the study of predator–prey system with defensive ability of prey and impulsive perturbations on...

Chaos, Solitons and Fractals 23 (2005) 631–643

www.elsevier.com/locate/chaos

The study of predator–prey system with defensive abilityof prey and impulsive perturbations on the predator

Shuwen Zhang a,b,*, Lingzhen Dong c, Lansun Chen a

a Department of Applied Mathematics, Dalian University of Technology, Liaoning, Dalian 116024, PR Chinab Institute of Biomathematics, Anshan Normal University, Liaoning Anshan 114005, PR China

c Department of Mathematics, Taiyuan University of Technology, Shanxi, Taiyuan 030024, PR China

Accepted 5 May 2004

Abstract

Predator–prey system with non-monotonic functional response and impulsive perturbations on the predator is

established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-

eradication periodic solution is obtained when the impulsive period is less than the critical value. Otherwise, if the

impulsive period is larger than the critical value, the system is permanent. Further, using numerical simulation method

the influences of the impulsive perturbations on the inherent oscillation are investigated. With the increasing of the

impulsive value, the system displays a series of complex phenomena, which include (1) quasi-periodic oscillating, (2)

period-doubling, (3) period-halfing, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor

crisis and (6) chaotic bands with periodic windows.

� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

In population dynamics, a functional response of the predator to the prey density refers to the change in the density

of prey attached per unit time per predator as the prey density changes. Holling [1] gave three different kinds of

functional response, which are monotonic in the first quadrant. But some experiments and observations indicate that

the non-monotonic response occur at a level: when the nutrient concentrations reaches a high level an inhibitory effect

on the specific growth rate may occur. To model such an inhibitory effect, Andrews [2] suggested a function

* Co

E-m

0960-0

doi:10.

pðxÞ ¼ mxaþ bxþ x2

;

called the Monod–Haldane function, or Holling type-IV function. Sokol and Howell [3] proposed a simplified Monod–

Haldane function of the form:

pðxÞ ¼ mxaþ x2

;

which describes the phenomenon of group defense whereby predation is decreased, or even prevented altogether, due to

the increased ability of the prey to better defend or disguise themselves when their numbers are large enough. An

example of this phenomenon is introduced by Tener [4].

rresponding author.

ail addresses: [email protected] (S. Zhang), [email protected] (L. Chen).

779/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

1016/j.chaos.2004.05.044

mail to: [email protected]

632 S. Zhang et al. / Chaos, Solitons and Fractals 23 (2005) 631–643

To study the predator–prey interaction when the prey exhibits group defense, Ruan and Xiao [5] investigated the

following model

x0ðtÞ ¼ rxðtÞ 1� xðtÞk

� �� xðtÞyðtÞ

a1þx2ðtÞ ;

y0ðtÞ ¼ yðtÞ lxðtÞa1þx2ðtÞ � d

� �;

8><>: ð1:1Þ

where xðtÞ and yðtÞ are functions of time representing population densities of prey and predator, all parameters arepositive constants, k is the carrying capacity of the prey and d is the death of the predator, l is the maximum predationrate, and a1 is the so-called half-saturation constant. For system (1.1), some useful results in [5] for us will be shown inSection 2.

With the model (1.1), let us take into account the possible effects of the human’s exploitative activities. In the papers

[6,7], the authors assumed the human’s activities occur continuously. However, in practical situations, human activities

always happen in a short time or instantaneous. For example, biological control is the using of a specially chosen living

organism to control a particular pest, which a component of an integrated pest management strategy [8]. The chosen

organism might be a predator [9–11], the natural enemy can be stocked in fixed moments to eradiate the pest or regulate

it to densities below the threshold for economical damage. One of the first successful cases of biological control in

greenhouses was the use of the parasitiod Encarsia formosa against the greenhouse whitefly Trialeurodes vaporariorum

on tomaroes and cucumbers [10,12]. Probably, the reasonable model which describes this phenomena might be a

predator–prey differential equation with impulsive perturbation. Besides, fishery management for another important

example. There is a spate of interest in bioeconomic analysis of exploitation or renewable resources fishery manage-

ment. In China, in order to make fish resource develop persistently, the harvest of such recourses have been restricted in

a number of ways. It includes area of take, time of year, and the take of specific species. Time-area closures are used

extensively to control human activity. These closures may be temporary or permanent. This implies that the population

may be harvested and removed from the system, or stocked and added into the system continuously or instantaneously.

Once human activities occur in regular pulses, studying predator–prey system with impulsive perturbations will be more

significant.

Although impulsive perturbations make the differential system more intractable, some impulsive systems have been

recently studied in population dynamics in relation with: impulsive birth [13,14], impulsive vaccination [15,16], che-

motherapeutic treatment of disease [17,18] and population ecology [19].

