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A Holling II functional response food chain model with impulsive perturbations
Chaos, Solitons and Fractals 24 (2005) 1269–1278 www.elsevier.com/locate/chaos
A Holling II functional response food chain model with impulsive perturbations q Shuwen Zhang b
a,*
, Lansun Chen
b
a Institute of Biomathematics, Anshan Normal University, Liaoning, Anshan 114005, PR China Department of Applied Mathematics, Dalian University of Technology, Liaoning, Dalian 116024, PR China
Accepted 15 September 2004
Abstract In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Biological control is, generally, manÕs use of a specially chosen living organism to control a particular pest, which is a component of an integrated pest management strategy. This chosen organism might be a predator, parasite, or disease which will attack the harmful insect. It is a form of manipulating nature to increase a desired effect. It may also be a more economical alternative to some insecticides. Some biological control measures can actually prevent economic damage to agricultural crops. Virtually all insect and mite pests have some natural enemies. One approach to biological control is augmentation, which is manipulation of existing natural enemies to increase their effectiveness. This can be achieved by mass production and periodic release of natural enemies of the pest, and by genetic enhancement of the enemies to increase their effectiveness at control. One of the first successful cases of biological control in greenhouses was that of the parasitoid Encarsia formosa against the greenhouse whitefly Trialeurodes vaporariorum on tomatoes and cucumbers [1,2]. Considering the exploited predator–prey system (harvesting or stocking) is very valuable, for it involves the human activities. It can be referred to papers [3,4], in which the human activities always happen in a short time or
q
This work is supported by National Natural Science Foundation of China (10171106). Corresponding author. E-mail addresses: [email protected] (S. Zhang), [email protected] (L. Chen).
*
0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.051
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instantaneously. The continuous action of human is then removed from the model, and replaced with an impulsive perturbation. These models are subject to short-term perturbations which are often assumed to be in the form of impulsive in the modelling process. Consequently, impulsive differential equations provide a natural description of such systems [5]. Equations of this kind are found in almost every domain of applied sciences. Numerous examples are given in BainovÕs and his collaboratorsÕ books [6]. Some impulsive equations have been recently introduced in population dynamics in relation to: impulsive vaccination [7], population ecology [8,9], chemotherapeutic treatment of disease [10,11] and impulsive birth [12,13]. In this paper, we will consider the following a Holling II functional responses food chain mathematical models of many biological control processes naturally call for differential systems with three equations describing the growth of plant, pest and top predator(control agent), respectively, and with periodic constant impulsive perturbation of top predator: 8 9 1 xðtÞyðtÞ x0 ðtÞ ¼ xðtÞð1 xðtÞÞ a1þb > > > > 1 xðtÞ > > > = > > a xðtÞyðtÞ a yðtÞzðtÞ 0 1 2 > > y ðtÞ ¼ d yðtÞ t 6¼ nT 1 > 1þb 1þb xðtÞ yðtÞ 1 2 > > > > > > < 0 ; a2 yðtÞzðtÞ z ðtÞ ¼ 1þb2 yðtÞ d 2 zðtÞ ð1:1Þ > 9 > > DxðtÞ ¼ 0 > > > = > > > > DyðtÞ ¼ 0 t ¼ nT > > > > ; : DzðtÞ ¼ p where x(t), y(t), z(t) are the densities of prey, predator and top predator at time t respectively, a1, a2, b1, b2, d1, d2 are positive constants. Dx(t) = x(t+) x(t), Dy(t) = y(t+) y(t), Dz(t) = z(t+) z(t), T is the period of the impulsive effect, n 2 N, N be the set of all non-negative integers. Observe that the simple relation of these three species: z prey on y and only on y, and y prey on x and nutrient recycling is not accounted for. This simple relation produces the so-called simple food chain system with impulsive perturbations. With model (1.1) we can take into account the possible exterior effects under which the population densities change very rapidly. An impulsive increase of the top predator population density is possible by artificial breeding of the species or release some species (p > 0). Firstly, we study the effects of impulsive perturbations generally, establish condition for extinction of predator and give boundary of the system. Secondly, by using the numerical method, we study the influences on the inherent oscillation caused by the impulsive perturbations. Recently, it is of interest to investigate the possible existence of chaos in biological population. Chaos in three chain system with classical Lotka–Volterra type interactions and Holling type II functional response was demonstrated in [14]. Hsu et al. [15] studied the three species food chain model with ratio-dependence and its applications to biological control. Gakkhar and Naji [16] investigate order and chaos in predator to prey ratio-dependent food chain. In this paper, we study influences of the impulsive perturbations on dynamics behavior of system (1.1). Numerical analysis shows that impulsive perturbations cause complex dynamics of system (1.1).
