Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population

Applied Mathematical Modelling 37 (2013) 4337–4349

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Effects of prey refuge on a ratio-dependent predator–prey modelwith stage-structure of prey population

Sapna DeviDST-Centre for Interdisciplinary Mathematical Sciences, Faculty of Science, Banaras Hindu University, Varanasi 221005, India

a r t i c l e i n f o

Article history:Received 11 October 2011Received in revised form 5 July 2012Accepted 18 September 2012Available online 29 September 2012

Keywords:Predator–preyStage-structureRefugeStabilityBifurcationPersistence

0307-904X/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.apm.2012.09.045

E-mail addresses: [email protected], s

a b s t r a c t

In this paper, a stage-structured predator–prey model is proposed and analyzed to studyhow the type of refuges used by prey population influences the dynamic behavior of themodel. Two types of refuges: those that protect a fixed number of prey and those that pro-tect a constant proportion of prey are considered. Mathematical analyses with regard topositivity, boundedness, equilibria and their stabilities, and bifurcation are carried out. Per-sistence condition which brings out the useful relationship between prey refuge parameterand maturation time delay is established. Comparing the conclusions obtained from ana-lyzing properties of two types of refuges using by prey, we observe that value of matura-tion time at which the prey population and hence predator population go extinct is greaterin case of refuges which protect a constant proportion of prey.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Refuge use is any strategy that decreases predation risk. The existence of refuges can clearly have important effects on theco-existence of predators and prey. In the existing literature, two types of refuges: those that protect a constant number ofprey and those that protect a constant proportion of prey have been considered [1]. McNair [2], Sih [3], Ruxton [4] and Schef-fer and de Boer [5] etc. proposed and analyzed predator–prey models incorporating prey refuges. Ruxton [4] proposed a con-tinuous-time predator–prey model taking into consideration that the rate at which prey move to the refuge is proportionalto predator density. Krivan [6] investigated the influence of refuges used by prey on the dynamics of predator–prey system intwo patch environment, where one patch is the refuge for prey while the other is an open habitat. The results showed thatoptimal antipredator behavior of prey leads to the persistence and reduction of oscillations in population densities. Taking acue from this, Gonzalez-Olivars and Ramos-Jiliberto [7], Kar [8] and Haung et al. [9] derived a predator–prey model with Hol-ling type III functional response incorporating prey refuges and evaluated the effects with regard to the stability of the inte-rior equilibrium. They obtained that the refuges used by prey can increase the stability of the interior equilibrium. Ma et al.[10] studied the effects of two types of refuges used by prey on a predator–prey interaction with a class of functional re-sponses. After analyzing the stability properties of two types of refuges using by prey, they noted that refuges which protecta constant number of prey have stronger stabilizing effect on the dynamics than the refuges which protect a constant pro-portion of prey, which is agreement with previous works [6,7]. Chen et al. [11] discussed the instability and global stabilityproperties of the equilibria and the existence and uniqueness of limit cycle of a predator–prey model with Holling type IIfunctional response incorporating a prey refuge. They found that dynamic behavior of the model very much depends onthe prey refuge parameter and increasing amount of refuge could increase prey densities and lead to predator outbreaks.

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4338 S. Devi / Applied Mathematical Modelling 37 (2013) 4337–4349

In the natural world, there are many species (particularly mammalian population) whose individual members have a lifehistory that takes them through two stages: immature and mature with a time lag. Aiello and Freedman [12], Aiello et al. [13]and Song and Chen [14] proposed and analyzed single species growth models with stage structure consisting of immatureand mature stages. Agarwal and Devi [15] studied the effects of toxicants on the dynamics of a single species growth modelwith stage structure consisting of immature and mature stages. They showed that equilibrium level of immature populationis more prone for the effects of toxicants in comparison to the mature population. Wang and Chen [16] and Song and Chen[17] proposed and analyzed predator–prey models with stage structure. Shi and Chen [18] studied a ratio-dependent pred-ator–prey model with stage structure in the prey and obtained sufficient conditions for the existence and stability of all equi-libria. Agarwal and Devi [19] proposed and analyzed a ratio-dependent predator–prey model where the prey population isstage-structured and the predator population is influenced by the resource biomass. They noted that the influence of re-source biomass on the predator population may lead to the extinction of prey population at a lesser value of maturity timein comparison to the absence of the resource biomass. However, as for as our knowledge goes, we note that effects refugesusing by prey on the dynamics of stage-structured predator–prey system have not been considered.

