- Email: [email protected]
Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion
Chaos, Solitons and Fractals 29 (2006) 474–482 www.elsevier.com/locate/chaos
Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion Shuwen Zhang b
a,*
, Dejun Tan c, Lansun Chen
q
b
a Institute of Biomathematics, Anshan Normal University, Anshan, Liaoning 114005, PR China Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China c Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114005, PR China
Accepted 18 August 2005
Abstract In the paper, we considered a predator–prey model with Beddington–DeAngelis functional response in periodic pulsed chemostat. We discussed the boundness of system and the stability of prey and predator-eradication periodic solution of system. Further, using numerical simulation method, we show that this impulsive system with periodically pulsed substrate display a series of complex phenomena, which include (1) period-doubling cascade, (2) period-halfing cascade, (3) chaos and (4) periodic window. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The goal of this paper is to study the system for a chemostat with predator, prey, and periodically pulsed substrate, which incorporate the Beddington–DeAngelis functional response. The model takes the form: 8 9 m1 sðtÞxðtÞ > > 0 > > s ; ðtÞ ¼ DsðtÞ > > > > k 1 A þ sðtÞ þ BxðtÞ > > > > > > > = > m1 sðtÞxðtÞ m2 xðtÞyðtÞ > 0 > > x ðtÞ ¼ DxðtÞ þ ; t 6¼ nT ; > > A þ sðtÞ þ BxðtÞ k 2 A1 þ xðtÞ þ B1 yðtÞ > > > < > > > m2 xðtÞyðtÞ > ð1:0Þ > ; ; y 0 ðtÞ ¼ DyðtÞ þ > > A þ xðtÞ þ B yðtÞ > 1 1 > > 9 > > sðnT þ Þ ¼ xðnT Þ þ p; > > > = > > > > t ¼ nT . xðnT þ Þ ¼ xðnT Þ; > > > : ; þ yðnT Þ ¼ yðnT Þ; q *
This work is supported by National Natural Science Foundation of China (10171106) and (40372111). Corresponding author. E-mail address: [email protected] (S. Zhang).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.026
S. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 474–482
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where s(t), x(t) and y(t) represent the concentration of limiting substrate, prey, and predator. D is the dilution rate; m1 and m2 are the uptake and predation constants of the prey and predator; k1 is the yield of prey per unit mass of substrate; k2 is the biomass yield of predator per unit mass of prey; T is the period of the impulsive effect. A, A1, B, B1 are positive constants. N be the set of non-negative integers. Many paper [1–3] studied a chemostat model with the Michaelis–Menten functional response. But, there are few papers which study a chemostat model with Beddington–DeAngelis functional response. The Beddington–DeAngelis functional response is different from the traditional monotone or non-monotone functional response. It was introduced by Beddington and DeAnGelis et al. [4]. It is similar to the well-known Holling type II functional response but has an extra term Bx(t) or B1x(t) in the denominator that models mutual interference in a species. It can be derived mechanistically via considerations of time utilization [5,6] or spatial limits on predation [7]. Harrison [8] showed that the Beddington–DeAngelis functional response (for intraspecific interference competition) was superior to functional response without such competition in a microbial predator–prey interaction. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is, jumps) in their values. Recently, equations of this kind are found in a almost every domain of applied sciences. Numerous examples are given in BainovÕs and his collaboratorÕs books [9,10]. Some impulsive differential equations have been recently introduced in population dynamics in relation to: impulsive birth [11,12], impulsive vaccination [13,14], chemotherapeutic treatment of disease [15,16] and population ecology [17]. Funasaki and Kot [18] studied a predator–prey model in a chemostat with predator, prey, and periodically pulsed substrate. They have investigated the existence and stability of the periodic solutions of the impulsive subsystem with substrate and prey. Further, they showed that impulsive invasion cause complex dynamics of system. Recently, it is of interest to investigate the possible existence of chaos in biological population. Tang and Chen [19] studied quasi-periodic solutions and chaos in a periodically forced predator–prey model with age structure for predators. Chaos in three species food chain system with Holling type II functional response and impulsive perturbations was demonstrated in [20]. Chaos in functional response host-parasitoid ecosystem models was demonstrated in [21]. In this paper, we will show that the impulsive effect cause complex phenomenon. For simplicity, set m ¼ mk11 ; n ¼ mk22 , we transform system (1.0) into 8 9 msðtÞxðtÞ > > 0 > > ; s ðtÞ ¼ DsðtÞ > > > > > > A þ sðtÞ þ BxðtÞ > > > > > > > > > = > m sðtÞxðtÞ nxðtÞyðtÞ > 1 0 > x ðtÞ ¼ DxðtÞ þ ; > t 6¼ nT ; > > A þ sðtÞ þ BxðtÞ A1 þ xðtÞ þ B1 yðtÞ > > > > > > > < > > > n1 xðtÞyðtÞ > 0 > y ; ðtÞ ¼ DyðtÞ þ ; > > þ xðtÞ þ B yðtÞ A 1 1 > > > > 9 > > sðnT þ Þ ¼ xðnT Þ þ p; > > > > > = > > > þ > t ¼ nT . Þ ¼ xðnT Þ; xðnT > > > > > > ; : þ yðnT Þ ¼ yðnT Þ;
ð1:1Þ
This paper is arranged as follows. In Section 2, some notations and lemmas are given. In Section 3, using the Floquet theory of impulsive equation and small amplitude perturbation skills, we prove the local stability of prey and predatoreradication periodic solution and give the boundness of the system (1.1). In Section 4, the results of numerical analysis are shown, moreover, these results are discussed briefly.