2. Preliminaries The model (1.1) we considered is based on the following three species food model: 8 0 1 xðtÞyðtÞ x ðtÞ ¼ xðtÞð1 xðtÞÞ a1þb > > 1 xðtÞ > < a1 xðtÞyðtÞ a2 yðtÞzðtÞ 0 y ðtÞ ¼ 1þb1 xðtÞ 1þb2 yðtÞ d 1 yðtÞ > > > : 0 2 yðtÞzðtÞ z ðtÞ ¼ a1þb d 2 zðtÞ 2 yðtÞ
ð2:1Þ
where x(t), y(t), z(t) are the densities of prey, predator and top predator at time t respectively, a1, a2, b1, b2, d1, d2 are positive constants. The three species food chain model (2.1) has at most five non-negative equilibrium: ! d1 a1 b1 d 1 d 1 Að0; 0; 0Þ; Bð1; 0; 0Þ; C ; ; 0 ; Di ðxi ; y i ; zi Þ; i ¼ 1; 2; a1 b1 d 1 ða1 b1 d 1 Þ2 where
S. Zhang, L. Chen / Chaos, Solitons and Fractals 24 (2005) 1269–1278
b1 1 xi ¼ þ ð1Þi1 2b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b1 d 2 ðb1 þ 1Þ2 a4a2 b 2 d2 2b1
;
yi ¼
d2 ; a2 b2 d 2
zi ¼
ða1 b1 d 1 Þxi d 1 ; ða2 b2 d 2 Þð1 þ b1 x1 Þ
1271
i ¼ 1; 2:
System (2.1) was studied completely in [14] and it is shown complex dynamics of system (2.1) using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of equilibrium where the species in the food chain is absent. Underside, we will give some definitions, notations and some lemmas which will be useful for our main results. Let R+ = [0, 1), R3þ ¼ fX 2 R3 j X P 0g. Denote f = (f1, f2, f3) the map defined by the right hand of the first, second and third 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 ;X Þ ¼ V ðnT þ ; X Þ exists. (2) V is locally Lipschitzian in X. Definition 2.1. Let V 2 V0, then for ðt; X Þ 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; X Þ ¼ limþ sup ½V ðt þ h; X þ hf ðt; xÞÞ V ðt; X Þ 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 limt!nT þ X ðtÞ exists. The smoothness properties of f guarantee the global existence and uniqueness of solution of system (1.1), for the details see book [17]. The following lemma is obvious. Lemma 2.1. Let X(t) is 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, t > 0 if X(0+) > 0. We will use an important comparison theorem on impulsive differential equation [17]. Lemma 2.2. Suppose V 2 V0. Assume that ( þ D V ðt; X Þ 6 gðt; V ðt; X ÞÞ t 6¼ nT V ðt; X ðtþ ÞÞ 6 wn ðV ðt; X ÞÞ
t ¼ nT
ð2:2Þ
where g : R+ · R+ ! R is continuous in (nT, (n + 1)T] · R+ and for u 2 R+,n 2 N, then limðt;vÞ!ðnT þ ;uÞ gðt; vÞ ¼ gðnT þ ; uÞ exists, wn : R+ ! R+ is non-decreasing. Let r(t) be 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 ð2:3Þ > > : 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 (1.1). When predator(pest) eradicate, we obtain two subsystems and give some basic properties about the following subsystems (2.4), (2.5) of system (1.1). 8 0 z ðtÞ ¼ d 2 zðtÞ t 6¼ nT > > < zðtþ Þ ¼ zðtÞ þ p t ¼ nT > > : þ zð0 Þ ¼ z0
ð2:4Þ
p expðd 2 ðt nT ÞÞ p , t 2 (nT, (n + 1)T], n 2 N, z ð0þ Þ ¼ is a positive periodic solution of 1 expðd 2 T Þ 1 expðd 2 T Þ p expðd 2 tÞ þ z ðtÞ is the solution of system (2.4) with initial value system (2.4). Since zðtÞ ¼ zð0þ Þ 1 expðd 2 T Þ Clearly z ðtÞ ¼
z0 P 0, where t 2 (nT, (n + 1)T], n 2 N, we get
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Lemma 2.3. For a positive periodic solution z*(t) of system (2.4) and every solution z(t) of system (2.4) with z0 P 0, have jz(t) z*(t)j ! 0, t ! 1. Remark. Lemma 2.3 tell us that impulse p avoid top predator extinction x0 ðtÞ ¼ xðtÞð1 xðtÞÞ
ð2:5Þ
Lemma 2.4. Subsystem (2.5) exists a globally asymptotically stable equilibrium x = 1 and unstable equilibrium x = 0. Therefore, we obtain the complete expression for the predator eradication periodic solution and prey predator eradication periodic solution of system (1.1). X ðtÞ ¼ ð1; 0; z ðtÞÞ ¼
X ðtÞ ¼ ð0; 0; z ðtÞÞ ¼
1; 0;
p expðd 2 ðt nT ÞÞ ; 1 expðd 2 T Þ
t 2 ðnT ; ðn þ 1ÞT
0; 0;
p expðd 2 ðt nT ÞÞ ; 1 expðd 2 T Þ
t 2 ðnT ; ðn þ 1ÞT
3. Extinction and boundary In this section, we study the stability of predator eradication periodic solution and prey predator eradication periodic solution of system (1.1). And we show that all solution of system (1.1) are uniformly upper bounded. Theorem 3.1. Let (x(t), y(t), z(t)) be any solution of system (1.1), then: (1) X*(t) = (0, 0, z*(t)) is unstable. pð1þb1 Þ (2) X*(t) = (1, 0, z*(t)) is locally asymptotically stable provided T < d 2 ðaa12 b . 1 d 1 d 1 Þ
Proof. (1) The local stability of periodic solution (0, 0, z*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. Define xðtÞ ¼ uðtÞ;
zðtÞ ¼ wðtÞ þ z ðtÞ
yðtÞ ¼ vðtÞ;
