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Stability analysis of a prey-predator model with Holling type III response

function incorporating a prey refuge

Article in Applied Mathematics and Computation · November 2006

DOI: 10.1016/j.amc.2006.04.030 · Source: DBLP

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 Applied Mathematics and Computation 182 (2006) 672–683
www.elsevier.com/locate/amc

Stability analysis of a prey–predator model with holling

type III response function incorporating a prey refuge
Yunjin Huang, Fengde Chen *, Li Zhong
College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China

Abstract

In this paper, we consider a prey–predator model with Holling type III response function incorporating a prey refuge.
The purpose of the work is to offer mathematical analysis of the model and to discuss some significant qualitative results
that are expected to arise from the interplay of biological forces. Some numerical simulations are carried out.
Ó 2006 Elsevier Inc. All rights reserved.

Keywords: Prey–predator; Refuge; Stability; Limit cycle

1. Introduction

The stability of ecological systems and the persistence of species within them are fundamental concerns in
ecology. Mathematical models of ecological systems, reflecting these concerns, have been sued to investigate
the stability of a variety of systems. For example, see [1,2,7–10]. The dynamic relationship between predator
and their prey has long been and will continue to be one of the dominant themes in both ecology and math-
ematical ecology due to its universal existence and importance [3]. Many excellent works have been done for
the Lotka–Volterra type predator–prey system. In [4], Holling proposed that there exist three functional
response of the predator which usually called Holling type I, Holling type II and Holling type III. He proposed
the form
mx
UðxÞ ¼
aþx
as a Holling type II response function, it usually describes the uptake of substrate by the microorganisms in
microbial dynamics kinetics [12]. If the predator is the invertebrate, it always the case. He also proposed the
the Holling type III response function in the following form:
mx2
UðxÞ ¼ :
a þ x2

*
Corresponding author.
E-mail addresses: hyjlc@sina.com (Y. Huang), fdchen@263.net (F. Chen), lizhong04108@163.com (L. Zhong).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2006.04.030
 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683 673

This case suits the vertebral predator. Recently, several scholars have pointed out that in many situations,
there was a constant proportion of prey which were protected from predation by refuge. Some mathematical
models and a number of experiments indicated that refugia had a stabilizing effect on prey–predator interac-
tions. In [5], Maynard Smith showed that the presence of a constant proportion refuge didn’t alter the dynam-
ical stability of the neutrally stable Lotka–Volterra model, while a constant number refuge of any size replaced
the neutrally stable behavior with a stable equilibrium. Tapan Kumar Kar has considered the prey–predator
model with Holling type II response function incorporation a prey refuge [6], that is, he considered the follow-
ing system:
dx  x bð1  mÞxy
¼ax 1   ;
dt k 1 þ að1  mÞx
dy cbð1  mÞxy
¼  cy þ ;
dt 1 þ að1  mÞx
where m 2 [0, 1) is constant and this leaves (1  m)x of the prey available to the predator. He obtained some
perfect results and one of his basic results is that if
c cb
06m61  ;
kðcb  caÞ kaðcb  caÞ
the system admits exactly one limit cycle which is globally asymptotically stable.
But, what is the case for the predator–prey model with Holling type III response function incorporating a
prey refuge? In this paper, we will consider the effect of the refuge on the Lotka–Volterra type predator–prey
model with Holling type III response function.
This model considered is based on the following prey–predator system [11]:
dx ax2 y
¼ ax  bx2  2 ;
dt b þ x2
ð1Þ
dy kax2 y
¼ cy þ 2 ;
dt b þ x2
where x, y denote prey and predator population respectively at any time t, and a, b, a, b, c, k are all positive
constants. Here a represents the intrinsic growth rate and a/b the carrying capacity of the prey; c is the death
rate of the predator; k is the conversion factor denoting the number of newly born predators for each captured
prey. The term (ax2/(b2 + x2)) denotes the functional response of the predator. This response function is
termed as Holling type III response function [4].
This paper extends the above model by incorporating a refuge protecting mx of the prey, where m 2 [0, 1) is
constant. This leaves (1  m)x of the prey available to the predator, and modifying system (1) accordingly
yields the system:
2
dx að1  mÞ x2 y
¼ ax  bx2  2 2
;
dt b þ ð1  mÞ x2
2
ð2Þ
dy kað1  mÞ x2 y
¼ cy þ 2 :
dt b þ ð1  mÞx2
This paper is organized as follows. Basic results are given in Section 2. In this section we study the existence of
equilibria and their dependence on the parameter m. We also study the stability and instability properties of
the equilibria and existence of limit cycle for the system (2). In Section 3, we present a numerical simulation to
illustrate the established results. Concluding remarks are presented in Section 4.

