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

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1 Hindawi Pblishing Corporation Discrete Dynamics in Natre and Society Volme 2008 Article ID pages doi:101155/2008/ Research Article Permanence of a Discrete Predator-Prey Systems with Beddington-DeAngelis Fnctional Response and Feedback Controls Xepeng Li and Wensheng Yang School of Mathematics and Compter Science Fjian Normal University Fzho China Correspondence shold be addressed to Wensheng Yang ywensheng@126com Received 20 Jly 2007; Accepted 21 Febrary 2008 Recommended by Leonid Berezansky We propose a discrete predator-prey systems with Beddington-DeAngelis fnctional response and feedback controls By applying the comparison theorem of difference eqation sfficient conditions are obtained for the permanence of the system Copyright q 2008 X Li and W Yang This is an open access article distribted nder the Creative Commons Attribtion License which permits nrestricted se distribtion and reprodction in any medim provided the original work is properly cited 1 Introdction Zhang and Wang 1 considered the following nonatonomos discrete predator-prey systems with the Beddington-DeAngelis fnctional response c k y k x k 1 x k exp a k b k x k α k β k x k γ k y k f k x k y k 1 y k exp d k α k β k x k γ k y k 11 By sing a contination theorem sfficient criteria are established for the existence of positive periodic soltions of the system 11 As we know permanence is one of the most important topics on the stdy of poplation dynamics One of the most interesting qestions in mathematical biology concerns the srvival of species in ecological models Biologically when a system of interacting species is persistent in a sitable sense it means that all the species srvive in the long term It is reasonable to ask for conditions nder which the system is permanent However Zhang and Wang 1 did not investigate this property of the system 11

2 2 Discrete Dynamics in Natre and Society As we know ecosystems in the real world are continosly distribted by npredictable forces which can reslt in changes in the biological parameters sch as srvival rates Of practical interest in ecology is the qestion of whether or not an ecosystem can withstand those npredictable distrbances which persist for a finite period of time In the langage of control variables we call the distrbance fnctions as control variables Already Gopalsamy and Weng 2 have stdied the Logistic growth model with feedback control To the athor knowledge there is few works dealt with system 11 with feedback control Therefore one objective of this paper is to stdy the following discrete predator-prey systems with Beddington-DeAngelis fnctional response and feedback controls x k 1 x k exp a k b k x k y k 1 y k exp d k Δ 1 k η 1 k 1 k q 1 k x k Δ 2 k η 2 k 2 k q 2 k y k c k y k α k β k x k γ k y k e 1 k 1 k f k x k α k β k x k γ k y k e 2 k 2 k where a k b k c k d k f k α k β k γ k e 1 k e 2 k η 1 k η 2 k q 1 k and q 2 k are all bonded nonnegative seqence For more biological backgrond of system 12 one cold refer to 1 and the references cited therein Throghot this paper we se the following notations for any bonded seqence a k : 12 a sp a k k N a l inf k N a k 13 and assme that 0 <η l 1 η 1 < 1 0 <ηl 2 η 2 < 1 The aim of this paper is by frther developing the analysis techniqe of Chen 3 to obtain a set of sfficient conditions which ensre the permanence of the system 12 We say that system 12 is permanent if there are positive constants M and m sch that for each positive soltion x k y k 1 k 2 k of system 12 satisfies m lim inf x k lim sp x k M m lim inf y k lim sp y k M m lim inf i k lim sp i k M i 1 2 For biological reasons we only consider soltion x k y k 1 k 2 k with x 0 > 0; y 0 > 0; i 0 > 0i 1 2 Then system 12 has a positive soltion x k y k 1 k 2 k passing throgh x 0 y Permanence In this section we establish a permanence reslt for system 12 First let s consider the first order difference eqation 14 y n 1 Ay n B n