In this paper, with the idea of group defence and impulsive perturbations, we will study the following predator–prey

system with periodic constant impulsive immigration or stock of predator.

x0ðtÞ ¼ rxðtÞ 1� xðtÞk

� �� xðtÞyðtÞ

a1þx2ðtÞ ;

y0ðtÞ ¼ yðtÞ � d þ lxðtÞa1þx2ðtÞ

� �;

9>=>; t 6¼ nT ;

xðnTþÞ ¼ xðnT Þ; yðnTþÞ ¼ yðnT Þ þ s;

X ð0þÞ ¼ x0 ¼ ðx0; y0ÞT ;

t ¼ nT ;

8>>>>>><>>>>>>:

ð1:2Þ

where T is the period of the impulsive immigration or stock of the predator, s > 0 is the size of immigration or stockof the predator. We obtain the conditions for extinction of prey and permanence of the system. Further, using

numerical simulation method, we analyze the complexities of system with the increasing of the size of the impulsive

immigration or stock, which includes (1) quasi-periodic oscillating, (2) period-doubling, (3) period-halfing, (4) non-

unique dynamics (meaning that several attractor coexist), (5) attractor crisis and(6) chaotic bands with periodic

windows.

This paper is arranged as follows. In Section 3, some notations and lemmas are given. In Section 4, using the Floquet

theory of impulsive equation and small amplitude perturbation skills, we prove the local stability of prey-eradication

periodic solution and give the condition of permanence of system (1.2). In Section 5, the results of numerical analysis

are shown, moreover, these results are discussed briefly.

2. Formulation of the model

System (1.1) was studied completely in [5], and it is shown that the system exhibits complex dynamics behavior.

Their main results are summarized in the following Lemmas.

S. Zhang et al. / Chaos, Solitons and Fractals 23 (2005) 631–643 633

System (1.1) has at most four equilibria, (0,0), ðk; 0Þ, and two interior equilibria ðx1; y2Þ, ðx2; y2Þ, where

x1 ¼l �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2 � 4ad2

p2d

; y1 ¼ r 1�

� x1k

�ðaþ x21Þ;

x2 ¼l þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2 � 4ad2

p2d

; y2 ¼ r 1�

� x2k

�ðaþ x22Þ:

Lemma 2.1. If 4ad2 < l2 < 163ad2 and x1 < k < x2, then system (1.1) has a globally asymptotically stable positive equi-

librium ðx1; y1Þ.

Lemma 2.2. If l2 > 163ad2 and x2 > k > x3, where x3 ¼ 2l�

ffiffiffiffiffiffiffiffiffiffiffiffil2�4ad2

p2d , then system (1.1) has three equilibria: two hyperbolic

saddles (0,0) and ðk; 0Þ and a positive unstable focus (or node) ðx1; y1Þ. Moreover, system (1.1) has a unique limit cycle,which is stable.

Lemma 2.3. If 163ad2 < l2 < 18þ2

ffiffi6

p

3ad2 and x3 < k < l

d, where x3 ¼2l�

ffiffiffiffiffiffiffiffiffiffiffiffil2�4ad2

p2d , then system (1.1) has four equilibriums: two

hyperbolic saddles (0,0) and ðx2; y2Þ, a hyperbolic stable node ðk; 0Þ, and an unstable focus (or node) ðx1; y1Þ. Moreover,there exists a unique limit cycle, which is stable.

Remark 2.1.When 18þ2ffiffi6

p

3ad26 l2 and k < x3, the dynamics of system (1.1) in the interior of the first quadrant could be

very complicated.

For simplicity, set x1 ¼ x, x2 ¼ 1a1y. We transform system (1.2) into

x01ðtÞ ¼ x1ðtÞða� bx1ðtÞÞ � x1ðtÞx2ðtÞ1þex2

1ðtÞ ;

x02ðtÞ ¼ x2ðtÞ �d þ mx1ðtÞ1þex2

1ðtÞ

� �;

9=; t 6¼ nT ;

x1ðnTþÞ ¼ x1ðnT Þ; x2ðnTþÞ ¼ x2ðnT Þ þ p;

xð0þÞ ¼ x0 ¼ ðx01; x02ÞT;

)t ¼ nT ;

8>>>>><>>>>>:

ð2:1Þ

where a ¼ r, b ¼ rk, m ¼ l

a1, e ¼ 1

a1, p ¼ 1

a1s.

We will consider the dynamics of system (2.1) and study the influences of the impulsive perturbation on the inherent

oscillation by numerical methods, and we get some interesting results.

3. Preliminaries

In this section, we will give some definitions, notations and lemmas which will be useful for our main results.

Let Rþ ¼ ½0;1Þ, R2þ ¼ fx 2 R2jxP 0g. Denote f ¼ ðf1; f2Þ the map defined by the right hand of the first twoequations of system (2.1), and N the set of all non-negative integers. Let V : Rþ � R2þ ! Rþ, then V is said to belong toclass V0 if

(1) V is continuous in ðnT ; ðnþ 1ÞT � � R2þ and for each x 2 R2þ; n 2 N , limðt;yÞ!ðnTþ ;xÞ ¼ V ðnTþ; xÞ exists.(2) V is locally Lipschitzian in x.

