Content uploaded by Tewa Jean Jules
Author content
All content in this area was uploaded by Tewa Jean Jules on Jan 28, 2016
Content may be subject to copyright.
This article appeared in a journal published by Elsevier. The attached
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
http://www.elsevier.com/copyright
Author's personal copy
Predator–Prey model with Holling response function of type II
and SIS infectious disease
Jean Jules Tewa
a,d,
⇑
, Valaire Yatat Djeumen
b,d
, Samuel Bowong
c,d
a
Department of Mathematics and Physics, National Advanced School of Engineering (Polytechnic), University of Yaounde I, P.O. Box 8390, Yaounde, Cameroon
b
University of Yaounde I, Faculty of Science, Department of Mathematics, P.O. Box 812, Yaounde, Cameroon
c
Laboratory of Applied Mathematics, University of Douala, Department of Mathematics and Computer Science, P.O. Box 24157, Douala, Cameroon
d
UMI 209 IRD/UPMC UMMISCO, University of Yaounde I, Cameroon GRIMCAPE Team Project, LIRIMA, Yaounde, Cameroon
article info
Article history:
Received 13 March 2012
Received in revised form 5 September 2012
Accepted 1 October 2012
Available online 12 October 2012
Keywords:
Predator
Prey
Infectious disease
Response function
Bifurcation
Global stability
abstract
We analyze the influence of a SIS infectious disease affecting Preys or both Predators and
Preys in a Predator–Prey model. The response function used here is Holling function type II.
Many thresholds are computed and used to investigate the global stability results. The dis-
ease can disappear from the community, persist in one or two populations of the commu-
nity. At least one population can disappear from the community because of disease. In
some cases, the model exhibits periodic solutions with persistence of the disease or with-
out disease. Numerical simulations are used with nonstandard numerical schemes to
illustrate our results.
Ó2012 Elsevier Inc. All rights reserved.
1. Introduction
It is well known that Epidemiology and Ecology are two major and distinct fields of research. There are many epidemi-
ological models [1–15] and many ecological models [16–31], but certainly few models of the two fields. The main questions
regarding population dynamics concern the effects of infectious diseases in regulating natural populations, decreasing their
population sizes, reducing their natural fluctuations, or causing destabilizations of equilibria into oscillations of the popula-
tion states. With the Holling function response of type II, it is well known that the mortality of Preys due to predation
increases as well as the number of Preys decreases and become constant at the end.
There has been many Predator–Prey models with infectious diseases: Anderson and May [16] model in which the
pathogen tends to destabilize the Prey-Predator interactions; Hadeler and Freedman model [17] in which the authors con-
sidered that Predators could only survive on the Prey if some of the Preys were more easily caught due to being diseased;
Venturino model [18] with mass action incidence in which an SI or SIS disease spreads among either the Preys or the
Predators or the model given in [19] where he consider similar SI and SIS models with disease spread among the Preys
when there is logistic growth of Preys and Predators; Hudson et al., model [21] in which they considered the macropar-
asitic infections in red grouse and looked at situations in which parasitic infections of Preys made them more vulnerable
to predation. It is assumed in [20] that the Predators are infected when swallowing the infected Preys, and that the Preys
0307-904X/$ - see front matter Ó2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.apm.2012.10.003
⇑
Corresponding author at: Department of Mathematics and Physics, National Advanced School of Engineering (Polytechnic), University of Yaounde I, P.O.
Box 8390, Yaounde, Cameroon. Tel.: +237 77 71 13 69; fax: +237 22 23 50 70.
E-mail addresses: tewajules@gmail.com (J.J. Tewa), yatatvalaire@yahoo.fr (V.Y. Djeumen), sbowong@gmail.com (S. Bowong).
Applied Mathematical Modelling 37 (2013) 4825–4841
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Author's personal copy
are infected by contacting the excrement of the infected Predators. Generally, there are more macroparasitic infections
which can affect only Preys, only Predators, Preys and Predators. There are many works of Haque and coworkers
[25,26,29] concerning transmissible diseases spreading among the Prey or Predator population in Predator–Prey models
or symbiotic communities but although these models have some similarities with our models, there are many differences
concerning for example the horizontal incidence (mass incidence or standard incidence) or the presence of disease in the
community (Prey, Predator, Prey and Predator). In [30], the authors investigate the effect of delay in a Lotka–Volterra type
Predator–Prey model with a transmissible disease in the Predator species.
In the models considered in this paper, the Holling function response of type II is used for interactions between Predators
and Preys. The authors of [25] used the same kind of model with standard incidence but the disease was only in the Predator
population. The novelties in this paper are: Horizontal incidence follows standard incidence which is more appropriated for
large and non constant population; the form of response function when there is a disease with the coefficient haffecting
mortality and recruitment; the dynamics around origin is complicated because of standard incidence and population reaches
the origin either by following the axis or in spiral pattern; some numerical analysis are with nonstandard schemes. We also
include the possibility that infectious disease can persist in the Predator population and can be acquired by the Predators
during the predation process. Moreover, we use a nonstandard numerical scheme for some simulations. It has been proved
in [1] that the simulations can be very different using a nonstandard numerical scheme.
2. The model formulation
The Holling function response of type II is defined by hðHÞ¼
B
x
0
H
1þB
x
1
H
, where Hdenotes the Prey population,
x
0
and
x
1
de-
note respectively the time taking by a Predator to search and capture Preys, Bis the predation rate per unit of time. Then, the
Predator–Prey model with Holling function of type II if Pdenotes the Predator population is
_
HðtÞ¼r1
H
K
HgðH;PÞ;
_
PðtÞ¼egðH;PÞ
c
P;
(ð1Þ
where rdenotes the intrinsic growth rate of the Preys, Kis the carrying capacity of the environment,
c
is the mortality rate of
Predators, eis the coefficient in conversing Prey into Predator and gðH;PÞ¼
B
x
0
HP
1þB
x
1
H
.