2. Preliminaries In this section, we will give some definitions, notations and lemmas which will be useful for our main results. Let R+ = [0, 1), R3þ ¼ fx 2 R3 jx P 0g. Denote f = (f1, f2, f3) the map defined by the right hand of the first, second and three equations of system (1.1). Let V : Rþ R3þ ! Rþ , then V is said to belong to class V0 if (1) V is continuous in ðnT ; ðn þ 1ÞT R3þ and for each x 2 R3þ ; n 2 N ; limðt;yÞ!ðnT þ ;xÞ ¼ V ðnT þ ; xÞ exists. (2) V is locally Lipschitzian in x.
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Definition 2.1. Let V 2 V0, then for ðt; xÞ 2 ðnT ; ðn þ 1ÞT R3þ , the upper right derivative of V(t, x) with respect to the impulsive differential system (1.1) is defined as 1 Dþ V ðt; xÞ ¼ limþ sup ½V ðt þ h; x þ hf ðt; xÞÞ V ðt; xÞ. h h!0 The solution of system (1.1) is a piecewise continuous function x : Rþ ! R3þ , x(t) is continuous on (nT, (n + 1)T], n 2 N and xðnT þ Þ ¼ limt!nT þ xðtÞ exists. The smoothness of f guarantees the global existence and uniqueness of the solution of system (1.1). For the details, it is referred to the books [10]. The following lemma is obvious. Lemma 2.2. Let X(t) = (s(t), x(t), y(t)) be a solution of system (1.1) with X(0+) P 0, then X(t) P 0 for all t P 0. And further X(t) > 0, for all t > 0 if X(0+) > 0. And we will use the following important comparison theorem on impulsive differential equation [10]: Lemma 2.3. Suppose V 2 V0. Assume that ( þ D V ðt; xÞ 6 gðt; V ðt; xÞÞ; t 6¼ nT ; V ðt; xðtþ ÞÞ 6 wn ðV ðt; xÞÞ;
t ¼ nT ;
ð2:1Þ
where g : R+ · R+ ! R is continuous in (nT, (n + 1)T] · R+ and for u 2 R+, n 2 N, limðt;yÞ!ðnT þ ;uÞ gðt; yÞ ¼ gðnT þ ; uÞ exists, wn : R+ ! R+ is non-decreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation 8 0 u ðtÞ ¼ gðt; uðtÞÞ; t 6¼ nT ; > < uðtþ Þ ¼ wn ðuðtÞÞ; t ¼ nT ; ð2:2Þ > : þ uð0 Þ ¼ u0 ; existing on [0, 1). Then V(0+, x0) 6 u0, implies that V(t, x(t)) 6 r(t), t P 0, where x(t) is any solution of (2.1). Finally, we give some basic properties about the following subsystem of system (1.1) in the absence of prey and predator. 8 0 t 6¼ nT ; > < s ðtÞ ¼ DsðtÞ; sðtþ Þ ¼ sðtÞ þ p; t ¼ nT ; ð2:3Þ > : þ sð0 Þ ¼ s0 . ÞÞ p ; t 2 ðnT ; ðn þ 1ÞT ; n 2 N ; s ð0þ Þ ¼ 1expðDT is a positive periodic solution of system (2.3). Clearly s ðtÞ ¼ p expðDðtnT 1expðDT Þ Þ þ p Since sðtÞ ¼ ðsð0 Þ 1expðDT ÞÞ expðDtÞ þ s ðtÞ is the solution of system (2.3) with initial value s0 P 0, where t 2 (nT, (n + 1)T], n 2 N, we get
Lemma 2.4. For a positive periodic solution s (t) of system (2.3) and every solution s(t) of system (2.3) with s0 P 0, we have js(t) s (t)j ! 0, when t ! 1. Therefore, we obtain the following periodic solution of system (1.1) p expðdðt nT ÞÞ ðs ðtÞ; 0; 0Þ ¼ ; 0; 0 . 1 expðdT Þ for t 2 (nT, (n + 1)T].
