DSpace at VNU: Dynamics of species in a model with two predators and one prey

DSpace at VNU: Dynamics of species in a model with two predators and one prey tài liệu, giáo án, bài giảng , luận văn, l...

Nonlinear Analysis 74 (2011) 4868–4881 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Dynamics of species in a model with two predators and one prey Ta Viet Ton a,∗ , Nguyen Trong Hieu b a Department of Applied Physics, Graduate School of Engineering, Osaka University, Suita Osaka 565-0871, Japan b Faculty of Mathematics, Mechanics and Informatics, Hanoi National University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam article abstract info Article history: Received May 2010 Accepted 27 April 2011 Communicated by Ravi Agarwal In this paper, we study a predator–prey model which has one prey and two predators with Beddington–DeAngelis functional responses Firstly, we establish a set of sufficient conditions for the permanence and extinction of species Secondly, the periodicity of positive solutions is studied Thirdly, by using Liapunov functions and the continuation theorem in coincidence degree theory, we show the global asymptotic stability of such solutions Finally, we give some numerical examples to illustrate the behavior of the model © 2011 Elsevier Ltd All rights reserved MSC: 34C27 34D05 Keywords: Predator–prey system Beddington–DeAngelis functional response Permanence Extinction Periodic solution Asymptotic stability Liapunov function Introduction The dynamical relationship between predators and prey has been studied by several authors for a long time In those researches, to represent the average number of prey killed per individual predator per unit of time, a functional, called the functional response, was introduced The functional response can depend on only the prey’s density or both the prey’s and the predator’s densities However, some biologists have argued that in many situations, especially when predators have to search for food, the functional response should depend on both the prey’s and the predator’s densities [1–6] One of the most popular functional responses is the fractional one as in the following prey–predator model It is called the Beddington–DeAngelis functional response:   x′1 = x1 (a1 − b1 x1 ) − c1 x1 x2 α + β x1 + γ x2 c2 x1 x2 ′  x2 = −a2 x2 + α + β x1 + γ x2 , In this model, xi (t ) represents the population density of species Xi at time t (i ≥ 1); X1 is the prey and X2 is the predator At time t , a1 (t ) is the intrinsic growth rate of X1 and (t ) is the death rate of X2 ; b1 (t ) measures the inhibiting effect of the environment on X1 This model was originally proposed by Beddington [7] and DeAngelis et al [8] independently Since the appearance of these two investigations, there have been many other ones for analogous systems with diffusion in a ∗ Corresponding author Tel.: +81 6879 4249 E-mail addresses: taviet.ton@ap.eng.osaka-u.ac.jp (T.V Ton), hieungt@vnu.edu.vn (N.T Hieu) 0362-546X/$ – see front matter © 2011 Elsevier Ltd All rights reserved doi:10.1016/j.na.2011.04.061 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 4869 constant environment [9–14] However, a constant environment is rarely the case in real life Most natural environments are physically highly variable, i.e., the coefficients in those models should depend on time [15–18] In order to continue studying such models, in this paper, we consider a predator–prey model of one prey and two predators with Beddington–DeAngelis functional responses:  c3 (t )x1 x3 c2 (t )x1 x2   − , x′1 = x1 [a1 (t ) − b1 (t )x1 ] −   α( t ) + β( t ) x + γ ( t ) x α( t ) + β(t )x1 + γ (t )x3  [ ]   d2 (t )x1 x′2 = x2 −a2 (t ) + − b2 (t )x3 , (1.1) α(t ) + β(t )x1 + γ (t )x2   [ ]   d3 (t )x1   x′3 = x3 −a3 (t ) + − b3 (t )x2 α(t ) + β(t )x1 + γ (t )x3 Here xi (t ) represents the population density of species Xi at time t (i ≥ 1), X1 is the prey and X2 , X3 are the predators Two predators share one prey and it is assumed that there are two types of competition between the two predators The first type is direct interference where individuals of each predator species act with aggression against individuals of the other predator species In our model, this type of competition is described by the coefficients b2 (t ) and b3 (t ) The second type of competition is interference competition that occurs during hunting because predators spend time interacting with each other rather than seeking prey Here we assume that there is no competition of that type between individuals of the two different predator d (t )x species Therefore, the Beddington–DeAngelis functional responses are of the form α(t )+β(it )x 1+γ (t )x (i = 2, 3) We use the i same coefficients α, β, γ in the functional responses of both predators, since it is assumed that both predators take the same time to handle a prey once they encounter it and that individuals of each predator species interfere with each other when hunting by exactly the same amount in both species This assumption is somewhat restrictive from the biological viewpoint, but it could be removed without greatly changing the analysis of system (1.1) Throughout this paper, it is assumed that the functions (t ), bij (t ), ci (t ), di (t ), α(t ), β(t ), γ (t ) (1 ≤ i, j ≤ 3) are continuous on R and bounded above and below by some positive constants This article is organized as follows Section provides some definitions and notation In Section 3, we state some results on invariant sets, and the permanence and extinction of system (1.1) Then, the asymptotic stability of solution is proved by using a Liapunov function In Section 4, we continue using other Liapunov functions and the continuation theorem in coincidence degree theory to show the existence and global stability of a positive periodic solution The final section illustrates the behavior of system (1.1) by some computational results and gives our conclusion Definitions and notation In this section we introduce some basic definitions and facts which will be used throughout this paper Let R3+ = {(x1 , x2 , x3 ) ∈ R3 | xi > (i ≥ 1)} Denote by x(t ) = (x1 (t ), x2 (t ), x3 (t )) the solution of system (1.1) with initial condition x0 = (x01 , x02 , x03 ) = (x1 (t0 ), x2 (t0 ), x3 (t0 )), t0 ≥ For biological reasons, throughout this paper, we only consider the solutions x(t ) with positive initial values, i.e., x0 ∈ R3+ Let g (t ) be a continuous function; for brevity, instead of writing g (t ) we write g If g is bounded on R, we denote g u = sup g (t ), t ∈R g l = inf g (t ), t ∈R ω and gˆ = ω g (t )dt, if g is a periodic function with period ω The global existence and uniqueness of solution of system (1.1) are guaranteed by the properties of the map defined by the right hand side of system (1.1) [19] We have the following lemma Lemma 2.1 Both the non-negative and positive cones of R3 are positively invariant for (1.1) Proof The solution x(t ) of (1.1) with initial value x0 satisfies ∫ t [ ]   c2 x2 c3 x3   x1 = x1 exp a1 − b x − − du ,   α + β x1 + γ x2 ]α + β x1 + γ x3   ∫t0t [  d2 x1 −a + x2 = x02 exp − b2 x3 du ,  α + β x1 + γ x2 t0 [  ∫ ]   t  d3 x1   x3 = x03 exp −a + − b3 x2 du α + β x1 + γ x3 t0 The conclusion follows immediately for all t ∈ [t0 , ∞) The proof is complete Definition 2.2 System (1.1) is said to be permanent if there exist some positive δj (j = 1, 2) such that δ1 ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ δ2 (i ≥ 1) t →∞ for all solutions of (1.1) t →∞ 4870 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 Definition 