Definition 3.1. Let V 2 V0, then for ðt; xÞ 2 ðnT ; ðnþ 1ÞT � � R2þ, the upper right derivative of V ðt; xÞ with respect to theimpulsive differential system (2.1) is defined as

DþV ðt; xÞ ¼ limh!0þ

sup1

h½V ðt þ h; xþ hf ðt; xÞÞ � V ðt; xÞ�:

Definition 3.2. System (2.1) is said to be permanent if there exist two positive constants m;M and T0, such that eachpositive solution ðx1ðtÞ; x2ðtÞÞ of the system (2.1) satisfies m6 xiðtÞ6M , for all t > T0; i ¼ 1; 2.

The solution of system (2.1) is a piecewise continuous function x : Rþ ! R2þ, xðtÞ is continuous on ðnT ; ðnþ 1ÞT �,n 2 N and xðnTþÞ ¼ limt!nTþ xðtÞ exists. The smoothness of f guarantee the global existence and uniqueness of thesolution of system (2.1). For the details, it is referred to the books [20,21].


The following lemma is obvious.

Lemma 3.1. Let xðtÞ be a solution of system (2.1) with xð0þÞP 0, then xðtÞP 0 for all tP 0. And further xðtÞ > 0, for allt > 0 if xð0þÞ > 0.

And we will use the following important comparison theorem on impulsive differential equation [20]:

Lemma 3.2. Suppose V 2 V0. Assume that

DþV ðt; xÞ6 gðt; V ðt; xÞÞ; t 6¼ nT ;V ðt; xðtþÞÞ6wnðV ðt; xÞÞ; t ¼ nT ;

�ð3:1Þ

where g : Rþ � Rþ ! R is continuous in ðnT ; ðnþ 1ÞT � � Rþ and for u 2 Rþ; n 2 N , limðt;yÞ!ðnTþ;uÞ gðt; yÞ ¼ gðnTþ; uÞ ex-ists, wn : Rþ ! Rþ is non-decreasing. Let rðtÞ be the maximal solution of the scalar impulsive differential equation

u0ðtÞ ¼ gðt; uðtÞÞ; t 6¼ nT ;uðtþÞ ¼ wnðuðtÞÞ; t ¼ nT ;uð0þÞ ¼ u0;

8<: ð3:2Þ

existing on ½0;1Þ. Then V ð0þ; x0Þ6 u0 implies that V ðt; xðtÞÞ6 rðtÞ, tP 0, where xðtÞ is any solution of (2.1).

Finally, we give some basic properties about the following subsystem of system (2.1).

y0ðtÞ ¼ �dyðtÞ; t 6¼ nT ;yðtþÞ ¼ yðtÞ þ p; t ¼ nT ;yð0þÞ ¼ y0:

8<: ð3:3Þ

Clearly

y�ðtÞ ¼ p expð�dðt � nT ÞÞ1� expð�dT Þ ; t 2 ðnT ; ðnþ 1ÞT �; n 2 N ; y�ð0þÞ ¼ p

1� expð�dT Þ

is a positive periodic solution of system (3.3). Since

yðtÞ ¼ yð0þÞ � p1� expð�dT Þ

� �expð�dtÞ þ y�ðtÞ

is the solution of system (3.3) with initial value y0 P 0, where t 2 ðnT ; ðnþ 1ÞT �; n 2 N , we get

Lemma 3.3. For a positive periodic solution y�ðtÞ of system (3.3) and every solution yðtÞ of system (3.3) with y0 P 0, wehave jyðtÞ � y�ðtÞj ! 0, when t ! 1.

Therefore, we obtain the complete expression for the prey-eradication periodic solution of system (2.1)

ð0; x�2ðtÞÞ ¼ 0;p expð�dðt � nT ÞÞ1� expð�dT Þ

� �:

for t 2 ðnT ; ðnþ 1ÞT �.

4. Extinction and permanence

Firstly, we study the stability of prey-eradication periodic solution.

Theorem 4.1. Let ðx1ðtÞ; x2ðtÞÞ be any solution of (2.1), then ð0; x�2ðtÞÞ is locally asymptotically stable provided that T < pad.