When there is no Predator, the dynamics of Prey population is governed by the logistic equation _
HðtÞ¼r1
H
K
H. The
function gðH;PÞcan also be written as gðH;PÞ¼
a
HP
1þaH
, where
a
¼B
x
0
denotes the Prey searching rate, a¼B
x
1
denotes
the satiety rate of Predators. Setting b¼e
a
, the System (1) becomes
_
HðtÞ¼r1
H
K
H
a
HP
1þaH
;
_
PðtÞ¼
bHP
1þaH
c
P:
(ð2Þ
The SIS compartmental model in epidemiology with standard incidence is given by
_
S¼b
l
Sk
IS
N
þ
r
I;
_
I¼k
IS
N
r
I
l
I;
(ð3Þ
where Sdenotes the susceptible population, Ithe infectious population,
r
is the recover rate of infectious individuals
to become susceptible such that N¼SþIis the total population,
l
is the mortality death rate. We assume that all
recruitments are in susceptible compartment at a constant rate b;kis the adequate contact rate between susceptibles
and infectious. If
r
¼0, the model (3) becomes a simple SI model. The incidence is assumed to be standard
incidence.
Our task here is to combine the preceding models (2) and (3), in order to analyze the influence of SIS infectious disease in
a Predator–Prey community. The following hypothesis hold in our models.
1. (H1) In the absence of infection and predation, the Prey population grows logistically.
2. (H2) In the presence of infection, the Prey population are divided into two disjoint classes, namely, susceptible popula-
tion, and infected population.
3. (H3) The mode of disease transmission follows the standard incidence. The disease is spread among the Prey population
without Predators for the first model, with Predators for the second model and in the Prey and Predator populations for
the third and last model.
4. (H4) The disease is not genetically inherited. The infected population do not recover or become immune.
5. (H5) It is assumed that Predator cannot distinguish the infected and healthy Prey.
6. (H6) We assume that only susceptible Prey is capable of reproducing and contributing to its carrying capacity.
4826 J.J. Tewa et al./ Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
3. Mathematical analysis
3.1. The Predator–Prey model (2) without disease
The corresponding diagram for the Predator–Prey model when there is no disease in the community is given by Fig. 1
which yields the set of differential equation (1). The System (1) can also be written as System (2). Let us give the mathemat-
ical results concerning System (2).
Lemma 1. The following results hold for System (2).
1. The nonnegative orthant R
2
þ
is positively invariant by (2).
2. The set D ¼fðH;PÞ2R
2
þ
=H6Kgis a compact forward and absorbing set.
3. There exists three equilibria for System (2):E
0
¼ð0;0Þand E
1
¼ðK;0Þwhich always exists, and E
2
¼ðH
H
;P
H
Þ¼
c
ba
c
;
r
a
ð1þaH
H
Þ1
H
H
K
which is ecologically acceptable when the threshold R
1
¼
bK
c
ð1þaKÞ
is such that R
1
>1.
The following theorem holds concerning the stability of these equilibria.
Theorem 1. The following results hold for System (2).
1. E
0
is a saddle-node with unstability for Prey population and stability for Predator population.
2. E
1
is a globally asymptotically stable (GAS) node when R
1
<1, a saddle-node with stability for Prey population and unstability
for Predator population when R
1
>1.E
1
is a GAS node when R
1
¼1. The center manifold in this case is given by W
c
¼
fx¼hðyÞ:hð0Þ¼K;h
0
ð0Þ¼a
1
g, where hðyÞ¼Kþa
1
yþa
2
y
2
þOðy
3
Þ;a
1
¼
a
K
rð1þaKÞ
;a
2
¼
a
2
1
2raþ
rþ
c
K
ðÞ
þ
a
a
1
rð1þaKÞ
.
3. E
2
is not ecologically acceptable when R
1
61;if1<R
1
61þ
b
a
c
ð1þaKÞ
;E
2
is a GAS focus and when R
1
>1þ
b
a
c
ð1þaKÞ
;E
2
is an
unstable focus and there exists a stable limit cycle for (2). This phenomenon corresponds to a supercritical Hopf bifurcation.
3.2. Mathematical analysis of the models with disease only in the Prey population
3.2.1. The model when there is no Predator
When there is no Predator, the dynamics of Preys is governed by the logistic equation _
H¼r1
H
K
H. Setting Xthe sus-
ceptible Preys and Ythe infectious Preys such that H¼XþYis the total Prey population. It should be pointed out that the
density dependence affects not only the birth but also the death of the populations. Hence, we need to separate the effects of
the density dependence. Parameters band
l
are the natural birth and death rates coefficients, the parameter his such that
b
hrH
K
is the birth rate coefficient,
l
þ
ð1hÞrH
K
is the mortality rate, r¼b
l
is the intrinsic growth rate. The new system is
_
H¼rH 1
H
K
;
_
X¼b
hrH
K
H
ð1hÞrH
K
X
l
X
kXY
H
þ
r
Y;
_
Y¼
kXY
H
r
Y
l
Y
ð1hÞrH
K
Y:
8
>
>
<
>
>
:
ð4Þ
Setting I¼
Y
H
and S¼
X
H
¼1I, the proportions of infected Preys and susceptible Preys in the Prey population, System (4)
becomes
_
H¼r1
H
K
H;
_
I¼kð1IÞIbþ
r
hrH
K
I:
(ð5Þ
Let us consider the thresholds R
2
¼
k
bþ
r
rh
and R
3
¼
k
bþ
r
. The equilibria of System (5) are E
0
¼ð0;0Þ;E
1
¼ðK;0Þ;
E
2
¼0;1
1
R
3
which is acceptable if R
3
P1, and E
3
¼K;1
1
R
2
which is acceptable if R
2
P1.
The following lemma holds.
Lemma 2. The following results hold for System (5).
1. The domain D ¼fðH;IÞ2R
2
þ
j06I61;06H6Kgis positively invariant and absorbing for (5).
2. lim
t!þ1
ðH;IÞ¼E
0
if Ið0Þ>0;lim
t!0
HðtÞ¼0and R
3
61.
3. lim
t!þ1
ðH;IÞ¼E
2
if Ið0Þ>0;lim
t!0
HðtÞ¼0and R
3
>1.
4. lim
t!þ1
ðH;IÞ¼E
1
if Ið0Þ¼0and lim
t!0
HðtÞ>0.
5. lim
t!þ1
ðH;IÞ¼E
1
if Ið0Þ>0;lim
t!0
HðtÞ>0and R
2
61.
