Dynamic analysis of an impulsively controlled predator-prey system

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 21, No. 19, 1-14; Dynamic analysis of an impulsively controlled predator-prey system Hunki Baek Abstract In this paper, we study an impulsively controlled predator-prey model with Monod- Haldane functional response. By using the Floquet theory, we prove that there exists a stable prey-free solution when the impulsive period is less than some critical value, and give the condition for the permanence of the system. In addition, we show the existence and stability of a positive periodic solution by using bifurcation theory. Key words: Predator-prey model, Monod-Haldane type functional response, impulsive differential equation, Floquet theory. 2 AMS Subject Classification. 34A37, 34D23, 34H5, 92D25 1 Introduction In population dynamics, one of central goals is to understand the dynamical relationship between predator and prey. One important component of the predator-prey relationship is the predator s rate of feeding on prey, i.e., the so-called predator s functional response. Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. Holling [7] gave three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. These functional responses are monotonic in the first quadrant. But, some eriments and observations indicate that a non-monotonic response occurs at a level: when the nutrient concentration reaches a high level an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, the authors in [1, 14] suggested a function px = αx b + x 2 Department of Mathematics, Kyungpook National University, Daegu 72-71, South Korea, hkbaek@knu.ac.kr. This research was supported by Basic Science Research Program through the National Research Foundation of KoreaNRF funded by the Ministry of Education, Science and Technology No EJQTDE, 21 No. 19, p. 1

2 called the Monod-Haldane function, or Holling type-iv function. We can write a predatorprey model with Monod-Haldane type functional response as follows. x t = axt1 xt K cxtyt b + x 2 t, y t = Dyt + mxtyt 1 b + x 2 t, where xt and yt represent population densities of prey and predator at time t. All parameters are positive constants. Usually, a is the intrinsic growth rate of the prey, K is the carrying capacity of the prey, the constant D is the death rate of the predator, m is the rate of conversion of a consumed prey to a predator and b measures the level of prey interference with predation. As Cushing [5] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally osed for example, those due to seasonal effects of weather, food supply, mating habits or harvesting seasons and so on. Such perturbations were often treated continually. But, there are still some other perturbations such as fire, flood, etc, that are not suitable to be considered continually. These impulsive perturbations bring sudden changes to the system. In this paper, with the idea of impulsive perturbations, we consider the following predator-prey model with periodic constant impulsive immigration of the predator and periodic harvesting on the prey. x t = axt1 xt K cxtyt b + x 2 t, } y t = Dyt + mxtyt t nt, b + x 2 t }, 2 xt + = 1 p 1 xt, yt + t = nt, = 1 p 2 yt + q, x +, y + = x, where T is the period of the impulsive immigration or stock of the predator, p i p i < 1, = 1, 2 are the harvesting control parameters, q is the size of immigration or stock of the predator. This model is an example of impulsive differential equations whose theories and applications were greatly developed by the efforts of Bainov and Lakshmikantham et al. [4, 8]. Recently, many researchers have intensively investigated systems with impulsive perturbations cf, [2, 3, 1, 11, 12, 13, 15, 16, 17, 18, 19, 2, 21]. Most of such systems have dealt with impulsive harvesting and immigration of predators at different fixed times. On the contrary, here we consider the impulsive harvesting and immigration at the same time in our model which has not been studied well until now. EJQTDE, 21 No. 19, p. 2