there may be written 0 1 0 1 uðtÞ uð0Þ B C B C @ vðtÞ A ¼ UðtÞ@ vð0Þ A 0 6 t < T wðtÞ
wð0Þ
where U(t) satisfies 0 1 0 dU B ¼ @ 0 a2 z ðtÞ d 1 dt 0 a2 z ðtÞ
0
1
C 0 AUðtÞ d 2
and U(0) = I, the identity matrix. The linearization of the fourth, fifth and sixth equations of system (1.1) become 1 10 1 0 0 uðnT Þ 1 0 0 uðnT þ Þ C CB C B B @ vðnT þ Þ A ¼ @ 0 1 0 A@ vðnT Þ A wðnT þ Þ
0 0 1
Hence, if both eigenvalues of 1 0 1 0 0 C B M ¼ @ 0 1 0 AUðT Þ 0 0 1
wðnT Þ
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have absolute value less than one, then the periodic solution (0, 0, z*(t)) is locally stable. But one Floquet multiplier is l1 = exp(T) > 1, according to Floquet theory, the solution X*(t) is unstable. (2) The local stability of periodic solution (1, 0, z*(t)) may be similar discussed. So we define xðtÞ ¼ uðtÞ þ 1;
yðtÞ ¼ vðtÞ;
there may be written 0 1 0 1 uðtÞ uð0Þ B C B C @ vðtÞ A ¼ UðtÞ@ vð0Þ A wðtÞ
zðtÞ ¼ wðtÞ þ z ðtÞ
06t
wð0Þ
where U(t) satisfies 0 1 dU B ¼@ 0 dt 0
a1 1þb 1 a1 1þb1
a2 z ðtÞ d 1 a2 z ðtÞ
0
1
C 0 AUðtÞ d 2
and U(0) = I, the identity matrix. The linearization of the fourth, fifth and sixth equations of system (1.1) become 0 1 0 10 1 1 0 0 uðnT Þ uðnT þ Þ B C B CB C @ vðnT þ Þ A ¼ @ 0 1 0 A@ vðnT Þ A 0 0 1 wðnT Þ wðnT þ Þ the periodic solution (1, 0, z*(t)) is determined by the eigenvalues of 1 0 0 C 1 0 AUðT Þ 0 1 Z T a1 ð a2 z ðtÞ d 1 Þdt , l3 = exp(d2T) < 1 according to Floquet theory, which are l1 = exp(T) < 1, l2 ¼ exp 1 þ b1 0 a2 pð1 þ b1 Þ the solution X*(t) = (1, 0, z*(t)) is locally stable if jl2j < 1, i.e., T < . This completes the proof. h d 2 ða1 b1 d 1 d 1 Þ The stability of 0 1 B M ¼ @0 0
Theorem 3.2. There exists a constant M > 0, such that x(t) 6 M, y(t) 6 M, z(t) 6 M for each solution X(t) = (x(t), y(t), z(t)) of system (1.1) with all t large enough. Proof. Define function V(t, X(t)) such that V ðt; X ðtÞÞ ¼ xðtÞ þ yðtÞ þ zðtÞ then V 2 V0. We calculate the upper right derivative of V(t, X) along a solution of system (1.1) and get the following impulsive differential equation: þ D V ðtÞ þ LV ðtÞ ¼ ð1 þ LÞx x2 þ ðL d 1 Þy þ ðL d 2 Þz t 6¼ nT ð3:1Þ V ðtþ Þ ¼ V ðtÞ þ p t ¼ nT Let 0 < L < min{d1, d2}, then D+V(t) + LV(t) is bounded. Select L0 and L1 such that þ D V ðtÞ 6 L0 V ðtÞ þ L1 t ¼ 6 nT V ðtþ Þ ¼ V ðtÞ þ p
t ¼ nT
where L0 and L1 are two positive constant. According to Lemma 2.2, we have L1 P ð1 expðnL0 T ÞÞ L1 V ðtÞ 6 V ð0þ Þ expðL0 T Þ expðL0 ðt nT ÞÞ þ expðL0 tÞ þ L0 L0 expðL0 T Þ 1 where t 2 (nT, (n + 1)T]. Hence lim V ðtÞ 6
t!1
L1 p expðL0 T Þ þ L0 expðL0 T Þ 1
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Therefore V(t, x) is ultimately bounded, and we obtain that each positive solution of system (1.1) is uniformly ultimately bounded. This completes the proof. h
4. Numerical analysis In this section we will study the influence of impulsive perturbation p on inherent oscillation. For A, a1 = 4, a2 = 2, b1 = 0.8, b2 = 1, d1 = 0.1, d2 = 0.9, we know system (2.1) does not have positive equilibrium, but exists equilibrium C. For B, a1 = 2, a2 = 2, b1 = 1.5, b2 = 1, d1 = 0.1, d2 = 0.4, we know system (2.1) has unique positive equilibrium D1. For C, a1 = 1.51, a2 = 1, b1 = 1.2, b2 = 1, d1 = 0.05, d2 = 0.4, we know system (2.1) has two positive equilibriums D1, D2.
Fig. 1. Bifurcation diagrams of system (1.1) with a1 = 4, a2 = 2, b1 = 0.8, b2 = 1, d1 = 0.1, d2 = 0.9, T = 6, 0.001 < p 6 6 and initial values x0 = 1, y0 = 2, z0 = 4.
Fig. 2. Period doubling cascade: (a) phase portrait of T-periodic solution for p = 0.1; (b) phase portrait of 2T-periodic solution for p = 0.49.
Fig. 3. Period halfing cascade: (a) phase portrait of 4T-period solution for p = 4.19; (b) phase portrait of 2T-period solution for p = 4.51.
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Since the corresponding continuous system (2.1) 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 p may be documented by stroboscopically sampling one of the variables over a range of p values. Stroboscopic map is a special case of the poincare´ map for periodically forced system or periodically pulsed system. Fixed points of the stroboscopic map corresponds to periodic solutions of system (1.1) which has the same period as the pulsing term; periodic points of period k about stroboscopic map corresponds to entrained periodic solutions of system (1.1) which has exact k times the period of the pulse.