3. Boundness Theorem h3.1. Let (s(t), x(t), y(t)) i be any solution of (1.1), then (s (t), 0, 0) is locally asymptotically stable provided that
1 T >m ln D2
AþpA expðDT Þ Aþp expðDT ÞA expðDT Þ
.
Proof. The local stability of periodic solution (s (t), 0, 0) may be determined by considering the behavior of small amplitude perturbations of the solution. Define sðtÞ ¼ uðtÞ þ s ðtÞ;
xðtÞ ¼ vðtÞ;
yðtÞ ¼ wðtÞ
S. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 474–482
there may be written 0 1 0 1 uðtÞ uð0Þ B C B C @ vðtÞ A ¼ UðtÞ@ vð0Þ A; wðtÞ wð0Þ
477
0 6 t < T;
where U(t) satisfies 1 0 ms ðtÞ 0 C B D A þ s ðtÞ C dU B C B ¼B m1 s ðtÞ CUðtÞ C B 0 D þ dt 0 A @ A þ s ðtÞ 0 0 D and U(0) = I, the identity 0 1 0 1 uðnT þ Þ B C B @ vðnT þ Þ A ¼ @ 0 0 wðnT þ Þ
matrix. The linearization of the third and fourth equations of system (1.1) becomes 1 10 uðnT Þ 0 0 C CB 1 0 A@ vðnT Þ A. wðnT Þ 0 1
Hence, if both eigenvalues of 0 1 1 0 0 B C M ¼ @ 0 1 0 AUðT Þ; 0 0 1 have absolute values less than one, then the periodic solution (s (t), 0, 0) is locally stable. Since all eigenvalues of M are Z T m1 s ðtÞ dt ; l3 ¼ expðDT Þ < 1 l1 ¼ expðDT Þ < 1; l2 ¼ exp D þ A þ s ðtÞ 0
Fig. 1. Bifurcation diagrams of system (1.1) with A = A1 = 1, B = 0.09, B1 = 0.1, m = m1 = 9, n = n1 = 5, D = 1, T = 6. (a), (b) and (c) s, x, y are plotted for p over [10, 100] respectively.
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Fig. 2. Period-doubling cascade to chaos: (a) phase portrait of 2T-periodic solution for p = 20, (b) phase portrait of 4T-periodic solution for p = 28, (c) phase portrait of chaos solution for p = 60.
Fig. 3. Bifurcation diagrams of system (1.1) with A = A1 = 1, k1 = 1, B = 0.09, B1 = 0.1, n = n1 = 5, D = 1, T = 6, p = 50. (a), (b) and (c) s, x, y are plotted for m over [1, 9] respectively.
AþpA expðDT Þ jl2j < 1 if and only if T > mD12 ln½Aþp expðDT . According to Floquet theory of impulsive differential equation, the ÞA expðDT Þ prey-eradication solution (s (t), 0, 0) is locally stable. This completes the proof. h
Theorem 3.2. There exists a constant M > 0, such that s(t) 6 M, x(t) 6 M, y(t) 6 M for each solution (s(t), x(t), y(t)) of system (1.1) with all t large enough.
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Fig. 4. Period-halfing cascade to chaos: (a) phase portrait of T-periodic solution for m = 2, (b) phase portrait of T-periodic solution for m = 2.16, (c) phase portrait of chaos solution for m = 3.
Fig. 5. Bifurcation diagrams of system (1.1) with A = A1 = 1, k2 = 1, B = 0.09, B1 = 0.1, m = m1 = 9, D = 1, T = 6, p = 50. (a), (b) and (c) s, x, y are plotted for n over [1, 8.5] respectively.
Proof. Define V(t) as m mn yðtÞ. V ðtÞ ¼ sðtÞ þ xðtÞ þ m1 m1 n1 It is clear that V 2 V0. We calculate the upper right derivative of V(t) along a solution of system (1.1) and get the following impulsive differential equation
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Fig. 6. Bifurcation diagrams of system (1.1) with A = A1 = 1, B1 = 0.1, m = m1 = 9, n = n1 = 5, D = 1, T = 6, p = 40. (a), (b) and (c) s, x, y are plotted for B over [0.1, 5] respectively.
(
Dþ V ðtÞ þ LV ðtÞ ¼ ðD LÞsðtÞ þ
V ðt Þ ¼ V ðtÞ þ p;
m m1
D L xðtÞ mmn D L yðtÞ; 1 n1
t 6¼ nT ;
ð3:1Þ
t ¼ nT .