2.3 A set A ⊂ R3+ is called an ultimately bounded region of system (1.1) if for any solution x(t ) of (1.1) with positive initial values, there exists T1 > such that x(t ) ∈ A for all t ≥ t0 + T1 Definition 2.4 A bounded non-negative solution x∗ (t ) of (1.1) is said to be globally asymptotically stable (or globally ∑3 attractive) if any other solution x(t ) of (1.1) with positive initial values satisfies limt →∞ i=1 |xi (t ) − x∗i (t )| = Remark 2.5 It is easy to see that if a solution of (1.1) is globally asymptotically stable, then so are all solutions In this case, system (1.1) is also said to be globally asymptotically stable The model with general coefficients Let ϵ ≥ be sufficiently small Put M1ϵ = mϵ1 = mϵi = au1 bl1 Miϵ = + ϵ, al1 γ l − c2u − c3u bu1 γ l dui M1ϵ − ali α l ali γ l , − ϵ, (3.1) dli mϵ1 − (aui + bui Mjϵ )(β u mϵ1 + α u ) (aui + bui Mjϵ )γ u (i, j ≥ 2, i ̸= j), then Miϵ > mϵi (i ≥ 1) We will show that max{m0i , 0} (i ≥ 1) are the lower bounds for the limiting bounds of species Xi as time t tends to infinity This is obvious when m0i ≤ Therefore, it is assumed that m0i > Hypothesis 3.1 m0i > (i ≥ 1) Theorem 3.2 Under Hypothesis 3.1, for any sufficiently small ϵ > such that mϵi > (i ≥ 1), a set Γϵ defined by Γϵ = {(x1 , x2 , x3 ) ∈ R3 | mϵi < xi < Miϵ (i ≥ 1)} is positively invariant with respect to system (1.1) Proof Throughout this proof, we use the fact that the solution to the equation X ′ (t ) = A(t , X )X (t )[B − X (t )] (B ̸= 0) is given by X (t ) = BX exp  X exp  t t0  t t0 BA(s, X (s))ds    BA(s, X (s))ds − + B , where t0 ≥ and X = X (t0 ) Consider the solution of system (1.1) with an initial value x0 ∈ Γϵ From Lemma 2.1 and from the first equation of (1.1), we have x′1 (t ) ≤ x1 (t )[a1 (t ) − b1 (t )x1 (t )] ≤ x1 (t )[au1 − bl1 x1 (t )] = bl1 x1 (t )(M10 − x1 ) Using the comparison theorem gives x1 ( t ) ≤ ≤ x01 M10 exp{au1 (t − t0 )} x01 [exp{au1 (t − t0 )} − 1] + M10 x01 M1ϵ exp{au1 (t − t0 )} x01 [exp{au1 (t − t0 )} − 1] + M1ϵ < M1ϵ , t ≥ t0 It follows from the third equation of (1.1) and from (3.2) that x′2 ≤ −al2 x2 + ≤ −al2 x2 + du2 x1 x2 α l + β l x1 + γ l x2 du2 M1ϵ x2 α l + γ l x2 (3.2) T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 x2 (du2 M1ϵ − al2 α l ) − al2 γ l x2  = = 4871  α l + γ l x2 al2 γ l α + γ l x2 l x2 (M2ϵ − x2 ) Putting C2 ( t ) = al2 γ l α l + γ l x2 (t ) , (3.3) and using the comparison theorem again yields  x2 (t ) ≤ M2ϵ x02 exp M2ϵ   x02 exp M2ϵ t t0 t t0 C2 (s)ds    C2 (s)ds − + M2ϵ < M2ϵ , t ≥ t0 (3.4) Similarly, x3 (t ) < M3ϵ for every t ≥ t0 Now, by the first equation of (1.1), it implies that x′1 (t ) ≥ x1  al1 − c2u + c3u γl − bu1 x1  = bu1 x1 (m01 − x1 ) Since x01 > mϵ1 , by the comparison theorem, we obtain x1 (t ) ≥ x01 m01 exp{bu1 m01 (t − t0 )} x01 [exp{bu1 m01 (t − t0 )} − 1] + m01 > mϵ1 for all t ≥ t0 Similarly, for i, j ≥ (i ̸= j), x′i = −ai xi + di x1 xi α + β x1 + γ xi − bi xi xj d l m ϵ xi i ≥ −(aui + bui Mjϵ )xi + u α + β u mϵ1 + γ u xi   l ϵ di m1 − (aui + bui Mjϵ )(β u mϵ1 + α u ) xi − (aui + bui Mjϵ )γ u x2i = α u + β u mϵ1 + γ u xi u u ϵ u (ai + bi Mj )γ = u xi (mϵi − xi ), α + β u mϵ1 + γ u xi from which follows that xi (t ) > mϵi for all t ≥ t0 We complete the proof In the next theorem, the permanence of system (1.1) is shown A treatment called practical persistence to prove the permanence of models and its application to various types of models can be seen in [20,9,21] Theorem 3.3 Under Hypothesis 3.1, for any sufficiently small ϵ > such that mϵi > 0, mϵi ≤ lim inf xi (t ) ≤ lim sup xi (t ) ≤ Miϵ t →∞ t →∞ (i ≥ 1) Consequently, system (1.1) is permanent Proof According to the proof of Theorem 3.2 we have x1 (t ) ≤ x01 M10 exp{au1 (t − t0 )} x01 [exp{au1 (t − t0 )} − 1] + M10 Thus, lim supt →∞ x1 (t ) ≤ M10 , i.e., there