3 X Li and W Yang 3 where A B are positive constants Following Lemma 21 is a direct corollary of Theorem 62 of LWangandMQWang 4 page 125 Lemma 21 Assme that A < 1 for any initial vale y 0 there exists a niqe soltion yn) of 21 which can be expressed as follows: y n A n y 0 y ) y 22 where y B/ 1 A Ths for any soltion y n of system 21 lim y n n y 23 Following Comparison Theorem of difference eqation is Theorem 21 of 4 page 241 Lemma 22 Let k N k 0 k 0 k 0 1k 0 lr 0 For any fixed k g k r is a nondecreasing fnction with respect to r and for k k 0 the following ineqalities hold: y k 1 g k y k k 1 g k k 24 If y k 0 k 0 theny k k for all k k 0 Now let s consider the following single species discrete model: N k 1 N k expa k b k N k 25 where a k and b k are strictly positive seqences of real nmbers defined for k N and 0 <a l a 0 <b l b Similarly to the proof of 5 Propositions 1 and 3 we can obtain the following Lemma 23 Any soltion of system 25 with initial condition N 0 > 0 satisfies m lim inf N k lim spn k M 26 where M 1 b l exp a 1 m al b exp a l b M 27 Lemma 24 see 6 Let x n and b n be nonnegative seqences defined on N and c 0 is a constant If n 1 x n c b s x s for n N 28 s 0

4 4 Discrete Dynamics in Natre and Society Then Proposition 25 Assme that holds then n 1 x n c 1 b s for n N 29 s 0 d l f > where lim sp x k M 1 lim sp y k M 2 lim sp i k W i i 1 2 M 1 1 b exp a 1 l M 2 exp 2 d l f ) W i q i M i i 1 2 η l i Proof Let s k x k y k 1 k 2 k be any positive soltion of system 12 ; from 12 we have x k 1 x k expa k b k x k 213 By applying Lemmas 22 and 23 it immediately follows that lim sp x k 1 b exp a 1 : M l 1 From the second eqation of the system 12 we can obtain y k 1 y k exp d k f k β k y k exp d l f Let y k exp k then k 1 k d l f ) k b s s d l f ) s 0 216

5 X Li and W Yang 5 where 0 0 s k 1 b s 1 s k 217 Condition 210 shows that Lemma 24 cold be applied to 216 and so by applying Lemma 24 it immediately follows that k 1 2 d l f ) 218 This is lim sp y k exp 2 d l f ) : M For any positive constant ε small enogh it follows from 214 and 219 that there exists enogh large K 0 sch that x k M 1 ε y k M 2 ε k K From the third and forth eqations of the system 12 and 220 we can obtain Δ 1 k η 1 k 1 k q 1 k M 1 ε ) Δ 2 k η 2 k 2 k q 2 k M 2 ε ) 221 So 1 k 1 1 η l 1) 1 k q 1 M1 ε ) 2 k 1 1 η l 2) 2 k q 2 M2 ε ) By applying Lemmas 21 and 22 it immediately follows that lim sp 1 k q 1 M1 ε ) η l 1 lim sp 2 k q 2 M2 ε ) η l Setting ε 0 in the above ineqality leads to lim sp 1 k q 1 M 1 η l 1 lim sp 2 k q 2 M 2 η l This completes the proof of Proposition 25

6 6 Discrete Dynamics in Natre and Society Now we are in the position of stating the permanence of the system 12 Theorem 26 In addition to 210 assme frther that a l c e γ l 1 W 1 > 0 d f l m 1 e 2 W 2 > then system 12 is permanent where m 1 al c /γ l e 1 W 1 b exp a l c e γ l 1 W 1 b M Proof By applying Proposition 25 we see that to end the proof of Theorem 26 itisenoghto show that nder the conditions of Theorem 26 lim inf x k m 1 lim inf y k m 2 lim inf i k w i i From Proposition 25 for all ε>0 there exists a K 1 > 0K 1 N for all k>k 1 x k M 1 ε y k M 2 ε; i k W i ε i From the first eqation of systems 12 and 228 wehave x k 1 x k exp a k b k x k c k γ k e 1 k W 1 ε ) x k exp a k c k γ k e 1 k W 1 ε ) 229 b k x k for all k>k 1 Condition 225 shows that Lemmas 22 and 23 coldbeappliedto 229 andsoby applying Lemmas 22 and 23 to 229 it immediately follows that lim inf x k al c /γ l e 1 b W1 ε ) exp a l c e γ l 1 W1 ε ) b M Setting ε 0in 230 leads to lim inf x k al c /γ l e 1 W 1 b exp a l c e γ l 1 W 1 b M 1 : m Then for any positive constant ε small enogh from 231 we know that there exists an enogh large K 2 >K 1 sch that x k m 1 ε k k 2 232

