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

Volume 2008, Article ID 149267,8pages doi:10.1155/2008/149267

Research Article

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

Xuepeng Li and Wensheng Yang

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China

Correspondence should be addressed to Wensheng Yang,[email protected] Received 20 July 2007; Accepted 21 February 2008

Recommended by Leonid Berezansky

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.

Copyrightq2008 X. Li and W. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Zhang and Wang1considered the following nonautonomous discrete predator-prey systems with the Beddington-DeAngelis functional response

xk1 xkexp

ak−bkxk− ckyk

αk βkxk γkyk

,

yk1 ykexp

−dk fkxk

αk βkxk γkyk

.

1.1

By using a continuation theorem, sufficient criteria are established for the existence of positive periodic solutions of the system1.1.

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 Wang1did not investigate this property of the system1.1.

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 prac- tical 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 2have studied the Logistic growth model with feedback control. To the author knowledge, there is few works dealt with system1.1with 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

xk1 xkexp

ak−bkxk− ckyk

αk βkxk γkyk−e₁ku1k ,

yk1 ykexp

−dk fkxk

αk βkxk γkyk−e₂ku2k

, Δu1k −η1ku1k q1kxk,

Δu2k −η2ku2k q2kyk,

1.2

whereak, bk, ck, dk, fk, αk, βk, γk, e1k, e2k,η₁k, η2k,q₁k, andq₂kare all bounded nonnegative sequence. For more biological background of system1.2, one could refer to1and the references cited therein.

Throughout this paper, we use the following notations for any bounded sequence{ak}:

a^u sup

k∈Nak, a^l inf

k∈Nak, 1.3

and assume that 0< η^l₁≤η^u₁ <1,0< η^l₂≤η₂^u<1.

The aim of this paper is, by further developing the analysis technique of Chen3, to obtain a set of sufficient conditions which ensure the permanence of the system1.2.

We say that system1.2is permanent if there are positive constantsMandmsuch that for each positive solutionxk, yk, u1k, u2kof system1.2satisfies

m≤ lim

k→∞infxk≤ lim

k→∞supxk≤M,

m≤ lim

k→∞infyk≤ lim

k→∞supyk≤M,

m≤ lim

k→∞infu_ik≤ lim

k→∞supu_ik≤M, i 1,2.

1.4

For biological reasons, we only consider solutionxk, yk, u1k, u2kwithx0 >

0;y0>0;u_i0>0, i 1,2.Then system1.2has a positive solutionxk, yk, u1k, u2k passing throughx0, y0, u10, u20.

2. Permanence

In this section, we establish a permanence result for system1.2.

First, let us consider the first order difference equation

yn1 Ayn B, n 1,2, . . . , 2.1

whereA, Bare positive constants. FollowingLemma 2.1is a direct corollary of Theorem 6.2 of L. Wang and M. Q. Wang4, page 125.

Lemma 2.1. Assume that|A| < 1, for any initial valuey0, there exists a unique solution y(n) of 2.1which can be expressed as follows:

yn Aⁿ

y0−y^∗

y^∗, 2.2

wherey^∗ B/1−A.Thus, for any solution{yn}of system2.1,

n→∞limyn y^∗. 2.3

Following Comparison Theorem of difference equation is Theorem 2.1 of4, page 241.

Lemma 2.2. Letk∈N_k

0 {k0, k₀1, . . . , k0l, . . .}, r≥0. For any fixedk, gk, ris a nondecreasing function with respect tor, and fork≥k0, the following inequalities hold:

yk1≤gk, yk, uk1≥gk, uk. 2.4

Ifyk0≤uk0, thenyk≤ukfor allk≥k0.

Now let us consider the following single species discrete model:

Nk1 Nkexp{ak−bkNk}, 2.5

where{ak}and{bk}are strictly positive sequences of real numbers defined fork ∈N {0,1,2, . . .}and 0< a^l ≤a^u,0< b^l≤b^u. Similarly to the proof of5, Propositions 1 and 3, we can obtain the following.