3 The main purpose of this paper is to study the dynamics of the system 2. The organization of this paper is as follows. In the next section, we introduce some notations and lemmas related to impulsive differential equations which are used in this paper. In Section 3, we show the stability of prey-free periodic solutions and give a sufficient condition for the permanence of system 2 by applying the Floquet theory and the comparison theorem. In Section 4, we show the existence of nontrivial periodic solutions via the bifurcation theorem. Finally, in conclusion, we give a bifurcation diagram that shows the system has various dynamic aspects including chaos. 2 Preliminaries Let R + = [, and R 2 + = {x = x, y R 2 : x, y }. Denote N the set of all of nonnegative integers and f = f 1, f 2 T the right hand of 2. Let V : R + R 2 + R +, then V is said to be in a class V if 1V is continuous onnt, n + 1T] R 2 +, and 2V is locally Lipschitz in x. lim t,y nt,x t>nt V t,y = V nt +,x exists. Definition 2.1. Let V V, t,x nt, n + 1T] R 2 +. The upper right derivatives of V t,x with respect to the impulsive differential system 2 is defined as D + V t,x = lim sup [V t + h,x + hft,x V t,x]. h h + 1 Remarks The solution of the system 2 is a piecewise continuous function x : R + R 2 +, xt is continuous on nt, n + 1T], n N and xnt + = lim t nt + xt exists. 2 The smoothness properties of f guarantee the global existence and uniqueness of solution of the system 2. See [8] for the details. We will use the following important comparison theorem on an impulsive differential equation [8]. Lemma 2.2. Comparison theorem Suppose V V and { D + V t,x gt, V t,x, t nτ V t,xt + ψ n V t,x, t = nτ, 3 g : R + R + R is continuous on nτ, n + 1τ] R + and for ut R +, n N, lim t,y nτ +,u gt, y = gnτ +, u exists, ψ n : R + R + is non-decreasing. Let rt be EJQTDE, 21 No. 19, p. 3

4 the maximal solution of the scalar impulsive differential equation u t = gt, ut, t nτ, ut + = ψ n ut, t = nτ, u + = u, existing on [,. Then V +,x u implies that V t,xt rt, t, where xt is any solution of 3. Note that dx dt = dy = whenever xt = yt =, t nt. So, we can easily show dt the following lemma. Lemma 2.3. Let xt = xt, yt be a solution of 2. 1 If x + then xt for all t. 2 If x + > then xt > for all t. Now, we show that all solutions of 2 are uniformly ultimately bounded. Lemma 2.4. There is an M > such that xt, yt M for all t large enough, where xt, yt is a solution of 2. Proof. Let xt = xt, yt be a solution of 2 and let V t,x = mxt + cyt. Then V V, and if t nt D + V + βv = ma K x2 t + ma + βxt + cβ Dyt. 5 Clearly, the right hand of 5, is bounded when < β < D. When t = nt, V nt + = mxnt + + cynt + = 1 p 1 mxnt + 1 p 2 cynt + cq V nt + cq. So we can choose < β < D and M > such that { D + V β V + M, t nt, 6 V nt + V nt + cq. From Lemma 2.2, we can obtain that V t V + M β t + cq1 n + 1β T β t nt+ M β 1 β T β for t nt, n + 1T]. Therefore, V t is bounded by a constant for sufficiently large t. Hence there is an M > such that xt M, yt M for a solution xt, yt with all t large enough. Now, we consider the following impulsive differential equation. y t = Dyt, t nt, yt + = 1 p 2 yt + q, t = nt, y + = y. We can easily obtain the following results. 4 7 EJQTDE, 21 No. 19, p. 4

5 Lemma y t = q Dt nt, t nt, n + 1T], n N is a 1 1 p 2 DT q 1 1 p 2 DT positive periodic solution of 7 with the initial value y + =. 2 yt = 1 p 2 y n+1 + q DT Dt + y t is the 1 1 p 2 DT solution of 7 with y, t nt, n + 1T] and n N. 3 All solutions yt of 2 with y tend to y t. i.e., yt y t as t. It is from Lemma 2.5 that the general solution yt of 7 can be identified with the positive periodic solution y t of 7 for sufficiently large t and we can obtain the complete ression for the prey-free periodic solution of 2, y q Dt nt t =, for t nt, n + 1T]. 1 1 p 2 DT 3 Extinction and permanence Now, we present a condition which guarantees local stability of the prey-free periodic solution, y t. Theorem 3.1. The prey-free solution, y t is locally asymptotically stable if at + ln1 p 2 < cq1 DT bd1 1 p 2 DT. Proof. The local stability of the periodic solution, y t of 2 may be determined by considering the behavior of small amplitude perturbations of the solution. Let xt, yt be any solution of 2. Define xt = ut, yt = y t + vt. Then they may be written as where Φt satisfies ut vt dφ dt = = Φt u v, t T, 8 a c b y t m Φt 9 D b y t and Φ = I, the identity matrix. The linearization of the third and fourth equation of 2 becomes unt + 1 p1 unt vnt + =. 1 1 p 2 vnt EJQTDE, 21 No. 19, p. 5