7 X Li and W Yang 7 From the second eqation of systems and 232 wehave 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 ) e 2 k 2 k y k exp d k f k β k f k ) α k β k α k β k m 1 ε ) f k ) γ k y k β k α k β k m 1 ε ) e 2 k W 2 ε ) y k exp d k f k m 1 ε ) e 2 k W 2 ε ) f k γ k β k [ α k β k m 1 ε )]y k 233 for all k>k 2 Condition 225 shows that Lemmas 22 and 23 coldbeappliedto 233 andsoby applying Lemmas 22 and 23 to 233 it immediately follows that [ βl α l m 1 ε )][ d f l m 1 ε ) e lim inf y k 2 W2 ε )] f γ exp d f l m 1 ε ) e2 W2 ε ) f γ [ α l m 1 ε )]M Setting ε 0in 234 leads to ) βl α l m 1 d f l m 1 e 2 lim inf y k W 2) f γ exp d f l m 1 e 2 W 2 f γ )M α l 2 : m 2 m Withot loss of generality we may assme that ε< 1/2 minm 1 m 2 For any positive constant ε small enogh it follows from 231 and 235 that there exists enogh large K 3 >K 2 sch that x k m 1 ε y k m 2 ε k K From the third and forth eqations of the system 12 and 236 we can obtain that Δ 1 k η 1 k 1 k q 1 k m 1 ε ) Δ 2 k η 2 k 2 k q 2 k m 2 ε ) 237 So 1 k 1 1 η 1) 1 k q l 1 m1 ε ) 2 k 1 1 η 2) 2 k q l 2 m2 ε ) 238

8 8 Discrete Dynamics in Natre and Society By applying Lemmas 21 and 22 it immediately follows that lim inf 1 k ql 1 m1 ε ) η 1 lim inf 2 k ql 2 m2 ε ) Setting ε 0 in the above ineqality leads to η This completes the proof of Theorem 26 lim inf 1 k ql 1 m 1 η : w 1 1 lim inf 2 k ql 2 m 2 η : w To check the conditions of Theorem 26 we give an example We consider the following discrete predator-prey systems with Beddington-DeAngelis fnctional response and feedback controls x k 1 x k exp 1 x k y k 1 y k exp 001 Δ 1 k 08 1 k x k Δ 2 k 05 2 k y k 08y k 1 02x k 2y k k 01x k 1 02x k 2y k k 241 One cold easily obtain that the conditions of Theorem 26 are satisfied Hence by Theorem 26 we see that system 241 is permanent Acknowledgment This work is spported by the Fondation of Edcation Department of Fjian Province Grant no JA05204 and the Fondation of Science and Technology Department of Fjian Province Grant no 2005K027 References 1 J Zhang and J Wang Periodic soltions for discrete predator-prey systems with the Beddington- DeAngelis fnctional response Applied Mathematics Letters vol 19 no 12 pp K Gopalsamy and P-X Weng Feedback reglation of logistic growth International Jornal of Mathematics and Mathematical Sciences vol 16 no 1 pp F Chen Permanence of a discrete n-species food-chain system with time delays Applied Mathematics and Comptation vol 185 no 1 pp L Wang and M Q Wang Ordinary Difference Eqation Xinjiang University Press China F Chen Permanence and global attractivity of a discrete mltispecies Lotka-Volterra competition predator-prey systems Applied Mathematics and Comptation vol 182 no 1 pp Y Takechi Global Dynamical Properties of Lotka-Volterra Systems World Scientific River Edge NJ USA F Chen Permanence and global attractivity of a discrete mltispecies Lotka-Volterra competition predator-prey systems Applied Mathematics and Comptation vol 182 no 1 pp