Two Modifications of the Beverton Holt Equation

International Journal of Difference Equations ISSN 0973-5321 Volume 4 Number 1 pp. 115 136 2009 http://campus.mst.edu/ijde Two Modifications of the Beverton Holt Equation G. Papaschinopoulos. J. Schinas and G. Stefanidou Democritus University of Thrace School of Engineering 67100 Xanthi Greece gpapas@env.duth.gr Abstract In this paper we study the boundedness the persistence of the positive solutions the existence of a unique positive equilibrium the convergence of the positive solutions to the positive equilibrium and the stability of two systems of rational difference equations which are modifications of the Beverton Holt equation. AMS Subject lassifications: 39A10. Keywords: Beverton Holt equation difference equation boundedness persistence convergence stability. 1 Introduction Difference equations have many applications in several applied sciences such as biology ecology economics population dynamics genetics etc. see [4811 13152128] and the references cited therein. For this reason there exist an increasing interest in studying difference equations and systems of difference equations see [1 11 13 21 23 25 27 30] and the references cited therein. In [13] the authors studied some discrete competition models. It is known that if b > 1 then all solutions of the Beverton Holt equation x n+1 = bx n c 11 x n n = 0 1... where x 0 > 0 tend monotonically to the equilibrium x = b 1 c 11. This difference equation is the discrete analog of the logistic differential equation studied in [12]. The Received September 30 2008; Accepted October 15 2008 ommunicated by Gerasimos Las

116 G. Papaschinopoulos. J. Schinas and G. Stefanidou Leslie/Gower difference equation competition model see [21] x n+1 = b 1 x n c 11 x n + c 12 y n y n+1 = b 2 y n c 21 x n + c 22 y n is a modification of the Beverton Holt equation which has played a key role in theoretical ecology. In [29] the authors studied the global stability of the rational difference equation x n+1 = bx n b 0 x n + b 1 x n + b k x n k n = 0 1... where b b i i = 0 1... k are positive constants and the initial values x i i = k k + 1... 0 are positive numbers. In this paper we consider two systems of rational difference equations which are modifications of the Beverton Holt equation of the form x n+1 = y n+1 = ax n p m b i x n i + c i y n i dy n e i y n i + n = 0 1... s k i x n i 1.1 x n+1 = y n+1 = ay n m c i y n 2i + dx n s k i x n 2i + p b i x n 2i 1 n = 0 1... e i y n 2i 1 1.2 where a d b i i = 0 1... p c i i = 0 1... m e i i = 0 1... q k i = 0 1... s are nonnegative constants the initial values of 1.1 x i i = π π + 1... 0 y i i = τ τ + 1... 0 π = max{p s} τ = max{m q} are positive real numbers and the initial values of 1.2 x i i = λ λ + 1... 0 y i i = µ µ + 1... 0 λ = max{2p+1 2s} µ = max{2q +1 2m} are also positive real numbers. More precisely we study the boundedness the persistence of the positive solutions of 1.1 and 1.2 the existence of the unique positive equilibrium of 1.1 and 1.2 the convergence of the positive solutions of 1.1 to the unique positive equilibrium of 1.1. In dition

Two Modifications of the Beverton Holt Equation 117 we study the convergence of the positive solutions of the system ay n x n+1 = p c 0 y n + b i x n 2i 1 y n+1 = dx n n = 0 1... k 0 x n + e i y n 2i 1 1.3 where the constants c 0 k 0 b i i = 0 1... p e i i = 0 1... q are nonnegative real numbers and the initial values are also positive real numbers to the unique positive equilibrium of 1.3. Finally we study the global asymptotic stability of ax n x n+1 = bx n + cy n 1 y n+1 = dy n ey n + kx n 1 n = 0 1... 1.4 ay n x n+1 = cy n + bx n 1 1.5 dx n y n+1 = n = 0 1... kx n + ey n 1 where a b c d e k are positive constants and the initial values are also positive real numbers. 2 Study of System 1.1 In this section we study system 1.1. First we find conditions so that system 1.1 has a unique positive equilibrium. Proposition 2.1. onsider system 1.1 where Suppose that either or hold where B = a > 1 d > 1. 2.1 a 1E > d 1 d 1B > a 1K 2.2 a 1E < d 1 d 1B < a 1K 2.3 p b i = m c i E = e i K = Then system 1.1 has a unique positive equilibrium x ȳ. s k i. 2.4