6 1 p1 Note that all eigenvalues of S = ΦT are µ 1 p 1 = DT < 1 and 2 µ 2 = 1 p 2 T a c b y tdt. Since we have T y tdt = q1 DT D1 1 p 2 DT, cq1 DT µ 2 = 1 p 2 at. bd1 1 p 2 DT By Floquet Theory in [4],, y t is locally asymptotically stable if µ 2 < 1.i.e., at < cq1 DT bd1 1 p 2 DT + ln 1 1 p 2. Definition 3.1. The system 2 is permanent if there exist M m > such that, for any solution xt, yt of 2 with x >, m lim t inf xt lim t sup xt M and m lim t inf yt lim t sup yt M. Theorem 3.2. The system 2 is permanent if at + ln1 p 2 > cq1 DT bd1 1 p 2 DT. Proof. Let xt, yt be any solution of 2with x >. From Lemma 2.4, we may assume that xt M, yt M, t and M > ab c. Let m q DT 2 = 1 1 p 2 DT ǫ 2, ǫ 2 >. From Lemma 2.5, clearly we have yt m 2 for all t large enough. Now we shall find an m 1 > such that xt m 1 for all t large enough. We will do this in the following two steps. Step 1 Since at > cq1 DT bd1 1 p 2 DT + ln 1, 1 p 2 we can choose m 3 >, ǫ 1 > small enough such that δ = mm 3 < D and R = b + m 3 1 p 2 at a K Tm cq1 DT 3 bd1 1 p 2 DT cǫ 1 b T > 1. Suppose that xt < m 3 for all t. Then we get y t yt D + δ from above assumptions. EJQTDE, 21 No. 19, p. 6

7 By Lemma 2.2, we have yt ut and ut u t, t, where ut is the solution of u t = D + δut, t nt, ut + = 1 p 2 ut + q, t = nt, 11 u + = y, and u q D + δt nt t = 1 1 p 2 D + δt, t nt, n+1t]. Then there exists T 1 > such that yt ut u t + ǫ 1 and x t = xt a a K xt cyt b + x 2 t xt a a K m 3 c b yt xt a a K m 3 c b u t + ǫ 1 for t T 1. Let N 1 N and N 1 T T 1. We have, for n N 1 x t xt a a K m 3 c b u t + ǫ 1, t nt, xt + = 1 pxt, t = nt. 12 Integrating 12 on nt, n + 1T]n N 1, we obtain n+1t xn + 1T xnt + a a K m 3 c b u t + ǫ 1 dt = xntr. nt Then xn 1 +kt xn 1 TR k as k which is a contradiction. Hence there exists a t 1 > such that xt 1 m 3. Step 2 If xt m 3 for all t t 1, then we are done. If not, we may let t = inf t>t1 {xt < m 3 }.Then xt m 3 for t [t 1, t ] and, by the continuity of xt, we have xt = m 3. In this step, we have only to consider two possible cases. Case 1 t = n 1 T for some n 1 N. Then 1 p 1 m 3 xt + = 1 p 1 xt < m 3. Select n 2, n 3 N such that n 2 1T > ln ǫ 1 M+q d + δ and 1 p 1 n 2 R n 3 n 2 σt > 1 p 1 n 2 R n 3 n 2 +1σT > 1, where σ = a a K m 3 c b M <. Let T = n 2 T+n 3 T. In this case we will show that there exists t 2 t, t + T ] such that xt 2 m 3. Otherwise, by 11 with ut + = yt +, we have ut = 1 p 2 n 1+1 ut + q D + δt D+δt t +u t 1 1 p 2 D + δt EJQTDE, 21 No. 19, p. 7