6. lim
t!þ1
ðH;IÞ¼E
3
if Ið0Þ>0;lim
t!0
HðtÞ>0and R
2
>1.
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4827
Author's personal copy
Proof. In Appendix A. h
Remark 1. We have observed in this section that the disease can persist in the Prey population when HðtÞgoes to the car-
rying capacity K(stability of E
3
) or can cause extinction of the Prey population (stability of E
2
). There is a saddle-node bifur-
cation when each equilibrium E
2
or E
3
loses stability.
3.2.2. The model when the Predators are present
When the disease spreads only among the Preys but in presence of Predators, the new system is
_
H¼rH 1
H
K
a
XþqY
1þaH
P;
_
X¼b
hrH
K
H
ð1hÞrH
K
X
l
X
kXY
H
þ
r
Y
a
XP
1þaH
;
_
Y¼
kXY
H
r
Y
l
Y
ð1hÞrH
K
Y
a
q
YP
1þaH
;
_
P¼b
XþqY
1þaH
P
c
P;
8
>
>
>
>
>
<
>
>
>
>
>
:
ð6Þ
where Xdenotes the susceptible Prey population, Ythe infectious Prey population and qP1 is to taking into account the fact
that infected Preys are more vulnerable to predation. The Predators can not be infected here.
Setting I¼
Y
H
and S¼
X
H
¼1I, the proportions of infected Preys and susceptible Preys in the Prey population, the System
(6) becomes
_
H¼rH 1
H
K
a
1þðq1ÞI
1þaH
PH;
_
I¼kð1IÞI
r
þb
hrH
K
I
a
ðq1Þð1IÞ
1þaH
PI;
_
P¼b
1þðq1ÞI
1þaH
HP
c
P:
8
>
>
<
>
>
:
ð7Þ
This step consists in finding the equilibria E
i
¼ðH
i
;I
i
;P
i
Þ. These possible equilibria are E
0
¼ð0;0;0Þ;E
1
¼0;1
1
R
3
;0
which
exists if R
3
>1;E
2
¼ðK;0;0Þ;E
4
¼K;1
1
R
2
;0
which exists if R
2
>1,
E
3
¼
c
ba
c
;0;1
c
Kðba
c
Þ
rb
a
ðba
c
Þ
;E
5
¼ðH
H
;I
H
;P
H
Þ;where H
H
¼
c
c
abbA
1
I
H
;P
H
¼kI
H
A
3
c
A
4
A
2
bA
1
I
H
bþbA
1
I
H
A
6
þA
7
I
H
þA
8
I
H2
and
I
H
>0 is given by
A
13
I
H4
þA
14
I
H3
þA
15
I
H2
þA
16
I
H
þA
17
¼0;ð8Þ
where A
1
¼q1;A
2
¼
c
ab;A
3
¼kð
r
þbÞ;A
4
¼
rh
K
;A
5
¼
r
K
;A
6
¼
a
A
2
;A
7
¼
a
A
1
ðA
2
bÞ;A
8
¼
a
bA
2
1
;A
9
¼
a
bk A
2
1
;
A
10
¼A
9
a
A
1
ðkA
2
þbA
1
A
3
Þ;A
11
¼
a
A
1
ðkA
2
þbA
1
A
3
Þþ
a
A
1
ð
c
A
4
þA
2
A
3
Þ;A
12
¼
a
A
1
ðA
2
A
3
c
A
4
Þ;A
13
¼bA
1
A
9
;A
14
¼A
2
A
9
bA
1
A
10
rbA
1
A
8
;A
15
¼A
2
A
10
bA
1
A
11
c
A
5
A
8
þrA
2
A
8
rbA
1
A
7
;A
16
¼A
2
A
11
bA
1
A
12
c
A
5
A
7
þrA
2
A
7
rbA
1
A
6
;A
17
¼A
2
A
12
c
A
5
A
6
þrA
2
A
6
:
Setting R
4
¼R
1
1þðq1Þ1
1
R
2
. The following lemma holds.
Lemma 3. Let us consider the System (7).
1. The nonnegative orthant R
3
þ
is positively invariant by (7).
2. The set D ¼fðH;I;PÞ2R
3
þ
=06H6K;06I61gis a compact forward and absorbing set.
Theorem 2. The following properties hold for System (7).
HP
rH(1- )
H
K
αP
1+aH
1+aH
1+aH
βP
γ
(α−β)P
Fig. 1. Interaction diagram for the Predator Prey model when there is no disease.
4828 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
1. The equilibrium E
0
is unstable and E
1
which exists when R
3
>1is also unstable.
2. If R
1
61and R
2
61, then lim
t!þ1
ðH;I;PÞ¼E
2
. The equilibrium E
2
is stable.
3. If R
2
61and 1<R
1
61þ
b
a
c
ð1þa
c
Þ
, then lim
t!þ1
ðH;I;PÞ¼E
3
. The equilibrium E
3
is stable.
4. If R
2
>1and R
4
61, then lim
t!þ1
ðH;I;PÞ¼E
4
. The equilibrium E
4
is stable.
5. If R
1
>1þ
b
a
c
ð1þa
c
Þ
and R
2
61, then there exists a stable limit cycle in ðH;PÞplane for System (7).
6. If R
1
>1þ
b
a
c
ð1þa
c
Þ
and R
2
>1, then there exists a stable limit cycle in ðH;I;PÞspace for System (7).
Proof. In Appendix B. h
Remark 2. In this section, we have observed that the disease can persist and become endemic with extinction of Predators.
We can also have oscillations of the trajectories of Preys and Predators while the disease disappear or not. There is a saddle-
node bifurcation when each equilibrium E
2
or E
3
loses stability as we prove in part 2 and part 3 of Appendix B, and Hopf
bifurcation when equilibrium E
4
loses stability as we prove in part 4 of Appendix B. The numerical continuation method
can also be used in order to compute the limit cycles as in [27,28].