Proof. The local stability of periodic solution ð0; x�2ðtÞÞ may be determined by considering the behavior of smallamplitude perturbations of the solution. Define

x1ðtÞ ¼ uðtÞ; x2ðtÞ ¼ x�2ðtÞ þ vðtÞ:

there may be written


uðtÞvðtÞ

� �¼ UðtÞ uð0Þ

vð0Þ

� �; 06 t < T ;

where UðtÞ satisfies

dUdt

¼ a� x�2ðtÞ 0

mx�2ðtÞ �d

� �UðtÞ;

and Uð0Þ ¼ I , the identity matrix. The linearization of the third and fourth equations of system (2.1) becomes

uðnTþÞvðnTþÞ

� �¼ 1 0

0 1

� �uðnT ÞvðnT Þ

� �:

Hence, if both eigenvalues of

M ¼ 1 0

0 1

� �UðT Þ;

have absolute values less than one, then the periodic solution ð0; x�2ðtÞÞ is locally stable. Since all eigenvalues of M are

l1 ¼ expð�dT Þ < 1; l2 ¼ expZ T

0

ða�

� x�2ðtÞÞdt�;

j l2 j< 1 if and only if T < pad. According to Floquet theory of impulsive differential equation, the prey-eradication

solution ð0; x�2ðtÞÞ is locally stable. This completes the proof. h

Remark. If the period of pulses T is more than Tmax ¼ pad, the prey-eradication solution becomes unstable.

Theorem 4.2. There exists a constant M > 0, such that xiðtÞ6M , i ¼ 1; 2 for each solution xðtÞ ¼ ðx1ðtÞ; x2ðtÞÞ of system(2.1) with all t large enough.

Proof. Define V ðt; xÞ as
V ðt; xÞ ¼ mx1ðtÞ þ x2ðtÞ:
It is clear that V 2 V0. We calculate the upper right derivative of V ðt; xÞ along a solution of system (2.1) and get the

following impulsive differential equation

DþV ðtÞ þ LV ðtÞ ¼ mðaþ LÞx1 � bmx21 þ ðL� dÞx2; t 6¼ nT ;V ðtþÞ ¼ V ðtÞ þ p; t ¼ nT :

�ð4:1Þ

Let 0 < L < d, then mðaþ LÞx1 � bmx21 þ ðL� dÞx2 is bounded. Select L0 and L1 such that

DþV ðtÞ6 � L0V ðtÞ þ L1; t 6¼ nT ;V ðtþÞ ¼ V ðtÞ þ p; t ¼ nT :

�

where L0; L1 are two positive constant. According to Lemma 3.2, we have

V ðtÞ6 V ð0þÞ�

� L1L0

�expð�L0tÞ þ

Pð1� expð�nL0T ÞÞexpðL0T Þ � 1

expðL0T Þ expð�L0ðt � nT ÞÞ þ L1L0

:

where t 2 ðnT ; ðnþ 1ÞT �. Hence

limt!1

V ðtÞ6 L1L0

þ p expðL0T ÞexpðL0T Þ � 1

:

Therefore V ðt; xÞ is ultimately bounded. We obtain that each positive solution of system (2.1) is uniformly ultimately

bounded. This completes the proof. h

In the following, we investigate the permanence of system (2.1).


Theorem 4.3. System (2.1) is permanent if T > pad.

Proof. Suppose xðtÞ is a solution of system (2.1) with x0 > 0. From Theorem 4.2 we may assume xiðtÞ6M , i ¼ 1; 2 andM > a

b, tP 0. Let m2 ¼ p expð�dT Þ1�expð�dT Þ � e2, e2 > 0. According to Lemmas 3.2 and 3.3, we have x2ðtÞ > m2, for all t large

enough. In the following, we want to find m1 > 0 such that x1ðtÞPm1 for all t large enough. We will do it in thefollowing two steps for convenience.

Step 1. Since T > pad, we can select m3 > 0, e1 > 0 small enough such that m3 < minfab, 1ffiffi

ep g, d ¼ mm3

1þem23

< d,

r ¼ aT � bm3T � pd�d � e1T > 0. We will prove there exists t1 2 ð0;1Þ such that x1ðt1ÞPm3. Otherwise, according to the

above assumption, we get

x02ðtÞ6 x2ðtÞð�d þ dÞ:

By Lemmas 3.2 and 3.3, we get x2ðtÞ6 yðtÞ and yðtÞ ! yðtÞ, where yðtÞ ¼ p expðð�dþdÞðt�nT ÞÞ1�expðð�dþdÞT Þ , t 2 ðnT ; ðnþ 1ÞT �, and yðtÞ is

the solution of the following equation

y0ðtÞ ¼ yðtÞð�d þ dÞ; t 6¼ nT ;

yðtþÞ ¼ yðtÞ þ p; t ¼ nT ;

yð0þÞ ¼ x02 > 0:

8><>: ð4:2Þ

Therefore there exists a T1 > 0 such that

x2ðtÞ6 yðtÞ6 yðtÞ þ e1;

x01ðtÞP x1ðtÞða� bm3 � ðyðtÞ þ e1ÞÞ: ð4:3Þ

Let N1 2 N and N1T P T1. Integrating (4.3) on ðnT ; ðnþ 1ÞT �ðnPN1Þ, we have

x1ððnþ 1ÞT ÞP x1ðnT Þ expZ ðnþ1ÞT

nTða

�� bm3 � ðyðtÞ þ e1ÞÞdt

�¼ x1ðnT Þ expðrÞ:

