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

Authors: Xuepeng Li; Wensheng Yang
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## 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.


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## Body

## 1. Introduction

Zhang and Wang [1] considered the following
nonautonomous discrete predator-prey systems with the Beddington-DeAngelis
functional response (1.1)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)}. 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(1.2)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), 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)}: (1.3)au=supk∈Na(k),al=infk∈Na(k), and assume that 0<η1l≤η1u<1,0<η2l≤η2u<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 (1.4)m≤limk→+∞infx(k)≤limk→+∞supx(k)≤M,m≤limk→+∞infy(k)≤limk→+∞supy(k)≤M,m≤limk→+∞infui(k)≤limk→+∞supui(k)≤M,i=1,2.For biological reasons, we only consider solution(x(k),y(k),u1(k),u2(k)) with x(0)>0;y(0)>0;ui(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(2.1)y(n+1)=Ay(n)+B,n=1,2,…, 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: (2.2)y(n)=An(y(0)−y*)+y*, where y*=B/(1−A). Thus, for any solution {y(n)} of system
(2.1),
(2.3)limn→+∞y(n)=y*.Following Comparison Theorem of difference equation is Theorem 2.1 of [4, page 241].
Lemma 2.2.
Letk∈Nk0+={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: (2.4)y(k+1)≤g(k,y(k)),u(k+1)≥g(k,u(k)). If y(k0)≤u(k0), then y(k)≤u(k) for all k≥k0.Now let us consider the following single species
discrete model:(2.5)N(k+1)=N(k)exp{a(k)−b(k)N(k)}, where {a(k)} and {b(k)} are strictly
positive sequences of real numbers defined for k∈N={0,1,2,…} and 0<al≤au,0<bl≤bu. 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 (2.6)m≤limk→+∞infN(k)≤limk→+∞supN(k)≤M, where (2.7)M=1blexp{au−1},m=albuexp{al−buM}.Lemma 2.4 (see [6]).
Letx(n) and b(n) be nonnegative
sequences defined on N and c≥0 is a constant.
If (2.8)x(n)≤c+∑s=0n−1b(s)x(s),forn∈N. Then (2.9)x(n)≤c∏s=0n−1[1+b(s)],forn∈N.Proposition 2.5.
Assume that(2.10)−dl+fuβl>0 holds, then (2.11)limk→+∞supx(k)≤M1,limk→+∞supy(k)≤M2,limk→+∞supui(k)≤Wi,i=1,2, where (2.12)M1=1blexp{au−1},M2=exp{2(−dl+fuβl)},Wi=qiuMiηil,i=1,2.Proof.
Lets(k)=(x(k),y(k),u1(k),u2(k)) be any positive
solution of system (1.2); from  (1.2), we
have (2.13)x(k+1)≤x(k)exp{a(k)−b(k)x(k)}. By applying Lemmas 2.2
and 2.3, it immediately follows that (2.14)limk→+∞supx(k)≤1blexp{au−1}:=M1. From the second equation of the
system (1.2), we can obtain (2.15)y(k+1)≤y(k)exp{−d(k)+f(k)β(k)}≤y(k)exp{−dl+fuβl}. Let y(k)=exp{u(k)}, then (2.16)u(k+1)≤u(k)+(−dl+fuβl)=∑s=0kb(s)u(s)+(−dl+fuβl), where (2.17)b(s)={0,0≤s≤k−1,1,s=k. 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 (2.18)u(k+1)≤2(−dl+fuβl). This is (2.19)limk→+∞supy(k)≤exp{2(−dl+fuβl)}:=M2. For any positive constant ε small enough,
it follows from (2.14) and (2.19) that there exists enough large K0 such