Fig. 4. A strange attractor: (d) phase portrait of system (1.1) of p = 3, (a)–(c) time series of x, y, z solution initial values x0 = 1, y0 = 2, z0 = 4.
Fig. 5. Bifurcation diagrams of system (1.1) with a1 = 2, a2 = 2, b1 = 1.5, b2 = 1, d1 = 0.1, d2 = 0.4, T = 6, and initial values x0 = 1, y0 = 2, z0 = 4, where (a)–(c) 0.001 < p < 1, (a 0 )–(c 0 ) 0.001 < p < 0.12.
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Fig. 6. Quasi-periodic oscillate: (a) phase portrait of system (1.1) of p = 0; (b) phase portrait of system (1.1) of p = 0.03.
Fig. 7. A strange attractor: (a) phase portrait of system (1.1) of p = 0.1; (b) phase portrait of T-period solution for p = 0.4.
Fig. 8. Bifurcation diagrams of system (1.1) with a1 = 1.51, a2 = 1, b1 = 1.2, b2 = 1, d1 = 0.05, d2 = 0.4, T = 6, and initial values x0 = 1, y0 = 2, z0 = 4, where (a)–(c) 0.001 < p < 1.6, (a 0 )–(c 0 ) 0.001 < p < 0.1.
For A, system (2.1) does not has positive equilibrium, but bifurcation diagrams (Fig. 1) for bifurcation parameters p, clearly show that: with p increasing from 0.001 to 6, the system experiences process of periodic doubling cascade ! chaos ! periodic halfing cascade. When p < p1 0.29, system (1.1) exists stable periodic solution of the period of the impulsive perturbation. When p > p1, it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos (Fig. 2), which is followed by a cascade of periodic halfing bifurcations from chaos to cycles (Fig. 3). A typical chaotic oscillation is captured when p = 3 (Fig. 4). This periodic-doubling route to chaos is the hallmark of the logistic and Ricker maps [18,19] and has been studied extensively by Mathematicians [20,21]. For the predator–prey system, chaotic behaviors are usually obtained by continuous system with periodic forcing [22]. Periodic halfing is the flip bifurcation in the opposite direction, which is also observed in [23,24].
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Fig. 9. Quasi-periodic oscillate: (a) phase portrait of system (1.1) of p = 0; (b) phase portrait of system (1.1) of p = 0.01.
Fig. 10. A strange attractor: (a) phase portrait of 3T-periodic solution for p = 0.06; (b) phase portrait of system (1.1) of p = 0.09.
Fig. 11. Extinction: (d) phase portrait of system (1.1) of p = 1.6, (a)–(c) time series of x, y, z solution initial values x0 = 1, y0 = 2, z0 = 4.
For B, system (2.1) has unique positive equilibrium D1, but bifurcation diagrams (Fig. 5) show that: the system experiences process of quasi-periodic ! chaos ! chaotic crisis ! periodic oscillate. When p is sufficiently small (p < p2 0.05), system (1.1) shows quasi-periodic oscillate (Fig. 6(b)), if p tends to zero, quasi-periodic oscillate tends to space periodic solution (Fig. 6(a)). When p > p2, system (1.1) comes into chaos regions (Fig. 7(a)), when p 0.12, chaotic attractors suddenly disappear and T-periodic solution appear (Fig. 7(b)) (that is chaotic crisis). For C, system (2.1) has two positive equilibrium D1, D2, but bifurcation diagrams (Fig. 8) clearly show that: the system experiences process of quasi-periodic ! chaos ! chaotic crisis ! periodic oscillate. When p is sufficiently small (p < p3 0.02), system (1.1) shows quasi-periodic oscillate (Fig. 9(b)), if p tends to zero, quasi-periodic oscillate tends to plane periodic solution (Fig. 9(a)). When p > p3, system (1.1) comes into chaos regions (Fig. 10(b)) with periodic windows (Fig. 10(a)), when p 0.1, chaotic attractors suddenly disappear and T-periodic solution appear (that is chaotic 1 d 1 d 1 Þ crisis). If p < Td 2 ðaa12 b , as Theorem 3.1 shows, predator(pest) will go extinct y(t) ! 0 as t ! 1 (Fig. 11). ð1þb1 Þ 5. Conclusion In this paper, we have investigated three species food chain system with Holling type II functional response and periodic constant impulsive perturbation of top predator. Using Floquet theorem and small amplitude perturbation skills, we have proved the prey and predator eradication periodic solution (0, 0, z*(t)) is unstable, but predator eradication pð1þb1 Þ periodic solution (1, 0, z*(t)) is locally asymptotically stable when T < T max ¼ d 2 ðaa12 b . Otherwise the impulsive 1 d 1 d 1 Þ
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period T > Tmax, predator eradication periodic solution (1, 0, z*(t)) becomes unstable. Moreover, we have obtained the system is boundary. Numerical analysis indicates that the complex dynamics of system (1.1) depends on the values of impulsive perturbations p and all parameters. by choosing impulsive perturbations p as bifurcation parameter, we have obtained bifurcation diagrams (Figs. 1, 5 and 8). Bifurcation diagrams haven shown that there exists complexity for system (1.1) including periodic doubling cascade, periodic windows, periodic halfing cascade, quasi-periodic oscillate, chaos and chaos crisis. All these results show that dynamical behavior of system (1.1) becomes more complex under periodically impulsive perturbations.