Let 0 < L < minfD; mm1 D; mmn Dg, then ðD LÞsðtÞ ðmm1 D LÞxðtÞ ðmmn D LÞyðtÞ is bounded. Select h0 and h1 1 n1 1 n1 such that þ D V ðtÞ 6 h0 V ðtÞ þ h1 ; t 6¼ nT ; V ðtþ Þ ¼ V ðtÞ þ p;
t ¼ nT ;
where h0, h1 are two positive constant. According to Lemma 2.3, we have h1 P ð1 expðnh0 T ÞÞ h1 V ðtÞ 6 V ð0þ Þ expðh0 T Þ expðh0 ðt nT ÞÞ þ ; expðh0 tÞ þ expðh0 T Þ 1 h0 h0 where t 2 (nT, (n + 1)T]. Hence lim V ðtÞ 6
t!1
h1 p expðh0 T Þ . þ h0 expðh0 T Þ 1
Fig. 7. Period-halfing cascade: (a) phase portrait of 4T-periodic solution for B = 4, (b) phase portrait of 2T-periodic solution for B = 4.7.
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Therefore V(t) is ultimately bounded. We obtain that each positive solution of system (1.1) is uniformly ultimately bounded. This completes the proof. h
4. Chemostat chaos In this section we will study the influence of impulsive perturbation p on inherent oscillation. We will give some bifurcation diagrams about different bifurcation parameters. Since the corresponding continuous system (1.1) (p = 0) cannot be solved explicitly and system (1.1) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. However, the influence of bifurcation parameter may be documented by stroboscopically sampling one of the variables over a range of bifurcation parameter. Stroboscopic map is a special case of the Poincare´ map for periodically forced system or periodically pulsed system. Fixing points of the stroboscopic map correspond to periodic solutions of system (1.1) having the same period as the pulsing term; periodic points of period k about stroboscopic map correspond to entrained periodic solutions of system (1.1) having exactly k times the period of the pulsing; invariant circles correspond to quasi-periodic solutions of system (1.1); system (1.1) possibly appears chaotic (strange) attractors. Let A = A1 = 1, B = 0.09, B1 = 0.1, m = m1 = 9, n = n1 = 5, D = 1, T = 6. Bifurcation diagram (Fig. 1) shows the effect of impulsive perturbations with p increasing from 10 to 100. The system experiences process of periodic doubling cascade (Fig. 2) ! chaos ! periodic window (68.6 6 p 6 75) ! chaos. Figs. 3 and 5 show that the effect of bifurcation parameter m and n, respectively. The resulting bifurcation diagram (Fig. 3) clearly shows: (1) the first period-doubling at m = 2.07, (2) a cascade of period doubling (Fig. 4(a) and (b)), (3) chaotic solutions, (4) periodic windows within the chaotic regime (6.94 < m < 7.54). Bifurcation diagram (Fig. 5) shows the complexity are almost sameness with the behave of bifurcation diagram (Fig. 3). Figs. 6 and 8 show that the effect of bifurcation parameter B and B1, respectively. The complexity that they show are almost sameness. For bifurcation diagram (Fig. 6), when B < 3.24, system (1.1) incarnate the chaotic regions with periodic window. When B > 3.24, there is cascade of period halfing bifurcations from chaos to T-periodic solution (Fig. 7). The periodic-doubling route to chaos is the hallmark of the Logistic and Ricker maps [22,23] and has been studied extensively by Mathematicians [24,25]. Periodic halfing is the flip bifurcation in the opposite direction, which is also observed in [26,27].
Fig. 8. Bifurcation diagrams of system (1.1) with A = A1 = 1, B = 0.09, m = m1 = 9, n = n1 = 5, D = 1, T = 6, p = 50. (a), (b) and (c) s, x, y are plotted for B1 over [0.01, 0.55] respectively.
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5. Conclusion In this paper, we have investigated a model for a chemostat with predator, prey, Beddington–DeAngelis functional response and periodically pulsed substrate. We have proved prey and predator eradication periodic solution (s (t), 0, 0) is locally asymptotically stable and obtained the system is boundary. We have obtained some bifurcation diagrams (Figs. 1, 3, 5, 6, 8) with different bifurcation parameters. Numberical analysis indicates that the system (1.1) exhibits the rich dynamics, which include: (1) periodic solution, (2) periodic doubling cascade, (3) chaos (chaotic region with periodic window), (4) periodic-halfing cascade. All these numberical results show that dynamical behavior of system (1.1) becomes more complex under periodically impulsive perturbations.