118 G. Papaschinopoulos. J. Schinas and G. Stefanidou Proof. In order x ȳ to be a positive equilibrium for 1.1 we must have or equivalently x = a x B x + ȳ ȳ = dȳ Eȳ + K x B x + ȳ = a 1 K x + Eȳ = d 1. So using 2.1 and since either 2.2 or 2.3 hold we get x = a 1E d 1 BE K > 0 ȳ = d 1B a 1K BE K > 0. This completes the proof. In the following proposition we study the boundedness and persistence of the positive solution of 1.1. Proposition 2.2. onsider system 1.1 where relations 2.1 and a 1e 0 > d 1 d 1b 0 > a 1K 2.5 hold. Then every positive solution of 1.1 is bounded and persists. Proof. Let x n y n be an arbitrary solution of 1.1. Since from 2.5 b 0 0 e 0 0 then from 1.1 we have x n a b 0 y n d e 0 n = 1 2... 2.6 and so x n y n is a bounded solution. We prove now that x n y n persists. Suppose that x n does not persists. Then we may suppose that there exists a subsequence x nr of x n such that lim x n r = 0 x nr = min{x s 0 s n r }. 2.7 r Firstly suppose that there exists a v N such that Then from 1.1 and 2.8 we obtain y v+1 = dy v e i y v i + y v d 1 e 0. 2.8 s k i x v i dy v e 0 y v d 1 d e 0 = d 1 e 0 e 0 d 1 e 0

Two Modifications of the Beverton Holt Equation 119 and working inductively we have Moreover since from 1.1 y n d 1 e 0 n v. 2.9 x nr = ax nr 1 p b i x nr i m c i y nr i 1 2.10 and x nr+1 = ax nr p b i x nr i + then using 2.6 and 2.7 it follows that m c i y nr i Then working inductively we have lim x n r 1 = 0 r lim x nr+1 = 0. r lim x n r+τ = 0 τ =.. 1 0 1.... 2.11 r Furthermore from 2.5 there exists sufficiently small positive ɛ such that Using 2.11 for sufficiently large n r we have a 1e 0 d 1 ɛbe 0 > 0. 2.12 x nr j ɛ j = 1 2... p + 1 2.13 where ɛ satisfies 2.12. Therefore from 2.9 2.10 2.12 and 2.13 for sufficiently large n r it follows that x nr ax nr 1 d 1 > x nr 1 Bɛ + e 0 which contricts to 2.7. Therefore x n persists in the case where 2.8 holds. Suppose now y n > d 1 e 0 n = 1 2.... 2.14

120 G. Papaschinopoulos. J. Schinas and G. Stefanidou From 1.1 and 2.14 we have y n+1 y n = = dy n y s n e i y n i + k i x n i y n d 1 e 0 y n e i y n i i=1 e i y n i + s k i x n i < 0. s k i x n i 2.15 So from 2.14 and 2.15 there exists the lim n y n and it is different from zero. Let Then from 2.11 2.16 and since from 1.1 lim y n = l 0. 2.16 n y nr+1 = dy nr e i y nr i + s k i x nr i we can prove that l = d 1 E. 2.17 Moreover from 2.5 there exists a sufficiently small ɛ > 0 such that a 1E d 1 ɛeb + > a 1e 0 d 1 ɛeb + > 0. 2.18 From 2.16 and 2.17 there exists a sufficiently large n r such that 2.13 and y nr 1 i d 1 E hold. Then from 2.10 2.13 2.18 and 2.19 we have x nr + ɛ i {0 1... m} 2.19 ax nr 1 d 1 > x nr 1 Bɛ + E + ɛ which contricts again to 2.7. Therefore x n persists. Working similarly we can prove that y n persists. This completes the proof. In the following proposition we study the convergence of the positive solution of 1.1 to the unique positive equilibrium of 1.1.

