₁(𝑘+1)≥(1−𝜂^𝑢₁)𝑢₁(𝑘)+𝑞^𝑙₁(𝑚₁𝑢−𝜀),₂(𝑘+1)≥(1−𝜂^𝑢₂)𝑢₂(𝑘)+𝑞^𝑙₂(𝑚₂−𝜀). (2.38) By applying Lemmas 2.1 and 2.2, it immediately follows that 𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑢₁𝑞(𝑘)≥^𝑙1(𝑚₁−𝜀)𝜂^𝑢₁,𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑢₂𝑞(𝑘)≥^𝑙2(𝑚₂−𝜀)𝜂^𝑢₂. (2.39) Setting 𝜀→0 in the above inequality leads to 𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑢₁𝑞(𝑘)≥^𝑙1𝑚₁𝜂^𝑢₁∶=𝑤₁,𝑙𝑖𝑚_𝑘→+∞𝑖𝑛𝑓𝑢₂𝑞(𝑘)≥^𝑙2𝑚₂𝜂^𝑢₂∶=𝑤₂. (2.40) This completes the proof of Theorem 2.6.

To check the conditions of Theorem 2.6, we give an example. We consider the following discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls 𝑥(𝑘+1)=𝑥(𝑘)exp{1−𝑥(𝑘)−0.8𝑦(𝑘)1+0.2𝑥(𝑘)+2𝑦(𝑘)−0.001𝑢₁(𝑘)},𝑦(𝑘+1)=𝑦(𝑘)exp{−0.01+0.1𝑥(𝑘)1+0.2𝑥(𝑘)+2𝑦(𝑘)−0.001𝑢₂(𝑘)},Δ𝑢₁(𝑘)=−0.8𝑢₁(𝑘)+𝑥(𝑘),Δ𝑢₂(𝑘)=−0.5𝑢₂(𝑘)+𝑦(𝑘). (2.41) One could easily obtain that the conditions of Theorem 2.6 are satisfied. Hence, by Theorem 2.6, we see that system (2.41) is permanent.

Acknowledgment

This work is supported by the Foundation of Education Department of Fujian Province (Grant no. JA05204), and the Foundation of Science and Technology Department of Fujian Province (Grant no. 2005K027).