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Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion Shuwen Zhang b
a,*
, Dejun Tan c, Lansun Chen
q
b
a Institute of Biomathematics, Anshan Normal University, Anshan, Liaoning 114005, PR China Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China c Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114005, PR China
Accepted 18 August 2005
Abstract In the paper, we considered a predator–prey model with Beddington–DeAngelis functional response in periodic pulsed chemostat. We discussed the boundness of system and the stability of prey and predator-eradication periodic solution of system. Further, using numerical simulation method, we show that this impulsive system with periodically pulsed substrate display a series of complex phenomena, which include (1) period-doubling cascade, (2) period-halfing cascade, (3) chaos and (4) periodic window. 2005 Elsevier Ltd. All rights reserved.
1. Introduction The goal of this paper is to study the system for a chemostat with predator, prey, and periodically pulsed substrate, which incorporate the Beddington–DeAngelis functional response. The model takes the form: 8 9 m1 sðtÞxðtÞ > > 0 > > s ; ðtÞ ¼ DsðtÞ > > > > k 1 A þ sðtÞ þ BxðtÞ > > > > > > > = > m1 sðtÞxðtÞ m2 xðtÞyðtÞ > 0 > > x ðtÞ ¼ DxðtÞ þ ; t 6¼ nT ; > > A þ sðtÞ þ BxðtÞ k 2 A1 þ xðtÞ þ B1 yðtÞ > > > < > > > m2 xðtÞyðtÞ > ð1:0Þ > ; ; y 0 ðtÞ ¼ DyðtÞ þ > > A þ xðtÞ þ B yðtÞ > 1 1 > > 9 > > sðnT þ Þ ¼ xðnT Þ þ p; > > > = > > > > t ¼ nT . xðnT þ Þ ¼ xðnT Þ; > > > : ; þ yðnT Þ ¼ yðnT Þ; q *
This work is supported by National Natural Science Foundation of China (10171106) and (40372111). Corresponding author. E-mail address: [email protected] (S. Zhang).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.026
S. Zhang et al. / Chaos, Solitons and Fractals 29 (2006) 474–482
475
where s(t), x(t) and y(t) represent the concentration of limiting substrate, prey, and predator. D is the dilution rate; m1 and m2 are the uptake and predation constants of the prey and predator; k1 is the yield of prey per unit mass of substrate; k2 is the biomass yield of predator per unit mass of prey; T is the period of the impulsive effect. A, A1, B, B1 are positive constants. N be the set of non-negative integers. Many paper [1–3] studied a chemostat model with the Michaelis–Menten functional response. But, there are few papers which study a chemostat model with Beddington–DeAngelis functional response. The Beddington–DeAngelis functional response is different from the traditional monotone or non-monotone functional response. It was introduced by Beddington and DeAnGelis et al. [4]. It is similar to the well-known Holling type II functional response but has an extra term Bx(t) or B1x(t) in the denominator that models mutual interference in a species. It can be derived mechanistically via considerations of time utilization [5,6] or spatial limits on predation [7]. Harrison [8] showed that the Beddington–DeAngelis functional response (for intraspecific interference competition) was superior to functional response without such competition in a microbial predator–prey interaction. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is, jumps) in their values. Recently, equations of this kind are found in a almost every domain of applied sciences. Numerous examples are given in BainovÕs and his collaboratorÕs books [9,10]. Some impulsive differential equations have been recently introduced in population dynamics in relation to: impulsive birth [11,12], impulsive vaccination [13,14], chemotherapeutic treatment of disease [15,16] and population ecology [17]. Funasaki and Kot [18] studied a predator–prey model in a chemostat with predator, prey, and periodically pulsed substrate. They have investigated the existence and stability of the periodic solutions of the impulsive subsystem with substrate and prey. Further, they showed that impulsive invasion cause complex dynamics of system. Recently, it is of interest to investigate the possible existence of chaos in biological population. Tang and Chen [19] studied quasi-periodic solutions and chaos in a periodically forced predator–prey model with age structure for predators. Chaos in three species food chain system with Holling type II functional response and impulsive perturbations was demonstrated in [20]. Chaos in functional response host-parasitoid ecosystem models was demonstrated in [21]. In this paper, we will show that the impulsive effect cause complex phenomenon. For simplicity, set m ¼ mk11 ; n ¼ mk22 , we transform system (1.0) into 8 9 msðtÞxðtÞ > > 0 > > ; s ðtÞ ¼ DsðtÞ > > > > > > A þ sðtÞ þ BxðtÞ > > > > > > > > > = > m sðtÞxðtÞ nxðtÞyðtÞ > 1 0 > x ðtÞ ¼ DxðtÞ þ ; > t 6¼ nT ; > > A þ sðtÞ þ BxðtÞ A1 þ xðtÞ þ B1 yðtÞ > > > > > > > < > > > n1 xðtÞyðtÞ > 0 > y ; ðtÞ ¼ DyðtÞ þ ; > > þ xðtÞ þ B yðtÞ A 1 1 > > > > 9 > > sðnT þ Þ ¼ xðnT Þ þ p; > > > > > = > > > þ > t ¼ nT . Þ ¼ xðnT Þ; xðnT > > > > > > ; : þ yðnT Þ ¼ yðnT Þ;
ð1:1Þ
This paper is arranged as follows. In Section 2, some notations and lemmas are given. In Section 3, using the Floquet theory of impulsive equation and small amplitude perturbation skills, we prove the local stability of prey and predatoreradication periodic solution and give the boundness of the system (1.1). In Section 4, the results of numerical analysis are shown, moreover, these results are discussed briefly.