Two Modifications of the Beverton Holt Equation 121 Proposition 2.3. onsider system 1.1 where relations 2.1 and 2.5 are satisfied. Suppose also that b 0 > B 1 e 0 > E 1 b 0 B 1 e 0 E 1 > K B 1 = p b i E 1 = i=1 e i. 2.20 i=1 Then every positive solution of 1.1 tends to the unique positive equilibrium of 1.1. Proof. From Proposition 2.2 there exist Then from 1.1 we get L 1 = lim sup x n < n L 2 = lim sup y n < n l 1 = lim inf x n > 0 n l 2 = lim inf y n > 0. n 2.21 or equivalently L 1 L 2 al 1 b 0 L B 1 l l 2 l 1 dl 2 e 0 L 2 + E 1 l 2 + Kl 1 l 2 al 1 b 0 l B 1 L L 2 dl 2 e 0 l 2 + E 1 L 2 + KL 1 b 0 L B 1 l l 2 a 1 b 0 l B 1 L L 2 e 0 L 2 + E 1 l 2 + Kl 1 d 1 e 0 l 2 + E 1 L 2 + KL 1. 2.22 Therefore from 2.22 we get b 0 B 1 L 1 l 1 L 2 l 2 e 0 E 1 L 2 l 2 KL 1 l 1. 2.23 Hence using 2.20 and multiplying both sides of 2.23 we obtain b 0 B 1 e 0 E 1 L 1 l 1 L 2 l 2 KL 1 l 1 L 2 l 2. 2.24 So relations 2.20 and 2.24 imply that L 1 l 1 L 2 l 2 0. Therefore either L 1 = l 1 or L 2 = l 2. If L 1 = l 1 resp. L 2 = l 2 then from 2.23 L 2 = l 2 resp. L 1 = l 1. This completes the proof. In the last proposition of this section we study the global asymptotic stability of the positive equilibrium x ȳ of 1.4. We need the following lemma which has been proved in [22]. For reers convenience we state it here without its proof. Lemma 2.4. onsider the equation x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 2.25

122 G. Papaschinopoulos. J. Schinas and G. Stefanidou where a i i = 0 1 2 3 are positive constants. Then every solution of 2.25 is of modulus less than 1 if and only if the following conditions are satisfied: K 1 > 0 K 2 > 0 K 3 > 0 K 5 > 0 K 4 K 2 K 3 K4K 2 1 > K2K 2 5 2.26 where K 1 = a 0 + a a 2 + a 3 K 2 = 4 + 2a 3 2a 1 4a 0 K 3 = 6 2a 2 + 6a 0 K 4 = 4 2a 3 + 2a 1 4a 0 K 5 = 1 a 3 + a 2 a a 0. After some calculations we can see that K 4 K 2 K 3 K 2 4K 1 K 2 2K 5 = a 3 0 + a 0 a 1 a 3 + a 1 a 3 + 2a 0 a 2 + 1 a 0 a 2 a 2 0 a 2 1 a 2 0a 2 a 0 a 2 3. In dition if a 1 = 0 then K 4 K 2 K 3 K 2 4K 1 K 2 2K 5 = a 0 1 2 a 0 a 2 a 0 a 2 3. 2.27 Proposition 2.5. onsider system 1.4 where 2.1 and the relations a 1e > d 1c d 1b > a 1k 2.28 hold. Then the unique positive equilibrium x ȳ of 1.4 is globally asymptotically stable. Proof. We prove that x ȳ is locally asymptotically stable. The linearized system about the positive equilibrium of 1.4 is x n+1 = y n+1 = System 2.29 is equivalent to system w n+1 = Aw n A = a cȳ cȳ + b x x ac x 2 n cȳ + b x y n 1 2 d k x eȳ + k x y dkȳ 2 n eȳ + k x x n 1. 2 H 0 0 T 0 M N 0 1 0 0 0 0 1 0 0 w n = x n y n x n 1 y n 1 2.29 H = M = a cȳ cȳ + b x T = ac x 2 cȳ + b x 2 d k x k x + eȳ 2 N = dkȳ k x + eȳ 2.