3.3. Mathematical analysis of the model (9) with disease in the two populations
The disease is present now in the two populations. The variables S
1
and I
1
denote respectively the susceptible and infec-
tious in Prey population, S
2
and I
2
denote respectively the susceptible and infectious in Predator population. The correspond-
ing diagram for the Predator–Prey model when the disease is present in the community is given by Fig. 2 which yields the set
of differential equations
_
H¼rH 1
H
K
a
HP
1þaH
;
_
S
1
¼b
hrH
K
H
ð1hÞrH
K
S
1
l
S
1
k
1
S
1
I
1
H
þ
r
1
I
1
a
S
1
P
1þaH
;
_
I
1
¼
k
1
S
1
I
1
H
r
1
I
1
l
I
1
ð1hÞrH
K
I
1
a
I
1
P
1þaH
;
_
P¼b
H
1þaH
P
c
P;
_
S
2
¼b
H
1þaH
Pd
S
2
I
1
P
c
S
2
k
2S
2
I
2
P
þ
r
2
I
2
;
_
I
2
¼k
2S
2
I
2
P
c
I
2
þd
S
2
I
1
P
r
2
I
2
;
8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
ð9Þ
where H¼S
1
þI
1
;P¼S
2
þI
2
.
H
P
S1
S2
I1
I2
λ2I2
σ1
σ2
γγ
γ
1+aH
βP
1+aH
αP
1+aH
αP
1+aH
αP
rθH
K
b - K
(1−θ)rH
μ + K
(1−θ)rH
μ +
λ1I1
H
P
1+aH
βH
rH(1- )
H
K
Fig. 2. Interaction diagram for the Predator–Prey model when the disease is present in the two populations.
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4829
Author's personal copy
Using the fact that H¼S
1
þI
1
;P¼S
2
þI
2
, System (9) becomes
_
H¼rH 1
H
K
a
HP
1þaH
;
_
I
1
¼
kðHI
1
ÞI
1
H
r
1
I
1
l
þ
ð1hÞrH
K
hi
I
1
a
I
1
P
1þaH
;
_
P¼b
H
1þaH
P
c
P;
_
I
2
¼k
2ðPI
2
ÞI
2
P
c
I
2
þd
ðPI
2
ÞI
1
P
r
2
I
2
:
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
ð10Þ
Lemma 4. The following results hold for System (10).
1. The nonnegative orthant R
4
þ
is positively invariant by (10).
2. The set D ¼fðH;I
1
;P;I
2
Þ2R
4
þ
=06I
1
6H6K;06I
2
6Pgis a compact forward and absorbing set.
This step consists in finding the equilibria E
i
¼ðH
i
;I
i
1
;P
i
;I
i
2
Þ. Setting R
01
¼
k
1
r
1
þbrh
;R
5
¼
k
2
r
2
þ
c
. These equilibria are
E
0
¼ð0;0;0;0Þ;E
1
¼ðK;0;0;0Þwhich always exists; E
2
¼K;K1
1
R
01
;0;0
which is ecologically acceptable if
R
01
>1;E
3
¼H
H
¼
c
ba
c
;0;P
H
¼
r
a
ð1þaH
H
Þ1
H
H
K
;0
which is ecologically acceptable if R
1
>1;E
4
¼ðH
H
;0;P
H
;I
2E
Þwith
I
2E
¼1
1
R
5
P
H
, which is acceptable if R
1
>1 and R
5
>1;E
5
¼ðH
H
;I
1EE
;P
H
;I
2EE
Þwith
I
1EE
¼1
1
R
6
P
H
;R
6
¼
k
1
r
1
þ
l
þð1hÞ
rH
H
K
þ
a
PH
1þaH
H
and I
2EE
is the positive solution of equation
x
2
þ1
R
5
1
P
H
þdI
1EE
k
2
xdI
1EE
k
2
P
H
¼0:ð11Þ
This equilibrium is acceptable if R
1
>1 and R
6
>1.
Theorem 3. The following properties hold for System (10).
1. If lim
t!0
HðtÞ¼0, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
0
;
2. If R
1
61;R
01
61and lim
t!0
HðtÞ>0, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
1
;
3. If R
1
61;R
01
>1and lim
t!0
HðtÞ>0, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
2
;
4. If 1<R
1
61þ
b
a
c
ð1þaKÞ
;R
5
61;R
6
61;lim
t!0
HðtÞ>0and Pð0Þ>0, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
3
;
5. If 1<R
1
61þ
b
a
c
ð1þaKÞ
;R
5
>1;R
6
61;lim
t!0
HðtÞ>0and I
2
ð0Þ>0, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
4
;
6. If 1<R
1
61þ
b
a
c
ð1þaKÞ
;R
5
61;R
6
>1;I
1
ð0Þ>0and Pð0Þ>0, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
5
;
Table 2
Numerical values for the parameters of the System (9).
Parameters Description Estimated value/range References
rIntrinsic birth rate 1 per day [22]
bBiomass conversion rate 0:015 per day [22]
aPredator’s satiety rate 0:01 per unit designated area [22]
KCarrying capacity 490 per unit designated area [22]
a
Prey searching rate 0:1 per day [22]
c
Predator’s mortality rate 1 per day [22]
k
1
¼k
2
Contact rates between susceptible and infectious individuals 1 per day [22]
r
1
¼
r
2
Recovery rates for infectious Preys and infectious Predators 0:15 per day [22]
hPredator’s mortality coefficient 0:8 per day [22]
dContact rate between susceptible Predators and infectious Preys 0:02 per day [22]
Table 1
Numerical values for the parameters of systems (2)–(6).
Parameter Description Estimated value/range Reference
rIntrinsic birth rate 2 per day [22]
bBiomass conversion rate 0:015 Per day [22]
aPredator’s satiety rate 0:01 per unit designated area [22]
KCarrying capacity 500–600 per unit designated area [22]
a
Prey searching rate 0:1 per day [22]
c
Predator’s mortality rate 1 per day [22]
4830 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
7. If R
1
>1þ
b
a
c
ð1þaKÞ
;R
01
61and R
5
61then there exists a stable limit cycle in ðH;PÞplane for (10);
8. If R
1
>1þ
b
a
c
ð1þaKÞ
;R
01
61and R
5
>1then there exists a stable limit cycle in the space I
1
¼0for (10);
9. If R
1
>1þ
b
a
c
ð1þaKÞ
;R
01
>1and R
5
>1then there exists a stable limit cycle for (10).