2. Preliminaries In this section, we will give some definitions, notations and lemmas which will be useful for our main results. Let R+ = [0, 1), R3þ ¼ fx 2 R3 jx P 0g. Denote f = (f1, f2, f3) the map defined by the right hand of the first, second and three equations of system (1.1). Let V : Rþ R3þ ! Rþ , then V is said to belong to class V0 if (1) V is continuous in ðnT ; ðn þ 1ÞT R3þ and for each x 2 R3þ ; n 2 N ; limðt;yÞ!ðnT þ ;xÞ ¼ V ðnT þ ; xÞ exists. (2) V is locally Lipschitzian in x.
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Definition 2.1. Let V 2 V0, then for ðt; xÞ 2 ðnT ; ðn þ 1ÞT R3þ , the upper right derivative of V(t, x) with respect to the impulsive differential system (1.1) is defined as 1 Dþ V ðt; xÞ ¼ limþ sup ½V ðt þ h; x þ hf ðt; xÞÞ V ðt; xÞ. h h!0 The solution of system (1.1) is a piecewise continuous function x : Rþ ! R3þ , x(t) is continuous on (nT, (n + 1)T], n 2 N and xðnT þ Þ ¼ limt!nT þ xðtÞ exists. The smoothness of f guarantees the global existence and uniqueness of the solution of system (1.1). For the details, it is referred to the books [10]. The following lemma is obvious. Lemma 2.2. Let X(t) = (s(t), x(t), y(t)) be a solution of system (1.1) with X(0+) P 0, then X(t) P 0 for all t P 0. And further X(t) > 0, for all t > 0 if X(0+) > 0. And we will use the following important comparison theorem on impulsive differential equation [10]: Lemma 2.3. Suppose V 2 V0. Assume that ( þ D V ðt; xÞ 6 gðt; V ðt; xÞÞ; t 6¼ nT ; V ðt; xðtþ ÞÞ 6 wn ðV ðt; xÞÞ;
t ¼ nT ;
ð2:1Þ
where g : R+ · R+ ! R is continuous in (nT, (n + 1)T] · R+ and for u 2 R+, n 2 N, limðt;yÞ!ðnT þ ;uÞ gðt; yÞ ¼ gðnT þ ; uÞ exists, wn : R+ ! R+ is non-decreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation 8 0 u ðtÞ ¼ gðt; uðtÞÞ; t 6¼ nT ; > < uðtþ Þ ¼ wn ðuðtÞÞ; t ¼ nT ; ð2:2Þ > : þ uð0 Þ ¼ u0 ; existing on [0, 1). Then V(0+, x0) 6 u0, implies that V(t, x(t)) 6 r(t), t P 0, where x(t) is any solution of (2.1). Finally, we give some basic properties about the following subsystem of system (1.1) in the absence of prey and predator. 8 0 t 6¼ nT ; > < s ðtÞ ¼ DsðtÞ; sðtþ Þ ¼ sðtÞ þ p; t ¼ nT ; ð2:3Þ > : þ sð0 Þ ¼ s0 . ÞÞ p ; t 2 ðnT ; ðn þ 1ÞT ; n 2 N ; s ð0þ Þ ¼ 1expðDT is a positive periodic solution of system (2.3). Clearly s ðtÞ ¼ p expðDðtnT 1expðDT Þ Þ þ p Since sðtÞ ¼ ðsð0 Þ 1expðDT ÞÞ expðDtÞ þ s ðtÞ is the solution of system (2.3) with initial value s0 P 0, where t 2 (nT, (n + 1)T], n 2 N, we get
Lemma 2.4. For a positive periodic solution s (t) of system (2.3) and every solution s(t) of system (2.3) with s0 P 0, we have js(t) s (t)j ! 0, when t ! 1. Therefore, we obtain the following periodic solution of system (1.1) p expðdðt nT ÞÞ ðs ðtÞ; 0; 0Þ ¼ ; 0; 0 . 1 expðdT Þ for t 2 (nT, (n + 1)T].