exists t1 ≥ t0 such that x1 (t ) < M1ϵ for all t ≥ t1 By the same arguments as made for (3.4), it follows that  x2 (t ) ≤ M2ϵ x12 exp M2ϵ   x12 exp M2ϵ t t1 t t1 C2 (s)ds    C2 (s)ds − + M2ϵ , (3.5) from which it is implied that < x2 (t ) ≤ max{M2ϵ , x12 } for all t ≥ t1 , where x12 = x2 (t1 ) Then from (3.3), inft ≥t1 C2 (s) > By using (3.5), we have lim supt →∞ x2 (t ) ≤ M2ϵ Similarly, lim supt →∞ x3 (t ) ≤ M3ϵ and lim inft →∞ xi (t ) ≥ mϵi (i ≥ 1) The permanence follows from Definition 2.2 The proof is complete 4872 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 Theorem 3.4 Let i ∈ {2, 3} If Mi0 < then limt →∞ xi (t ) = 0, i.e., the ith predator goes to extinction Proof It follows from Mi0 < that Miϵ < with a sufficiently small ϵ Similarly to the proof of Theorem 3.2 we have x′i (t ) ≤ ali γ l α + γ l xi l xi (Miϵ − xi ) < (3.6) Thus, there exists C ≥ such that limt →∞ xi (t ) = C and C ≤ xi (t ) ≤ x0i for all t ≥ t0 If C > then from (3.6) there exists µ > such that x′i (t ) < −µ for all t ≥ t0 We therefore have xi (t ) < −µ(t − t0 ) + x0i and then limt →∞ xi (t ) = −∞, which contradicts xi (t ) > for all t ≥ t0 Hence, limt →∞ xi (t ) = In order to consider the global asymptotic stability of system (1.1), we need the following result called Barbalat’s lemma Lemma 3.5 (See [22]) Let h be a real number and f be a non-negative function defined on [h, +∞) such that f is integrable and uniformly continuous on [h, +∞) Then limt →∞ f (t ) = Theorem 3.6 Suppose that Hypothesis 3.1 holds and let ϵ > be sufficiently small such that mϵi > (i ≥ 1) Let x∗ be a solution of system (1.1) satisfying β c2 M2ϵ + d2 (α + γ M2ϵ ) β c3 M3ϵ + d3 (α + γ M3ϵ ) + u2 (mϵ1 , M2ϵ ) u3 (mϵ1 , M3ϵ ) t →∞   ϵ ϵ di γ m1 ci (α + β M1 ) − < 0, lim sup bj + ϵ ϵ ui (M1 , mi ) ui (mϵ1 , Miϵ ) t →∞   lim sup −b1 + < 0, (3.7) where ui (a, b) = (α + β x∗1 + γ x∗i )(α + β a + γ b) (i, j ≥ 2, i ̸= j) Then x∗ is globally asymptotically stable Proof Let x be the other solution of (1.1) From Theorem 3.3, Γϵ is an ultimately bounded region of (1.1) Then there exists ∑3 ∗ T1 > such that x, x∗ ∈ Γϵ for all t ≥ t0 + T1 Consider a Liapunov function defined by V (t ) = i=1 | ln xi − ln xi |, t ≥ t0 + A direct calculation of the right derivative D V (t ) of V (t ) along the solution of (1.1) gives D+ V (t ) = − sgn(xi − x∗i )  i=1  = sgn(x1 − x∗1 ) −c2  x′i xi  − x∗i ′  x∗i x2 α + β x1 + γ x2 x3 x∗3 − x∗2  α + β x∗1 + γ x∗2   − − b1 (x1 − x1 ) α + β x1 + γ x3 α + β x∗1 + γ x∗3 [ ] x∗1 x1 b2 ∗ + d2 sgn(x2 − x∗2 ) − ( x − x ) − 3 α + β x1 + γ x2 α + β x∗1 + γ x∗2 d2 [ ] b3 x1 x∗1 ∗ + d3 sgn(x3 − x∗3 ) − − ( x − x ) 2 α + β x1 + γ x3 α + β x∗1 + γ x∗3 d3 ∗ ∗ ∗ α(x2 − x2 ) + β(x1 x2 − x1 x2 ) ≤ −b1 |x1 − x∗1 | − c2 sgn(x1 − x∗1 ) u2 (x1 , x2 ) ∗ ∗ ∗ ∗ α(x3 − x3 ) + β(x1 x3 − x1 x3 ) − c3 sgn(x1 − x1 ) u3 (x1 , x3 ) ∗ ∗ ∗ α( x − x 1 ) + γ (x1 x2 − x2 x1 ) + d2 sgn(x2 − x∗2 ) + b2 |x3 − x∗3 | u2 (x1 , x2 ) α(x1 − x∗1 ) + γ (x1 x∗3 − x3 x∗1 ) + d3 sgn(x3 − x∗3 ) + b3 |x2 − x∗2 | u3 (x1 , x3 ) − c3 ∗ It follows from x, x∗ ∈ Γϵ for t ≥ t0 + T1 and x1 x∗i − x∗1 xi = x1 (x∗i − xi ) + xi (x1 − x∗1 ) (i = 2, 3) that (α + β x1 )(x2 − x∗2 ) − β x2 (x1 − x∗1 ) u2 (x1 , x2 ) ∗ (α + β x )( x − x ) − β x ( x1 − x∗1 ) 3 − c3 sgn(x1 − x∗1 ) u3 (x1 , x3 ) (α + γ x2 )(x1 − x∗1 ) − γ x1 (x2 − x∗2 ) + d2 sgn(x2 − x∗2 ) + b2 |x3 − x∗3 | u2 (x1 , x2 ) D+ V (t ) ≤ −b1 |x1 − x∗1 | − c2 sgn(x1 − x∗1 ) T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 (α + γ x3 )(x1 − x1 ) − γ x1 (x3 − x3 ) + d3 sgn(x3 − x∗3 ) + b3 |x2 − x∗2 | u2 (x1 , x3 ) ] [ β c2 M2ϵ + d2 (α + γ M2ϵ ) β c3 M3ϵ + d3 (α + γ M3ϵ ) + |x1 − x∗1 | ≤ −b + u2 (mϵ1 , M2ϵ ) u3 (mϵ1 , M3ϵ ) [ ] c2 (α + β M1ϵ ) d2 γ mϵ1 + b3 + − |x2 − x∗2 | u2 (M1ϵ , mϵ2 ) u2 (mϵ1 , M2ϵ ) [ ] c3 (α + β M1ϵ ) d3 γ mϵ1 + b2 + − |x3 − x∗3 | for t ≥ t0 + T1 u3 (M1ϵ , mϵ3 ) u3 (mϵ1 , M3ϵ ) ∗ 4873 ∗ (3.8) Combining (3.7) and (3.8) gives the existence of a positive number µ > and of T2 ≥ t0 + T1 such that D+ V (t ) ≤ −µ − |xi − x∗i | for every t ≥ T2 (3.9) i =1 Integrating both sides of (3.9) from T2 to t yields V (t ) + µ ∫ t − T2  Then t T2  |xi − xi | ds ≤ V (T2 ) < ∞ for every t ≥ T2 ∗ i=1  ∑3 i =1 |xi − x∗i | ds ≤ µ−1 V (T2 ) < ∞ for every t ≥ T2 Hence, ∑3 i=1 |xi − x∗i | ∈ L1 ([T2 , ∞)) On the other hand, it follows from x, x∗ ∈ Γϵ for all t ≥ t0 + T1 and from the equations of (1.1) that the derivatives ∑3 ∗ of xi (t ), x∗i (t )(i ≥ 1) are bounded on [T2 , ∞) As a consequence i=1 |xi − xi | is uniformly continuous on [T2 , ∞) By Lemma 3.5 we have limt →∞ ∑3 i=1 |xi − x∗i | = 0, which completes the proof The model with periodic coefficients In this section, we assume that the coefficients in system (1.1) are ω-periodic in t and bounded above and below by some positive constants We study the existence and stability of a periodic solution of this system To this, we will employ an alternative approach to establish some criteria in terms of the average of the related functions over an interval of the common period That is continuation theorem in coincidence degree theory, which has been successfully used to establish criteria for the existence of positive periodic solutions of some mathematical models of predator–prey type; we refer the reader to [23–26] To this end, we shall summarize in the following a few concepts and results from [27] that will be basic for this section Let X and Y be two Banach spaces, let L:Dom L ⊂ X → Y be a linear mapping, and let N :X → Y be a continuous mapping The mapping L will be called a Fredholm mapping of index zero if the following conditions hold: (i) Im L is closed; (ii) dim Ker L = codim Im L < ∞ If L is a Fredholm mapping of index zero and there exist continuous projections P :X → X and Q :Y → Y such that Im P = Ker L, Im L = Ker Q = Im(I − Q ), it follows that Lp = L|Dom L∩Ker P :(I − P )X → Im L is invertible We denote by Kp the inverse of that map If Ω is an open bounded subset of X, the mapping N will be called ¯ if the mapping QN : Ω ¯ → Y is continuous and bounded, and Kp (I − Q )N : Ω ¯ → X is compact, i.e., L-compact on Ω ¯ ) is relatively compact Since Im Q is isomorphic to Ker L, there exists an isomorphism it is continuous and Kp (I − Q )N (Ω J :Im Q → Ker L The following continuation theorem is from [27] Lemma 4.1 (Continuation Theorem) Let X and Y be two Banach spaces and L a Fredholm mapping of index zero Assume that ¯ → Y is L-compact on Ω ¯ with Ω is open and bounded in X Furthermore, assume that N :Ω (a) for each λ ∈ (0, 1), x ∈ ∂ Ω ∩ Dom L, Lx ̸= λNx; (b) for each x ∈ ∂ Ω ∩ Ker L, QNx ̸= 0; (c) deg{QNx, Ω ∩ Ker L, 0} ̸= 0; ¯ then the operator equation Lx = Nx has at least one solution in Dom L ∩ Ω 4874 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 We now put aˆ L11 = ln L12 bˆ 1 , H11 = ln aˆ bˆ + 2aˆ ω,    c2 + c3 = ln aˆ − , γ Li1 = ln aˆ (dˆ i − aˆ i β l ) exp{2aˆ ω} − aˆ i bˆ α l aˆ i bˆ γ l    di Hi1 = Li2 = ln H12 = L12 − 2aˆ ω, β , ω + Li1 , di exp{H12 } − (α u + β u exp{H12 })(ˆai + bˆ i exp{Hj1 }) γ u (ˆai + bˆ i exp{Hj1 })    Hi2 = Li2 − di β , ω (i, j ≥ 2, i ̸= j) The convention here is that ln x = −∞ if x ≤ In the next theorem, a sufficient condition for existence of an ω-periodic solution of (1.1) is presented Theorem 4.2 If Li2 > −∞ (i ≥ 1) then system (1.1) has at least one positive ω-periodic solution Proof Put xi (t ) = exp{ui (t )} (i ≥ 1), then system (1.1) becomes  c3 exp{u3 } c2 exp{u2 }   u′1 = a1 − b1 exp{u1 } − − ,   α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 }   d2 exp{u1 } − b2 exp{u3 }, u′2 = −a2 + (4.1)  α + β exp{u1 } + γ exp{u2 }    ′ d3 exp{u1 }  u3 = −a3 + − b3 exp{u2 } α + β exp{u1 } + γ exp{u3 } ∑3 Let X = Y = {u = (u1 , u2 , u3 )T ∈ C1 (R, R3 ) | ui (t ) = ui (t + ω) (i ≥ 1)} with ‖u‖ = i=1 maxt ∈[0,T ] |ui (t )|, u ∈ X Then X, Y are both Banach spaces with the above norm ‖ · ‖ Let     u1 N1 (t ) N u2 = N2 (t ) u3 N (t )   c2 exp{u2 } c3 exp{u3 } a1 − b1 exp{u1 } − −  α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 }    d2 exp{u1 }   − b2 exp{u3 } −a + =  ,   α + β exp{u1 } + γ exp{u2 }   d3 exp{u1 } − b3 exp{u2 } −a + α + β exp{u1 } + γ exp{u3 }   ∫ ω u1 (t )dt             ω ∫0  1 ω u′1 u1 u1 u1 u1   ′ u2 ∈ X u2 (t )dt  , L u2 = u2  , P u2 = Q u2 =   ω ∫0  u3 u3 u3 u3 u′3  1 ω u3 (t )dt ω ω   T Then Ker L = {u ∈ X | u = (h1 , h2 , h3 ) ∈ R }, Im L = u ∈ Y | ui (t )dt = (i ≥ 1) , and dim Ker L = = codim Im L Since Im L is closed in Y, L is a Fredholm mapping of index zero It is easy to show that P , Q are continuous projections such that Im P = Ker L, Im L = Ker Q = Im (I − Q ) Furthermore, the generalized inverse (to L) KP :Im L → Dom L ∩ Ker P exists and is given by t ∫ u1 u2 u3   KP u1 (s)ds − ∫0  t  =  u2 (s)ds − ∫0  t u3 (s)ds − ω ∫ ∫ t T ∫0 ∫0 ω t u1 (s)dsdt     u2 (s)dsdt   T ∫0 ∫0  ω t u3 (s)dsdt T 0 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 4875 ¯ with any open bounded Obviously, QN and KP (I − Q )N are continuous It is easy to see that N is L-compact on Ω set Ω ⊂ X Now we will find an appropriate open, bounded subset Ω for application of the continuation theorem Corresponding to the operator equation Lu = λNu, λ ∈ (0, 1), we have  u1 = λ a1 − b1 exp{u1 } − c2 exp{u2 } ′  u = λ −a + d2 exp{u1 } ′ α + β exp{u1 } + γ exp{u2 }  u = λ −a + α + β exp{u1 } + γ exp{u2 } − α + β exp{u1 } + γ exp{u3 }  , − b2 exp{u3 } , (4.2)  d3 exp{u1 } ′  c3 exp{u3 } α + β exp{u1 } + γ exp{u3 } − b3 exp{u2 } Suppose that (u1 , u2 , u3 ) ∈ X is an arbitrary solution of system (4.2) for a certain λ ∈ (0, 1) Integrating both sides of (4.2) over the interval [0, ω], we obtain aˆ ω = ∫ ω c3 exp{u3 } c2 exp{u2 }  + dt , α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 } ∫ ω ∫ ω di exp{u1 }dt aˆ i ω + bi exp{uj }dt = α + β exp{u1 } + γ exp{ui } 0   ∫ ω  di di dt = ω (i, j ≥ 2, i ̸= j) ≤ β β b1 exp{u1 } + (4.3) It follows from (4.2) and (4.3) that for i, j ≥ (i ̸= j), ω ∫ ∫ |u1 (t ) |dt ≤ λ ′ ω ∫ a1 dt + ∫ + 0 ω ω ∫ b1 exp{u1 }dt + 0 ∫ ω c2 exp{u2 } dt α + β exp{u1 } + γ exp{u2 }  c3 exp{u3 } dt α + β exp{u1 } + γ exp{u3 } < 2aˆ ω,   ω  di |ui (t )′ |dt < ω β Since u ∈ X, there exist ξi , ηi ∈ [0, ω] such that ui (ξi ) = ui (t ), ui (ηi ) = max ui (t ) t ∈[0,ω] t ∈[0,ω] (i ≥ 1) From the first equation of (4.3) and (4.4), we obtain aˆ ω ≥ u1 (ξ1 ) < L11 Hence u1 (t ) ≤ u1 (ξ1 ) + ω ∫ (4.4) ω b1 exp{u1 (ξ1 )}dt = bˆ ω exp{u1 (ξ1 )}, from which follows |u′1 (t )|dt < L11 + 2aˆ ω = H11 for all t ≥ 0 On the other hand, from the first equation of (4.3) and (4.4), we also have aˆ ω ≤ ω ∫ b1 exp{u1 (η1 )}dt + ω ∫ c2 (t ) + c3 (t )  = bˆ exp{u1 (η1 )} +    c2 + c3 ω γ Then for any t ≥ 0, u1 (t ) ≥ u1 (η1 ) − ω ∫ |u′1 (t )|dt     c2 + c3 ≥ ln aˆ − − 2aˆ ω γ = H12 γ (t ) dt 4876 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 From the arguments above, we have H12 ≤ u1 (t ) ≤ H11 for all t ∈ [0, ω] It then follows from the second equation of (4.3) and (4.4) that ω ∫ di exp{u1 }dt α + β exp{u1 } + γ exp{ui } ∫ ω di exp{H11 }dt ≤ l l α + β exp{H11 } + γ l exp{ui (ξi )} ωdˆ i exp{H11 } = l , l α + β exp{H11 } + γ l exp{ui (ξi )} aˆ i ω ≤ from which it is implied that ui (ξi ) ≤ ln (dˆ i − aˆ i β l ) exp{H11 } − aˆ i α l aˆ i γ l aˆ (dˆ i − aˆ i β l ) exp{2aˆ ω} − aˆ i bˆ α l = ln aˆ i bˆ γ l , and then ui (t ) ≤ ui (ξi ) + ω ∫ |u′i (t )|dt ≤ ln aˆ (dˆ i − aˆ i β l ) exp{2aˆ ω} − aˆ i bˆ α l aˆ i bˆ γ l    +2 di β ω = Hi1 (i ≥ 2) Similarly, for i, j ≥ (i ̸= j) and t ≥ 0, we have aˆ i ω = ω ∫ [ di exp{u1 } ] − bi exp{uj } dt α + β exp{u1 } + γ exp{ui } ] di exp{H12 } ≥ − bi exp{Hj1 } dt α u + β u exp{H12 } + γ u exp{ui (ηi )}   dˆ i exp{H12 } = − bˆ i exp{Hj1 } ω, α u + β u exp{H12 } + γ u exp{ui (ηi )} ∫ ω [ from which it is implied that ui (ηi ) ≥ ln di exp{H12 } − (α u + β u exp{H12 })(ˆai + bˆ i exp{Hj1 }) γ u (ˆai + bˆ i exp{Hj1 }) , and ui (t ) ≥ ui (ηi ) − ω ∫ |u′i (t )|dt ≥ ln di exp{H12 } − (α u + β u exp{H12 })(ˆai + bˆ i exp{Hj1 }) γ u (ˆai + bˆ i exp{Hj1 })    di −2 ω β = Hi2 Put Bi = max{|Hi1 |, |Hi2 |} (i ≥ 1), then maxt ∈[0,ω] |ui | ≤ Bi Thus, for any solution u ∈ X of (4.2), we have ‖u‖ ≤ and, clearly, Bi (i ≥ 1) are independent of λ Taking B = ∑4 ∑3 i=1 Bi , i=1 Bi where B4 is taken sufficiently large such that B4 ≥ i=1 j=1 |Lij | and letting Ω = {u ∈ X| ‖u‖ < B}, then Ω satisfies the condition (a) of Lemma 4.1 To compute the Brouwer degree, let us consider the homotopy ∑3 ∑2 Hµ (u) = µQN (u) + (1 − µ)G(u), µ ∈ [0, 1], T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 4877 where G : R3 → R3 , ˆ aˆ − ∫ bω1 exp{u1 } d2 (t ) exp{u1 }dt   −ˆa2 − bˆ exp{u3 } + G(u) =    ω −ˆa3 − bˆ exp{u2 } + ω ∫0 ω    α + β exp{u2 } + γ exp{u2 }    d3 (t ) exp{u1 }dt α + β exp{u3 } + γ exp{u3 } We have ]  ∫ ω[ µc2 exp{u2 } µc3 exp{u3 } ˆ + dt aˆ − b1 exp{u1 } −   ω α + β exp{u1 } + γ exp{u2 } α + β exp{u1 } + γ exp{u3 }   ∫ ω   d exp { u } dt   ˆ Hµ (u) =  −ˆa2 − b2 exp{u3 } +  ω α + β exp{u1 } + γ exp{u2 }   ∫ ω   d3 exp{u1 }dt −ˆa3 − bˆ exp{u2 } + ω α + β exp{u1 } + γ exp{u3 }  By carrying out similar arguments to those above, one can easily show that any solution u∗ of the equation Hµ (u) = ∈ R3 with µ ∈ [0, 1] satisfies Li1 ≤ u∗i ≤ Li2 (i ≥ 1) Thus, ̸∈ Hµ (∂ Ω ∩ Ker L) for µ ∈ [0, 1], and then QN (∂ Ω ∩ Ker L) ̸= Note that the isomorphism J can be the identity mapping I; since Im P = Ker L, by the invariance property of homotopy, we have deg(JQN , Ω ∩ Ker L, 0) = deg(QN , Ω ∩ Ker L, 0) = deg(QN , Ω ∩ R3 , 0) = deg(G, Ω ∩ R3 , 0)    −bˆ exp{u1 } 0      ∂ f2 (u1 , u2 )   ∂ f2 (u1 , u2 )   ˆ  − b2 exp{u3 } = sgn det   ∂ u ∂ u   ∂ f (u 1, u )   ∂ f3 (u1 , u3 )  3     −bˆ exp{u2 } ∂ u1 ∂ u3    ∂ f2 (u1 , u2 ) ∂ f3 (u1 , u3 ) ˆ ˆ = −sgn bˆ exp{u1 } + b2 b3 exp{u2 + u3 } , ∂ u2 ∂ u3 (4.5) where deg(·, ·, ·) is the Brouwer degree [28] and f ( u1 , u2 ) = f ( u1 , u3 ) = ω ω ∫ ∫0 ω ω d2 exp{u1 }dt , α + β exp{u1 } + γ exp{u2 } d3 exp{u1 }dt α + β exp{u1 } + γ exp{u3 } It is easy to see that the functions fi (u1 , ui ) are decreasing in ui ∈ R (i ≥ 2) Then ∂ f2 (u1 , u2 ) ∂ f3 (u1 , u3 ) > ∂ u2 ∂ u3 (4.6) Combining (4.5) and (4.6) gives deg(JQN , Ω ∩ Ker L, 0) = −1 ̸= By now we have proved that Ω verifies all requirements ¯ , i.e., (4.1) has at least one ω-periodic solution u∗ in of Lemma 4.1, then Lu = Nu has at least one solution in Dom L ∩ Ω ¯ Set x∗i = exp{u∗i }(i ≥ 1), then x∗ is an ω-periodic solution of system (1.1) with strictly positive components We Dom L ∩ Ω complete the proof Corollary 4.3 If the ω-periodic solution x∗ in Theorem 4.2 satisfies the assumptions in Theorem 3.6, then x∗ is globally asymptotically stable Proof The proof of this corollary is derived directly from Theorems 3.6 and 4.2 Numerical examples and conclusion In this section, we present some numerical examples As a first example, we consider the case a1 = 4.7 + sin(π t ), b1 = 2.4 − cos(2.7t ), c2 = 10.1 + 2.2 sin(1.4t ), c3 = 9.3 + cos(2.8t ), a2 = (1.1 − cos(2π t ))/2.5, b2 = 1.4 + sin(0.8t ), d2 = 9.9 − 0.4 sin(0.6t ), a3 = (2.2 − cos(1.7t ))/6, b3 = (2.3 + 1.3 sin(3.2t ))/2, d3 = 8.5 + sin(0.9π t ), α = (1.2 − cos(2t ))/4, β = (2.3 + cos(1.2t ))/5, γ = 5.5 − 0.5 sin(t ) By (3.1), Mi0 > 0, m0i > (i ≥ 1) then Hypothesis 3.1 holds and system (1.1) 4878 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 