8 for n 1T < t nt and n 1 +1 n n 1 +1+n 2 +n 3. So we get ut u t M+ q D + δt t < ǫ 1 and yt ut u t + ǫ 1 for t + n 2 T t t + T. Thus we obtain the same equation as 12 for t [t + n 2 T, t + T ]. As in step 1, we have Since yt M, we have xt + T xt + n 2 TR n 3. { x t xt a a K m 3 c b M = σxt, t nt, xt + = 1 p 1 xt, t = nt, 13 for t [t, t + n 2 T]. Integrating 13 on [t, t + n 2 T] we have xt + n 2 T m 3 σn 2 T m 3 1 p 1 n 2 σn 2 T > m 3. Thus xt + T m 3 1 p 1 n 2 σn 2 TR n 3 > m 3 which is a contradiction. Now, let t = inf t>t {xt m 3 }. Then xt m 3 for t t < t and x t = m 3. For t [t, t, suppose t t +k 1T, t +kt], k N and k n 2 +n 3. So, we have, for t [t, t, from 13 we obtain xt xt + 1 p 1 k 1 k 1σT σt t + k 1T m 3 1 p 1 k kσt m 3 1 p 1 n 2+n 3 σn 2 + n 3 T m 1. Case 2 t nt, n N. Then xt m 3 for t [t 1, t and xt = m 3. Suppose that t n 1 T, n 1 + 1T for some n 1 N. There are two possible cases. Case2a xt < m 3 for all t t, n 1 + 1T]. In this case we will show that there exists t 2 [n 1 + 1T, n 1 + 1T + T ] such that x 2 t 2 m 3. Suppose not, i.e., xt < m 3, for all t [n 1 + 1T, n n 2 + n 3 T]. Then xt < m 3 for all t t, n n 2 + n 3 T]. By 11 with un 1 + 1T + = yn 1 + 1T +, we have ut = un 1 +1T + q D + δ D+δt n p 2 D + δ +1T+u t for t nt, n + 1T], n n n 1 + n 2 + n 3. By a similar argument as in step 1, we have xn n 2 + n 3 T x 2 n n 2TR n 3. It follows from 13 that xn n 2T m 3 1 p n 2+1 σn 2 + 1T. Thus xn n 2 + n 3 T m 3 1 p n2+1 σn 2 + 1TR n 3 > m 3 which is a contradiction. Now, let t = inf t>t {xt m 3 }. Then xt m 3 for t t < t and x t = m 3. For t [t, t, suppose t n 1T + k + 1T, n 1T + k T], k N, EJQTDE, 21 No. 19, p. 8

9 k 1 + n 2 + n 2, we have xt m 3 1 p 1+n 2+n 3 σ1 + n 2 + n 3 T m 1. Since m 1 < m 1, so xt m 1 for t t, t. Case 2b There is a t t, n 1+1T] such that x 2 t m 3. Let ˆt = inf t>t {xt m 3 }. Then xt m 3 for t [t, ˆt and xˆt = m 3. Also, 13 holds for t [t, ˆt. Integrating the equation on [t, tt t ˆt, we can get that xt xt σt t m 3 σt m 1. Thus in both cases a similar argument can be continued since xt m 1 for some t > t 1. This completes the proof. Remarks 3.3. Define GT = at + ln1 p 2 G = ln1 p 2 <, lim T GT = and cq1 DT bd1 1 p 2 DT. Since G T = cqp 2 DT1 + 1 p 2 DT b 4 D1 1 p 2 DT 3 >, 14 so we have that GT = has a unique positive solution T. From Theorem 3.1 and Theorem 3.2, we know that the prey-free periodic solution is locally asymptotically stable if T < T and otherwise, the prey and predator can coexist. Thus T plays the role of a critical value that discriminates between stability and permanence. 4 Existence and stability of a positive periodic solution Now, we deal with a problem of the bifurcation of the nontrivial periodic solution of the system 2, near, y t. The following theorem establishes the existence of a positive periodic solution of the system 2 near, y t. Theorem 4.1. The system 2 has a positive periodic solution which is supercritical if T > T. Proof. We will use the results of [6, 9, 11] to prove this theorem. To use theorems of [6, 9, 11], it is convenient for the computation to exchange the variables of x and y and change the period T to τ. Thus the system 2 becomes as follows x t = Dxt + mytxt b + y 2 t, } y t = ayt1 yt K cytxt t nτ, b + y 2 t, 15 } xt + = 1 p 2 xt + q, yt + t = nτ. = 1 p 1 yt, Let Φ be the flow associated with 15. We have Xt = Φt,x, < t τ, where x = x, y. The flow Φ applies up to time τ. So, Xτ = Φτ,x. We will use EJQTDE, 21 No. 19, p. 9