3. Boundness Theorem h3.1. Let (s(t), x(t), y(t)) i be any solution of (1.1), then (s (t), 0, 0) is locally asymptotically stable provided that
1 T >m ln D2
AþpA expðDT Þ Aþp expðDT ÞA expðDT Þ
.
Proof. The local stability of periodic solution (s (t), 0, 0) may be determined by considering the behavior of small amplitude perturbations of the solution. Define sðtÞ ¼ uðtÞ þ s ðtÞ;
xðtÞ ¼ vðtÞ;
yðtÞ ¼ wðtÞ
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there may be written 0 1 0 1 uðtÞ uð0Þ B C B C @ vðtÞ A ¼ UðtÞ@ vð0Þ A; wðtÞ wð0Þ
477
0 6 t < T;
where U(t) satisfies 1 0 ms ðtÞ 0 C B D A þ s ðtÞ C dU B C B ¼B m1 s ðtÞ CUðtÞ C B 0 D þ dt 0 A @ A þ s ðtÞ 0 0 D and U(0) = I, the identity 0 1 0 1 uðnT þ Þ B C B @ vðnT þ Þ A ¼ @ 0 0 wðnT þ Þ
matrix. The linearization of the third and fourth equations of system (1.1) becomes 1 10 uðnT Þ 0 0 C CB 1 0 A@ vðnT Þ A. wðnT Þ 0 1
Hence, if both eigenvalues of 0 1 1 0 0 B C M ¼ @ 0 1 0 AUðT Þ; 0 0 1 have absolute values less than one, then the periodic solution (s (t), 0, 0) is locally stable. Since all eigenvalues of M are Z T m1 s ðtÞ dt ; l3 ¼ expðDT Þ < 1 l1 ¼ expðDT Þ < 1; l2 ¼ exp D þ A þ s ðtÞ 0
Fig. 1. Bifurcation diagrams of system (1.1) with A = A1 = 1, B = 0.09, B1 = 0.1, m = m1 = 9, n = n1 = 5, D = 1, T = 6. (a), (b) and (c) s, x, y are plotted for p over [10, 100] respectively.
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Fig. 2. Period-doubling cascade to chaos: (a) phase portrait of 2T-periodic solution for p = 20, (b) phase portrait of 4T-periodic solution for p = 28, (c) phase portrait of chaos solution for p = 60.
Fig. 3. Bifurcation diagrams of system (1.1) with A = A1 = 1, k1 = 1, B = 0.09, B1 = 0.1, n = n1 = 5, D = 1, T = 6, p = 50. (a), (b) and (c) s, x, y are plotted for m over [1, 9] respectively.
AþpA expðDT Þ jl2j < 1 if and only if T > mD12 ln½Aþp expðDT . According to Floquet theory of impulsive differential equation, the ÞA expðDT Þ prey-eradication solution (s (t), 0, 0) is locally stable. This completes the proof. h
Theorem 3.2. There exists a constant M > 0, such that s(t) 6 M, x(t) 6 M, y(t) 6 M for each solution (s(t), x(t), y(t)) of system (1.1) with all t large enough.
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Fig. 4. Period-halfing cascade to chaos: (a) phase portrait of T-periodic solution for m = 2, (b) phase portrait of T-periodic solution for m = 2.16, (c) phase portrait of chaos solution for m = 3.
Fig. 5. Bifurcation diagrams of system (1.1) with A = A1 = 1, k2 = 1, B = 0.09, B1 = 0.1, m = m1 = 9, D = 1, T = 6, p = 50. (a), (b) and (c) s, x, y are plotted for n over [1, 8.5] respectively.
Proof. Define V(t) as m mn yðtÞ. V ðtÞ ¼ sðtÞ þ xðtÞ þ m1 m1 n1 It is clear that V 2 V0. We calculate the upper right derivative of V(t) along a solution of system (1.1) and get the following impulsive differential equation
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Fig. 6. Bifurcation diagrams of system (1.1) with A = A1 = 1, B1 = 0.1, m = m1 = 9, n = n1 = 5, D = 1, T = 6, p = 40. (a), (b) and (c) s, x, y are plotted for B over [0.1, 5] respectively.
(
Dþ V ðtÞ þ LV ðtÞ ¼ ðD LÞsðtÞ þ
V ðt Þ ¼ V ðtÞ þ p;
m m1
D L xðtÞ mmn D L yðtÞ; 1 n1
t 6¼ nT ;
ð3:1Þ
t ¼ nT .