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A Holling II functional response food chain model with impulsive perturbations q Shuwen Zhang b
a,*
, Lansun Chen
b
a Institute of Biomathematics, Anshan Normal University, Liaoning, Anshan 114005, PR China Department of Applied Mathematics, Dalian University of Technology, Liaoning, Dalian 116024, PR China
Accepted 15 September 2004
Abstract In this paper, we investigate a three trophic level food chain system with Holling II functional responses and periodic constant impulsive perturbations of top predator. Conditions for extinction of predator as a pest are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solution. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows the rich dynamics (for example: period doubling, period halfing, chaos crisis) in the positive octant. The dynamics behavior is found to be very sensitive to the parameter values and initial value. 2004 Elsevier Ltd. All rights reserved.
1. Introduction Biological control is, generally, manÕs use of a specially chosen living organism to control a particular pest, which is a component of an integrated pest management strategy. This chosen organism might be a predator, parasite, or disease which will attack the harmful insect. It is a form of manipulating nature to increase a desired effect. It may also be a more economical alternative to some insecticides. Some biological control measures can actually prevent economic damage to agricultural crops. Virtually all insect and mite pests have some natural enemies. One approach to biological control is augmentation, which is manipulation of existing natural enemies to increase their effectiveness. This can be achieved by mass production and periodic release of natural enemies of the pest, and by genetic enhancement of the enemies to increase their effectiveness at control. One of the first successful cases of biological control in greenhouses was that of the parasitoid Encarsia formosa against the greenhouse whitefly Trialeurodes vaporariorum on tomatoes and cucumbers [1,2]. Considering the exploited predator–prey system (harvesting or stocking) is very valuable, for it involves the human activities. It can be referred to papers [3,4], in which the human activities always happen in a short time or
q
This work is supported by National Natural Science Foundation of China (10171106). Corresponding author. E-mail addresses: [email protected] (S. Zhang), [email protected] (L. Chen).
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0960-0779/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.051
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instantaneously. The continuous action of human is then removed from the model, and replaced with an impulsive perturbation. These models are subject to short-term perturbations which are often assumed to be in the form of impulsive in the modelling process. Consequently, impulsive differential equations provide a natural description of such systems [5]. Equations of this kind are found in almost every domain of applied sciences. Numerous examples are given in BainovÕs and his collaboratorsÕ books [6]. Some impulsive equations have been recently introduced in population dynamics in relation to: impulsive vaccination [7], population ecology [8,9], chemotherapeutic treatment of disease [10,11] and impulsive birth [12,13]. In this paper, we will consider the following a Holling II functional responses food chain mathematical models of many biological control processes naturally call for differential systems with three equations describing the growth of plant, pest and top predator(control agent), respectively, and with periodic constant impulsive perturbation of top predator: 8 9 1 xðtÞyðtÞ x0 ðtÞ ¼ xðtÞð1 xðtÞÞ a1þb > > > > 1 xðtÞ > > > = > > a xðtÞyðtÞ a yðtÞzðtÞ 0 1 2 > > y ðtÞ ¼ d yðtÞ t 6¼ nT 1 > 1þb 1þb xðtÞ yðtÞ 1 2 > > > > > > < 0 ; a2 yðtÞzðtÞ z ðtÞ ¼ 1þb2 yðtÞ d 2 zðtÞ ð1:1Þ > 9 > > DxðtÞ ¼ 0 > > > = > > > > DyðtÞ ¼ 0 t ¼ nT > > > > ; : DzðtÞ ¼ p where x(t), y(t), z(t) are the densities of prey, predator and top predator at time t respectively, a1, a2, b1, b2, d1, d2 are positive constants. Dx(t) = x(t+) x(t), Dy(t) = y(t+) y(t), Dz(t) = z(t+) z(t), T is the period of the impulsive effect, n 2 N, N be the set of all non-negative integers. Observe that the simple relation of these three species: z prey on y and only on y, and y prey on x and nutrient recycling is not accounted for. This simple relation produces the so-called simple food chain system with impulsive perturbations. With model (1.1) we can take into account the possible exterior effects under which the population densities change very rapidly. An impulsive increase of the top predator population density is possible by artificial breeding of the species or release some species (p > 0). Firstly, we study the effects of impulsive perturbations generally, establish condition for extinction of predator and give boundary of the system. Secondly, by using the numerical method, we study the influences on the inherent oscillation caused by the impulsive perturbations. Recently, it is of interest to investigate the possible existence of chaos in biological population. Chaos in three chain system with classical Lotka–Volterra type interactions and Holling type II functional response was demonstrated in [14]. Hsu et al. [15] studied the three species food chain model with ratio-dependence and its applications to biological control. Gakkhar and Naji [16] investigate order and chaos in predator to prey ratio-dependent food chain. In this paper, we study influences of the impulsive perturbations on dynamics behavior of system (1.1). Numerical analysis shows that impulsive perturbations cause complex dynamics of system (1.1).