2. Basic results

To ensure the existence and uniqueness of system (2), we seek the solution in R2þ ¼ fðx; yÞ : x > 0; y > 0g, so
that all the standard results of existence, uniqueness and continuous dependence on initial conditions are evi-
674 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683

dently satisfied. Considering the existence of the equilibria of system (2) and feasibility of system (2), through-
out this paper, we assume that c < ka < 2c. By using the following change of variable
rffiffiffiffiffiffiffiffiffiffiffiffiffi
c b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cð1  mÞ2x2 þ ka  c
x¼ bx; y ¼ cðka  cÞy ; dt ¼ dt
ka  c a cðka  cÞ

and rewriting x; y ; t as x, y, t we obtain another form of system (2)

dx 2
¼ xðA0 þ A1 x þ A2 x2 þ A3 x3 Þ  ð1  mÞ x2 y;
dt ð3Þ
dy 2
¼ y þ ð1  mÞ x2 y;
dt
where
rffiffiffiffiffiffiffiffiffiffiffiffiffi
a bb c
A0 ¼ > 0; A1 ¼  < 0;
c c ka  c
2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1  mÞ a ð1  mÞ bb c
A2 ¼ > 0; A3 ¼  < 0:
ka  c ka  c ka  c

2.1. Equilibria

We study all possible equilibria of system (3).

(i) The trivial equilibrium P0(0, 0).

(ii) Equilibrium in the absence of predator P1(x*, 0), where
A2
x ¼  :
A3
We know that x* is the positive root of the equation

A3 x 3 þ A2 x 2 þ A1 x þ A0 ¼ 0
qffiffiffiffiffiffiffi
1 kac
and it is unique. Because the other two roots are x ¼  ð1mÞ c
i.
(iii) The interior (positive) equilibrium P2(x0, y0), where
 rffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 ka bb c
x0 ¼ ; y0 ¼ a : ðH1 Þ
1m ð1  mÞ cðka  cÞ 1  m ka  c

To ensure the equilibrium P2(x0, y0) to be positive we further assume m satisfies

rffiffiffiffiffiffiffiffiffiffiffiffiffi
bb c
06m<1 :
a ka  c

2.2. Dynamic behavior

In this subsection we discuss the stability properties of the equilibria P0, P1 and P2. The Jacobian of the
system (3) about the equilibrium point P0(0, 0) is given by
 
A0 0
:
0 1

Therefore, P0(0, 0) is a saddle point. Jacobian matrix for P1(x*, 0) is given by

 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683 675
0 1
A2 2 A22
 A22  A0 ð1  mÞ A23
B 3 C
@ A: ðH2 Þ
2 A2
0 1 þ ð1  mÞ A22
3
A2 A2
The eigenvalues of the matrix are  A22  A0 and 1 þ ð1  mÞ2 A22 . Hence P1(x*, 0) is locally asymptotically sta-
3 3
2 A2
ble when 1 þ ð1  mÞ A22 < 0, that is,
3
rffiffiffiffiffiffiffiffiffiffiffiffiffi
bb c
m>1
a ka  c
2 A22
and unstable (saddle) when 1 þ ð1  mÞ A23
> 0, that is,
rffiffiffiffiffiffiffiffiffiffiffiffiffi
bb c
m<1 :
a ka  c
It is observed that, when P2 exists, P1 is unstable (saddle). Jacobian about P2 is given by
 
X Y
;
Z 0
where
!
A3 A2 A1 A2 A3
X ¼2 3
þ 2
 A0 ; Y ¼ 1; Z ¼ 2 A0 þ þ þ :
ð1  mÞ ð1  mÞ 1  m ð1  mÞ2 ð1  mÞ3

Now X will be negative if

rffiffiffiffiffiffiffiffiffiffiffiffiffi
2bcb c
m>1þ ;
aðka  2cÞ ka  c
in this case, P2 is locally asymptotically stable; and if
rffiffiffiffiffiffiffiffiffiffiffiffiffi
2bcb c
m<1þ ;
aðka  2cÞ ka  c
then P2 is unstable in the xy-plane.