Then

x1ððN1 þ kÞT ÞP x1ðN1T Þ expðkrÞ ! 1; k ! 1;

which is a contradiction to the boundedness of x1ðtÞ.Step 2. If x1ðtÞPm3; tP t1, then our aim is obtained. Otherwise, if x1ðtÞ < m3 for some t > t1, we may set

t� ¼ inf tP t1fx1ðtÞ < m3g. We have x1ðtÞPm3, for t 2 ½t; t�Þ. It is easy to see x1ðt�Þ ¼ m3 since x1ðtÞ is continuous. Supposet� 2 ½n1T ; ðn1 þ 1ÞT Þ; n1 2 N . Select n2; n3 2 N such that

n2T > T2 ¼ lne1

M þ p

� ��ð�d þ dÞ;

expððn2 þ 1Þr1T Þ expðn3rÞ > 1;

where r1 ¼ a� bm3 �M < 0. Set T ¼ n2T þ n3T . We claim that there must exist a t0 2 ððn1 þ 1ÞT ; ðn1 þ 1ÞT þ T � suchthat x1ðt0ÞPm3. Otherwise, x1ðtÞ < m3; t 2 ððn1 þ 1ÞT ; ðn1 þ 1ÞT þ T �. Considering (4.2) with yððn1 þ 1ÞTþÞ ¼x2ððn1 þ 1ÞTþÞ, we have

yðtÞ ¼ yððn1 þ 1ÞTþÞ � p1� expðð�d þ dÞT Þ

� �expðð�d þ dÞðt � ðn1 þ 1ÞT ÞÞ þ yðtÞ

t 2 ðnT ; ðnþ 1ÞT �; n1 þ 16 n < n1 þ 1þ n2 þ n3. Then

j yðtÞ � yðtÞ j< ðM þ pÞ expð�ðd � dÞn2T Þ < e1;

x2ðtÞ6 yðtÞ6 yðtÞ þ e1;

for ðn1 þ n2 þ 1ÞT 6 t6 ðn1 þ 1ÞT þ T , which implies (4.3) holds on ½ðn1 þ 1þ n2ÞT ; ðn1 þ 1ÞT þ T �. As in step 1,we have
x1ððn1 þ 1þ n2 þ n3ÞT ÞP x1ððn1 þ 1þ n2ÞT Þ expðn3rÞ:
There are two possible cases for t 2 ðt�; ðn1 þ 1ÞT �.Case a. If x1ðtÞ < m3 for t 2 ðt�; ðn1 þ 1ÞT �, then x1ðtÞ < m3 for all t 2 ðt�; ðn1 þ 1þ n2ÞT �. System (2.1) gives

x01ðtÞP x1ðtÞða� bm3 �MÞ ¼ r1x1ðtÞ: ð4:4Þ


Integrating (4.4) on ½t�; ðn1 þ 1þ n2ÞT � yields

x1ððn1 þ 1þ n2ÞT ÞPm3 expðr1ðn2 þ 1ÞT Þ;

Then

x1ððn1 þ 1þ n2 þ n3ÞT ÞPm3 expðr1ðn2 þ 1ÞT Þ expðn3rÞ > m3;

which is a contradiction. Let t ¼ inftP t�

fx1ðtÞPm3g, then x1ðtÞ ¼ m3 and (4.4) holds for t 2 ½t�; tÞ. Integrating (4.4) on½t�; tÞ yields

x1ðtÞP x1ðt�Þ expðr1ðt � t�ÞÞPm3 expðr1ð1þ n2 þ n3ÞT Þ:

Let m3 expðr1ð1þ n2 þ n3ÞT Þ ¼ m1. For t > t the same argument can be continued since x1ðtÞPm3; tP t1, hencex1ðtÞPm1 for all t > t1.

Case b. There exists a t0 2 ðt�; ðn1 þ 1ÞT � such that x1ðt0ÞPm3. Let t ¼ inft>t�

fx1ðtÞPm3g, then x1ðtÞ < m3 for t 2 ½t�; tÞand xðtÞ ¼ m3. For t 2 ½t�; tÞ (4.4) holds. Integrating (4.4) on ½t�; tÞ, we have

x1ðtÞP x1ðt�Þ expðr1ðt � t�ÞÞPm3 expðr1T Þ > m1:

This process can be continued since x1ðtÞPm3, and we have x1ðtÞPm1 for t > t1. Thus in both cases we get x1ðtÞPm1for tP t1.This completes the proof. h

5. Numerical analysis

In this section we will study the influence of impulsive perturbation p on inherent oscillation.For (A) a ¼ 10, b ¼ 0:25, d ¼ 0:1, m ¼ 0:007, e ¼ 0:001, p ¼ 0, T ¼ 8, by Lemma 2.1 we know system (2.1) has a

globally asymptotically stable positive equilibrium.

