Mathematical Biosciences 240 (2012) 35–44
Contents lists available at SciVerse ScienceDirect
Mathematical Biosciences
journal homepage: www.elsevier.com/locate/mbs
Discrete-time models with mosquitoes carrying genetically-modified bacteria q
Jia Li ⇑
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, United States
a r t i c l e
i n f o
Article history:
Received 29 January 2012
Received in revised form 21 May 2012
Accepted 29 May 2012
Available online 6 July 2012
Keywords:
Mosquito population models
Discrete-time models
Genetically modified bacteria
Ricker-type nonlinearity
Stability
Period-doubling bifurcation
a b s t r a c t
We formulate a homogeneous model and a stage-structured model for the interactive wild mosquitoes
and mosquitoes carrying genetically-modified bacteria. We establish conditions for the existence and stability of fixed points for both models. We show that a unique positive fixed point exists and is asymptotically stable if the two boundary fixed points are both unstable. The unique positive fixed point exists and
is unstable if the two boundary fixed points are both locally asymptotically stable. Using numerical
examples, we demonstrate the models undergoing a period-doubling bifurcation.
Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction
Mosquito-borne diseases, such as malaria, are considerable
public health concerns worldwide. These diseases are transmitted
between humans by blood-feeding mosquitoes. To prevent and
control malaria, genetically-altered or transgenic mosquitoes provide a new and potentially effective weapon [6,8,12]. Nevertheless,
while those transgenic mosquitoes resistant to malaria have been
successfully produced in laboratories and can be eventually introduced into the field, there still exist substantial hurdles, including
drive mechanisms, transposon stability, and multiple mosquito
subspecies, that need to be overcome [17,18]. Given time these
obstacles will eventually be surmounted, but, in the meantime,
new control measures are needed quickly because the effectiveness of current measures is decreasing. The paratransgenic
approach then may lead to a speedier solution [1,4,13,17,18].
Paratransgenesis is a technique that attempts to eliminate
pathogens from vector populations through transgenesis of a symbiont of the vector [1,4]. The first step is to identify proteins that
prevent the vector species from transmitting the pathogens. The
genes coding for these proteins are then introduced into the
symbiont, so that they can be expressed in the vector [15]. Such
a technique, by utilizing bacteria, such as Escherichia coli and
Enterobacter agglomerans, capable of colonizing the mosquito
q
The author thanks two anonymous referees for their careful reading and
valuable comments and suggestions. This research was supported partially by U.S.
National Science Foundation Grant DMS-1118150.
⇑ Tel.: +1 2568246470; fax: +1 2568246173.
E-mail address: [email protected]
0025-5564/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.mbs.2012.05.012
midgut to produce effector molecules that kill the Plasmodium
parasite or inhibit its development, has been applied to controlling
and preventing the malaria transmission [17,18]. This approach
addresses several of the problems associated with the release of
genetically-modified mosquitoes: (1) It is compatible with traditional control strategies such as insecticide treatment. (2) It is
likely that the bacteria will be capable of colonizing a wide range
of mosquito species, allowing for the treatment of reproductively
isolated strains. (3) The production of sufficient recombinant bacteria is trivial in comparison with rearing sufficient transgenic
mosquitoes, and safety guidelines for field trials of recombinant
bacteria, although stringent, have already been established [20].
This approach has several advantages that (1) the feasibility of
releasing genetically-modified, nonpathogenic bacteria has been
or is being tested in several different systems; (2) a contained field
trial was performed after risk assessment and approval by the US
Environmental Protection Agency’s Office of Pollution Prevention
and Toxics; (3) the logistics of producing sufficient numbers of bacteria for control is trivial compared with producing the numbers of
mosquitoes required for population replacement, and bacteria can
also be readily engineered to express multiple effector molecules;
and (4) the bacterial approach does not require the release of biting
insects that may pose local safety or nuisance problems [17,20].
The transmission mechanism of the genetically-modified bacteria (called transgenic bacteria for short) is different from the genetically-altered mosquitoes, and is complex. The transmission can be
vertically as well horizontally. The transgenic bacteria come with
the eggs reproduced vertically from the mosquitoes carrying
the transgenic bacteria or from wild mosquitoes mating with the
mosquitoes carrying the transgenic bacteria. They can also be
36
J. Li / Mathematical Biosciences 240 (2012) 35–44
transmitted horizontally between the larvae of the two types of
mosquitoes through water during the first three metamorphic
stages when they are in the same breeding sites. To better understand the transmission of the transgenic bacteria, we formulate
simple discrete-time mathematical models for interacting wild
and transgenic mosquito populations based on systems of difference equations in this paper. We first assume homogeneous mosquito populations without including the metamorphic stages of
mosquitoes in Section 2. After basically analyzing the simplified
model and using it as a basis, we then include mosquito metamorphic stages into our model formulation in Section 3. To make the
mathematical analysis more tractable, however, we combine the
three aquatic metamorphic stages of mosquitoes as one group such
that there are only two groups for the wild mosquitoes as well for
the mosquitoes carrying genetically-modified bacteria. We investigate the existence of fixed points and their stability for both models, and provide numerical examples to demonstrate our findings.