Two Modifications of the Beverton Holt Equation 123 The characteristic equation of A is λ 4 λ 3 H + M + λ 2 HM NT = 0. 2.30 Using Lemma 2.4 all the roots of 2.30 are of modulus less than 1 if and only if 2.26 are satisfied where K 1 = 1 H M + HM NT K 2 = 4 2H + M + 4NT K 3 = 6 2HM 6NT K 4 = 4 + 2H + M + 4NT K 5 = H + M + HM NT. Since x ȳ is a positive equilibrium of 1.4 we have cȳ + b x = a eȳ + k x = d 2.31 and so ea 1 cd 1 x = ȳ = be ck In dition from 2.31 we get bd 1 ka 1. 2.32 be ck H = cȳ T = c x a a M = k x N = kȳ d d. 2.33 Using 2.28 2.31 2.32 2.33 and after some calculations we have K 1 = 1 d 1a c1 dȳ + k1 a x 1 = ea 1 cd 1 bd 1 ka 1 > 0 be ck K 3 = 2 = 2 = 2 K 2 = 2 = 2 = 2 da ad 1 cdȳ ak x + 2ck xȳ dcȳ + b x + aeȳ + k x cdȳ ak x + 2ck xȳ db x + aeȳ + 2ck xȳ > 0 3 1 k x cȳ 4ck xȳ 3 cȳ + b x eȳ + k x 1 k x cȳ 4ck xȳ 2 + 3b x + 3eȳ + 2k x + 2cȳ + 3ceȳ 2 + 3bk x 2 + 3be ck xȳ > 0

124 G. Papaschinopoulos. J. Schinas and G. Stefanidou K 5 = cȳ + k x + a d Moreover from 2.27 and 2.31 we have k x + cȳ > 0. K 4 K 2 K 3 K 2 4K 1 K 2 2K 5 ck xȳ 2 = + 1 1 ck xȳ = 1 ck xȳ 2 + ck xȳ cȳ + k x = 1 a ck xȳ cȳ + ck xȳ a cȳ k x + ck xȳ cȳ a + k x 2 d cȳ + b x eȳ + k x ck xȳ cȳ k x 2 d 2 eȳ + b x + ceȳ 2 + bk x 2 + be ck xȳ + k x 2 > 0. d Then from Lemma 2.4 all the roots of 2.30 are of modulus less than 1 and so the positive equilibrium x ȳ of 1.4 is locally asymptotically stable. So from Proposition 2.3 x ȳ is globally asymptotically stable. This completes the proof. 3 Study of System 1.2 In the first proposition we study the existence of a positive equilibrium of 1.2. Proposition 3.1. onsider system 1.2 such that Then system 1.2 has a unique positive equilibrium. Proof. We consider the system of algebraic equations x = > 1. 3.1 ay y + Bx y = dx Kx + Ey 3.2 where the constants B K E are defined in 2.4. System 3.2 is equivalent to the system fx = BEB Kx 3 + 2BE + KBa K B d 2 x 2 +ak + ab + 2 + Ex + a1 da = 0 3.3 y = Bx2 + x a x. 3.4

Two Modifications of the Beverton Holt Equation 125 Suppose that From 3.1 and 3.3 it follows that a f = B2 Ea 3 3 + 2BEa2 2 Moreover from 3.3 and 3.5 we have EB K < 0. 3.5 + Ea > 0 f0 = a1 da < 0. 3.6 f > 0 f < 0. 3.7 Therefore from 3.6 and 3.7 it follows that equation 3.3 has one solution in the interval 0 one solution in 0 a a and one solution in the interval. Therefore equation 3.3 has a unique solution in the interval 0 a and so system 3.2 has a unique positive solution x ȳ such that x 0 a and ȳ satisfies 3.4. Now suppose that EB K > 0. 3.8 Then either inequality or E > 3.9 B > K 3.10 holds. Firstly consider that 3.9 is satisfied. Using 3.6 we have that equation 3.3 has a solution in the interval 0 a. We prove that 3.3 has a unique solution in 0 a Let. We set θ = EB 2 BK λ = 2BE + KBa K B d 2 µ = ak + ab + 2 + E ν = a1 da. λ 0. 3.11 Then if p 1 p 2 p 3 are the roots of 3.3 from 3.1 3.8 and 3.11 we get p p 2 + p 3 = λ θ 0 p 1p 2 p 3 = ν θ > 0 and so equation 3.3 has a unique solution in the interval Suppose now that onsider equation 0 a. λ < 0. 3.12 f x = 3θx 2 + 2λx + µ = 0. 3.13