For (B) a ¼ 6, b ¼ 619, d ¼ 0:1, m ¼ 0:025, e ¼ 0:01, p ¼ 0, T ¼ 8, by Lemma 2.2 we know system (2.1) has an

unstable positive equilibrium and a unique stable limit cycle.

For (C) a ¼ 5, b ¼ 522, d ¼ 0:2, m ¼ 0:05, e ¼ 0:01, p ¼ 0, T ¼ 8, by Lemma 2.3 we know system (2.1) has two

positive equilibrium and a unique stable limit cycle.

For (D) a ¼ 10, b ¼ 6, d ¼ 0:1, m ¼ 0:95, e ¼ 0:001, p ¼ 0, T ¼ 8, by Remark 2.1 we know the dynamics of system(2.1) in the interior of the first quadrant could be very complicated.

Since the corresponding continuous system (2.1) (p ¼ 0) cannot be solved explicitly and system (2.1) cannot be

rewritten as equivalent difference equations, it is difficult to study them analytically. However, the influence of p may bedocumented by stroboscopically sampling one of the variables over a range of p values. Stroboscopic map is a specialcase of the Poincar�e map for periodically forced system or periodically pulsed system. Fixing points of the stroboscopicmap correspond to periodic solutions of system (2.1) having the same period as the pulsing term; periodic points of

period k about stroboscopic map correspond to entrained periodic solutions of system (2.1) having exactly k times theperiod of the pulsing; invariant circles correspond to quasi-periodic solutions of system (2.1); system (2.1) possibly

appears chaotic (strange) attractors.

For (A), the system (2.1) incarnates T -periodic solution with p increasing from 0.001 to 7. When p > 7, x1 will goextinct. Complexity doesn’t occur with respect to system (2.1) in this case.

For (B) and (C), from bifurcation diagrams (Figs. 1 and 2) or called final state diagrams in [22], we can easily see that

the dynamical behavior of these two cases is very complicated, which includes quasi-periodic oscillating, many chaotic

bands, narrow periodic windows, wide periodic windows, and cries (the phenomenon of ‘‘crisis’’ in chaotic attractors

can suddenly appear or disappear, or change size discontinuously as a parameter smoothly varies, was first extensively

analogized by Grebogi et al., (see [23]).

For (B), when p is sufficiently small (p < p1 � 0:025), system (2.1) experiences quasi-periodic oscillating (Fig. 3(a)

limit cycle), and if p tends to zero, the annular region (Fig. 3(b)) will shrink to limit cycle. However, when p > p1 quasi-periodic oscillating is destroyed and 2T -periodic solution occurs (Fig. 4(a)) and is stable if p < p2 � 0:09. When p > p2,it is unstable (chaotic crises happen) and it comes in chaotic area (Fig. 5) with periodic windows. When p > p3 � 0:8,the chaos suddenly disappear and appear T -periodic solution. Whereas the parameter p further increases, 2T -periodicsolution occurs at p4 ¼ 2:8 (Fig. 6) again.For (C), when p is sufficiently small (p < p1 � 0:085), system (2.1) experiences quasi-periodic oscillating (Fig. 7(a)

limit cycle), if p tends to zero, the annular region (Fig. 7(b)) will shrink to limit cycle. However, when p > p1 quasi-periodic oscillating will be destroyed and it comes in chaotic area (Fig. 8(a)) with periodic windows (Fig. 9). When

Fig. 2. Bifurcation diagrams of system (2.1) with a ¼ 5, b ¼ 522, d ¼ 0:2, m ¼ 0:05, e ¼ 0:01, T ¼ 8. (a) and (c) x1 are plotted for p over

[0.001,8], [0,0.7], (b) and (d) x2 are plotted for p over [0.001,8], [0,0.7]. Show the effect of parameter p on the dynamical behavior.

Fig. 1. Bifurcation diagrams of system (2.1) with a ¼ 6, b ¼ 619, d ¼ 0:1, m ¼ 0:025, e ¼ 0:01, T ¼ 8. (a) and (c) x1 are plotted for p over

[0.001,5], [0,0.8], (b) and (d) x2 are plotted for p over [0.001,5], [0,0.8]. Show the effect of parameter p on the dynamical behavior.


p ¼ p2 � 0:71 the chaos suddenly disappear and appear 2T -periodic solution (Fig. 10(a)). When p ¼ p3 � 2:17 the chaosabruptly appears (Fig. 8(b)) and p ¼ p4 � 2:25 the chaos abruptly disappears.

Fig. 4. Periodic windows. (a) Phase portrait of 2T -periodic solution for p ¼ 0:06, (b) phase portrait of 5T -periodic solution forp ¼ 0:66.

Fig. 5. Strange attractor. (a) Phase portrait of system (2.1) for p ¼ 0:3, (b) phase portrait of system (2.1) for p ¼ 0:76:

Fig. 3. Phase portraits of system (2.1). The system (2.1) are plotted with initial value x0 ¼ ð1; 2:4Þ, (a) p ¼ 0, (b) p ¼ 0:02.