Brief discussions are given in Section 4.
2. The homogeneous model
We assume homogeneous mosquito populations and let xðnÞ
and yðnÞ be the sizes of the wild mosquitoes and the mosquitoes
carrying genetically-modified bacteria (called transgenic mosquitoes for short), respectively. The dynamics of the interaction between the mosquitoes can be described by the following system:
xnþ1 ¼ ð1 Cðxn ; yn ÞÞB1 ðxn ; yn Þxn S1 ðxn ; yn Þ;
ynþ1 ¼ ðB2 ðxn ; yn Þyn þ Cðxn ; yn ÞB1 ðxn ; yn Þxn ÞS2 ðxn ; yn Þ;
ð2:1Þ
where Bi and Si ; i ¼ 1; 2, are the birth and survival functions of the
two types of mosquitoes, respectively, and C is the horizontal transmission rate that wild mosquitoes becoming carrying bacteria. Notice that the vertical transmission is characterized by the birth
functions.
The horizontal transmission depends on the mixing between
wild and transgenic mosquitoes and the available bacterial loads
per wild mosquitoes. Let wn :¼ yn =xn . Then we assume that the
horizontal transmission rate is a function of w, such that
C ¼ CðwÞ, and satisfies the following conditions
1
CðwÞ 2 C ½0; 1Þ;
Cð0Þ ¼ 0;
0
C ðwÞ > 0;
lim CðwÞ ¼ c 6 1;
w!1
ð2:2Þ
where the nonnegative constant c is the maximum horizontal
transmission rate.
More specifically, we take a simple form for C, that satisfies
condition (2.2), as
CðwÞ ¼
cw
:
1þw
ynþ1
cyn a1 xn þ a2 yn kðxn þyn Þ
xnþ1 ¼ 1 xn e
;
xn þ yn
xn þ y n
b1 xn þ b2 yn
cxn a1 xn þ a2 yn
yn ekðxn þyn Þ
þ
ynþ1 ¼
xn þ y n
xn þ y n xn þ y n
ð2:3Þ
for xn P 0; yn P 0, and ðxn ; yn Þ – ð0; 0Þ, where we assume a1 > 1
and b2 > 1. Notice that a1 and b2 are the net reproductive numbers
of the wild and transgenic mosquitoes, respectively.
2.1. Boundary fixed points
System (2.3) has boundary fixed points E1 :¼ lnka1 ; 0 , if a1 > 1,
and E2 :¼ 0; lnkb2 , if b2 > 1. The Jacobian matrices at E1 and E2 are
given, respectively, by
1 ln a1
b1
a1
0
!
þc
ð2:4Þ
;
ð1 cÞ ab22
0
1 ln b2
!
ð2:5Þ
:
Thus, boundary fixed point E1 is locally asymptotically stable if
b1
< a1 < e2
1c
ð2:6Þ
and is unstable if
a1 > e2 ;
or 1 c <
b1
:
a1
ð2:7Þ
Boundary fixed point E2 is locally asymptotically stable if
a2 ð1 cÞ < b2 < e2
ð2:8Þ
and is unstable if
b2 > e2 ;
or
b2
< 1 c:
a2
ð2:9Þ
2.2. Positive fixed points
We then investigate the existence of positive fixed points whose
two components are both positive and their stability.
Following the line as in [9], we assume that there is no fitness
distinguish between the two types of mosquitoes. Then the interaction is governed by the following system
xnþ1 ¼ 1 mosquito, respectively, d is the density-independent death rate,
and k characterizes the carrying capacity. If x0 P 0; y0 P 0, and
ðx0 ; y0 Þ – ð0; 0Þ; xn P 0 and yn P 0, and ðxn ; yn Þ – ð0; 0Þ, for all
n P 1.