126 G. Papaschinopoulos. J. Schinas and G. Stefanidou If = λ 2 3θµ 0 then f x > 0 for every x and so equation 3.3 has a unique solution in the interval 0 a. Suppose now that > 0. 3.14 After some calculations and using 3.8 and 3.9 we take a ab + abe K + 2aBE + E f = > 0. 3.15 2 Let q 1 q 2 be the roots of 3.13 such that q 1 < q 2. Then using 3.15 we have either relation a < q 1 3.16 or q 2 < a 3.17 holds. If 3.16 is satisfied then f x > 0 for every x unique solution in the interval 0 a. 0 a and so 3.3 has a We prove that 3.17 does not hold. Suppose on the contrary that 3.17 is true. Then we must have λ + 3θa > 0. 3.18 Then from 3.12 and 3.18 we have P = P < d < Q 3.19 BE K + BE + KBa 2 3aB + BE K + BE + KBa Q =. 3 After some calculations we get d = λ 2 3θµ = 3BBE K + E + ab + 2 + ak + 2 d + K + B 2EB abk 2. Moreover using 3.8 3.9 and 3.20 we have 3.20 P 3BBE K abe K + 2aBE + ab + E + 2KBa 2 = < 0 3.21

Two Modifications of the Beverton Holt Equation 127 3BaB + BE K abe K + 2aBE + E Q = < 0. 2 3.22 Therefore from 3.21 and 3.22 it follows that d < 0 P < d < Q which contricts to 3.14. So 3.17 is not true which means that if relations 3.8 and 3.9 are satisfied then equation 3.3 has a unique solution in 0 a. Therefore system 3.2 has a unique solution x ȳ such that x 0 a and ȳ satisfies 3.4. Suppose now that 3.10 holds. System 3.2 is equivalent to the system EEB Ky 3 + 2BE + Ed EK K ak 2 y 2 +de + d K + 2K + By + d1 da = 0 3.23 x = Ey2 + y d Ky. 3.24 Then arguing as above we can prove that system 3.2 has a unique solution x ȳ such that relations 3.23 and 3.24 are satisfied. Finally suppose that EB K = 0. 3.25 Then equation 3.3 becomes hx = ζx 2 + µx + ν = 0 ζ = BE + KBa B d 2. 3.26 If ζ 0 then using a h = a2 BE + ae 2 + a3 BK + a2 K > 0 h0 = ν < 0 2 we have that equation 3.26 has a unique solution in 0 a. Finally suppose that ζ = 0. Then since a h = a2 K + a2 B + ae + a2 d > 0 h0 = ν < 0 equation 3.26 has a unique solution in 0 a. This completes the proof. In the following proposition we study the boundedness and persistence of the positive solutions of system 1.2. Proposition 3.2. onsider system 1.2 where 3.1 holds and c 0 0 k 0 0. 3.27 Then every positive solution of 1.2 is bounded and persists.

128 G. Papaschinopoulos. J. Schinas and G. Stefanidou Proof. Let x n y n be an arbitrary solution of 1.2. Then using 1.2 and 3.27 it is obvious that x n a c 0 y n d k 0 n = 1 2... 3.28 which implies that x n y n is a bounded solution. We prove that x n y n persists. Suppose that x n does not persist. Without loss of generality we may assume that there exists a subsequence n r such that relations 2.7 are satisfied. Hence from 1.2 x nr = Then from 2.7 and 3.28 we get ay nr 1 m c i y nr 1 2i +. p b i x nr 2i 2 In dition since from 1.2 lim y n r 1 = 0. 3.29 r y nr 1 = dx nr 2 s k i x nr 2 2i + relation 3.29 implies that lim x n r 2 = 0. r Working inductively we can prove that lim x n r 2i = 0 r e i y nr 2i 3 lim y nr 2i 1 = 0 i = 0 1.... 3.30 r Therefore from 3.1 and 3.30 for sufficiently large n r we get x nr 2j ɛ j {1 2... φ + 1} y nr 2w 1 ɛ w {0 1... ψ + 1} 3.31 where φ = max{p s} ψ = max{m q} and ɛ is a sufficiently small positive number such that > 1. 3.32 ɛ + B ɛk + E Moreover from 1.4 we have x nr = m c i y nr 1 2i + x nr 2 p b i x nr 2i 2 s k i x nr 2 2i +. e i y nr 2i 3 3.33