2.3. Globally asymptotically stable of the equilibrium and the uniqueness of limit cycle

We define two functions

U1 ðx; yÞ ¼ ðA0 þ A1 x þ A2 x2 þ A3 x3 Þ  ð1  mÞ2 xy;
2
U2 ðxÞ ¼ 1 þ ð1  mÞ x2
and rewrite system (3) in the form
dx
¼ xU1 ðx; yÞ;
dt ð4Þ
dy
¼ yU2 ðxÞ:
dt
pffiffiffiffiffiffiffi
ffi
Theorem 1. If m > 1  bb
a
c
kac, P1 is globally asymptotically stable in {(x, y):x > 0, y > 0}.

Proof. From Eqs. (H1) and (H2) we know that when

rffiffiffiffiffiffiffiffiffiffiffiffiffi
bb c
m>1 :
a ka  c
676 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683

P1 is the unique equilibrium in {(x, y):x > 0, y P 0} and locally asymptotically. From (4) we know that when
x* < x < x0,
dx dy
< 0; <0
dt dt
and when 0 < x < x*,
dx dy
> 0; < 0:
dt dt
So, each solution (x(t), y(t)) with initial value (x(0), y(0)) 2 {(x, y):0 < x < x0, y > 0} satisfies (x(t), y(t)) ! P1 as
t ! +1. To finish the proof of Theorem 1, it is enough to show that all the solutions (x(t), y(t)) with
(x(0), y(0)) in G = {(x, y):x > x0, y > 0} must cross the line x = x0 as t ! +1.
We assume solution (x(t), y(t)) with initial value (x(0), y(0)) in G = {(x, y):x > x0, y > 0}. Noticing that when
x > x0
dx dy
< 0; >0
dt dt
and
dU2 ðxÞ 2 dx
¼ 2xð1  mÞ < 0:
dt dt
Therefore, U2(x) = 1 + (1  m)2x2 < k2 where,
k 2 ¼ U2 ðxð0ÞÞ ¼ 1 þ ð1  mÞ2 xð0Þ2 > 0:
With similar method we obtain the following result:
U1 ðx; yÞ < k 1 ;
where k1 = U(x(0), y(0)) > 0.
Now, we consider the following function:
vðx; yÞ ¼ xs y;
where s ¼ kk21 > 0. Calculating the upper-right derivative of v(x, y) along the solution of system (4), we obtain
dv dx dy
¼ sxs1 y þ xs ¼ xs y½sU1 þ U2  6 0:
dt dt dt
Therefore,
s
xs y 6 xð0Þ yð0Þ;

that is,
s
xð0Þ yð0Þ
y6 :
xs
If the solution does not cross the line x = x0, this denotes solution (x(t), y(t)) is bounded in the limit field. By
Poincaré–Bendixson Theorem there must exist one equilibrium in the field. But we know this is not true, P1 is
the uniqueness. This completes the proof of Theorem 1. h
2bcb pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi
Theorem 2. When 1 þ aðka2cÞ c
kac
< m < 1  bb
a
c
kac
; P 2 ðx0 ; y 0 Þ is globally asymptotically stable.

Proof. In this case, we know that P2 is locally asymptotically stable. Next, we will prove there exists no limit
cycle for system (3). We consider the Dulac function
Bðx; yÞ ¼ x2 y r1 ;
 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683 677

where r is will be defined in the following. We define

oðxU1 BÞ oðyU2 BÞ
W ¼ þ ¼ x2 y r1 ½2A3 x3 þ ½A2 þ rð1  mÞ2 x2  ðA0 þ rÞ ¼ x2 y r1 Uðx; rÞ:
ox oy
Following we will prove that there exists a r which satisfies
Uðx; rÞ 6 0ðx > 0Þ:

If A2 x20  A0 < 0, we only need to define r ¼ A2 x20 . Then

Uðx; rÞjr¼A2 x2 ¼ 2A3 x3  ðA0  A2 x20 Þ < 0:

0

If A2 x20  A0 > 0, we notice that

oUðx; rÞ 2
¼ 2x½3A3 x þ A2 þ rð1  mÞ :
ox
2
So, when x = 0 and x ¼  A2 þrð1mÞ
3A3
we have

2x½3A3 x þ A2 þ rð1  mÞ2  ¼ 0:

In this case, we could choose a r which satisfies

A2 þ rð1  mÞ2
 >0
3A3
and
2 3 2
½A2 þ rð1  mÞ  ½A2 þ rð1  mÞ  A2
max Uðx; rÞ ¼ Uðx; rÞj A þrð1mÞ2 ¼ 2
 2
þ 2
 A0 ¼ 0:
0<x<þ1 x¼ 2 3A 27A3 ð1  mÞ ð1  mÞ
3