10 all notations in [9]. Note that Then F 1 x, y = Dx + myx b + y 2, F 2x, y = ay1 y K cyx b + y 2, Θ 1 x, y = 1 p 2 x + q, Θ 2 x, y = 1 p 1 y, ζt = y t,, Φ 1 t,x t F 1 ζr = dr, Φ 2t,x t = x x Φ 1 t,x t t F 1 ζr F1 ζν ν = dr x ν F 2 ζr dr, F 2 ζr dr Θ 1 = Θ 2 x =, Θ 1 x = 1 p 2, Θ 2 = 1 p 1, 2 Θ i x =, i = 1, 2 and 2 Θ 2 2 =. Now, we can compute d Θ2 = 1 Φ 2 τ,x τ = 1 1 p 1 a c b y rdr, where τ is the root of d =. Actually, we know that τ = T. a = 1 Θ1 x Φ 1 = 1 1 p 2 Dτ >, x ν τ,x b = Θ1 x Φ 1 + Θ 1 Φ 2 Φ1 = τ,x τ,x τ τ F 1 ζr m ν F 2 ζr = dr x b y ν dr dν <, 2 Φ 2 t,x t t F 2 ζr 2 F 2 ζν = dr x ν x 2 Φ 2 τ,x = c τ τ F 2 ζr ν dr x b ν 2 Φ 2 t,x t t F 2 ζr 2 F 2 ζν = dr 2 ν 2 t [ t ] F 2 ζr 2 F 2 ζν + dr ν x [ ν ν F 1 ζr F1 ζθ dr x θ ν ν dν. F 2 ζr dr dν, dν <, F 2 ζr dr θ F 2 ζr dr dν ] F 2 ζr dr dθ dν, EJQTDE, 21 No. 19, p. 1

11 2 Φ 2 τ,x 2 τ ν [ = 2a τ F 2 ζr ν dr K cm τ τ ] F 2 ζr dr b 2 ν [ ν ν F 1 ζr θ dr y F 2 ζr θ θ x 2 Φ 2 t,x = F t 2ζt F 2 x p r, dr, τ 2 Φ 2 τ,x = F 2ζτ τ F 2 x p r, dr τ = a c τ b y τ a c b y r dr, F 2 ζr dr dν ] dr dθ dν <, Φ 1 τ,x q Dτ = y τ τ = D1 1 p 2 Dτ <, C = 2 2 Θ 2 b Φ 1τ,x + Φ 1τ,x Φ2 τ,x x a x 2 2 Θ 2 Φ2 τ,x + 2 Θ 2 2 b 2 Φ 2 τ,x Θ 2 a x 2 Φ 2 τ,x 2 = 21 p 1 b 2 Φ 2 a x 1 p 1 2 Φ 2 >, 2 B = 2 Θ 2 Φ1 τ,x + Φ 1τ,x 1 Θ 1 x τ x a x Φ 1τ, X Φ2 τ,x τ Θ 2 2 Φ 2 τ,x 1 Θ 1 x a x Φ 1τ,x + 2 Φ 2 τ,x τ τ 2 Φ 2 τ,x = 1 p 1 1 Φ 1τ,x x a τ + a c τ b y τ a c b y r dr. To determine the sign of B, let φt = a c b y t. Then we obtain that φ t = and so φt is strictly increasing. Since τ φtdt = at cdq Dt b1 1 p 2 Dτ > cq1 Dτ = ln1 p >, bd1 1 p 2 Dτ EJQTDE, 21 No. 19, p. 11