Two Modifications of the Beverton Holt Equation 129 Therefore from relations 3.31 3.33 it follows that x nr > x nr 2 + Bɛ K + Eɛ > x n r 2 which contricts to 2.7. Therefore x n persists. Using the same argument we can easily prove that also y n persists. This completes the proof. In the next proposition we study the convergence of the positive solutions of the system 1.3 to the unique positive equilibrium. Proposition 3.3. Suppose that relations 3.1 3.27 and c 0 E k 0 B 3.34 hold. Then every positive solution of 1.3 tends to the unique positive equilibrium of 1.3. Proof. Suppose that either c 0 E or k 0 B holds. Using Proposition 3.2 we have that 2.21 are satisfied. Then from 1.3 we take L 1 L 2 Then relations 3.35 imply that al 2 c 0 L 2 + Bl 1 l 1 dl 1 k 0 L El 2 l 2 al 2 c 0 l 2 + BL 1 dl 1 k 0 l EL 2. 3.35 L 1 L 2 L 1 L 2 c 0 L 2 + Bl 1 k 0 L El 2 which implies that l 1 l 2 l 1 l 2 c 0 l 2 + BL 1 k 0 l EL 2 c 0 L 2 + Bl 1 k 0 L El 2 c 0 l 2 + BL 1 k 0 l EL 2. 3.36 So from 3.34 3.36 we have k 0 BL 1 l 1 + c 0 EL 2 l 2 + c 0 k 0 BEL 1 L 2 l 1 l 2 0 which implies that L 1 = l 1 L 2 = l 2. Now suppose that c 0 = E k 0 = B. 3.37 Then from 3.35 we get d l 1 l 2 Bl c 0 L 2 a L 2 L 1 a l 2 l 1 c 0 l 2 + BL 1 d L 1 L 2

130 G. Papaschinopoulos. J. Schinas and G. Stefanidou from which we take From 3.35 3.37 and 3.38 we get L 1 l 2 dl 1 L 1 = al 2 L 2. 3.38 al 2 l 2 c 0 L 2 + Bl 1 = dl 1 l 1 c 0 L 2 + Bl 1 l 1 L 2 L 2 l 1 al 2 l 2 c 0 l 2 + BL 1 = dl 1 l 1 BL c 0 l 2 = dl 1 l 1 c 0 l 2 + BL 1 al 2 l 2 BL c 0 l 2 3.39 L 1 l 2 dl 1 l 1 Bl c 0 L 2 = Then relations 3.35 3.37 and 3.39 imply that l 2 = l 1 = dl 1 c 0 L 2 + Bl 1 L 2 = al 2 BL c 0 l 2 L 1 = al 2 l 2 Bl c 0 L 2. dl 1 c 0 l 2 + BL 1 al 2 Bl c 0 L 2. 3.40 Without loss of generality we may assume that there exists a subsequence n r such that From 1.3 we have and so from 3.41 we get lim x n r = L 1 lim x nr i = M i i = 2 3... 2p + 2 r r lim y n r i = R i i = 1 2... 2q + 3. r L 1 = x nr = Thus from 3.40 and 3.42 we have In dition from 1.3 we get c 0 y nr ay nr 1 p b i x nr 2i 2 ar 1 p c 0 R b i M 2i+2 3.41 al 2 Bl c 0 L 2. 3.42 L 2 = R 1 M 2i+2 = l 1 i = 0 1... p. 3.43 y nr 1 = dx nr 2 k 0 x nr 2 + e i y n 2i 3

Two Modifications of the Beverton Holt Equation 131 and so from 3.41 we have L 2 = dm 2 k 0 M 2 + e i R 2i+3 dl 1 k 0 L El 2. 3.44 Then from 3.40 and 3.44 we get Therefore from 3.43 and 3.45 we take M 2 = L 1 R 2i+3 = l 2 i = 0 1... q. 3.45 Finally using 3.40 and 3.46 it is obvious that This completes the proof. M 2 = L 1 = l 1. 3.46 L 2 = l 2. In the last proposition we study the global asymptotic stability of the positive equilibrium of 1.5. We need the following lemma which has been proved in [26]. For reers convenience we state it here without its proof. Lemma 3.4. onsider the algebraic equation x 2 + a 1 x + a 0 = 0. 3.47 Then all roots of 3.47 are of modulus less than 1 if and only if a 1 < a 0 + 1 < 2. 3.48 Proposition 3.5. onsider system 1.5 where 3.1 holds. Suppose also that c > e k > b. 3.49 Then the unique positive equilibrium x ȳ of 1.5 is globally asymptotically stable. Proof. We prove that x ȳ is locally asymptotically stable. The linearized system of 1.5 about x ȳ is abȳ x n+1 = cȳ + b x x a b x 2 n cȳ + b x y n 2 y n+1 = d eȳ k x + eȳ x ed x 2 n k x + eȳ y n 1. 2 3.50