Lemma 2.3. Any solution of system2.5with initial conditionN0>0 satisfies

m≤ lim

k→∞infNk≤ lim

k→∞supNk≤M, 2.6

where

M 1

b^lexp a^u−1

, m a^l b^uexp

a^l−b^uM

. 2.7

Lemma 2.4 see 6. Let xn andbn be nonnegative sequences defined onN and c ≥ 0 is a constant. If

xn≤cⁿ⁻¹

s 0

bsxs, forn∈N. 2.8

Then

xn≤c

n−1 s 0

1bs, forn∈N. 2.9 Proposition 2.5. Assume that

−d^lf^u

β^l >0 2.10

holds, then

k→∞limsupxk≤M₁,

k→∞limsupyk≤M₂,

k→∞limsupuik≤Wi, i 1,2,

2.11

where

M₁ 1 b^lexp

a^u−1 , M2 exp

2

−d^lf^u β^l

,

Wi

q^u_iMi

η^l_i , i 1,2.

2.12

Proof. Letsk xk, yk, u1k, u2kbe any positive solution of system1.2; from1.2, we have

xk1≤xkexp{ak−bkxk}. 2.13

By applying Lemmas2.2and2.3, it immediately follows that

k→∞limsupxk≤ 1 b^lexp

a^u−1

: M1. 2.14

From the second equation of the system1.2, we can obtain yk1≤ykexp

−dk fk βk

≤ykexp

−d^lf^u β^l

.

2.15

Letyk exp{uk}, then

uk1≤uk

−d^lf^u β^l

k

s 0bsus

−d^lf^u β^l

,

2.16

where

bs

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

uk1≤2

−d^lf^u β^l

. 2.18

This is

k→∞limsupyk≤exp

2

−d^lf^u β^l

: M₂. 2.19

For any positive constantεsmall enough, it follows from2.14and2.19 that there exists enough largeK₀such that

xk≤M₁ε, yk≤M₂ε, ∀k≥K₀. 2.20 From the third and fourth equations of the system1.2and2.20, we can obtain

Δu1k≤ −η1ku1k q1k

M1ε , Δu2k≤ −η2ku2k q₂k

M₂ε

. 2.21

So

u1k1≤ 1−η^l₁

u1k q^u₁

M1ε , u₂k1≤

1−η^l₂

u₂k q^u₂

M₂ε

. 2.22

By applying Lemmas2.1and2.2, it immediately follows that

k→∞limsupu₁k≤q^u₁

M1ε η^l₁ ,

k→∞limsupu₂k≤q^u₂

M2ε η^l₂ .

2.23

Settingε→0 in the above inequality leads to

k→∞limsupu₁k≤ q^u₁M1

η^l₁ ,

k→∞limsupu2k≤ q^u₂M₂ η^l₂ .

2.24

This completes the proof ofProposition 2.5.

Now we are in the position of stating the permanence of the system1.2.

Theorem 2.6. In addition to2.10, assume further that a^l−c^u

γ^l −e^u₁W₁>0,

−d^uf^lm₁−e^u₂W₂>0,

2.25

then system1.2is permanent, where m1

a^l−c^u/γ^l−e₁^uW₁

b^u exp

a^l−c^u

γ^l −e^u₁W1−b^uM1

. 2.26

Proof. By applyingProposition 2.5, we see that to end the proof ofTheorem 2.6, it is enough to show that under the conditions ofTheorem 2.6,

k→∞liminfxk≥m1,

k→∞liminfyk≥m₂,

k→∞liminfuik≥wi, i 1,2.

2.27

FromProposition 2.5, for allε >0, there exists aK₁>0, K₁∈N, for allk > K₁,

xk≤M1ε, yk≤M2ε; uik≤Wiε, i 1,2. 2.28 From the first equation of systems1.2and2.28, we have

xk1≥xkexp

ak−bkxk−ck

γk−e₁k

W₁ε , xkexp

ak−ck

γk−e1k W1ε

−bkxk

2.29

for allk > K1.