Fig. 6. Periodic solution. (a) Phase portrait of T -periodic solution for p ¼ 1:6, (b) phase portrait of 2T�periodic solution for p ¼ 2:8.


Fig. 7. Phase portraits of system (2.1). The system are ploted with initial value x0 ¼ ð1; 2:4Þ, (a) p ¼ 0, (b) p ¼ 0:05.

Fig. 8. Strange attractor. (a) Phase portrait of system (2.1) for p ¼ 0:61, (b) phase portrait of system (2.1) for p ¼ 2:18.

Fig. 9. Periodic windows. (a) Phase portrait of 3T -periodic solution for p ¼ 0:3, (b) phase portrait of 4T -periodic solution for p ¼ 0:47.

Fig. 10. Periodic solution. (a) Phase portrait of 2T -periodic solution for p ¼ 1:2, (b) phase portrait of 2T -periodic solution forp ¼ 2:48.



For (D), the bifurcation diagrams (Fig. 11) clearly show that: with p increasing from 1 to 8, the system (2.1)

experiences the process of periodic doubling cascadefi chaosfi periodic halfing, which is characterized by (1) periodic

doubling, (2) period halfing, (3) non-unique dynamics, (4) periodic windows.

When p < p1 � 3:63, system (2.1) incarnates a 2T -periodic solution of the period of the impulsive perturbation and itis stable. when p > p1, it is unstable and there is a cascade of period doubling bifurcations (Fig. 12) leading to chaos(Fig. 13). which is followed by a cascade of periodic halfing bifurcations from chaos to T -periodic solution (Fig. 14). A

Fig. 11. Bifurcation diagrams of system (2.1) with a ¼ 10, b ¼ 6, d ¼ 0:1, m ¼ 0:95, e ¼ 0:001, T ¼ 8. (a, a1) x1 are plotted for p over[1,8], [6.4,8], (b, b1) x2 are plotted for p over [1,8], [6.4,8]. Show the effect of parameter p on the dynamical behavior.

Fig. 12. Period doubling cascade. (a) Phase portrait of T -periodic solution for p ¼ 3, (b) phase portrait of 2T -periodic solution forp ¼ 4.

Fig. 13. A strange attractor. (a) Phase portrait of system (2.1) of p ¼ 6:6, (b) and (c) time series of x1 and x2.

Fig. 14. Period halfing cascade. (a) Phase portrait of 8T -period solution for p ¼ 7:09, (b) phase portrait of 4T -period solution forp ¼ 7:12.

Fig. 15. Coexistence of 2T -period solution with a stranger attractor when p ¼ 7:32. (a) Solution with x0 ¼ ð1:2; 3Þ will finally tend to a2T -period solution, (b) solution with x0 ¼ ð0:2; 7:5Þ will tend to a strange attractor.

Fig. 16. (a) Phase portrait and (b), (c) time series of solution with x0 ¼ ð0:5; 7Þ when p > adT ¼ 8. x1ðtÞ ! 0 as t ! 0.


typical chaos oscillation is captured when p ¼ 5:71. This periodic-doubling route to chaos is the hallmark of the logisticand Ricker maps [24,25] and has been studied extensively by Mathematicians [26,27]. For the predator–prey system,

chaotic behaviors are usually obtained by continuous system with periodic forcing [28,29]. Periodic halfing is the flip

bifurcation in the opposite direction, which is also observed in [30,31]. However, when p ¼ 7:32, it appears thatattractor is non-unique[32]: different attractors can co-exist, obviously, which one of the attractors is reached depends

on the initial values. For example in Fig. 15 2T -periodic solution and a strange attractor co-exist.If p > adT ¼ 8, by Theorem 4.1 we know that the system will go extinct with x1ðtÞ ! 0 as t ! 0 (Fig. 16).

6. Conclusion

In this paper, we have investigated predator–prey system with defensive ability of prey and impulsive perturbations

on the predator. Using Floquet theorem and small amplitude perturbation skills, we have proved that prey-eradication

periodic solution ð0; x�2ðtÞÞ is locally asymptotically stable when the impulsive period T < Tmax ¼ pad (critical value).

Otherwise, if the impulsive period T > Tmax, the prey-eradication periodic solution becomes unstable and the system ispermanent. That is, we have established conditions guaranteeing the system to be permanent and driving the prey to be

extinct.


Numerical analysis indicates that the complex dynamics of system (2.1) depends on the values of impulsive per-

turbations p and all parameters. By choosing impulsive perturbations p as bifurcation parameter, we have obtainedbifurcation diagrams (Figs. 1, 2, 11). Figs. 1 and 2 have shown that there exists complexity for system (2.1) including

quasi-periodic oscillating, many chaotic bands, narrow periodic windows, wide periodic windows and cries. Moreover,

bifurcation diagram Fig. 11 displays richer structure, which consists of periodic doubling cascade, chaotic bands with

periodic windows, periodic halfing cascade and non-unique dynamics. All these results show that dynamical behavior of

system (2.1) becomes more complex under periodically impulsive perturbations.