We assume a1 > ed and b2 > ed , so that the origin ð0; 0Þ is a
repeller. For convenience, we merge ed into ai and bi and, hereafter, consider the system
cyn
a1 xn þ a2 yn
xn edkðxn þyn Þ ;
xn þ yn
xn þ y n
b 1 xn þ b 2 y n
cyn a1 xn þ a2 yn dkðxn þyn Þ
¼
yn edkðxn þyn Þ þ
xn e
xn þ y n
xn þ yn xn þ yn
b 1 xn þ b 2 y n
cxn a1 xn þ a2 yn
yn edkðxn þyn Þ
¼
þ
xn þ y n
xn þ y n xn þ y n
for xn P 0; yn P 0, and ðxn ; yn Þ – ð0; 0Þ, where a1 and a2 are the
numbers of wild offspring that a wild mosquito produces, through
a mating with a wild and a transgenic mosquito, respectively, b1
and b2 are the numbers of transgenic offspring produced per transgenic mosquito, through a mating with a wild and a transgenic
2.2.1. Existence of positive fixed points
It follows from (2.3) that a positive fixed point, ðx > 0; y > 0Þ,
satisfies
cy a1 x þ a2 y
;
ekðxþyÞ ¼ 1 xþy
xþy
b1 x þ b2 y
cx a1 x þ a2 y
þ
;
ekðxþyÞ ¼
xþy
xþy xþy
ð2:10Þ
which leads to
ð1 cÞða1 x þ a2 yÞ ¼ b1 x þ b2 y:
Hence there exists a positive solution of (2.10) only if
ðð1 cÞa1 b1 Þðb2 ð1 cÞa2 Þ > 0;
ð2:11Þ
37
J. Li / Mathematical Biosciences 240 (2012) 35–44
that is, either
Proof. We first have [11]
b2
b1
<1c< ;
a2
a1
ð2:12Þ
ðb1 þ b2 wn Þð1 þ wn Þ ð1 cÞð1 þ wn Þða1 þ a2 wn Þ
wn
ð1 þ ð1 cÞwn Þða1 þ a2 wn Þ
ð1 þ wn Þwn
¼ ðb1 þ b2 wn ð1 cÞða1 þ a2 wn ÞÞ
ð1 þ ð1 cÞwn Þða1 þ a2 wn Þ
ð1 cÞa1 b1
ð1 þ wn Þwn
¼ ðb2 ð1 cÞa2 Þ wn b2 ð1 cÞa2 ð1 þ ð1 cÞwn Þða1 þ a2 wn Þ
ð1 þ wn Þwn
¼ ðb2 ð1 cÞa2 Þðwn w Þ
:
ð1 þ ð1 cÞwn Þða1 þ a2 wn Þ
ð2:20Þ
wnþ1 wn ¼
or
b2
b1
>1c> :
a2
a1
ð2:13Þ
Thus, we notice, from (2.6)–(2.9), that a positive fixed point exists only if the two boundary fixed points, E1 and E2 are both stable
or both unstable.
Suppose either condition (2.12) or (2.13) holds. Writing y in
terms of x, we have
y¼
ð1 cÞa1 b1
x :¼ Ax:
b2 ð1 cÞa2
ð2:14Þ
Substituting (2.14) into (2.10) yields
ð1 þ AÞ2 ekð1þAÞx ¼ ð1 þ ð1 cÞAÞða1 þ a2 AÞ:
Suppose (2.13) holds, that is, b2 > ð1 cÞa2 . If w0 > w , then
w1 > w0 , and consequently wnþ1 > wn , for all n P 1. Thus fwn g is
an increasing sequence, going away from w . If w0 < w , then
w1 < w0 and consequently wnþ1 < wn . Thus fwn g is a decreasing
sequence, also going away from w . That is; w is an unstable repeller for (2.19).
On the other hand, if (2.12) is satisfied, that is, b2 < ð1 cÞa2 ,
then fwn g is an increasing sequence if w0 < w , and a decreasing
sequence if w0 > w .
We next consider
ð2:15Þ
ðb1 þ b2 wn Þð1 þ wn Þ þ cða1 þ a2 wn Þ
wn
ð1 þ ð1 cÞwn Þða1 þ a2 wn Þ
ðb1 þ b2 w Þð1 þ w Þ þ cða1 þ a2 w Þ w
ð1 þ ð1 cÞw Þða1 þ a2 w Þ
ðb1 þ b2 wn Þwn
1 þ wn
¼
a1 þ a2 wn 1 þ ð1 cÞwn
ðb1 þ b2 w Þw
1 þ w
cwn
þ
ða1 þ a2 w Þ 1 þ ð1 cÞw 1 þ ð1 cÞwn
cw
:
1 þ ð1 cÞw
wnþ1 w ¼
Then, solving (2.15) for x, we obtain
x¼
1
ð1 þ ð1 cÞAÞða1 þ a2 AÞ
:
ln
kð1 þ AÞ
ð1 þ AÞ2
ð2:16Þ
Define
q :¼ 1 cA
1þA
a1 þ a2 A
;
1þA
ð2:17Þ
that is, by substituting A into (2.17),
q¼
Notice that
ðb1 þ b2 wn Þwn
a1 þ a2 wn
b2 ð1 cÞb1 þ ð1 cÞðð1 cÞa1 a2 Þ a1 b2 a2 b1
:
b2 ð1 cÞa2
b2 ð1 cÞa2
ð2:18Þ
Then, under condition (2.12) or (2.13), x > 0 if and only if q > 1.