0 50 100 150 200 250 300 350 400 450 500
0
100
200
300
400
500
600
prey and predator trajectories
t (days)
H(t), P(t)
prey
predator
0 100 200 300 400 500 600
0
10
20
30
40
50
60
phase portrait
H
P
Fig. 4. Dynamics of the trajectories of System (2) with periodic solutions (stable limit cycle).
0 50 100 150 200 250 300 350 400 450 500
0
100
200
300
400
500
prey and predator trajectories
t (days)
H(t), P(t)
prey
predator
0 50 100 150 200 250 300 350 400 450 500
0
10
20
30
40
50
phase portrait
H
P
Fig. 3. Global asymptotic stability of the coexisting equilibrium of System (2).
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4831
Author's personal copy
Proof. In Appendix C. h
Remark 3. There is Hopf bifurcation with limit cycles as we prove in part 7, part 8 and part 9 of Appendix C. The numerical
continuation method can also be used in order to compute these limit cycles as in [27,28].
0 10 20 30 40 50 60 70 80 90 100
0
0,05
0,1
0,15
0,2
0,25
0,3
0.32258
0,35
prey trajectories and infected prey trajectories
t (days)
H(t), I(t)
prey
infected prey
0 10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
35
40
45
50
prey trajectories and infected prey trajectories
t (days)
H(t), I(t)
prey
infected prey
(a)
(b)
Fig. 5. (a) Global asymptotic stability of the equilibrium E
1
of System (5). (b) Global asymptotic stability of the equilibrium E
2
of System (5).
4832 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
4. Conclusion
The thresholds R
01
;R
5
and R
6
can be considered respectively as the basic reproduction rate of Preys when there is no
Predator, the basic reproduction rate of Predators when there is no Prey, the basic reproduction rate of Predators and Preys
when the two populations coexist.
For a classic Predator–Prey model, when there is no Predator, the Prey population disappears or goes to the carrying
capacity K. When the Predators are present, Predators and Preys can coexist. In this work, we have observed that when
the disease appears in Prey population, only the Preys can disappear; only the Predator can disappear; the uninfected Preys
and Predators can disappear and then, only infected Preys persist. The disease can disappear or persist in the community.
The analysis of our models gives us the following conclusions: The biological interpretation of Hopf bifurcation is that the
Predator coexists with the susceptible Prey and the infected Prey, exhibiting oscillatory balance behavior. To avoid extinction
of the species, one should look carefully some parameters, namely, the rate of infection, the death rate of the infective pop-
ulation and the growth rate of the susceptible population. In the absence of disease, the systems are unstable around the
origin. Thus, there is no possibility of extinction of the populations. However, this is not true for the systems with infection.
Some of these results deal with the results in [25–29,31] but since the authors in this paper used mass action incidence, they
did not have problems with the behavior around the origin. The behavior around the origin was analyzed and the dynamics
0 10 20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
30
35
40
45
50
prey trajectories
t (days)
H(t)
prey
0 10 20 30 40 50 60 70 80 90 100
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0.75
0,8
infected prey trajectories
t (days)
I(t)
infected prey
Fig. 6. Global asymptotic stability of the equilibrium E
3
of System (5).
0 50 100 150 200 250 300 350 400 450 500
0
200
400
600
prey trajectories
t (days)
H(t)
prey
0 50 100 150 200 250 300 350 400 450 500
0
0.05
0.1
infected prey trajectories
t (days)
I(t)
infected prey
0 50 100 150 200 250 300 350 400 450 500
0
10
20
30
predator trajectories
t (days)
P(t)
predator
0 50 100 150 200 250 300 350 400 450 500
0
200
400
600
model trajectories
t (days)
H(t), I(t), P(t)
prey
infected prey
predator
0100 200 300 400 500
0
0.05
0.1
0
20
40
H
phase prortrait
I
P
Fig. 7. Global asymptotic stability of the equilibrium E
3
of System (7).
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4833
Author's personal copy
around origin is complicated because of standard incidence and population reaches the origin either by following the axis or
in spiral pattern.
5. Numerical simulations
Using the same nonstandard numerical scheme as in [1], we have these simulations where all parameter values have been
chosen in such a way that they are realistic and at the same time obey the conditions for stability or bifurcation. For Pred-
0 100 200 300 400 500
0
200
400
600
prey trajectories
t (days)
H(t)
prey
0100 200 300 400 500
0
0.05
0.1
infected prey trajectories
t (days)
I(t)
infected prey
0 100 200 300 400 500
0
20
40
predator trajectories
t (days)
P(t)
predator
0100 200 300 400 500
0
200
400
600
model trajectories
t (days)
H(t), I(t), P(t)
prey
infected prey
predator
0200 400 600
0
0.05
0.1
0
20
40
H
phase prortrait
I
P
Fig. 8. Oscillations of the Prey and Predator trajectories of System (7). The disease will disappear.
0 100 200 300 400 500
0
200
400
600
prey trajectories
t (days)
H(t)
prey
0100 200 300 400 500
0
0.5
1
infected prey trajectories
t (days)
I(t)
infected prey
0 100 200 300 400 500
0
20
40
60
predator trajectories
t (days)
P(t)
predator
0100 200 300 400 500
0
200
400
600
model trajectories
t (days)
H(t), I(t), P(t)
prey
infected prey
predator
0200 400 600
0
0.5
1
0
50
H
phase prortrait
I
P
Fig. 9. Oscillations of the Prey and Predator trajectories of System (7). The disease will persist.
4834 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
ator–Prey model without disease and Predator–Prey model with disease spreads among the Preys, we have these parameters
given in Table 1.
For the Predator–Prey model with disease spreads in the community among both Predators and Preys, we have these
parameters in Table 2 for the global stability of coexisting equilibrium.
The simulations given by Figs. 3 and 4are realized for the Predator–Prey model (2) without disease.
Fig. 10. Global asymptotic stability of the equilibrium E
3
of System (10).
Fig. 11. Global asymptotic stability of the equilibrium E
4
of System (10).
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4835
Author's personal copy
The simulations given by Figs. 5 and 6are realized for the Predator–Prey model (5) with disease in Prey population and
without Predator.