Acknowledgements

This Work is Supported by the National Natural Science Foundation of China (10171106).

References

[1] Holling CS. The functional response of predator to prey density and its role in mimicry and population regulation. Mem Ent Sec

Can 1965;45:1–60.

[2] Andrews JF. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol

Bioeng 1968;10:707–23.

[3] Sugie J, Howell JA. Kinetics of phenol oxidation by washed cell. Biotechnol Bioeng 1980;23:2039–49.

[4] Tener JS. Muskoxen. Ottawa: Queen’s Printer; 1965.

[5] Ruan S, Xiao D. Global analysis in a predator–prey system with nonmonotonic functional response. SIAM J Appl Math

2001;61:1445–72.

[6] Brauer F, Soudack AC. Coexistence properties of some predator–prey systems under constant rate harvesting and stocking. J

Math Biol 1981;12:101–14.

[7] Brauer F, Soudack AC. Constant-rate stocking of predator–prey systems. J Math Biol 1981;11:1–14.

[8] Croft BA. Arthropod biological control agents and pesticides. New York: John Wiley Sons; 1990.

[9] Debach P, Rosen D. Biological control by natural enemies. 2nd ed. Cambridge: Cambridge University Press; 1991.

[10] Van lenteren JC. Measures of success in biological of anthrop pods by augmentation of natural enemies. In: Wratten S, Gurr G,

editors. Measures of success in biological control. Dordrdcht: Kluwer Academic Publishers; 2000. p. 77–89.

[11] Luff ML. The potential of predators for pest control. Agri Ecos Environ 1983;10:159–81.

[12] Van Lenteren JC, Woets J. Biological and integrated pest control in greenhouses. Ann Rev Ent 1988;33:239–50.

[13] Roberts MG, Kao RR. The dynamics of an infectious disease in a population with birth pulses. Math Biosci 1998;149:23–36.

[14] Tang SY, Chen LS. Density-dependent birth rate,birth pulse and their population dynamic consequences. J Math Biol

2002;44:185–99.

[15] Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model. Bull Math Biol 1998;60:1–26.

[16] D’Onofrio A. Stability properties of pulse vaccination strategy in SEIR epidemic model. Math Comput Modell 1997;26:59–72.

[17] Panetta JC. A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive

environment. Bull Math Biol 1996;58:425–47.

[18] Lakmeche A, Arino O. Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic

treatment. Dyn Continuous Discrete Impulsive Syst 2000;7:165–87.

[19] Ballinger G, Liu X. Permanence of population growth models with impulsive effects. Math Comput Modell 1997;26:59–72.

[20] Lakshmikantham V, Bainov DD, Simeonov PC. Theory of impulsive differential equations. Singapore: World Scientific; 1989.

[21] Bainov DD, Simeonov DD. Impulsive differential equations:periodic solutions and applications. England: Longman; 1993.

[22] Davies B. Exploring chaos, theory and experiment. Reading, MA: Perseus Books; 1999.

[23] Grebogi C, Ott E, Yorke JA. Crises,sudden changes in chaotic attractors and chaotic transients. Physick D 1983;7:181–200.

[24] May RM. Biological population with nonoverlapping generations: stable points, stable cycles, and chaos. Science 1974;186:645–57.

[25] May RM, Oster GF. Bifurcations and dynamic complexity in simple ecological models. Am Nature 1976;110:573–99.

[26] Collet P, Eeckmann JP. Iterated maps of the inter val as dynamical systems. Boston: Birkhauser; 1980.

[27] Venkatesan A, Parthasarathy S, Lakshmanan M. Occurrence of multiple period-doubling bifurcation route to chaos in

periodically pulsed chaotic dynamical systems. Chaos, Solitons & Fractals 2003;18:891–8.

[28] Vandermeer J, Stone L, Blasius B. Categories of chaos and fractal basin boundaries in forced predator–prey models. Chaos,

Solitons & Fractals 2001;12:265–76.

[29] Gakkhar S, Naji RK. Chaos in seasonally perturbed ratio-dependent prey–predator system. Chaos, Solitons & Fractals

2000;11:107–18.

[30] Neubert MG, Caswell H. Density-dependent vital rates and their population dynamic consequences. J Math Biol 2000;41:103–21.

[31] Wikan A. From chaos to chaos. An analysis of a discrete age-structured prey–predator model. J Math Biol 2001;43:471–500.

[32] Tang SY, Chen LS. Chaos in functional response host–parasitoid ecosystem models. Chaos, Solitons & Fractals 2002;13:875–84.

the study of predator–prey system with defensive ability of prey and impulsive perturbations on...

Documents