In this paper, a model describing the ratio-dependent predator–prey interaction with stage structure of prey populationconsisting of immature and mature stages with constant time of maturation delay is proposed. The effects of refuges used bymature prey population on the dynamics of considered model is investigated. This paper is concerned with questions of sta-bility, bifurcation and persistence of populations. But, the main question of this paper is: How prey refuges affect the criticalvalue of maturation time (the value of maturation time at which stability change occurs). The stability theory of delay dif-ferential equations is used to analyze the model. To accomplish our analytical findings and to observe the effects of impor-tant parameters on the dynamics of the system, computer simulations are carried out.

2. The mathematical model

In this paper, we consider a model given by the set of following differential equations:

_xiðtÞ ¼ axmðtÞ � cxiðtÞ � ae�csxmðt � sÞ;

_xmðtÞ ¼ ae�csxmðt � sÞ � bx2mðtÞ �

ðxmðtÞ � xmrðtÞÞyðtÞðxmðtÞ � xmrðtÞÞ þ yðtÞ ; ð2:1Þ

_yðtÞ ¼ kðxmðtÞ � xmrðtÞÞyðtÞðxmðtÞ � xmrðtÞÞ þ yðtÞ � dyðtÞ;

where xiðtÞ and xmðtÞ represent the densities of immature and mature prey populations, respectively at time t: yðtÞ is the den-sity of predator population at a time t.

Model (2.1) is derived under the following assumptions:(H1): The prey population: we first assume that the prey population is divided in two stages: immature and mature.a > 0; the birth rate of the immature population, c > 0; the death rate of immature population , b > 0; the death rateof the mature population, s > 0; the length of time from birth to maturity. e�cs denotes the surviving rate of immaturestage to reach maturity. The term ae�csxmðt � sÞ represents the immature prey individuals who were born at timeðt � sÞ and still survive at time t; and represents the transformation of immature population to mature population.(H2): Growth rate of predator population is wholly dependent on prey population and it is assumed that the predatorsfeed only on the mature prey population because for a number of animals immature prey population concealed in themountain cave depends on their parents. k > 0 is the efficiency with which predators convert consumed prey intonew predators. d > 0 is the death rate of predators.(H3): The quantity xmr is considered because of two alternative points of view: (i) xmr ¼ p; where p > 0 denotes the fixedquantity of mature prey. (ii) xmr ¼ fxm, the quantity of mature prey population using refuges is proportional to the exist-ing mature prey population with a proportionality constant 0 < f < 1.

2.1. Case when a constant number of prey using refuges

When a fixed quantity of prey using refuges, the model (2.1) takes the following form:



ðxmðtÞ � pÞyðtÞðxmðtÞ � pÞ þ yðtÞ ; ð2:2Þ

_yðtÞ ¼ kðxmðtÞ � pÞyðtÞðxmðtÞ � pÞ þ yðtÞ � dyðtÞ;

xmðtÞ ¼ /mðtÞP 0; �s 6 t < 0 and xið0Þ > 0; xmð0Þ > p; yð0Þ > 0:

S. Devi / Applied Mathematical Modelling 37 (2013) 4337–4349 4339

2.2. Case when constant proportion of prey using refuges

When considering xmr ¼ fxm; the model (2.1) is given as follows:



ðxmðtÞ � fxmðtÞÞyðtÞðxmðtÞ � fxmðtÞÞ þ yðtÞ ; ð2:3Þ

_yðtÞ ¼ kðxmðtÞ � fxmðtÞÞyðtÞðxmðtÞ � fxmðtÞÞ þ yðtÞ � dyðtÞ;

xmðtÞ ¼ /mðtÞP 0; �s 6 t < 0 and xið0Þ > 0; xmð0Þ > 0; yð0Þ > 0:

Here, we first analyze model (2.2).For continuity of initial conditions, we require

xið0Þ ¼Z 0

�saecs/mðsÞds; ð2:4Þ

the total survivors of those prey members who were born between time �s and 0.With the help of (2.4), the solution of first equation of system (2.2) can be written in terms of solution of xmðtÞ as follows:

xiðtÞ ¼Z t

t�sae�cðt�sÞxmðsÞds: ð2:5Þ

Now, from Eqs. (2.4) and (2.5) we note that, mathematically no information on the past history of xiðtÞ is needed for thesystem (2.2), because the properties of xiðtÞ can be obtained from (2.4) and (2.5) if we know the properties of xmðtÞ.