2. Preliminaries The model (1.1) we considered is based on the following three species food model: 8 0 1 xðtÞyðtÞ x ðtÞ ¼ xðtÞð1 xðtÞÞ a1þb > > 1 xðtÞ > < a1 xðtÞyðtÞ a2 yðtÞzðtÞ 0 y ðtÞ ¼ 1þb1 xðtÞ 1þb2 yðtÞ d 1 yðtÞ > > > : 0 2 yðtÞzðtÞ z ðtÞ ¼ a1þb d 2 zðtÞ 2 yðtÞ
ð2:1Þ
where x(t), y(t), z(t) are the densities of prey, predator and top predator at time t respectively, a1, a2, b1, b2, d1, d2 are positive constants. The three species food chain model (2.1) has at most five non-negative equilibrium: ! d1 a1 b1 d 1 d 1 Að0; 0; 0Þ; Bð1; 0; 0Þ; C ; ; 0 ; Di ðxi ; y i ; zi Þ; i ¼ 1; 2; a1 b1 d 1 ða1 b1 d 1 Þ2 where
S. Zhang, L. Chen / Chaos, Solitons and Fractals 24 (2005) 1269–1278
b1 1 xi ¼ þ ð1Þi1 2b1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 b1 d 2 ðb1 þ 1Þ2 a4a2 b 2 d2 2b1
;
yi ¼
d2 ; a2 b2 d 2
zi ¼
ða1 b1 d 1 Þxi d 1 ; ða2 b2 d 2 Þð1 þ b1 x1 Þ
1271
i ¼ 1; 2:
System (2.1) was studied completely in [14] and it is shown complex dynamics of system (2.1) using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of equilibrium where the species in the food chain is absent. Underside, we will give some definitions, notations and some lemmas which will be useful for our main results. Let R+ = [0, 1), R3þ ¼ fX 2 R3 j X P 0g. Denote f = (f1, f2, f3) the map defined by the right hand of the first, second and third 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 ;X Þ ¼ V ðnT þ ; X Þ exists. (2) V is locally Lipschitzian in X. Definition 2.1. Let V 2 V0, then for ðt; X Þ 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; X Þ ¼ limþ sup ½V ðt þ h; X þ hf ðt; xÞÞ V ðt; X Þ 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 limt!nT þ X ðtÞ exists. The smoothness properties of f guarantee the global existence and uniqueness of solution of system (1.1), for the details see book [17]. The following lemma is obvious. Lemma 2.1. Let X(t) is 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, t > 0 if X(0+) > 0. We will use an important comparison theorem on impulsive differential equation [17]. Lemma 2.2. Suppose V 2 V0. Assume that ( þ D V ðt; X Þ 6 gðt; V ðt; X ÞÞ t 6¼ nT V ðt; X ðtþ ÞÞ 6 wn ðV ðt; X ÞÞ
t ¼ nT
ð2:2Þ
where g : R+ · R+ ! R is continuous in (nT, (n + 1)T] · R+ and for u 2 R+,n 2 N, then limðt;vÞ!ðnT þ ;uÞ gðt; vÞ ¼ gðnT þ ; uÞ exists, wn : R+ ! R+ is non-decreasing. Let r(t) be 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 ð2:3Þ > > : 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 (1.1). When predator(pest) eradicate, we obtain two subsystems and give some basic properties about the following subsystems (2.4), (2.5) of system (1.1). 8 0 z ðtÞ ¼ d 2 zðtÞ t 6¼ nT > > < zðtþ Þ ¼ zðtÞ þ p t ¼ nT > > : þ zð0 Þ ¼ z0
ð2:4Þ
p expðd 2 ðt nT ÞÞ p , t 2 (nT, (n + 1)T], n 2 N, z ð0þ Þ ¼ is a positive periodic solution of 1 expðd 2 T Þ 1 expðd 2 T Þ p expðd 2 tÞ þ z ðtÞ is the solution of system (2.4) with initial value system (2.4). Since zðtÞ ¼ zð0þ Þ 1 expðd 2 T Þ Clearly z ðtÞ ¼
z0 P 0, where t 2 (nT, (n + 1)T], n 2 N, we get
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Lemma 2.3. For a positive periodic solution z*(t) of system (2.4) and every solution z(t) of system (2.4) with z0 P 0, have jz(t) z*(t)j ! 0, t ! 1. Remark. Lemma 2.3 tell us that impulse p avoid top predator extinction x0 ðtÞ ¼ xðtÞð1 xðtÞÞ
ð2:5Þ
Lemma 2.4. Subsystem (2.5) exists a globally asymptotically stable equilibrium x = 1 and unstable equilibrium x = 0. Therefore, we obtain the complete expression for the predator eradication periodic solution and prey predator eradication periodic solution of system (1.1). X ðtÞ ¼ ð1; 0; z ðtÞÞ ¼
X ðtÞ ¼ ð0; 0; z ðtÞÞ ¼
1; 0;
p expðd 2 ðt nT ÞÞ ; 1 expðd 2 T Þ
t 2 ðnT ; ðn þ 1ÞT
0; 0;
p expðd 2 ðt nT ÞÞ ; 1 expðd 2 T Þ
t 2 ðnT ; ðn þ 1ÞT
3. Extinction and boundary In this section, we study the stability of predator eradication periodic solution and prey predator eradication periodic solution of system (1.1). And we show that all solution of system (1.1) are uniformly upper bounded. Theorem 3.1. Let (x(t), y(t), z(t)) be any solution of system (1.1), then: (1) X*(t) = (0, 0, z*(t)) is unstable. pð1þb1 Þ (2) X*(t) = (1, 0, z*(t)) is locally asymptotically stable provided T < d 2 ðaa12 b . 1 d 1 d 1 Þ
Proof. (1) The local stability of periodic solution (0, 0, z*(t)) may be determined by considering the behavior of small amplitude perturbations of the solution. Define xðtÞ ¼ uðtÞ;