The simulations given by Figs. 7,8and 9are realized for the Predator–Prey model (7) with disease in Prey population and
with Predators.
Fig. 12. Global asymptotic stability of the equilibrium E
5
of System (10).
Fig. 13. Oscillations of the Prey and Predator trajectories of System (10). The disease will persist in the two populations.
4836 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
The simulations given by Figs. 10–16 are realized for the Predator–Prey model (10) with disease in Prey population and
Predator population.
Remark 4. We can observe that Figs. 7–9 illustrate the Hopf bifurcation for System (7) and Figs. 13–15 illustrate the Hopf
bifurcation for System (10).
Fig. 14. Oscillations of the Prey and Predator trajectories of System (10). The disease will persist in the Predator population.
Fig. 15. Oscillations of the Prey and Predator trajectories of System (10). The disease will disappear from the populations.
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4837
Author's personal copy
Appendix A. The proof of lemma 2
1. Let f
1
the function defined by f
1
ðH;IÞ¼HKand Fthe function defined by
FðH;IÞ¼ rH 1
H
K
Ikð1IÞ bþ
r
hrH
K
!
:
Then, hrf
1
jFi¼rH 1
H
K
60onD, where hji is the usual scalar product. Therefore, Dis positively invariant and absorbing
for (5).
2. The limit system of System (5) is such that _
HðtÞ¼0 and _
IðtÞ6k1
1
R
3
I. Since R
3
61;lim
t!þ1
IðtÞ¼0 and then
lim
t!þ1
ðH;IÞ¼ð0;0Þ¼E
0
.
3. The jacobian matrix at equilibrium E
2
has two eigenvalues: r>0 and k
R
3
1
R
3
<0. The limit system is such that
lim
t!þ1
IðtÞ¼1
1
R
3
and then lim
t!þ1
ðH;IÞ¼E
2
.
4. The set
g
1
¼fðH;IÞ2R
2
þ
jI¼0gis positively invariant by the flow of System (5). This implies lim
t!þ1
IðtÞ¼0. Moreover,
Hð0Þ>0 implies lim
t!þ1
HðtÞ¼Kand then, lim
t!þ1
ðH;IÞ¼E
1
.
5. Hð0Þ>0 implies lim
t!þ1
HðtÞ¼K. The variable Isatisfies equation _
I6k1
1
R
2
I. Since R
2
61;lim
t!þ1
IðtÞ¼0 and then
lim
t!þ1
ðH;IÞ¼E
1
.
6. Hð0Þ>0 implies lim
t!þ1
HðtÞ¼K. Then, lim
t!þ1
IðtÞ¼1
1
R
2
since R
2
>1 and therefore, lim
t!þ1
ðH;IÞ¼E
3
.
Appendix B. The proof of theorem 2
1. The jacobian matrix at equilibrium E
0
gives us the eigenvalues: r>0;
c
<0 and kðbþ
r
Þ. Then, E
0
is unstable. The
jacobian matrix at equilibrium E
1
gives us the eigenvalues: r>0;
c
<0 and k1
1
R
3
. Then, E
1
is unstable when it
exists.
2. The jacobian matrix at equilibrium E
2
gives us the eigenvalues: r<0;
c
ðR
1
1Þand k1
1
R
2
. Then, E
2
is stable when
R
1
<1 and R
2
<1.
When R
1
<1 and R
2
¼1, the equilibrium E
2
is non hyperbolic. Let us consider the Lyapunov candidate function
VðH;I;PÞ¼
1
2
ððHKÞ
2
þI
2
þP
2
Þ. The derivative of VðH;I;PÞgives us
_
VðH;I;PÞ¼ðHKÞr
KðKHÞ
a
ð1þðq1ÞIÞP
1þaH
HþI
2
kIrh
KðKHÞ
a
ðq1Þð1IÞP
1þaH
þP
2
bHð1þðq1ÞIÞ
1þaH
c
:
Fig. 16. Global asymptotic stability of the coexisting equilibrium of System (10) when R
2
>1 and R
4
>1.
4838 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
Then, _
VðK;0;0Þ¼0 and _
V60 on the set
X
0
¼ðH;I;PÞj06H6min K;
c
b
c
aþbðq1ÞI
;06P6m
0
;06I61
;
where m
0
¼
rðKHÞð1þaHÞ
K
a
ð1þðq1ÞIÞ
. Then, E
2
is locally stable.
When R
1
¼1 and R
2
<1, the equilibrium E
2
is non hyperbolic. Let us consider the Lyapunov candidate function
VðH;I;PÞ¼
1
2
ððHKÞ
2
þI
2
þP
2
Þ. The derivative of VðH;I;PÞgives us
_
VðH;I;PÞ¼ðHKÞr
KðKHÞ
a
ð1þðq1ÞIÞP
1þaH
HþI
2
kkIðbþ
r
ÞþrhH
K
a
ðq1Þð1IÞP
1þaH
þP
2
bHð1þðq1ÞIÞ
1þaH
c
:
Then, _
VðK;0;0Þ¼0 and _
V60 on the set
X
1
¼ðH;I;PÞj06H6
c
K
c
þbKðq1Þ;06P6rðKHÞð1þaHÞ
K
a
ð1þðq1ÞIÞ;06I61
:
Then, E
2
is locally stable.
When R
1
¼1 and R
2
¼1, the equilibrium E
2
is non hyperbolic. Let us consider the Lyapunov candidate function
VðH;I;PÞ¼
1
2
ððHKÞ
2
þI
2
þP
2
Þ. The derivative of VðH;I;PÞgives us
_
VðH;I;PÞ¼ðHKÞr
KðKHÞ
a
ð1þðq1ÞIÞP
1þaH
HþI
2
kIrh
KðKHÞ
a
ðq1Þð1IÞP
1þaH
þP
2
bHð1þðq1ÞIÞ
1þaH
c
:
Then, _
VðK;0;0Þ¼0 and _
V60 on the set
X
1
¼ðH;I;PÞj06H6
c
K
c
þbKðq1Þ;06P6rðKHÞð1þaHÞ
K
a
ð1þðq1ÞIÞ;06I61
:
Then, E
2
is locally stable.