Therefore, in the rest of this chapter, we need only to consider the following model:


ðxmðtÞ � pÞyðtÞðxmðtÞ � pÞ þ yðtÞ ;

_yðtÞ ¼ kðxmðtÞ � pÞyðtÞðxmðtÞ � pÞ þ yðtÞ � dyðtÞ; ð2:6Þ

xmðtÞ ¼ /mðtÞP 0; �s 6 t < 0 and xmð0Þ > p; yð0Þ > 0:

3. Positivity of solutions

Theorem 3.1. All solutions of system (2.6) with positive initial data will remain positive for all times t > 0.

Proof. First, we prove the positivity of yðtÞ. We prove it by contradiction. Let there exists a first time t1 such thatxmðt1Þyðt1Þ ¼ 0. Assume that yðt1Þ ¼ 0. Then xmðtÞP 0 for all t 2 ½0; t1�. Now, define

A ¼ min06t6t1

ðxmðtÞ � pÞðxmðtÞ � pÞ þ yðtÞ � d� �

:

Then, for t 2 ½0; t1�,
_yðtÞP AyðtÞ:
Therefore, yðt1ÞP yð0ÞeAt1 > 0.This is a contradiction. Thus yðtÞ > 0 for all t > 0.By the same argument it can be proved that xmðtÞ > 0 for all t > 0. Suppose not. Let t1 be the first time when again

xmðt1Þyðt1Þ ¼ 0. Assume that xmðt1Þ ¼ 0. Then yðtÞP 0 for all t 2 ½0; t1�. Then from the first equation of system (2.6), we have

_xmðtÞjt¼t1P ae�csxmðt1 � sÞ � ðxmðt1Þ � pÞyðt1Þ

¼ ae�csxmðt1 � sÞ þ pyðt1Þ > 0:

Since xmð0Þ > p > 0, for xmðt1Þ ¼ 0 we must have _xmðt1Þjt¼t16 0, which is a contradiction. Thus xmðtÞ > 0 for all t > 0. h


4. Boundedness of solutions

In some of the subsequent analysis, we need the following result [20].

Lemma 4.1. Consider the following equation:

_xðtÞ ¼ axðt � sÞ � bxðtÞ � cx2ðtÞ;

where a; b; c; s are positive constants, and xðtÞ > 0 for �s 6 t 6 0; then we have

(i) If a > b; then limt!1

xðtÞ ¼ ða�bÞc .

(ii) If a < b; then limt!1

xðtÞ ¼ 0.

Theorem 4.1. Assume that k > d and ae�cs > bp. Then all solutions of system (2.6) are bounded within region X,

X ¼ fðxm; yÞ : xmðtÞ 6 xmmax ; yðtÞ 6 ymaxg;

where xmmax ¼ ae�cs

b and ymax ¼ðk�dÞðxmmax�pÞ

d .

Proof. From the first equation of system (2.6), we have

_xmðtÞ 6 ae�csxmðt � sÞ � bx2mðtÞ:

Using Lemma 4.1 and comparison principle, we obtain

lim supt!1

xmðtÞ 6ae�cs

b¼ xmmax ðsayÞ:

Now, from the second equation of system (2.6), we obtain

_yðtÞ 6 kðxmmax � pÞyðtÞðxmmax � pÞ þ yðtÞ � dyðtÞ:

Again, using comparison principle in above inequality, we obtain

lim supt!1

yðtÞ 6 ðk� dÞðxmmax � pÞd

¼ ymax ðsayÞ:

This completes Proof of Theorem 4.1. h

5. Equilibrium points and their stabilities

Now, we analyze system (2.6) by finding its equilibria and studying their linear stability. Steady states of system (2.6)satisfy the following system of equations:

ae�csxm � bx2m �

ðxm � pÞyðxm � pÞ þ y

¼ 0;

kðxm � pÞyðxm � pÞ þ y

� dy ¼ 0: ð5:1Þ

It is easy to check that system (2.6) may have following equilibria for certain parameter values:

(i) E0ð0;0Þ,(ii) E1ð�xm;0Þ, where �xm ¼ ae�cs

b , and(iii) E2ðxm; yÞ.

