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J Math Anal Appl 362 (2010) 427–437 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Survival of three species in a nonautonomous Lotka–Volterra system Ta Viet Ton Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, Viet Nam a r t i c l e i n f o a b s t r a c t Article history: Received 10 December 2008 Available online August 2009 Submitted by C.V Pao In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra system of three species with constant interaction coefficients In this paper, we study a nonautonomous Lotka–Volterra model with one predator and two preys The explorations involve the persistence, extinction and global asymptotic stability of a positive solution © 2009 Elsevier Inc All rights reserved Keywords: Predator–prey model Survival Extinction Persistence Asymptotic stability Liapunov function Introduction We consider a Lotka–Volterra model of one predator and two preys ⎧ ⎪ ⎨ x1 (t ) = x1 (t ) a1 (t ) − b11 (t )x1 (t ) − b12 (t )x2 (t ) − b13 (t )x3 (t ) , x2 (t ) = x2 (t ) a2 (t ) − b21 (t )x1 (t ) − b22 (t )x2 (t ) − b23 (t )x3 (t ) , ⎪ ⎩ x3 (t ) = x3 (t ) −a3 (t ) + b31 (t )x1 (t ) + b32 (t )x2 (t ) − b33 (t )x3 (t ) (1.1) Here xi (t ) represents the population density of species X i at time t (i = 1, 2, 3), x1 (t ), x2 (t ) are the two preys and they interact other and x3 (t ) is the predator (t ), b i j (t ) (i , j = 1, 2, 3) are continuous on R and bounded above and below function by positive constants At time t, (t ) is the intrinsic growth rate of prey species X i (i = 1, 2), a3 (t ) is the death rate of the predator species X , b i j (t ) measures the amount of competition between the prey X i and X j (i = j, i , j = 1, 2), b3i (t ) b i3 (t ) denotes the coefficient in conversing prey species X i into new individual of predator species X (i = 1, 2) and b ii (t ) (i = 1, 2, 3) measures the inhibiting effect of environment on the ith population This paper is organized as follows Section provides some definitions and notations In Section we state some results about invariant set and asymptotic stability for problem (1.1) Section is special case of Section when the coefficient b i j (t ) is constant and Section is special case of Section when the coefficient (t ) is constant (i , j = 1, 2, 3) Definition and notation In this section we summarize the basic definitions and facts which are used later Let R3+ := {(x1 , x2 , x3 ) ∈ R3 | xi i = 1, 2, 3} For a bounded continuous function g (t ) on R, we use the following notation: g u := sup g (t ), t ∈R g l := inf g (t ) t ∈R E-mail address: tontvmath@yahoo.com 0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.07.053 © 2009 Elsevier Inc All rights reserved 0, 428 T.V Ton / J Math Anal Appl 362 (2010) 427–437 The global existence and uniqueness of the solutions of system (1.1) can be found in [3] From the uniqueness theorem, it is easy to prove that Lemma 2.1 Both the nonnegative and positive cones of R3 are positively invariant for (1.1) In the remainder of this paper, for biological reasons, we only consider the solutions (x1 (t ), x2 (t ), x3 (t )) with positive initial values, i.e, xi (t ) > 0, i = 1, 2, Definition 2.2 System (1.1) is said to be permanent if there exist positive constants δ , with < δ < such that for all i = 1, 2, 3, lim inft →∞ xi (t ) δ , lim supt →∞ xi (t ) for all solutions of (1.1) with positive initial values System (1.1) is called persistent if for all i = 1, 2, 3, lim supt →∞ xi (t ) > 0, ∀i = 1, 2, and strongly persistent if lim inft →∞ xi (t ) > for all solutions with positive initial values Definition 2.3 A set A is called to be an ultimately bounded region of system (1.1) if for any solution (x1 (t ), x2 (t ), x3 (t )) of (1.1) with positive initial values, there exists T > such that (x1 (t ), x2 (t ), x3 (t )) ∈ A for all t t + T Definition 2.4 A bounded nonnegative solution (x∗1 (t ), x∗2 (t ), x∗3 (t )) of (1.1) is said to be globally asymptotically stable (or globally attractive) if any other solution (x1 (t ), x2 (t ), x3 (t )) of (1.1) with positive initial values satisfies xi (t ) − x∗i (t ) = lim t →∞ i =1 Remark 2.5 It is easy to see that if system (1.1) has a solution is globally asymptotically stable, then any solution of (1.1) is also globally asymptotically stable Lemma 2.6 (See [2].) Let h be a real number and f be a nonnegative function defined on [h, +∞) such that f is integrable on [h, +∞) and is uniformly continuous on [h, +∞), then limt →∞ f (t ) = The model with general coefficients Theorem 3.1 If mi > 0, i = 1, 2, 3, then set Γ defined by Γ = (x1 , x2 , x3 ) ∈ R3 mi M i , i = 1, 2, x (3.1) is positively invariant with respect to system (1.1), where a1u M := bl11 −al3 M := m2 := and + , M := a2u bl22 u u + b31 M + b32 M2 bl33 u u al2 − b21 M − b23 M3 u b22 + , , m1 := m3 := , u u al1 − b12 M − b13 M3 u b11 , −a3u + bl31m1 + bl32m2 u b33 (3.2) is constant Proof First, we know that the logistic equation X (t ) = A (t ) X (t ) B − X (t ) ( B = 0) has a unique solution X (t ) = B X exp B X exp B t t0 t t0 A (s) ds A (s) ds − + B (3.3) where X := X (t ) Next, we consider the solution of system (1.1) with the initial values (x10 , x20 , x30 ) ∈ Γ By Lemma 2.1, we have xi (t ) > for all t t and i = 1, 2, We have x1 (t ) x1 (t ) a1 (t ) − b11 (t )x1 (t ) x1 (t ) a1u − bl11 x1 (t ) = bl11 x1 (t ) M 10 − x1 (t ) T.V Ton / J Math Anal Appl 362 (2010) 427–437 429 Using the comparison theorem, we obtain that x1 (t ) Because