132 G. Papaschinopoulos. J. Schinas and G. Stefanidou It is obvious that system 3.50 is equivalent to the system 0 H T 0 w n+1 = Aw n A = M 0 0 N 1 0 0 0 0 1 0 0 H = w n = a b x cȳ + b x T = abȳ 2 cȳ + b x 2 d eȳ M = k x + eȳ N = ed x 2 k x + eȳ. 2 Then the characteristic equation of A is x n y n x n 1 y n 1 λ 4 λ 2 HM + N + T + NT = 0. 3.51 Since x ȳ is the positive equilibrium of 1.5 we have Hence H = b x x2 aȳ 2 1 cȳ + b x = x aȳ 1 k x + eȳ = ȳ d x. 3.52 T = b x2 aȳ Relations 3.49 3.52 and 3.53 imply that NT = be be xȳ < Moreover from 3.1 3.52 we have HM + N + T NT = 1 eȳȳ2 M = N = eȳ2 d x 2 d x. 3.53 a d c k = be ck eȳ + b x bd x2 eaȳ2 ȳ x < 1. 3.54 = 1 eȳ + b x b x bk x 2 be xȳ eȳ ceȳ 2 eb xȳ = 1 1 bk x 2 2be xȳ ceȳ 2 < 1 HM + N + T + NT + 1 = 1 eȳ + b x + 2be xȳ bd x2 eaȳ2 ȳ x = 1 eȳ + b x + 2be xȳ eȳ ceȳ 2 eb xȳ b x bk x 2 be xȳ = 1 + 1 bk x 2 ceȳ 2 = 1 cȳ + b x k x + eȳ + 1 bk x 2 ceȳ 2 > 0. + 1 + 1 3.55 3.56

Two Modifications of the Beverton Holt Equation 133 Therefore relations 3.54 3.55 and 3.56 imply that all conditions of Lemma 3.4 are satisfied. Therefore all the roots of equation 3.51 are of modulus less than 1 which implies that x ȳ is locally asymptotically stable. Using Proposition 3.3 x ȳ is globally asymptotically stable. This completes the proof. 4 onclusion In this paper we considered two systems of rational difference equations of the form 1.1 and 1.2. These systems are modifications of the Beverton Holt equation which is the discrete analog of the logistic differential equation studied in [12]. Systems of this form are worthwhile studying since many authors studied discrete competition models see [13] and the references cited therein. The main results of this paper were presented in two sections. In Section 2 we studied system 1.1 and in Section 3 system 1.2. Summarizing the results of Sections 2 and 3 we get the following statements concerning both systems. i We studied the existence and the uniqueness of the positive equilibrium of the systems. ii We found conditions so that every positive solution of the systems is bounded and persists. iii We investigated the convergence of the positive solutions of the system 1.1 and system 1.3 which is a special case of system 1.2. iv We studied the global asymptotic stability to the unique positive equilibrium of systems 1.4 and 1.5 which are special cases of systems 1.1 and 1.2 respectively. Finally we state the following open problems. Open Problem 4.1. onsider the systems of difference equations 1.1 and 1.2 where a d b i i = 0 1... p c i i = 0 1... m e i i = 0 1... q k i 0 1... s are nonnegative constants the initial values of 1.1 x i i = π π + 1... 0 y i i = τ τ + 1... 0 π = max{p s} τ = max{m q} are positive real numbers and the initial values of 1.2 x i i = λ λ + 1... 0 y i i = µ µ + 1... 0 λ = max{2p + 1 2s} µ = max{2q + 1 2m} are also positive real numbers. Prove that: I. If relations 2.1 and 2.5 are satisfied then every positive solution of 1.1 tends to the unique positive equilibrium of 1.1. II. If relations 3.1 and 3.27 hold then every positive solution of 1.2 tends to the unique positive equilibrium of 1.2.

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