Existence of E0 and E1 is obvious. We prove the existence of E2ðxm; yÞ as follows:Existence of E2ðxm; yÞ: Here xm and y are obtained by solving following equations:

ae�csxm � bx2m �

ðxm � pÞyðxm � pÞ þ y

¼ 0; ð5:2Þ


kðxm � pÞðxm � pÞ þ y

� d ¼ 0: ð5:3Þ

From Eq. (5.3), we get

y ¼ ðk� dÞðxm � pÞd

: ð5:4Þ

Substituting this value of y in Eq. (5.2), we have

bx2m þ ð

ðk� dÞk

� ae�csÞxm �pðk� dÞ

k¼ 0: ð5:5Þ

Using Descartes rule of change of sign, we see that Eq. (5.5) has a unique positive root. Value of y can be obtained bysubstituting this value of xm to Eq. (5.4). Therefore, E2ðxm; yÞ exists if and only if k > d.

General variational matrix of the system (2.6) is given by

VðEÞ ¼ae�ðkþcÞs � 2bxm � y2

ððxm�pÞþyÞ2� ðxm�pÞ2

ððxm�pÞþyÞ2

ky2

ððxm�pÞþyÞ2kðxm�pÞ2

ððxm�pÞþyÞ2� d

264

375:

The variational matrix corresponding to equilibrium point E0 is given by

VðE0Þ ¼ae�ðkþcÞs �1

0 ðk� dÞ

" #:

For which one of the eigenvalues is positive because the graphs of W ¼ k and W ¼ ae�ðkþcÞs must intersect at a positivevalue and other eigenvalue depends on the sign of ðk� dÞ. Therefore, E0 is completely unstable if k > d and a saddle pointif k < d.

The variational matrix corresponding to equilibrium point E1 is given by

VðE1Þ ¼ae�ðkþcÞs � 2ae�cs �1

0 ðk� dÞ

" #:

When s ¼ 0, eigenvalues of VðE1Þ are �a and ðk� dÞ. This implies that, E1 is unstable if E2 exists otherwise it is locallyasymptotically stable equilibrium point.

Now, when s – 0, one eigenvalue of VðE1Þ is ðk� dÞ and other eigenvalues are given by solutions of

k ¼ ae�ðkþcÞs � 2ae�cs:

This implies that,

Rek < 0:

This again implies that, equilibrium point E1 is unstable if k > d, i.e. if E2 exists otherwise it is locally asymptoticallystable.

In the next theorem, we will show that equilibrium point E1 is globally asymptotically stable if E2 does not exist, i.e. ifk < d.

Theorem 5.1. Assume that k < d. Then the equilibrium E1ð�xm;0Þ is globally asymptotically stable.

Proof. We know that if k < d, then E1 is locally asymptotically stable. Now, we show that

limt!1ðxmðtÞ; yðtÞÞ ¼ ð�xm;0Þ:

From the second equation of system (2.6), we have

_yðtÞ 6 ðk� dÞyðtÞ:

When ðk� dÞ < 0, we get yðtÞ ! 0 as t !1. Thus, for an arbitrary positive number e sufficiently small, there exists a timete such that yðtÞ 6 e for all t P te.

By the first equation of system (2.6), we have

_xmðtÞP ae�csxmðt � sÞ � bx2mðtÞ � e:

Further,

_xmðtÞ 6 ae�csxmðt � sÞ � bx2mðtÞ;


is obvious. Taking into account that e sufficiently small, by Lemma 4.1, we get that

xmðtÞ ! �xm as t !1:

This completes the proof of the Theorem 5.1. h

5.1. Stability analysis of non-trivial equilibrium point

5.1.1. Local stability analysisThe variational matrix corresponding to equilibrium point E2 is given by

VðE2Þ ¼ae�ðkþcÞs � 2bxm � ðk�dÞ2

k2 � d2

k2

ðk�dÞ2k � dðk�dÞ

k

24

35:

Characteristic equation of this variational matrix is given by

ðk2 þ A1kþ A2Þ � e�ksðB1kþ B2Þ ¼ 0; ð5:6Þ

where

A1 ¼ 2bxm þðk� dÞ2

k2 þ dðk� dÞk

!ð> 0Þ;

A2 ¼2bxmdðk� dÞ

dþ d2ðk� dÞ2

k3 þ dðk� dÞ3

k3

!ð> 0Þ;

B1 ¼ ae�cs ¼ bxm þðk� dÞðxm � pÞ

kxm

� �ð> 0Þ;

B2 ¼ae�csðk� dÞd

k¼ dðk� dÞ

kbxm þ

ðk� dÞðxm � pÞkxm

� �ð> 0Þ:

In the following, we consider the delay s as the parameter to study the local stability of the positive equilibrium point E2

and the Hopf bifurcation of the system (2.6). First consider the case when s ¼ 0, the characteristic equation (5.6) becomes

k2 þ ðA10 � B10Þkþ ðA20 � B20Þ ¼ 0; ð5:7Þ

where

A10 ¼ A1js¼0; A20 ¼ A2js¼0; B10 ¼ B1js¼0; B20 ¼ B2js¼0:

Hence all roots of (5.7) have negative real parts if ðA10 � B10Þ > 0 and ðA20 � B20Þ > 0.Let us assume that a purely imaginary solution of the form k ¼ im exists for Eq. (5.6). After substituting it into (5.6) and

separating real and imaginary parts, we get

ð�m2 þ A2Þ ¼ B2 cos msþ B1m sin ms;

A1m ¼ �B2 sin msþ B1m cos ms:

Squaring and adding above two equations, we get

m4 þ ðA21 � 2A2 � B2

1Þm2 þ ðA22 � B2

2Þ ¼ 0: ð5:8Þ

Let m2 ¼ x, then Eq. (5.8) becomes

x2 þ ðA21 � 2A2 � B2

1Þxþ ðA22 � B2

2Þ ¼ 0: ð5:9Þ

Again let

P1 ¼ A21 � 2A2 � B2

1; P2 ¼ A22 � B2

2;

then equation (5.9) becomes

x2 þ P1xþ P2 ¼ 0: ð5:10Þ

Here, we have the following theorem:


Theorem 5.2. The positive equilibrium point E2ðxm; yÞ of the system (2.6) is locally asymptotically stable if ðA10 � B10Þ > 0,ðA20 � B20Þ > 0 and coefficients of Eq. (5.10) satisfy the following conditions

P1 > 0; P2 > 0; and P21 > 4P2:

5.1.2. Bifurcation analysisSubstituting k ¼ aðsÞ þ ibðsÞ in (5.6) and separating real and imaginary parts, we obtain following transcendental

equations

a2 � b2 þ aA1 þ A2 � e�asðaB1 þ B2Þ cos bs� e�asB1b sin bs ¼ 0; ð5:11Þ

2abþ A1bþ e�asðaB1 þ B2Þ sin bs� e�asB1b cos bs ¼ 0; ð5:12Þ

where a and b are functions of s.Now, we are interested to see the change in the stability behavior of E2 which will occur at the values of s for which a ¼ 0

and b – 0. To see this, we examine the sign of dads as a crosses zero. If this derivative is positive (negative), then clearly sta-

bilization (destabilization) cannot take place at that value of s.Let s be the value of s for which aðsÞ ¼ 0 and bðsÞ ¼ b – 0. Then Eqs. (5.11) and (5.12) become

�b2 þ A2 � B2 cos bs� B1b sin bs ¼ 0; ð5:13Þ

A1bþ B2 sin bs� B1b cos bs ¼ 0: ð5:14Þ

Now, eliminating s from (5.13) and (5.14), we get

b4 þ ðA21 � 2A2 � B2

1Þb2 þ ðA22 � B2

2Þ ¼ 0: ð5:15Þ

Now, differentiating equations (5.11) and (5.12) with respect to s, then setting s ¼ s, a ¼ 0 and b ¼ b, we get

h1dads

��s¼sþ h2

dbds

��s¼s¼ g; ð5:16Þ

�h2dads

��s¼sþ h1

dbds

��s¼s¼ h; ð5:17Þ

where

h1 ¼ A1 þ ð�b2 þ A2Þs� B1 cos bs; ð5:18Þ

h2 ¼ �2b� B1 sin bs� A1bs; ð5:19Þ

g ¼ A1b2; ð5:20Þ

h ¼ �bð�b2 þ A2Þ: ð5:21Þ

Solving (5.16) and (5.17), we get

dadsjs¼s ¼

ðgh1 � hh2Þh2

1 þ h22

: ð5:22Þ

From (5.22), it is clear that dads

��s¼s has the same sign as ðgh1 � hh2Þ.