Condition2.25shows that Lemmas2.2and2.3 could be applied to2.29, and so by applying Lemmas2.2and2.3to2.29, it immediately follows that

k→∞liminfxk≥ a^l−c^u/γ^l−e₁^u W₁ε

b^u exp

a^l−c^u

γ^l −e^u₁ W1ε

−b^uM1

. 2.30

Settingε→0 in2.30leads to

k→∞liminfxk≥a^l−c^u/γ^l−e^u₁W1

b^u exp

a^l−c^u

γ^l −e₁^uW₁−b^uM₁

: m₁. 2.31 Then, for any positive constantεsmall enough, from2.31we know that there exists an enough largeK2> K1such that

xk≥m₁−ε, ∀k≥k₂. 2.32

From the second equation of systems1.2,2.28, and2.32, we have yk1≥ykexp

−dk fk βk −fk

βk

αk γkyk αk βkxk γkyk

−e2ku2k

≥ykexp

−dk fk βk −fk

βk

αk αk βk

m1−ε

−fk βk

γkyk αk βk

m1−ε

−e2k

W2ε

≥ykexp

−dk fk m1−ε

−e2k W2ε

− fkγk

βk

αk βk

m1−εyk

2.33 for allk > K2.

Condition2.25shows that Lemmas2.2and2.3 could be applied to2.33, and so by applying Lemmas2.2and2.3to2.33, it immediately follows that

k→∞liminfyk≥β^l α^lβ^l

m1−ε

−d^uf^l m1−ε

−e^u₂

W2ε f^uγ^u

×exp

−d^uf^l m1−ε

−e₂^u W2ε

− f^uγ^u β^l

α^lβ^l

m1−εM2

.

2.34

Settingε→0 in2.34leads to

k→∞liminfyk≥ β^l

α^lβ^lm1

−d^uf^lm1−e₂^uW2 f^uγ^u

×exp

−d^uf^lm₁−e^u₂W₂− f^uγ^u β^l

α^lβ^lm1

M₂

: m₂.

2.35

Without loss of generality, we may assume thatε < 1/2min{m1, m₂}. For any positive con- stantεsmall enough, it follows from2.31and2.35that there exists enough largeK₃ > K₂ such that

xk≥m1−ε, yk≥m2−ε, ∀k≥K3. 2.36 From the third and fourth equations of the system,1.2and2.36, we can obtain that

Δu1k≥ −η1ku1k q1k m1−ε

, Δu2k≥ −η2ku2k q2k

m2−ε

. 2.37

So

u₁k1≥ 1−η^u₁

u₁k q^l₁ m₁−ε

, u₂k1≥

1−η^u₂

u₂k q^l₂ m₂−ε

. 2.38

By applying Lemmas2.1and2.2, it immediately follows that

k→∞liminfu1k≥ q^l₁ m₁−ε

η^u₁ ,

k→∞liminfu₂k≥ q^l₂ m2−ε

η^u₂ .

2.39

Settingε→0 in the above inequality leads to

k→∞liminfu1k≥ q^l₁m₁ η^u₁ : w1,

k→∞liminfu₂k≥ q^l₂m2

η^u₂ : w₂.

2.40

This completes the proof ofTheorem 2.6.

To check the conditions ofTheorem 2.6, we give an example. We consider the following discrete predator-prey systems with Beddington-DeAngelis functional response and feedback controls

xk1 xkexp

1−xk− 0.8yk

10.2xk 2yk−0.001u₁k ,

yk1 ykexp

−0.01 0.1xk

10.2xk 2yk−0.001u2k

, Δu1k −0.8u1k xk,

Δu2k −0.5u2k yk.

2.41

One could easily obtain that the conditions ofTheorem 2.6are satisfied. Hence, byTheorem 2.6, we see that system2.41is permanent.

Acknowledgment

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

References

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