zðtÞ ¼ wðtÞ þ z ðtÞ
yðtÞ ¼ vðtÞ;
there may be written 0 1 0 1 uðtÞ uð0Þ B C B C @ vðtÞ A ¼ UðtÞ@ vð0Þ A 0 6 t < T wðtÞ
wð0Þ
where U(t) satisfies 0 1 0 dU B ¼ @ 0 a2 z ðtÞ d 1 dt 0 a2 z ðtÞ
0
1
C 0 AUðtÞ d 2
and U(0) = I, the identity matrix. The linearization of the fourth, fifth and sixth equations of system (1.1) become 1 10 1 0 0 uðnT Þ 1 0 0 uðnT þ Þ C CB C B B @ vðnT þ Þ A ¼ @ 0 1 0 A@ vðnT Þ A wðnT þ Þ
0 0 1
Hence, if both eigenvalues of 1 0 1 0 0 C B M ¼ @ 0 1 0 AUðT Þ 0 0 1
wðnT Þ
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have absolute value less than one, then the periodic solution (0, 0, z*(t)) is locally stable. But one Floquet multiplier is l1 = exp(T) > 1, according to Floquet theory, the solution X*(t) is unstable. (2) The local stability of periodic solution (1, 0, z*(t)) may be similar discussed. So we define xðtÞ ¼ uðtÞ þ 1;
yðtÞ ¼ vðtÞ;
there may be written 0 1 0 1 uðtÞ uð0Þ B C B C @ vðtÞ A ¼ UðtÞ@ vð0Þ A wðtÞ
zðtÞ ¼ wðtÞ þ z ðtÞ
06t
wð0Þ
where U(t) satisfies 0 1 dU B ¼@ 0 dt 0
a1 1þb 1 a1 1þb1
a2 z ðtÞ d 1 a2 z ðtÞ
0
1
C 0 AUðtÞ d 2
and U(0) = I, the identity matrix. The linearization of the fourth, fifth and sixth equations of system (1.1) become 0 1 0 10 1 1 0 0 uðnT Þ uðnT þ Þ B C B CB C @ vðnT þ Þ A ¼ @ 0 1 0 A@ vðnT Þ A 0 0 1 wðnT Þ wðnT þ Þ the periodic solution (1, 0, z*(t)) is determined by the eigenvalues of 1 0 0 C 1 0 AUðT Þ 0 1 Z T a1 ð a2 z ðtÞ d 1 Þdt , l3 = exp(d2T) < 1 according to Floquet theory, which are l1 = exp(T) < 1, l2 ¼ exp 1 þ b1 0 a2 pð1 þ b1 Þ the solution X*(t) = (1, 0, z*(t)) is locally stable if jl2j < 1, i.e., T < . This completes the proof. h d 2 ða1 b1 d 1 d 1 Þ The stability of 0 1 B M ¼ @0 0
Theorem 3.2. There exists a constant M > 0, such that x(t) 6 M, y(t) 6 M, z(t) 6 M for each solution X(t) = (x(t), y(t), z(t)) of system (1.1) with all t large enough. Proof. Define function V(t, X(t)) such that V ðt; X ðtÞÞ ¼ xðtÞ þ yðtÞ þ zðtÞ then V 2 V0. We calculate the upper right derivative of V(t, X) along a solution of system (1.1) and get the following impulsive differential equation: þ D V ðtÞ þ LV ðtÞ ¼ ð1 þ LÞx x2 þ ðL d 1 Þy þ ðL d 2 Þz t 6¼ nT ð3:1Þ V ðtþ Þ ¼ V ðtÞ þ p t ¼ nT Let 0 < L < min{d1, d2}, then D+V(t) + LV(t) is bounded. Select L0 and L1 such that þ D V ðtÞ 6 L0 V ðtÞ þ L1 t ¼ 6 nT V ðtþ Þ ¼ V ðtÞ þ p
t ¼ nT
where L0 and L1 are two positive constant. According to Lemma 2.2, we have L1 P ð1 expðnL0 T ÞÞ L1 V ðtÞ 6 V ð0þ Þ expðL0 T Þ expðL0 ðt nT ÞÞ þ expðL0 tÞ þ L0 L0 expðL0 T Þ 1 where t 2 (nT, (n + 1)T]. Hence lim V ðtÞ 6
t!1
L1 p expðL0 T Þ þ L0 expðL0 T Þ 1
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Therefore V(t, x) is ultimately bounded, and we obtain that each positive solution of system (1.1) is uniformly ultimately bounded. This completes the proof. h
4. Numerical analysis In this section we will study the influence of impulsive perturbation p on inherent oscillation. For A, a1 = 4, a2 = 2, b1 = 0.8, b2 = 1, d1 = 0.1, d2 = 0.9, we know system (2.1) does not have positive equilibrium, but exists equilibrium C. For B, a1 = 2, a2 = 2, b1 = 1.5, b2 = 1, d1 = 0.1, d2 = 0.4, we know system (2.1) has unique positive equilibrium D1. For C, a1 = 1.51, a2 = 1, b1 = 1.2, b2 = 1, d1 = 0.05, d2 = 0.4, we know system (2.1) has two positive equilibriums D1, D2.
Fig. 1. Bifurcation diagrams of system (1.1) with a1 = 4, a2 = 2, b1 = 0.8, b2 = 1, d1 = 0.1, d2 = 0.9, T = 6, 0.001 < p 6 6 and initial values x0 = 1, y0 = 2, z0 = 4.
Fig. 2. Period doubling cascade: (a) phase portrait of T-periodic solution for p = 0.1; (b) phase portrait of 2T-periodic solution for p = 0.49.
Fig. 3. Period halfing cascade: (a) phase portrait of 4T-period solution for p = 4.19; (b) phase portrait of 2T-period solution for p = 4.51.
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Since the corresponding continuous system (2.1) 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 p may be documented by stroboscopically sampling one of the variables over a range of p values. Stroboscopic map is a special case of the poincare´ map for periodically forced system or periodically pulsed system. Fixed points of the stroboscopic map corresponds to periodic solutions of system (1.1) which has the same period as the pulsing term; periodic points of period k about stroboscopic map corresponds to entrained periodic solutions of system (1.1) which has exact k times the period of the pulse.