x10 x3 (t ) x10 M 10 exp{a1u (t − t )} x10 M exp{a1u (t − t )} x10 [exp{a1u (t − t )} − 1] + M 10 x10 [exp{a1u (t − t )} − 1] + M M , we have x1 (t ) M for all t M for all t t and because t Now, from above results, we have u u u u x1 (t ) al1 − b12 M − b13 M − b11 x1 (t ) = b11 x1 (t ) m1 − x1 (t ) mi , i = 1, 2, 3, we get that From the comparison theorem and from xi0 x1 (t ) t Similarly, we get that x2 (t ) u u x3 (t ) −al3 + b31 M + b32 M − bl33 x3 (t ) = bl33 x3 (t ) M − x3 (t ) , we also get that x3 (t ) x1 (t ) M for all t (3.4) u m1 x10 exp{b11 m1 (t − t )} u x10 [exp{b11 m1 (t − t )} − 1] + m1 Similarly, we obtain that x2 (t ) m2 , x3 (t ) m1 m3 for all t for all t t0 t The proof is complete ✷ Theorem 3.2 If mi > 0, i = 1, 2, 3, then the set Γ is an ultimately bounded region, i.e., system (1.1) is permanent Proof From (3.4) we have lim sup x1 (t ) t →∞ M1 Similarly, lim sup x2 (t ) t →∞ M2 Thus there exists t t such that xi (t ) M i , i = 1, for all t t By the same argument in Theorem 3.1, we also get that lim supt →∞ x3 (t ) M Similarly, we claim that lim inft →∞ xi (t ) mi Then Γ is an ultimately bounded region ✷ Theorem 3.3 If M 30 < then limt →∞ x3 (t ) = Proof We see that if M 30 < then M < with x3 (t ) is sufficiently small Similar to the proof of Theorem 3.1 we get that bl33 x3 (t ) M − x3 (t ) < (3.5) Therefore, < x3 (t ) x3 (t ) for t t and there exists c such that limt →∞ x3 (t ) = c If c > then < c x3 (t ) x3 (t ), t t From (3.5), there exists ν > such that x3 (t ) < −ν for all t t It follows x3 (t ) < −ν (t − t ) + x3 (t ) and limt →∞ x3 (t ) = −∞ which contradicts our result that x3 (t ) > for all t t Hence, limt →∞ x3 (t ) = ✷ Theorem 3.4 Let (x∗1 (t ), x∗2 (t ), x∗3 (t )) be a solution of system (1.1) If mi > 0, i = 1, 2, and the following conditions hold ⎧ lim inf 2m1 b11 (t ) + m2 b12 (t ) + m3 b13 (t ) − M b21 (t ) − M b31 (t ) − a1 (t ) > 0, ⎪ ⎪ ⎪ t →∞ ⎨ lim inf 2m2 b22 (t ) + m1 b21 (t ) + m3 b23 (t ) − M b12 (t ) − M b32 (t ) − a2 (t ) > 0, t →∞ ⎪ ⎪ ⎪ ⎩ lim inf 2m b33 (t ) − M b13 (t ) + b31 (t ) − M b23 (t ) + b32 (t ) + a3 (t ) > 0, (3.6) t →∞ then (x∗1 (t ), x∗2 (t ), x∗3 (t )) is globally asymptotically stable Proof From (3.6), there exists t > t such that (3.6) holds when we replace lim inft →∞ in (3.6) by inft t1 Let (x1 (t ), x2 (t ), x3 (t )) be any solution of (1.1) with positive initial value Since Γ is an ultimately bounded region, there exists T > t such that (x1 (t ), x2 (t ), x3 (t )) ∈ Γ and (x∗1 (t ), x∗2 (t ), x∗3 (t )) ∈ Γ for all t T T For brevity, we denote xi (t ), x∗i (t ), (t ) Considering a Liapunov function defined by V (t ) = i =1 |xi (t ) − x∗i (t )|, t ∗ and b i j (t ) by xi , xi , and b i j , respectively A direct calculation of the right