Therefore, we only need to ensure there exists A2 + r(1  m)2 which satisfies
2 3 2
½A2 þ rð1  mÞ  þ p½A2 þ rð1  mÞ  þ q ¼ 0; ðH3 Þ

where
" #
27A23 A2
p¼ 2
< 0; q¼ 27A23 2
 A0 > 0:
ð1  mÞ ð1  mÞ

The discriminant of (H3) has the following form

8" #2 9
q2 p3 27A43 < A2 4A23 =
D¼ þ ¼  A0  :
4 27 4 : ð1  mÞ2 ð1  mÞ ;
6

Table 1
Nature of equilibria of system (2) with m
Parameters p0(0, 0) p1(x*, 0) p2(x0, y0) Phase portrait
m 2 [0, A) Unstable Unstable Unstable, limit cycles exists Fig. 1
m=A Unstable Unstable Unstable, limit cycles exists Fig. 2
m 2 (A, B) Unstable Unstable Asymptotically stable Fig. 3
m=B Unstable Unstable Does not exist Fig. 4
m 2 (B, 1) Unstable Stable Does not exist Fig. 5
678 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683

When 2A3 x30 þ A2 x20  A0 ¼ 0 we know D = 0. In this case, (H3) admits one negative real root and two positive
real roots. And when 2A3 x30 þ A2 x20  A0 < 0 (H3) admits three different real roots.
By Descartes’ rule of sign we know that positive real roots of (H3) are no more than two and the negative
real root is no more than one.
Under the assumption of the Theorem 2, we know 2A3 x30 þ A2 x20  A0 6 0. Hence (H3) must admit at least
one positive real root, that is, there always exists r which satisfies

oðxU1 BÞ oðyU2 BÞ
þ 6 0:
ox oy
By Dulac Theorem, there exists no limit cycle. Hence, P2 is globally asymptotically stable. This completes
the proof of Theorem 2. h
2bcb pffiffiffiffiffiffiffi
c
Theorem 3. If m < 1 þ aðka2cÞ kac
; system (3) admits at least one limit cycle.

Proof. Under the assumption of the Theorem 3, we know that P2 is unstable. At first, we define the functions

V 1 ¼ x  xB ¼ 0;
V 2 ¼ y  y D ¼ 0;
V 3 ¼ y  y E þ x  xB ¼ 0:

5
Predator

1
0 1 2 3 4 5 6 7 8 9
Prey

Fig. 1. The phase portrait of system (2) for m = 0.2.

 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683 679

We let x < xB ; maxx0 6x6xB fxðA0 þ 1 þ A1 x þ A2 x2 þ A3 x3 Þ; y 0 g < y E < y D ; Dðx0 ; y D Þ; EðxB ; y E Þ are as follows:

We know that 
dx 
V_ 1 jV 1 ¼0 ¼  6 0;
dt x¼xB

dy 
V_ 2 jV 2 ¼0 ¼  6 0;
dt y¼y D
V_ 3 jV 3 ¼0 ¼ ðy E þ xB Þ þ xðA0 þ 1 þ A1 x þ A2 x2 þ A3 x3 Þ 6 0:

14

12

10

8
Population

Predator
6

4
Prey

0
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Time

Fig. 2. Solution curves for m = 0.6625 with initial value (x(0), y(0)) = (6, 8).
680 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683

Therefore, by Poincaré–Bendixson Theorem, system (3) admits at least one limit cycle on region F. This com-
pletes the proof of Theorem 3. h
Let us rewrite system (3) in the following form:
dx
¼ xgðxÞ  ypðxÞxð0Þ > 0;
dt
ð5Þ
dy
¼ y½r þ qðxÞyð0Þ > 0;
dt
where g(x) = A0 + A1x + A2x2 + A3x3, p(x) = q(x) = (1  m)2x2, r = 1.

Lemma 1 [13]. Suppose in system (5)

" 0 0 ðxÞ#0
xg ðxÞ þ gðxÞ  xgðxÞ ppðxÞ
60
r þ qðxÞ

in 0 6 x < x0 and x0 < x 6 ab. Then system (5) admits exactly one limit cycle which is globally asymptotically sta-
ble with respect to the set {(x, y):x > 0,y > 0}{P2(x0, y0}.
By using Lemma 1, we can prove the following theorem.
2bcb pffiffiffiffiffiffiffi
Theorem 4. If 2A3 x30 þ A2 x20  A0 P 0, that is, 0 < m < 1 þ aðka2cÞ c
kac
, system (3) admits only one limit cycle
which is globally asymptotically stable.