Let 0 < L < minfD; mm1 D; mmn Dg, then ðD LÞsðtÞ ðmm1 D LÞxðtÞ ðmmn D LÞyðtÞ is bounded. Select h0 and h1 1 n1 1 n1 such that þ D V ðtÞ 6 h0 V ðtÞ þ h1 ; t 6¼ nT ; V ðtþ Þ ¼ V ðtÞ þ p;
t ¼ nT ;
where h0, h1 are two positive constant. According to Lemma 2.3, we have h1 P ð1 expðnh0 T ÞÞ h1 V ðtÞ 6 V ð0þ Þ expðh0 T Þ expðh0 ðt nT ÞÞ þ ; expðh0 tÞ þ expðh0 T Þ 1 h0 h0 where t 2 (nT, (n + 1)T]. Hence lim V ðtÞ 6
t!1
h1 p expðh0 T Þ . þ h0 expðh0 T Þ 1
Fig. 7. Period-halfing cascade: (a) phase portrait of 4T-periodic solution for B = 4, (b) phase portrait of 2T-periodic solution for B = 4.7.
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Therefore V(t) is ultimately bounded. We obtain that each positive solution of system (1.1) is uniformly ultimately bounded. This completes the proof. h
4. Chemostat chaos In this section we will study the influence of impulsive perturbation p on inherent oscillation. We will give some bifurcation diagrams about different bifurcation parameters. Since the corresponding continuous system (1.1) (p = 0) cannot be solved explicitly and system (1.1) cannot be rewritten as equivalent difference equations, it is difficult to study them analytically. However, the influence of bifurcation parameter may be documented by stroboscopically sampling one of the variables over a range of bifurcation parameter. Stroboscopic map is a special case of the Poincare´ map for periodically forced system or periodically pulsed system. Fixing points of the stroboscopic map correspond to periodic solutions of system (1.1) having the same period as the pulsing term; periodic points of period k about stroboscopic map correspond to entrained periodic solutions of system (1.1) having exactly k times the period of the pulsing; invariant circles correspond to quasi-periodic solutions of system (1.1); system (1.1) possibly appears chaotic (strange) attractors. Let A = A1 = 1, B = 0.09, B1 = 0.1, m = m1 = 9, n = n1 = 5, D = 1, T = 6. Bifurcation diagram (Fig. 1) shows the effect of impulsive perturbations with p increasing from 10 to 100. The system experiences process of periodic doubling cascade (Fig. 2) ! chaos ! periodic window (68.6 6 p 6 75) ! chaos. Figs. 3 and 5 show that the effect of bifurcation parameter m and n, respectively. The resulting bifurcation diagram (Fig. 3) clearly shows: (1) the first period-doubling at m = 2.07, (2) a cascade of period doubling (Fig. 4(a) and (b)), (3) chaotic solutions, (4) periodic windows within the chaotic regime (6.94 < m < 7.54). Bifurcation diagram (Fig. 5) shows the complexity are almost sameness with the behave of bifurcation diagram (Fig. 3). Figs. 6 and 8 show that the effect of bifurcation parameter B and B1, respectively. The complexity that they show are almost sameness. For bifurcation diagram (Fig. 6), when B < 3.24, system (1.1) incarnate the chaotic regions with periodic window. When B > 3.24, there is cascade of period halfing bifurcations from chaos to T-periodic solution (Fig. 7). The periodic-doubling route to chaos is the hallmark of the Logistic and Ricker maps [22,23] and has been studied extensively by Mathematicians [24,25]. Periodic halfing is the flip bifurcation in the opposite direction, which is also observed in [26,27].
Fig. 8. Bifurcation diagrams of system (1.1) with A = A1 = 1, B = 0.09, m = m1 = 9, n = n1 = 5, D = 1, T = 6, p = 50. (a), (b) and (c) s, x, y are plotted for B1 over [0.01, 0.55] respectively.
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5. Conclusion In this paper, we have investigated a model for a chemostat with predator, prey, Beddington–DeAngelis functional response and periodically pulsed substrate. We have proved prey and predator eradication periodic solution (s (t), 0, 0) is locally asymptotically stable and obtained the system is boundary. We have obtained some bifurcation diagrams (Figs. 1, 3, 5, 6, 8) with different bifurcation parameters. Numberical analysis indicates that the system (1.1) exhibits the rich dynamics, which include: (1) periodic solution, (2) periodic doubling cascade, (3) chaos (chaotic region with periodic window), (4) periodic-halfing cascade. All these numberical results show that dynamical behavior of system (1.1) becomes more complex under periodically impulsive perturbations.
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