x3 x1 x2 Fig Orbit of globally asymptotically stable system t-x1 1.5 1.26 x1 1.02 0.78 0.54 0.3 12 16 24 30 t Fig Population sizes X1 with respect to time t-x2 1.6 1.28 x2 0.96 0.64 0.32 0 12 18 24 30 t Fig Population sizes X2 with respect to time has an invariant set Fig is the orbit of solution with initial value x0 = (1.3, 1.4, 1.1); it seems to be very chaotic but it is permanent According to Theorem 3.6, it is not only permanent but also globally asymptotically stable In spite of different initial values, x0 = (1.3, 1.4, 1.1) and x¯ = (0.6, 1, 1.5), solutions xi and x¯ i (i ≥ 1) still tend to one trajectory (see Figs 2–4) In the next example, Theorem 3.4 will be illustrated by system (1.1) with coefficients a1 = + 1.2 sin(2.4t ), b1 = 2.4 + cos(π t ), c2 = 5.1 − 0.9 sin(1.7π t ), c3 = 4.4 − 1.2 cos(π t ), a2 = 1.1 − cos(1.9t ), b2 = (2.6 + sin(3π t ))/4, d2 = 4.3 − 1.7 sin(0.5π t ), a3 = 2.1 − 2.5 cos(1.4π t ), b3 = 0.8 − 1.3 sin(1.6t ), d3 = 2.9 + 2.3 sin(0.4t ), α = (1.8 − cos(5.7t ))/3, T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 4879 t-x3 1.56 1.32 x3 1.08 0.84 0.6 0.36 12 18 24 30 t Fig Population sizes X3 with respect to time x3 x2 x1 Fig Orbit of non-permanent system X1 X2 X3 2.4 1.8 1.2 0.6 0 12 16 20 t Fig Orbit of non-permanent system with respect to time β = − 0.2 cos(0.2π t ), γ = 3.4 − 1.6 sin(1.8t ) and initial condition x0 = (1.3, 2.1, 2.4) Since M30 < then the density of species X3 becomes extinct (see Figs 5–6) Therefore, the predator X3 vanishes and system (1.1) is not permanent For the model with periodic coefficients, we consider the last example concerning the numerical solutions of system (1.1) where a1 = + 1.3 sin(2t ), b1 = 2.2 + 1.9 cos(2t ), c2 = 2.8(2.2 − sin(2t )), c3 = (3.5 − cos(2t ))/2, a2 = − 0.6 cos(2t ), b2 = 1.2 + 0.5 sin(2t ), d2 = + 1.8 sin(2t ), a3 = (1.2 − cos(2t ))/3, b3 = 1.4 + 1.1 sin(2t ), d3 = 3.1 − 2.3 sin(2t ), α = 0.1(3.2 − cos(2t )), β = (2.1 + 1.8 cos(2t ))/5, γ = 3.3 − sin(2t ) and the initial value x0 = (1.1, 1.9, 1.4) Under π -periodic perturbation satisfying Theorem 4.2, system (1.1) has positive π -periodic solution (see Figs 7–8) Moreover, as the hypothesis of Theorem 3.6 also holds then it is globally asymptotically stable In conclusion, this work provides some results about the asymptotic behavior of a model of one prey and two predators with Beddington–DeAngelis functional responses The mathematical analysis presented in this model shows 4880 T.V Ton, N.T Hieu / Nonlinear Analysis 74 (2011) 4868–4881 x3 x1 x2 Fig Orbit of periodic system X1 X2 X3 1.6 1.2 0.8 0.4 0 3.6 7.2 10.8 14.4 18 t Fig Orbit of periodic system with respect to time that according to the values of the coefficients, one can make suitable predictions about the asymptotic behavior of the overall predator–prey system including the permanence, the periodicity, the global asymptotic stability and especially the extinction of species Those conclusions warn us to make timely decisions to protect species in our ecological system Further, the given conditions on coefficients can be easily numerically computed Acknowledgments The authors would like to thank the anonymous referees for their very helpful suggestions which improved the manuscript References [1] R Arditi, H.R Akcakaya, Underestimation of mutual interference of predators, 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Oscillation and global attractivity of hematopoiesis model with delay time, Appl Math Comput 136 (2003) 27–36 R.E Gaines, J.L Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer,... (Continuation Theorem) Let X and Y be two Banach spaces and L a Fredholm mapping of index zero Assume that ¯ → Y is L-compact on Ω ¯ with Ω is open and bounded in X Furthermore, assume that N