14

12

10

8
Population

6 Predator

Prey
4

0
0 50 100 150 200 250 300 350 400 450 500
Time

Fig. 3. Solution curves for m = 0.7 with initial value (x(0), y(0)) = (12, 10).
 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683 681

Proof. We easily obtain

" 0 0 ðxÞ#0
xg ðxÞ þ gðxÞ  xgðxÞ ppðxÞ 2 2
¼ ð1  mÞ A3 x3  3A3 x  A2 þ ð1  mÞ A0 : ðH4 Þ
r þ qðxÞ

Noticing that
" 0 0 ðxÞ#00
xg ðxÞ þ gðxÞ  xgðxÞ ppðxÞ
¼ 3ð1  mÞ2 A3 x3  3A3
r þ qðxÞ

and when x = ±x0

3ð1  mÞ2 A3 x3  3A3 ¼ 0:
This shows that x = ±x0 are maximum value points of (H4). Furthermore, we notice that
½ð1  mÞ2 A3 x3  3A3 x  A2 þ ð1  mÞ2 A0 jx¼0 < 0
Therefore, to prove (1  m)2A3x3  3A3x  A2 + (1  m)2A0 < 0, it is enough to show that
ð1  mÞ2 A3 x30  3A3 x0  A2 þ ð1  mÞ2 A0 6 0: ðH5 Þ
Already, we know that
1
x0 ¼
ð1  mÞ

14

12

Prey
10

8
Population

Predator

0
0 5000 10000 15000
Time

Fig. 4. Solution curves for m = 0.85 with initial value (x(0), y(0)) = (6, 6).
682 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683

and
2A3 x30 þ A2 x20  A0 P 0:
Hence, (H5) obviously holds. This completes the proof of Theorem 4. h

3. Numerical simulations

It is possible to prevent the extinction of the prey or the predator and drive the state of the considered sys-
tem to a stable state by choosing the refuge parameter m appropriately (see Table 1), here,
rffiffiffiffiffiffiffiffiffiffiffiffiffi
2bcb c
A ¼1 þ ;
aðka  2cÞ ka  c
rffiffiffiffiffiffiffiffiffiffiffiffiffi
bb c
B ¼1  :
a ka  c
Let a = 1, b = 0.1, a = 0.5, k = 0.2, c = 0.09, b = 0.5. For these values of parameters, we verify the existence
and stability properties of the equilibrium for the system.
In Fig. 1, we clearly observe that two distinct solutions of the system, one with initial value in the interior of
the limit cycle and the other with initial value in the exterior of the limit cycle, approaching the limit cycle.
In Fig. 2, there is a periodic solution around the equilibrium point P2.
In this condition, we know that the system (3) is globally asymptotically stable (see Fig. 3).
In this condition, we know that the system has no interior equilibrium and the boundary equilibrium point
(10, 0) is globally asymptotically stable (see Fig. 4).

14

12

Prey
10

8
Population

2
Predator

0
0 50 100 150 200 250 300 350 400 450 500
Time

Fig. 5. Solution curves for m = 0.9 with initial value (x(0), y(0)) = (10, 12).
 Y. Huang et al. / Applied Mathematics and Computation 182 (2006) 672–683 683

In this case, we notice that the predator is also extinction (see Fig. 5).
Then, we observe that a refuge takes good advantage for the prey. When 0 < m < 0.85 the system (3) is per-
manence and when m P 0.85 the predator is extinction.

4. Concluding remarks

In this paper we have considered a prey–predator system incorporating a prey refuge. We obtained a more
realistic model with Holling type III response function by incorporating a refuge in system (1). We have
proved that exactly one stable limit cycle occurred in this system when the positive equilibrium is unstable.
We also proved that local asymptotic stability of the positive equilibrium implies it is global asymptotic sta-
bility. All the results indicated that refuge had a stabilizing effect on prey–predator interactions. If adding a
small refuge to the model which we considered, the refuge did not alter the dynamical stability of the neutrally
stable Lotka–Volterra model. Adding a large refuge to the model replaced the oscillatory behavior with a sta-
ble equilibrium. We have given a numerical simulation to verify some of the key results we have obtained.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (10501007), the Foundation
of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian
Education Bureau (JA04156).

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