In summary, we have the following results.
Theorem 2.1. There exists a unique positive fixed point of system
(2.3), with x given by (2.16) and y ¼ Ax, where A is defined in (2.14), if
and only if q > 1 and (2.12) or (2.13) is satisfied.
2.2.2. Stability of the positive fixed point
We next study the stability of the unique positive fixed point if
it exists. Note that if x0 > 0; y0 > 0, the solutions of (2.3) have
xn > 0 and yn > 0, for all n P 1. Let wn :¼ yn =xn , for n P 0; x0 > 0,
y0 > 0, such that w0 > 0. It follows from (2.3) that wn satisfies
the following equation
wnþ1
b1 xn þ b2 yn
cxn a1 xn þ a2 yn
þ
xn þ yn
xn þ yn xn þ yn
¼
wn
yn
a1 xn þa2 yn
1 xncþy
xn þy
n
¼
n
ð2:19Þ
ðb1 þ b2 wn Þð1 þ wn Þ þ cða1 þ a2 wn Þ
wn :
ð1 þ ð1 cÞwn Þða1 þ a2 wn Þ
We have the following results for the global dynamics of Eq.
(2.19).
Lemma 2.2. The unique positive fixed point of Eq. (2.19), w ¼ A,
defined in (2.14), is globally asymptotically stable if condition (2.12)
holds. It is unstable if condition (2.13) holds.
and
1 þ wn
1 þ ð1 cÞwn
are increasing functions of wn , and
cwn
1 þ ð1 cÞwn
cw
1 þ ð1 c
Þw
¼
cðwn w Þ
:
ð1 þ ð1 cÞwn Þ:ð1 þ ð1 cÞw Þ
Then if wn > w ; wnþ1 > w , and if wn < w , wnþ1 < w , for all n P 0.
Together with the monotonicity of the sequence fwn g for either
w0 < w or w0 > w , we have shown that the positive fixed point
w of Eq. (2.19) is globally asymptotically stable. The proof is
complete. h
Remark 1. It follows from (2.6)–(2.9) that if one of the boundary
equilibria is stable and one is unstable, no positive fixed point
exists. If both boundary fixed points E1 and E2 are locally asymptotically stable, the positive fixed point may exist, but must be unstable. If both boundary fixed points E1 and E2 are unstable, the
positive fixed point may exist. It may be stable, or unstable. Further
analysis follows.
We now investigate the stability of the positive fixed point of
system (2.3). Let the positive fixed point of Eq. (2.19), denoted by
w, be globally asymptotically stable. Then the x-limit set of solutions with x0 > 0 and y0 > 0 lies in the line y ¼ Ax [5,19,21], and
by setting yn ¼ Axn , the two equations in (2.3) become
cA a1 þ a2 A kð1þAÞxn
xn e
xnþ1 ¼ 1 ;
1þA
1þA
b1 þ b2 A
c a1 þ a2 A
þ
yn ekð1þ1=AÞyn :
ynþ1 ¼
1þA
1þw 1þA
ð2:21Þ
38
J. Li / Mathematical Biosciences 240 (2012) 35–44
10
8
wild mosquitoes
transgenic mosquitoes
9
wild mosquitoes
transgenic mosquitoes
7
6
Mosquito Populations
Mosquito Populations
8
7
6
5
4
3
5
4
3
2
2
1
1
0
10
20
30
40
50
60
70
0
80
10
20
30
Time n
40
50
60
70
80
Time n
Fig. 1. Fix c ¼ 0:65 and k ¼ 0:2. With parameters given in (2.22), boundary fixed point E1 is stable and boundary fixed point E2 is unstable, as shown in the left figure. With
parameters given in (2.23), boundary fixed point E1 is unstable and boundary fixed point E2 is stable, as shown in the right figure. No positive fixed point exists in either case.