12 we have φτ >. This implies that B <. Hence BC <. Thus, from Theorem 2 of [9], the statement follows. 5 Conclusions In this paper, we have studied the influences of impulsive perturbations on a predatorprey system with the Monod-Haldane functional response. We have found out that there exists a threshold value that plays a key role on discriminating between the stability of the prey-free periodic solution and the permanence of the system via Floquet theory and the comparison theorem. Furthermore, we have shown that the system has a positive periodic solution which is supercritical under some conditions. a b y x T* 1 2 T T 3 39 Figure 1: The bifurcation diagrams of the system 2 with an initial value 2.5, 4. a x is plotted for T over [, 39]. b y is plotted for T over [, 39]. a b y x T T Figure 2: The bifurcation diagrams of the system 2 with an initial value 2.5, 4. a x is plotted for T over [21, 25.7]. b y is plotted for T over [21, 25.7]. In order to illustrate the dynamics of the system by a numerical example, we give bifurcation diagrams in Figures 1 and 2 when the parameters are fixed except the period T as follows: a = 4., K = 1.9, b =.6, c =.75, D =.25, m =.6, p1 =.3, p2 =.1 and q = 1.2. EJQTDE, 21 No. 19, p. 12

13 These figures point out that system 2 has various dynamical behaviors such as quasiperiodic, periodic windows, strange attractors and periodic doubling and halving phenomena etc. see Figure 2. In this case, we can obtain the critical value T 1.58 suggested in Theorems 3.1 and 3.2. As mentioned in Theorem 4.1, the value T plays an important part in the classification for the existence of a positive periodic solution as shown in Figure 1. Acknowledgements We would thank the referee for carefully reading the manuscript and for suggesting improvements. References [1] J. F. Andrews, A mathematical model for the continuous culture of macroorganisms untilizing inhibitory substrates, Biotechnol. Bioeng., 11968, [2] H. Baek, Dynamic complexites of a three - species Beddington-DeAngelis system with impulsive control strategy, Acta Appl. Math., 11121, [3] H. Baek, Qualitative analysis of Beddingtion-DeAngelis type impulsive predator-prey models, Nonlinear Analysis: Real World Applications, doi: 1.116/j.nonrwa [4] D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66, of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Science & Technical, Harlo, UK, [5] J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., , [6] S. Gakkhar and K. Negi, Pulse vaccination in SIRS epidemic model with nonmonotonic incidence rate, Chaos, Solitons and Fractals27, 3528, [7] C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulations, Mem. Ent. Sec. Can, , 1-6. [8] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publisher, Singapore, [9] A. Lakmeche and O. Arino, Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynamics of Continuous, Discrete and Impulsive Systems, 72, EJQTDE, 21 No. 19, p. 13

14 [1] B. Liu, Y. Zhang and L. Chen, Dynamic complexities in a Lotka-Volterra predatorprey model concerning impulsive control strategy, Int. J. of Bifur. and Chaos, 15225, [11] B. Liu, Y. Zhang and L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest mangement, Nonlinearity Analysis, 625, [12] Z. Li, W. Wang and H. Wang, The dynamics of a Beddington-type system with impulsive control strategy, Chaos, Solitons and Fractals, 2926, [13] Z. Lu, X. Chi and L. Chen, Impulsive control strategies in biological control and pesticide, Theoretical Population Biology, 6423, [14] W. Sokol and J. A. Howell, Kinetics of phenol oxidation by washed cell, Biotechnol. Bioeng., 23198, [15] H. Wang and W. Wang, The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect, Chaos, Solitons and Fractals, 3828, [16] Z. Xiang and X. Song, The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response, Chaos, Solitons and Fractals, 3929, [17] S. Zhang, L. Dong and L. Chen, The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, Solitons and Fractals, 2325, [18] S. Zhang, D. Tan and L. Chen, Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations, Chaos, Solitons and Fractals, 2726, [19] S. Zhang and L. Chen, A study of predator-prey models with the Beddington- DeAngelis functional response and impulsive effect, Chaos, Solitons and Fractals, 2726, [2] S. Zhang and L. Chen, Chaos in three species food chain system with impulsive perturbations, Chaos Solitons and Fractals, 2425, [21] S. Zhang and L. Chen, A Holling II functional response food chain model with impulsive perturbations, Chaos Solitons and Fractals, 2425, Received July 15, 29 EJQTDE, 21 No. 19, p. 14