derivative D + V (t ) of V (t ) along the solution of system (1.1) produces 430 T.V Ton / J Math Anal Appl 362 (2010) 427–437 D + V (t ) = sgn xi − x∗i xi − x∗i i =1 = xi a i − i =1 bi j x j − x∗i − j =1 + sgn x3 − x∗3 b i j x∗j sgn xi − x∗i j =1 x3 −a3 + b3 j x j − b33 x3 j =1 − b ii xi + x∗i = − x∗3 −a3 + b3 j x∗j − b33 x∗3 j =1 xi − x∗i − sgn xi − x∗i i =1 b i j xi x j − x∗i x∗j − a3 + b33 x3 + x∗3 x3 − x∗3 j =1 , j = i + sgn x3 − x∗3 b3 j x3 x j − x∗3 x∗j j =1 − b ii xi + x∗i − = i =1 bi j x j xi − x∗i − sgn xi − x∗i j =1 , j = i b i j x∗i x j − x∗j j =1 , j = i − a3 + b33 x3 + x∗3 − b31 x1 − b32 x2 x3 − x∗3 + sgn x3 − x∗3 b3 j x∗3 x j − x∗j j =1 Then D + V (t ) − b ii xi + x∗i − i =1 bi j x j xi − x∗i + j =1 , j = i b i j x∗i x j − x∗j j =1 , j = i − a3 + b33 x3 + x∗3 − b31 x1 − b32 x2 x3 − x∗3 + b3 j x∗3 x j − x∗j j =1 − 2b ii mi − i =1 bi j m j xi − x∗i + M i j =1 , j = i b i j x j − x∗j j =1 , j = i − a3 + 2b33m3 − b31 M − b32 M x3 − x∗3 + M b3 j x j − x∗j j =1 = M b21 + M b31 + a1 − 2m1 b11 − m2 b12 − m3 b13 x1 − x∗1 + M b12 + M b32 + a2 − 2m2 b22 − m1 b21 − m3 b23 x2 − x∗2 + M (b13 + b31 ) + M (b23 + b32 ) − 2m3 b33 − a3 x3 − x∗3 From (3.6) it follows that there exists a positive constant D + V (t ) −μ xi (t ) − x∗i (t ) for all t μ > such that T (3.8) i =1 Integrating on both sides of (3.8) from T to t produces t V (t ) + μ T1 xi (t ) − x∗i (t ) dt V ( T ) < +∞ for all t i =1 Then t T1 Hence, xi (t ) − x∗i (t ) i =1 i =1 |xi dt μ − x∗i | ∈ L ([ T , +∞)) V ( T ) < +∞ (3.7) for all t T T T.V Ton / J Math Anal Appl 362 (2010) 427–437 431 On the other hand, the ultimate boundedness of xi (t ) and x∗i imply that xi (t ) and x∗i , i = 1, 2, all have bounded derivatives for t T (from the equations satisfied by them) As a consequence on [ T , +∞) By Lemma 2.6 we have is uniformly continuous xi (t ) − x∗i (t ) = lim t →∞ ∗ i =1 |xi (t ) − xi (t )| i =1 ✷ which completes the proof The model with constant effects In this section, we assume that the coefficients b i j , shall assume that M [ai ] = lim in system (1.1) are positive constants Furthermore, we t0 +T T →∞ i, j (t ) dt T (4.1) t0 exists uniformly with respect to t in (−∞, ∞) First, we consider a predator–prey system x1 (t ) = x1 (t ) a1 (t ) − b11 x1 (t ) − b13 x3 (t ) , x3 (t ) = x3 (t ) −a3 (t ) + b31 x1 (t ) − b33 x3 (t ) (4.2) Put Z i (T ) = t0 +T zi (t ) dt , T t0 we have the following theorem Theorem 4.1 Assume that b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > Then inft t0 x1 (t ) > Furthermore, i) If M [a3 ] < then inft t0 b31 b11 M [a1 ] x3 (t ) > and lim X ( T ) = T →∞ lim X ( T ) = b33 M [a1 ] + b13 M [a3 ] b13 b31 + b11 b33 b31 M [a1 ] − b11 M [a3 ] b13 b31 + b11 b33 T →∞ , ii) If M [a3 ] > b31 b11 M [a1 ] then lim