Fig. 4. A strange attractor: (d) phase portrait of system (1.1) of p = 3, (a)–(c) time series of x, y, z solution initial values x0 = 1, y0 = 2, z0 = 4.
Fig. 5. Bifurcation diagrams of system (1.1) with a1 = 2, a2 = 2, b1 = 1.5, b2 = 1, d1 = 0.1, d2 = 0.4, T = 6, and initial values x0 = 1, y0 = 2, z0 = 4, where (a)–(c) 0.001 < p < 1, (a 0 )–(c 0 ) 0.001 < p < 0.12.
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Fig. 6. Quasi-periodic oscillate: (a) phase portrait of system (1.1) of p = 0; (b) phase portrait of system (1.1) of p = 0.03.
Fig. 7. A strange attractor: (a) phase portrait of system (1.1) of p = 0.1; (b) phase portrait of T-period solution for p = 0.4.
Fig. 8. Bifurcation diagrams of system (1.1) with a1 = 1.51, a2 = 1, b1 = 1.2, b2 = 1, d1 = 0.05, d2 = 0.4, T = 6, and initial values x0 = 1, y0 = 2, z0 = 4, where (a)–(c) 0.001 < p < 1.6, (a 0 )–(c 0 ) 0.001 < p < 0.1.
For A, system (2.1) does not has positive equilibrium, but bifurcation diagrams (Fig. 1) for bifurcation parameters p, clearly show that: with p increasing from 0.001 to 6, the system experiences process of periodic doubling cascade ! chaos ! periodic halfing cascade. When p < p1 0.29, system (1.1) exists stable periodic solution of the period of the impulsive perturbation. When p > p1, it is unstable and there is a cascade of periodic doubling bifurcations leading to chaos (Fig. 2), which is followed by a cascade of periodic halfing bifurcations from chaos to cycles (Fig. 3). A typical chaotic oscillation is captured when p = 3 (Fig. 4). This periodic-doubling route to chaos is the hallmark of the logistic and Ricker maps [18,19] and has been studied extensively by Mathematicians [20,21]. For the predator–prey system, chaotic behaviors are usually obtained by continuous system with periodic forcing [22]. Periodic halfing is the flip bifurcation in the opposite direction, which is also observed in [23,24].
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Fig. 9. Quasi-periodic oscillate: (a) phase portrait of system (1.1) of p = 0; (b) phase portrait of system (1.1) of p = 0.01.
Fig. 10. A strange attractor: (a) phase portrait of 3T-periodic solution for p = 0.06; (b) phase portrait of system (1.1) of p = 0.09.
Fig. 11. Extinction: (d) phase portrait of system (1.1) of p = 1.6, (a)–(c) time series of x, y, z solution initial values x0 = 1, y0 = 2, z0 = 4.
For B, system (2.1) has unique positive equilibrium D1, but bifurcation diagrams (Fig. 5) show that: the system experiences process of quasi-periodic ! chaos ! chaotic crisis ! periodic oscillate. When p is sufficiently small (p < p2 0.05), system (1.1) shows quasi-periodic oscillate (Fig. 6(b)), if p tends to zero, quasi-periodic oscillate tends to space periodic solution (Fig. 6(a)). When p > p2, system (1.1) comes into chaos regions (Fig. 7(a)), when p 0.12, chaotic attractors suddenly disappear and T-periodic solution appear (Fig. 7(b)) (that is chaotic crisis). For C, system (2.1) has two positive equilibrium D1, D2, but bifurcation diagrams (Fig. 8) clearly show that: the system experiences process of quasi-periodic ! chaos ! chaotic crisis ! periodic oscillate. When p is sufficiently small (p < p3 0.02), system (1.1) shows quasi-periodic oscillate (Fig. 9(b)), if p tends to zero, quasi-periodic oscillate tends to plane periodic solution (Fig. 9(a)). When p > p3, system (1.1) comes into chaos regions (Fig. 10(b)) with periodic windows (Fig. 10(a)), when p 0.1, chaotic attractors suddenly disappear and T-periodic solution appear (that is chaotic 1 d 1 d 1 Þ crisis). If p < Td 2 ðaa12 b , as Theorem 3.1 shows, predator(pest) will go extinct y(t) ! 0 as t ! 1 (Fig. 11). ð1þb1 Þ 5. Conclusion In this paper, we have investigated three species food chain system with Holling type II functional response and periodic constant impulsive perturbation of top predator. Using Floquet theorem and small amplitude perturbation skills, we have proved the prey and predator eradication periodic solution (0, 0, z*(t)) is unstable, but predator eradication pð1þb1 Þ periodic solution (1, 0, z*(t)) is locally asymptotically stable when T < T max ¼ d 2 ðaa12 b . Otherwise the impulsive 1 d 1 d 1 Þ
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period T > Tmax, predator eradication periodic solution (1, 0, z*(t)) becomes unstable. Moreover, we have obtained the system is boundary. Numerical analysis indicates that the complex dynamics of system (1.1) depends on the values of impulsive perturbations p and all parameters. by choosing impulsive perturbations p as bifurcation parameter, we have obtained bifurcation diagrams (Figs. 1, 5 and 8). Bifurcation diagrams haven shown that there exists complexity for system (1.1) including periodic doubling cascade, periodic windows, periodic halfing cascade, quasi-periodic oscillate, chaos and chaos crisis. All these results show that dynamical behavior of system (1.1) becomes more complex under periodically impulsive perturbations.
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