We deduce from the System (7) that the variable Isatisfies equation _
I6k1
1
R
2
I. Since R
2
61;lim
t!þ1
IðtÞ¼0 and then
lim
t!þ1
ðH;I;PÞ¼E
2
when R
1
61 and R
2
61.
3. The jacobian matrix of System (7) at E
3
gives us the eigenvalue corresponding to the variable I:
g
2
¼k1
bþ
r
Þ
rh
c
Kðb
c
aÞ
þrðq1Þ1
c
Kðb
c
aÞ
k
0
@1
A:
Then, the infected Preys are stable if
g
2
<0 and unstable if
g
>0. If
g
¼0, the equilibrium E
3
is non hyperbolic and _
I¼kI
2
.
Then, lim
t!þ1
IðtÞ¼0. Therefore, lim
t!þ1
ðH;I;PÞ¼E
3
when 1 <R
1
61þ
b
a
c
ð1þaKÞ
and R
2
61.
4. The solutions of System (7) satisfy _
H6r1
H
K
Hand _
I6kð1IÞðbþ
r
Þþ
rhH
K
I.IfR
2
>1, then lim sup
t!þ1
HðtÞ6Kand
lim sup
t!þ1
IðtÞ61
1
R
2
. Therefore, there exists T>0 such that HðtÞ6Kþ
e
and IðtÞ61
1
R
2
for t>T, where
e
is such that
bðKþ
e
Þ1þðq1Þ1
1
R
2
þ
e
1þaðKþ
e
Þ
c
<0:
The variable Psatisfies _
P6
bðKþ
e
Þ1þðq1Þ1
1
R2þ
e
1þaðKþ
e
Þ
c
2
43
5P. Then lim
t!þ1
PðtÞ¼0 and then lim
t!þ1
ðH;I;PÞ¼E
4
.
5. Since R
2
61;lim
t!þ1
IðtÞ¼0 and there exists a periodic stable solution in ðH;PÞplane.
6. This step has been obtained numerically.
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4839
Author's personal copy
Appendix C. The proof of theorem 3
1. The limit system of (10) is such that _
H¼0;_
I
1
¼0;_
P¼
c
Pand _
I
2
¼
k
2
ðPI
2
ÞI
2
P
c
I
2
r
2
I
2
. Then
lim
t!þ1
PðtÞ¼0;lim
t!þ1
I
2
ðtÞ¼0 and therefore lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
0
.
2. From System (10),_
H6r1
H
K
Hand then lim sup
t!þ1
HðtÞ6K. Since R
1
61, we consider
e
such that
bðKþ
e
Þ
c
ð1þaðKþ
e
ÞÞ
61 and
bðKþ
e
Þ
c
ð1þaðKþ
e
ÞÞ
c
61. There exists T>0 such that HðtÞ6Kþ
e
for t>T. Thus, _
P6
bðKþ
e
Þ
c
ð1þaðKþ
e
ÞÞ
c
hi
Pand lim
t!þ1
PðtÞ¼0. Since
06I
2
ðtÞ6PðtÞ;lim
t!þ1
I
2
ðtÞ¼0. Moreover,
HI
1
H
61 implies k
1
I
1HI
1
H
6k
1
I
1
. Then, _
I
1
6k
1
1
1
R
01
I
1
and since
R
01
61;lim
t!þ1
I
1
ðtÞ¼0 and therefore lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
1
.
3. Since R
1
61;lim
t!þ1
ðP;I
2
ÞðtÞ¼ð0;0Þ.IfR
01
>1;lim
t!þ1
ðH;I
1
ÞðtÞ¼ K;K1
1
R
01
. Then, lim sup
t!þ1
HðtÞ6Kand
lim sup
t!þ1
HðtÞ6K1
1
R
01
. Therefore lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
2
.
4. The solutions I
1
ðtÞand I
2
ðtÞof System (10) satisfy _
I
1
61
1
R
6
k
1
I
1
and _
I
2
61
1
R
5
k
2
I
2
.IfR
5
61 and
R
6
61;lim
t!þ1
I
1
ðtÞ¼0 and lim
t!þ1
I
2
ðtÞ¼0. Since 1 <R
1
61þ
b
a
c
ð1þaKÞ
;lim
t!þ1
HðtÞ¼H
H
;lim
t!þ1
PðtÞ¼P
H
and
therefore lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
3
.
5. Since R
6
61;lim
t!þ1
I
1
ðtÞ¼0. If 1 <R
1
61þ
b
a
c
ð1þaKÞ
;lim
t!þ1
ðH;PÞðtÞ¼ðH
H
;P
H
Þ. With the jacobian matrix at equilib-
rium ðH
H
;P
H
;I
2E
Þ;lim
t!þ1
I
2
ðtÞ¼I
2E
if R
5
>1. Therefore, if 1 <R
1
61þ
b
a
c
ð1þaKÞ
and R
5
>1, then lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
4
.
6. The solutions HðtÞand PðtÞof (10) satisfy lim
t!þ1
ðH;PÞðtÞ¼ðH
H
;P
H
Þ. With the jacobian matrix at equilibrium
E
5
;lim
t!þ1
I
1
ðtÞ¼I
1EE
if R
6
>1 and lim
t!þ1
I
2
ðtÞ¼I
2EE
if R
5
61. Therefore, lim
t!þ1
ðH;I
1
;P;I
2
Þ¼E
5
.
7. Since R
01
61 and R
5
61;lim
t!þ1
ðI
1
;I
2
ÞðtÞ¼ð0;0Þ. Since R
1
>1þ
b
a
c
ð1þaKÞ
, the System (10) has a stable limit cycle in
ðH;PÞplane.
8. The last two steps are obtained numerically.
Appendix D. Supplementary data
Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/
j.apm.2012.10.003.
References
[1] R. Anguelov, Y. Dumont, J.M-S. Lubuma, M. Shillor, Comparison of some standard and nonstandard numerical methods for the MSEIR epidemiological
model, in: Proceedings of the International Conference of Numerical Analysis and Applied Mathematics, American Institute of Physics Conference
Proceedings–AIP 1168, vol. 2, Crete, Greece, 2009, pp. 1209–1212.
[2] S. Bowong, J.J. Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Commun. Nonlinear Sci. Numer. Simul. 14 (2009)
4010–4021.