Now, from Eqs. (5.18), (5.19), (5.20), (5.21), it is clear that

ðgh1 � hh2Þ ¼ b2½2b2 þ ðA21 � 2A2 � B2

1Þ�: ð5:23Þ

Let

GðuÞ ¼ u2 þ ðA21 � 2A2 � B2

1Þuþ ðA22 � B2

2Þ: ð5:24Þ

From (5.23), we note that GðuÞ is the left hand side of Eq. (5.15) with b2 ¼ u.Therefore,

Gðb2Þ ¼ 0: ð5:25Þ

Now

G0ðb2Þ ¼ 2b2 þ ðA21 � 2A2 � B2

1Þ


¼ ðgh1 � hh2Þb2

¼ h21 þ h2

2

b2

dads

��s¼s:

This implies that,

dads

��s¼s¼ b2

h21 þ h2

2

G0ðb2Þ: ð5:26Þ

Hence, we have the following theorem:

Theorem 5.3. If one of the following conditions is satisfied

(i) P21 � 4P2 > 0,

(ii) P2 < 0, P21 � 4P2 P 0, P1 < 0, P2 > 0,

and E2 is asymptotically stable for s ¼ 0, it is impossible that it remain stable for s > 0. Hence, there exists a s > 0, such thatfor s < s, E2 is asymptotically stable and for s > s, E2 is unstable and as s increases together with s, E2 bifurcates into smallamplitude periodic solutions of Hopf type [21]. The value of s is given by the following equation:

s ¼ 1

bcos�1 ð�b2 þ A2ÞB2 þ A1B1b2

ðB22 þ B2

1b2Þ

!þ 2jp

" #; j ¼ 0; 1; 2; . . . :

5.1.3. Global stability analysis

Theorem 5.4. Let the following inequality holds in a region X

max bxm þyymin

ððxmmax � pÞ þ yÞððxm � pÞ þ yÞ þky

2pððxm � pÞ þ yÞ ;�

kðxm � pÞððxmmax � pÞ þ ymaxÞððxm � pÞ þ yÞ þ

ky2pððxm � pÞ þ yÞ

�>

ðxm � pÞ2ððxm � pÞ þ yÞ ; ð5:27Þ

and

s > 1c

loga

M1

� �;

where

ymin ¼ðk� dÞ

dae�cs � 1

b� p

� �; ðsee Section 6Þ;

and

M1 ¼ bxm þyymin


2pððxm � pÞ þ yÞ �ðxm � pÞ

2ððxm � pÞ þ yÞ ;

then the equilibrium point E2 is globally asymptotically stable.

Proof. To prove this theorem, we have to show that

limt!1

xmðtÞ ¼ xm; limt!1

yðtÞ ¼ y:

Let us consider the following function

W1ðxm; yÞ ¼12ðxm � xmÞ2 þ ðy� y� y ln

yyÞ:

The derivative of W1 with respect to time t is


_W1ðxm; yÞ ¼ ae�csðxm � xmÞðxm � xmÞðt � sÞ � bðxm þ xmÞðxm � xmÞ2

� ðxm � pÞðxm � pÞððxm � pÞ þ yÞððxm � pÞ þ yÞ ðxm � xmÞðy� yÞ

� yyððxm � pÞ þ yÞððxm � pÞ þ yÞ ðxm � xmÞ2

� kðxm � pÞððxm � pÞ þ yÞððxm � pÞ þ yÞ ðy� yÞ2

þ kyððxm � pÞ þ yÞððxm � pÞ þ yÞ ðxm � xmÞðy� yÞ:

Applying Cauchy–Schwartz inequality, we arrive at the following expression

_W1ðxm; yÞ 6ae�cs

2ðxm � xmÞ2 þ

ae�cs

2ðxm � xmÞ2ðt � sÞ � bðxm þ xmÞðxm � xmÞ2

þ ðxm � pÞðxm � pÞ2ððxm � pÞ þ yÞððxm � pÞ þ yÞ ðxm � xmÞ2

þ ðxm � pÞðxm � pÞ2ððxm � pÞ þ yÞððxm � pÞ þ yÞ ðy� yÞ2

� yyððxm � pÞ þ yÞððxm � pÞ þ yÞ ðxm � xmÞ2

� kðxm � pÞððxm � pÞ þ yÞððxm � pÞ þ yÞ ðy� yÞ2

þ ky2ððxm � pÞ þ yÞððxm � pÞ þ yÞ ðxm � xmÞ2

þ ky2ððxm � pÞ þ yÞððxm � pÞ þ yÞ ðy� yÞ2:

Arranging similar terms, maximizing right hand side and assuming that following inequality holds

max bxm þyymin


2pððxm � pÞ þ yÞ ;�

kðxm � pÞððxmmax � pÞ þ ymaxÞððxm � pÞ þ yÞ þ

ky2pððxm � pÞ þ yÞ

�>

ðxm � pÞ2ððxm � pÞ þ yÞ ;

we arrive at following inequality

_W1ðxm; yÞ 6 �M1ðxm � xmÞ2 �M2ðy� yÞ2 þ ae�cs

2ðxm � xmÞ2 þ

ae�cs

2ðxm � xmÞ2ðt � sÞ;

where

M1 ¼ bxm þyymin



2ððxm � pÞ þ yÞ ;

M2 ¼kðxm � pÞ

ððxmmax � pÞ þ ymaxÞððxm � pÞ þ yÞ þky


2ððxm � pÞ þ yÞ :

Now, we choose the Lyapunov function of the following form

W2ðxm; yÞ ¼W1ðxm; yÞ þae�cs

2

Z t

t�sðxm � xmÞ2ðhÞdh;

and, hence,

_W2ðxm; yÞ ¼ _W1ðxm; yÞ þae�cs

2ðxm � xmÞ2ðtÞ �

ae�cs

2ðxm � xmÞ2ðt � sÞ:

Substituting inequality for _W1ðxm; yÞ, we get

_W2ðxm; yÞ 6 �M1ðxm � xmÞ2 �M2ðy� yÞ2 þ ae�csðxm � xmÞ2:

Therefore,

_W2ðxm; yÞ 6 �ðM1 � ae�csÞðxm � xmÞ2 �M2ðy� yÞ2:


The last expression is negative definite provided that

s > 1c

loga

M1

� �:

A direct application of the Lyapunov–LaSalle type theorem [10] shows that

limt!1

xmðtÞ ¼ xm; limt!1

yðtÞ ¼ y:

This completes the proof of the theorem 5.4. h

6. Persistence of solutions

Definition 6.1. System (2.6) is said to be uniformly persistent if there is an g > 0 (independent on the initial data) such thatevery solution ðxmðtÞ; yðtÞÞ of system (2.6) satisfies lim inf

t!1xmðtÞP g, lim inf

t!1yðtÞP g.

Theorem 6.1. Assume that k > d. Then system (2.6) is uniformly persistent if the following inequality holds

ae�cs > 1þ bp: ð6:1Þ

Proof. From the first Eq. of (2.6), we get

_xmðtÞP ae�csxmðt � sÞ � bx2mðtÞ � ðxmðtÞ � pÞ

P ae�csxmðt � sÞ � bx2mðtÞ � xmðtÞ:

Using Lemma 4.1 and comparison principle in above inequality, we obtain

lim inft!1

xmðtÞPðae�cs � 1Þ

b¼ xmmin

ðsayÞ:

Now, from the second equation of system (2.6), we obtain

_yðtÞP kðxmmin� pÞyðtÞ

ðxmmin� pÞ þ yðtÞ � dyðtÞ:

Again, using comparison principle in above inequality, we obtain

lim inft!1

yðtÞP ðk� dÞðxmmin� pÞ

d¼ ymin ðsayÞ:

According to above arguments and Theorem 4.1, we have

xmmin6 lim inf

t!1xmðtÞ 6 lim sup

t!1xmðtÞ 6 xmmax ;

ymin 6 lim inft!1

yðtÞ 6 lim supt!1

yðtÞ 6 ymax:

This completes the proof of theorem 6.1. h

Remark 1. From inequality (6.1), we note that the upper bound of prey refuge parameter, p is ðae�cs�1Þb . This implies that if the

value of p exceeds it, then the condition of persistence will not be satisfied and populations tend to extinction. This conditionalso shows a very important relationship between the maturation time s, and the prey refuge parameter p.