X ( T ) = M [a1 ] T →∞ b11 , lim X ( T ) = T →∞ Proof To proof the first statement, we use the same proof as in Theorem 3.1 Let > be a sufficient small constant From the comparison theorem and from x1 (t ) x1 (t )[a1u − b11 x1 (t )], it is easy to get that lim sup x1 (t ) t →∞ a1u b11 432 T.V Ton / J Math Anal Appl 362 (2010) 427–437 Then there exists T > t such that x1 (t ) < P := x3 (t ) < x3 (t ) −al3 + b31 P − b33 x3 (t ) a1u b11 + for t for all t T Thus T (4.3) Consider two cases Case 1: There exists > such that −al3 + b31 P < From (4.3), it follows that limt →∞ x3 (t ) = Therefore, there exists T > T such that a1 (t ) − b13 x3 (t ) > 12 al1 It follows from the first equation of system (4.2) that x1 (t ) x1 (t ) al1 − b11 x1 (t ) for t T Using the comparison theorem, we obtain al1 lim inf x1 (t ) t →∞ 2b11 Case 2: −al3 + b31 P 10 It follows from (4.3) that P := lim sup x3 (t ) t →∞ −al3 + b31 P b33 Then, we can choose a sufficient positive small equation of system (4.2), we have x1 (t ) al1 − b13 P − b11 x1 (t ) x1 (t ) and T > T such that x1 (t ) for t P , x3 (t ) P for all t T Because of our assumption b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > 0, there exists a sufficient positive small b11 b13 al3 + b11 b33 al1 − b13 b31 a1u al1 − b13 P = b11 b33 − b13 b31 b33 Then lim inft →∞ x1 (t ) > From the conclusions of two above cases, we obtain that inft c < x1 (t ) < d1 for all t T From the first such that > t0 x1 (t ) > Then there exists c > such that t0 (4.4) To prove Part i), first, we show that it is impossible to have lim x3 (t ) = (4.5) t →∞ Assume the contrary, it follows from (4.4) and (4.5) that lim T →∞ lim T →∞ T ln x1 (t + T ) x1 (t ) = 0, t0 +T x3 (s) ds = T t0 Then, we have from the first equation of (4.2) that lim T →∞ t0 +T b11 x1 (s) ds = lim T T →∞ t0 Since (4.5) implies that T ln x3 (t + T ) x3 (t ) t →∞ If, contrary to the assertion of the theorem, inft t0 x3 (t ) = 0, then there exists a sequence of numbers {sn }∞ such that sn t , sn → ∞ as n → ∞ and x3 (sn ) → as n → ∞ Put c= t0 +T lim inf T →∞ x3 (t ) dt T t0 Since x3 (t ) > c for arbitrarily large values of t and since sn → ∞ and x3 (sn ) → as n → ∞, there exist sequences { pn }∞ , ∞ {qn }∞ 1, t < pn < τn < qn < pn+1 , x3 ( pn ) = x3 (qn ) = c and and {τn }1 such that for all n < x3 (τn ) < c n exp{−b31 d1n} ∗ ∞ From this we see that there exist sequences {tn }∞ and {tn }1 such that for n tn < τn < tn∗ , c x3 (tn ) = x3 tn∗ = , n c for t ∈ tn , tn∗ x3 (t ) n (4.7) Thus tn∗ 0< ∗ tn − tn c x3 (t ) dt n → as n → ∞ (4.8) tn We declare the following inequalities hold: tn∗ − tn > tn∗ − τn n for n (4.9) In fact, x3 (t ) = x3 (t ) −a3 (t ) + b31 x1 (t ) − b33 x3 (t ) < b31 d1 x3 (t ) for all t τn t , then for t t −a3 (s) + b31 x1 (s) − b33 x3 (s) ds x3 (t ) = x3 (τn ) exp τn c = n c n exp{−b31 d1n} exp b31 d1 (t − τn ) exp b31 d1 (t − τn − n) From (4.10) and (4.7), we obtain that tn∗ − τn (4.10) n It follows (4.9) that t∗ M [ai ] = lim ∗ n→∞ tn n − tn (t ) (i = 1, 3) tn From the first equation of system (4.2) we get that tn∗ − tn ln x1 (tn∗ ) x1 (tn ) = tn∗ − tn tn∗ tn∗ a1 (t ) dt − b11 tn tn∗ x1 (t ) dt − b13 tn x3 (t ) dt tn 434 T.V Ton / J Math Anal Appl 362 (2010) 427–437 Then, it follows from (4.4), (4.8) and (4.9) that lim ∗ n→∞ tn tn∗ − tn x1 (t ) dt = M [a1 ] tn b11 (4.11) Similarly, from the second equation of system (4.2) we have tn∗ − tn ln x3 (tn∗ ) = x3 (tn ) tn∗ a3 (t ) dt + b31 − tn∗ − tn tn∗ tn tn∗ x1 (t ) dt − b33 tn x3 (t ) dt tn From this and from (4.7), (4.8) and (4.11), we get that − M [a3 ] + b31 b11 M [a1 ] = Since this contradicts our assumption, we obtain that inft c < x3 (t ) < d3 for all t t0 x3 (t ) > Therefore, there exists c > such that t0 (4.12) Now, from system (4.2), for all T > 0, we have ⎧ x1 (t + T ) ⎪ ⎪ = A ( T ) − b11 X ( T ) − b13 X ( T ), ⎨ ln T x1 (t ) ⎪ ⎪ ⎩ ln x3 (t0 +T ) = − A ( T ) + b X ( T ) − b X ( T ) x3 (t ) T 31 33 x1 (t + T ) ] + b13 [ T1 x1 (t ) ln Then X1 (T ) = X3 (T ) = b33 [ A ( T ) − T ln x3 (t + T ) x3 (t ) + A ( T )] x3 (t + T ) x3 (t ) + A ( T )] b13 b31 + b11 b33 b31 [ A ( T ) − T ln x1 (t + T ) ] − b11 [ T1 x1 (t ) ln b13 b31 + b11 b33 , (4.13) It follows from (4.4) and (4.12) that lim T →∞ T ln xi (t + T ) xi (t ) = (i = 1, 3), then lim X ( T ) = T →∞ lim X ( T ) = T →∞ b33 M [a1 ] + b13 M [a3 ] b13 b31 + b11 b33 b31 M [a1 ] − b11 M [a3 ] b13 b31 + b11 b33 , To prove Part ii), first, we show that limt →∞ x3 (t ) = Assume the contrary, then there exist δ > and a sequence of numbers { T n }∞ , T n > 0, T n → ∞ (n → ∞) such that δ < x3 (t + T n ) < d3 for all n Then, from the second equation of (4.13), we get that lim X ( T n ) = n→∞ b31 M [a1 ] − b11 M [a3 ] b13 b31 + b11 b33 This contradiction follows that limt →∞ x3 (t ) = and then lim T →∞ X ( T ) = It follows from the first equation of (4.13) that limT →∞ X ( T ) = Mb[a1 ] ✷ 11 Now, we consider the following system ⎧ ⎪ ⎨ x1 (t ) = x1 (t ) a1 (t ) − b11 x1 (t ) − b12 x2 (t ) − b13 x3 (t ) , x2 (t ) = x2 (t ) a2 (t ) − b21 x1 (t ) − b22 x2 (t ) − b23 x3 (t ) , ⎪ ⎩ x3 (t ) = x3 (t ) −a3 (t ) + b31 x1 (t ) + b32 x2 (t ) − b33 x3 (t ) (4.14) T.V Ton / J Math Anal Appl 362 (2010) 427–437 435 Proposition 4.2 If the following conditions hold M [a1 ] > M [a3 ] < b12 b22 M [a2 ], M [a2 ] > b21 b11 M [a1 ] and (b21 b32 − b31 b22 ) M [a1 ] + (b12 b31 − b11 b32 ) M [a2 ] b12 b21 − b11 b22 (4.15) then lim supt →∞ x3 (t ) > Proof Assume the contrary, then limt →∞ x3 (t ) = Thus lim X ( T ) = (4.16) T →∞ By replacing t by a larger number, if necessary, we may assume that (t ) − b i3 x3 (t ) > for t for i = 1, 2, ⎧ t t0 , ⎨ (t ) − b i3 x3 (t ), (t ) = (t ) − (t − t + 1)b i3 x3 (t ), t − t < t , ⎩ (t ), t < t − 1, ∗ t − and i = 1, We put, (4.17) then a∗i is continuous on R, a∗i l > 0, a∗i u < ∞ Moreover, since limt →∞ x3 (t ) = 0, the limit ∗ M [ai ] = lim T →∞ t∗ +T ∗ (t ) dt = lim T t∗ T →∞ t∗ +T (t ) dt = M [ai ] T t∗ exists uniformly with respect to t ∗ ∈ R and i = 1, Then for t system t , (x1 (t ), x2 (t )) is a solution of the following competitive x1 (t ) = x1 (t ) a∗1 (t ) − b11 x1 (t ) − b12 x2 (t ) , x2 (t ) = x2 (t ) a∗2 (t ) − b21 x1 (t ) − b22 x2 (t ) This system has been studied in [1] (see Theorem 2.1) By condition (4.15), we have −b22 M [a1 ] + b12 M [a2 ] , T →∞ b12 b21 − b11 b22 b21 M [a1 ] − b11 M [a2 ] lim X ( T ) = T →∞ b12 b21 − b11 b22 lim X ( T ) = (4.18) From the third equation of system (4.14) we have T ln x3 (t + T ) x3 (t ) = − A ( T ) + b31 X ( T ) + b32 X ( T ) − b33 X ( T ) Then − A ( T ) + b31 X ( T ) + b32 X ( T ) − b33 X ( T ) < for T sufficiently large Let T → ∞ and using (4.16) and (4.18) we obtain that − M [a3 ] + (b21 b32 − b31 b22 ) M [a1 ] + (b12 b31 − b11 b32 ) M [a2 ] b12 b21 − b11 b22 which contradicts (4.15) This proves the proposition ✷ Proposition 4.3 If one of the following conditions holds ⎧ b31 ⎪ M [a3 ] < M [a1 ], ⎪ ⎪ ⎪ b11 ⎨ (b21 b33 + b23 b31 ) M [a1 ] + (b21 b13 − b11 b23 ) M [a3 ] M [a2 ] > , ⎪ ⎪ b13 b31 + b11 b33 ⎪ ⎪ ⎩ b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > 0, (4.19) 436 T.V Ton / J Math Anal Appl 362 (2010) 427–437 ⎧ b31 ⎪ M [a3 ] > M [a1 ], ⎪ ⎪ ⎪ b11 ⎨ b21 (4.20) M [a2 ] > M [a1 ], ⎪ ⎪ b11 ⎪ ⎪ ⎩ b11 b13 al3 + b11 b33 al1 − b13 b31 a1u > then lim supt →∞ x2 (t ) > Proof Similarly to Proposition 4.2, we assume the contrary, then t0 +T lim x2 (t ) = 0, x2 (t ) dt = lim t →∞ T →∞ (4.21) t0 and (x1 (t ), x3 (t )) is a solution of a predator–prey system x1 (t ) = x1 (t ) a∗1 (t ) − b11 x1 (t ) − b13 x3 (t ) , x3 (t ) = x3 (t ) −a∗3 (t ) + b31 x1 (t ) − b33 x3 (t ) where a∗1 (t ), a∗3 (t ) are defined as in (4.17) by replacing x3 (t ) by x2 (t ) and b i3 by b i2 First, if the condition (4.19) holds From Part i) of Theorem 4.1 and from (4.21) and t0 +T a2 (t ) dt − T i =1 t0 t0 +T b2i xi (t ) dt = T T ln t0 x2 (t + T ) x2 (t ) 0, (4.23) ⎧ b32 ⎪ M [a3 ] > M [a2 ], ⎪ ⎪ ⎪ b22 ⎨ b12 M [a1 ] > M [a2 ], ⎪ ⎪ b22 ⎪ ⎪ ⎩ b22 b23 al3 + b22 b33 al2 − b23 b32 a2u > then lim supt →∞ x1 (t ) > From Propositions 4.2–4.4, we obtain the main theorem in this section Theorem 4.5 If one of the following conditions holds A1 : (4.15), (4.20) and (4.23) hold, (4.24) T.V Ton / J Math Anal Appl 362 (2010) 427–437 437 A2 : (4.15), (4.20) and (4.24) hold, A3 : (4.15), (4.21) and (4.23) hold, A4 : (4.15), (4.21) and (4.24) hold then system (4.14) is persistent The model with the constant intrinsic growth rates In this section, we consider system (1.1) under the condition , b i j , ⎧ ⎪ ⎨ x1 (t ) = x1 (t ) a1 − b11 x1 (t ) − b12 x2 (t ) − b13 x3 (t ) , x2 (t ) = x2 (t ) a2 − b21 x1 (t ) − b22 x2 (t ) − b23 x3 (t ) , ⎪ ⎩ x3 (t ) = x3 (t ) −a3 + b31 x1 (t ) + b32 x2 (t ) − b33 x3 (t ) i, j are constants (5.1) Put x∗1 = x∗3 = a1 b33 + a3 b13 b13 b31 + b11 b33 a1 b31 − a3 b11 b13 b31 + b11 b33 , Theorem 5.1 If ⎧ ⎨ a < b31 a , b11 ⎩ a2 − b21 x∗1 − b23 x∗3 < (5.2) then the constant solution (x∗1 , 0, x∗3 ) of system (5.1) is locally asymptotically stable It means that if (x1 (t ), x2 (t ), x3 (t )) is any solution of (5.1) such that (x1 (t ), x3 (t )) is close to (x∗1 , x∗3 ) and x2 (t ) is sufficiently small and positive, then limt →∞ x1 (t ) = x∗1 , limt →∞ x3 (t ) = x∗3 , limt →∞ x2 (t ) = Proof It is easy to see that x∗1 > 0, x∗3 > and (x∗1 , 0, x∗3 ) is the constant solution of system (5.1) Put f (x1 , x2 , x3 ) = x1 (a1 − b11 x1 − b12 x2 − b13 x3 ), f (x1 , x2 , x3 ) = x2 (a2 − b21 x1 − b22 x2 − b23 x3 ), f (x1 , x2 , x3 ) = x3 (−a3 + b31 x1 + b32 x2 − b33 x3 ), then system (5.1) becomes xi = f i (x1 , x2 , x3 ), and f i (x∗1 , 0, x∗3 ) = 0, i ⎡ ∂ f1 ⎢ A=⎣ ∂ x1 ∂ f2 ∂ x1 ∂ f3 ∂ x1 ∂ f1 ∂ x2 ∂ f2 ∂ x2 ∂ f3 ∂ x2 i = 1, 2, 3, = 1, 2, Consider ∂ f1 ∂ x3 ∂ f2 ∂ x3 ∂ f3 ∂ x3 ⎤ ⎤ ⎡ −b12 x∗1 −b13 x∗1 −b11 x∗1 ⎥ ∗ ∗ ∗ ∗ ⎦ a2 − b21 x1 − b23 x3 ⎦ x1 , 0, x3 = ⎣ ∗ ∗ ∗ b32 x3 −b33 x3 b31 x3 It is easy to see that det( A − λ I ) = a2 − b21 x∗1 − b23 x∗3 − λ λ2 + b11 x∗1 + b33 x∗3 λ + (b11 b33 + b13 b31 )x∗1 x∗3 and all eigenvalues of A are less than zero Therefore (x∗1 , 0, x∗3 ) is locally asymptotically stable ✷ References [1] S Ahmad, I.M Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal (2004) 219–229 [2] I Barb˘alat, Systèmes dèquations diffèrentielles dosillations non linéaires, Rev Roumaine Math Pures Appl (1975) 267–270 [3] Y Xia, F Chen, A Chen, J Cao, Existence and global attractivity of an almost periodic ecological model, Appl Math Comput 157 (2004) 449–475 ... survival of species, Nonlinear Anal (2004) 219–229 [2] I Barb˘alat, Systèmes dèquations diffèrentielles dosillations non linéaires, Rev Roumaine Math Pures Appl (1975) 267–270 [3] Y Xia, F Chen, A Chen,... is easy to see that if system (1.1) has a solution is globally asymptotically stable, then any solution of (1.1) is also globally asymptotically stable Lemma 2.6 (See [2].) Let h be a real number... and all eigenvalues of A are less than zero Therefore (x∗1 , 0, x∗3 ) is locally asymptotically stable ✷ References [1] S Ahmad, I.M Stamova, Almost necessary and sufficient conditions for survival