[3] A. Iggidr, J.C. Kamgang, G. Sallet, J.J. TEWA, Global analysis of new malaria intrahost models with a competitive exclusion principle, SIAM J. Appl. Math.
67 (N1) (2006) 260–278.
[4] A. Fall, A. Iggidr, G. Sallet, J.J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom. 2 (1) (2006) 55–71.
[5] P. Adda, A. Iggidr, J.C. Kamgang, G. Sallet, J.J. Tewa, General models of host-parasite systems. Global analysis, Disc. Contin. Dyn. Syst. B 8 (N1) (2007) 1–
17.
[6] A. Iggidr, J. Mbang, G. Sallet, J.J. Tewa, Multi-compartment models, Disc. Contin. Dyn. Syst. B (Suppl.) (2007) 506–519.
[7] N. Bame, S. Bowong, J. Mbang, G. Sallet, J.J. Tewa, Global stability analysis for SEIS models with nlatent classes, Math. Biosci. Eng. 5 (N1) (2008) 20–33.
[8] Jean Jules Tewa, Samuel Bowong, C.S. Oukouomi Noutchie, Mathematical analysis of two-patch model of tuberculosis disease with staged progression,
Appl. Math. Model. 36 (2012) 5792–5807.
[9] J.J. Tewa, R. Fokouop, B. Mewoli, S. Bowong, Mathematical analysis of a general class of ordinary differential equations coming from within-hosts
models of malaria with immune effectors, Appl. Math. Comput. 218 (2012) 7347–7361.
[10] D.P. Moualeu, S. Bowong, J.J. Tewa, Y. Emvudu, Analysis of the impact of diabetes on the dynamical transmission of tuberculosis, Math. Model. Nat.
Phenom. 7 (3) (2012) 117–146, http://dx.doi.org/10.1051/mmnp/20127309.
[11] Jean Jules Tewa, Samuel Bowong, Boulchard Mewoli, Mathematical analysis of two-patch model for the dynamical transmission of tuberculosis, Appl.
Math. Model. 36 (2012) 2466–2485.
[12] Samuel Bowong, Jean Jules Tewa, Global analysis of a dynamical model for transmission of tuberculosis with general contact rate, Commun. Nonlinear
Sci. Numer. Simul. 15 (2010) 3621–3631.
[13] Jean Jules Tewa, Samuel Bowong, Boulchard Mewoli, Jurgen Kurths, Two-patch transmission of tuberculosis, Math. Popul. Studies (MPS) 18 (2) (2011)
189–205.
[14] Yves Emvudu, Boulchard Mewoli, Jean Jules Tewa, Jean Pierre Kouenkam, Epidemiological model for the spread of anti-tuberculosis resistance, Int. J.
Info. Syst. Sci. Inst. Sci. Comput. Info. 7 (4) (2011) 279–301.
[15] Samuel Bowong, Jean Jules Tewa, Jean Claude Kamgang, Stability analysis of the transmission dynamics of tuberculosis models, World J. Model. Simul.
(WJMS) 7 (2) (2011) 83–100.
[16] R.M. Anderson, R.M. May, The invasion persistence and spread of infectious diseases within animal and plant communities, Phil. Trans. R. Sot. Lond.
B314 (1986) 533–570.
[17] K.P. Hadeler, H.I. Freedman, Predator–Prey populations with parasitic infection, J. Math. Biol. 27 (1989) 609–631.
[18] E. Venturino, The influence of diseases on Lotka–Volterra systems, Rocky Mt. J. Math. 24 (1994) 381–402.
[19] E. Venturino, Epidemics in Predator-Prey models: disease in the Prey, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population
Dynamics: Analysis of Heterogeneity, Theory of Epidemics, vol. 1, Wuerz, Winnipeg, Canada, 1995, pp. 381–393.
[20] K.P. Hadeler, H.I. Freedman, Predator–Prey populations with parasitic infection, J. Math. Biol. 27 (1989) 609–631.
[21] P.J. Hudson, A.P. Dobson, D. Newborn, Do parasites make Prey more vulnerable to predation? Red grouse and parasites, J. Anim. Ecol. 61 (1992) 681–
692.
[22] Ying-Hen Hsieh, Chin-Kuei Hsiao, Predator–Prey model with disease infection in both populations, Math. Med. Biol. 25 (2008) 247–266.
[23] T.K. Kar, Selective harvesting in a Prey–Predator fishery with time delay, Appl. Math. Comput. 388 (2003) 449–458.
4840 J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841
Author's personal copy
[24] Shigui Ruan, Dongmei Xiao, Global analysis in a Predator–Prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (4) (2001)
1445–1472.
[25] M. Haque, A Predator–Prey model with disease in the Predator species only, Nonlinear Anal. Real World Appl. 11 (2010) 2224–2236.
[26] M. Haque, D. Greenhalgh, When Predator avoids infected Prey: a model based theoretical studies, IMA J. Math. Med. Biol. 27 (2009) 75–94.
[27] N. Bairagi, P.K. Roy, J. Chattopadhyay, Role of infection on the stability of a Predator–Prey system with several response functions – A comparative
study, J. Theor. Biol. 248 (1) (2007) 10–25. ISSN 0022-519.
[28] Krishna Padas Das, Kusumika Kundu, J. Chattopadhyay, A Predator–Prey mathematical model with both populations affected by diseases, Ecolo.
Complex. 8 (2011) 68–80.
[29] M. Haque, E. Venturino, Modelling disease spreading in symbiotic communities, Wildlife: Destruction, Conservation and Biodiversity, Nova Science
Publishers, USA, 2009. <www.novapublishers.com>.
[30] M. Haque, S. Sarwardi, S. Preston, E. Venturino, Effect of delay in a Lotka–Volterra type Predator–Prey model with a transmissible disease in the
Predator species, Math. Biosci. 234 (1) (2011) 47–57.
[31] M. Haque, J. Zhen, E. Venturino, Rich dynamics of Lotka–Volterra type Predator–Prey model system with viral disease in Prey species, Math. Methods
Appl. Sci. 32 (2009) 875–898.
J.J. Tewa et al. / Applied Mathematical Modelling 37 (2013) 4825–4841 4841