7. Numerical simulations

To substantiate our all analytical findings numerically, we consider the following set of parameter values

a ¼ 10; c ¼ 6; b ¼ 2; s ¼ 0:1; p ¼ 0:2; k ¼ 0:9; d ¼ 0:5: ð7:1Þ

Values of parameters are hypothetical and do not necessarily have a biological meaning.For the above set of parameter values, the equilibrium point E2 is given by

xm ¼ 2:5393; y ¼ 1:8715:

Table 1Equilibrium values of xm and y for different values of p

p xm y

0 2.5209 2.01750.8 2.5905 1.43241.4 2.6397 0.99182.0 2.6872 0.54982.4 2.7180 0.25442.74405 2.74405 0.0000

Fig. 1. Graph of xm; y versus t for different values of s when p ¼ 0 and other values of parameters are same as in (7.1)

Fig. 2. Graph of xm; y versus t for different values of s when p ¼ 0:2 and other values of parameters are same as in (7.1)


Here, we also note that all conditions of local stability, global stability and persistence are satisfied for the above set ofparameter values.

From Table 1, we note that, equilibrium value of mature prey population increases with p whereas the equilibrium valueof predator population decreases with p. Here, we also note that increase in xm is lesser than the decrease in y for increasingvalue of p. This Table also depicts that if the value of parameter p crosses a certain limit, then predator population will go toextinction and prey population will reach to maximum value. Biologically, we can interpret it as, if the value of prey refugeparameter increases, then the equilibrium density of predator population decreases due to lack of food resources because inour model we have assumed that predator population is wholly dependent on prey population.

Table 2Equilibrium values of xm and y for different values of f

f xm y

0 2.5209 2.01750.8 2.6987 0.43191.0 2.7421 0.0000

Fig. 3. Graph of xm versus y for different initial starts and values of parameters are same as in (7.1)

Fig. 4. Graph of xm versus t for different values of s when f ¼ 0:2 and other values of parameters are same as in (7.1)


Figs. 1 and 2 are the plots of xm; y versus t for different values of s when p ¼ 0 and p ¼ 0:2, respectively.Fig. 3 is the plot of xm versus y for different initial starts I, II, III, IV. From this Figure, it can be depicted that solutions con-

verge to the equilibrium point regardless of its initial values indicating the global stability of the equilibrium point xm.Comparing the results of Tables 1 and 2, it can be observed that, when constant proportion of prey using refuges then

predation risk for prey population decreases.Figs. 4 and 5, are the plots of mature prey and predator populations against t, respectively, when constant proportion of

prey using refuges. Here, the value of maturity time at which the prey population and hence predator population go extinct iss ¼ 0:56.

Comparing Figs. 1, 2, 4 and 5, we observe that, when constant proportion of mature prey population using refuges thenthe value of maturity time at which prey population and hence predator populations go extinct is greater than the values ofmaturity time when refuges are not used by prey population and when fixed number of prey using refuges.

Fig. 5. Graph of y versus t for different values of s when f ¼ 0:2 and other values of parameters are same as in (7.1)


8. Conclusions

In this paper, effects of prey refuges have been studied on the dynamics of predator–prey model when prey population isstage-structured. Two types of refuges: a fixed number of prey and a constant proportion of prey using refuges are consid-ered. Model is analyzed mathematically only for the case when a fixed number of prey using refuges but numerically resultshave been compared for both the cases. Our results show that the effects of prey refuges play an important role in determin-ing the stability behavior, bifurcation and the persistence of the system. When refuge use is high enough, the model predictsthat mature prey population reaches its maximum environmental carrying capacity and predators go extinct. We observethat, when constant proportion of mature prey population using refuges then the value of maturity time at which prey pop-ulation and hence predator populations go extinct is greater than the values of maturity time when refuges are not used byprey population or when fixed number of prey using refuges.

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Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population