that (2.20)x(k)≤M1+ε,y(k)≤M2+ε,∀k≥K0. From the third and fourth
equations of the system (1.2) and (2.20), we can obtain (2.21)Δu1(k)≤−η1(k)u1(k)+q1(k)(M1+ε),Δu2(k)≤−η2(k)u2(k)+q2(k)(M2+ε). So (2.22)u1(k+1)≤(1−η1l)u1(k)+q1u(M1+ε),u2(k+1)≤(1−η2l)u2(k)+q2u(M2+ε). By applying Lemmas 2.1 and 2.2,
it immediately follows that (2.23)limk→+∞supu1(k)≤q1u(M1+ε)η1l,limk→+∞supu2(k)≤q2u(M2+ε)η2l. Setting ε→0 in the above
inequality leads to (2.24)limk→+∞supu1(k)≤q1uM1η1l,limk→+∞supu2(k)≤q2uM2η2l. 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 (2.25)al−cuγl−e1uW1>0,−du+flm1−e2uW2>0, then system (1.2) is permanent,
where (2.26)m1=al−cu/γl−e1uW1buexp{al−cuγl−e1uW1−buM1}.Proof.
By applying Proposition2.5, we see that to end the proof of Theorem
2.6, it is enough to show that under the conditions of Theorem 2.6, (2.27)limk→+∞infx(k)≥m1,limk→+∞infy(k)≥m2,limk→+∞infui(k)≥wi,i=1,2. From Proposition 2.5, for all ε>0, there exists a K1>0,K1∈N, for all k>K1, (2.28)x(k)≤M1+ε,y(k)≤M2+ε;ui(k)≤Wi+ε,i=1,2. From the first equation of
systems (1.2) and (2.28), we have (2.29)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)} 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 (2.30)limk→+∞infx(k)≥al−cu/γl−e1u(W1+ε)buexp{al−cuγl−e1u(W1+ε)−buM1}. Setting ε→0 in (2.30) leads
to (2.31)limk→+∞infx(k)≥al−cu/γl−e1uW1buexp{al−cuγl−e1uW1−buM1}:=m1.
Then, for any positive constantε small enough,
from (2.31) we know that there exists an enough large K2>K1 such
that (2.32)x(k)≥m1−ε,∀k≥k2. From the second equation of
systems (1.2), (2.28), and (2.32), we have (2.33)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)} 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 (2.34)limk→+∞infy(k)≥βl[αl+βl(m1−ε)][−du+fl(m1−ε)−e2u(W2+ε)]fuγu×exp{−du+fl(m1−ε)−e2u(W2+ε)−fuγuβl[αl+βl(m1−ε)]M2}. Setting ε→0 in (2.34) leads
to (2.35)limk→+∞infy(k)≥βl(αl+βlm1)(−du+flm1−e2uW2)fuγu×exp{−du+flm1−e2uW2−fuγuβl(αl+βlm1)M2}:=m2. 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 (2.36)x(k)≥m1−ε,y(k)≥m2−ε,∀k≥K3. From the third and fourth
equations of the system, (1.2) and (2.36), we can obtain that (2.37)Δu1(k)≥−η1(k)u1(k)+q1(k)(m1−ε),Δu2(k)≥−η2(k)u2(k)+q2(k)(m2−ε). So (2.38)u1(k+1)≥(1−η1u)u1(k)+q1l(m1−ε),u2(k+1)≥(1−η2u)u2(k)+q2l(m2−ε). By applying Lemmas 2.1 and 2.2,
it immediately follows that (2.39)limk→+∞infu1(k)≥q1l(m1−ε)η1u,limk→+∞infu2(k)≥q2l(m2−ε)η2u. Setting ε→0 in the above
inequality leads to (2.40)limk→+∞infu1(k)≥q1lm1η1u:=w1,limk→+∞infu2(k)≥q2lm2η2u:=w2. This completes the proof of
Theorem 2.6.To check the conditions of Theorem2.6, we give an
example. We consider the following discrete predator-prey systems with
Beddington-DeAngelis functional response and feedback controls (2.41)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). 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.



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