25
25
wild mosquitoes
transgenic mosquitoes
wild mosquitoes
transgenic mosquitoes
20
Mosquito Populations
Mosquito Populations
20
15
10
5
0
15
10
5
10
20
30
40
50
60
70
80
0
10
20
30
Time n
40
50
60
70
80
Time n
Fig. 2. With parameters given in 2.24, both boundary fixed points E1 and E2 are locally asymptotically stable while a unique positive fixed point exists but is unstable. The
solution with initial value ðx0 ; y0 Þ ¼ ð14; 0:5Þ approaches E1 as shown in the left figure, and the solution with initial value ðx0 ; y0 Þ ¼ ð4; 22Þ approaches E2 , as shown in the right
figure.
The two equations in (2.21) are uncoupled and are Ricker-type
equations. Its positive fixed point ðx ; y Þ ¼ ðx ; Ax Þ, if it exists, is
globally asymptotically stable provided
ln
cA a1 þ a2 A
< 2;
1
1þA
1þA
that is, q < e2 . The positive fixed point is unstable if q > e2 [2,19].
We summarize the stability results for the positive fixed points as
follows.
Theorem 2.3. Suppose that the unique positive fixed point of system
(2.3), ðx ; Ax Þ, where x is given in (2.16) and A is given in (2.14),
exists. Then it is globally asymptotically stable if condition (2.12) holds
and q < e2 , where q is given in (2.18). This unique positive fixed point
is unstable if condition (2.13) holds, or q > e2 .
2.3. Numerical examples
We now give numerical examples in this section to verify the
dynamics of system (2.3).
Example 1. Fix parameters c ¼ 0:65 and k ¼ 0:2. With parameters
a1 ¼ 5;
a2 ¼ 11;
b1 ¼ 1;
b2 ¼ 3;
ð2:22Þ
conditions (2.6) and (2.9) are satisfied. Boundary fixed point E1 is locally asymptotically stable and boundary fixed point E2 is unstable.
No positive fixed point exists in this case. The dynamics of system
(2.3) are shown in the left figure in Fig. 1. With parameters
a1 ¼ 5;
a2 ¼ 8;
b1 ¼ 4;
b2 ¼ 3;
ð2:23Þ
conditions (2.7) and (2.8) are satisfied. Boundary fixed point E1 is
unstable and boundary fixed point E2 is locally asymptotically stable. No positive fixed point exists either. The dynamics of system
(2.3) are shown in the right figure in Fig. 1.
Example 2. We use the following parameters
a1 ¼ 7;
a2 ¼ 5;
b1 ¼ 1;
b2 ¼ 6;
k ¼ 0:1;
c ¼ 0:8;
ð2:24Þ
such that both boundary fixed points E1 ¼ ð19:4591; 0Þ and E2 ¼
ð0; 17:9176Þ are locally asymptotically stable. Since q ¼ 6:4458,
there exists a unique positive fixed point ð17:2540; 1:3803Þ, but it
39
J. Li / Mathematical Biosciences 240 (2012) 35–44
is unstable. The solution with initial value ðx0 ; y0 Þ ¼ ð14; 0:5Þ
approaches E1 and the solution with initial value ðx0 ; y0 Þ ¼ ð4; 22Þ
approaches E2 , as shown in Fig. 2.
Example 3. We use the following parameters
a1 ¼ 21;
a2 ¼ 8;
b1 ¼ 24;
b2 ¼ 3;
k ¼ 0:1;
ð2:25Þ
in this example, and let c vary. As c ¼ 0, the two boundary fixed
points are both unstable, and there exists a positive fixed point
ð20:6455; 8:2582Þ. However, since q ¼ 18 > e2 , the positive fixed
point is unstable. The behavior of the dynamics of system (2.3) is chaotic. We then increase c to 0:04; 0:1, and 0:22, respectively. A positive
fixed point still exists in each of these cases, but the values of q are
14:8679; 13:1103, and 10:0448, respectively, such that the fixed
60
40
wild mosquitoes
transgenic mosquitoes
Mosquito Populations
50
Mosquito Populations
wild mosquitoes
transgenic mosquitoes
35
40
30
20
30
25
20
15
10
10
5
0
10
20
30
40
50
60
70
0
80
10
20
30
Time n
40
50
30
Mosquito Populations
Mosquito Populations
30
25
20
15
10
25
20
15
10
5
5
10
20
30
40
50
60
70
0
80
10
20
30
40
50
60
70
80
Time n
Time n
25
25
wild mosquitoes
transgenic mosquitoes
wild mosquitoes
transgenic mosquitoes
20
Mosquito Populations
20
Mosquito Populations
80
wild mosquitoes
transgenic mosquitoes
wild mosquitoes
transgenic mosquitoes
15
10
15
10
5
5
0
70
35
35
0
60
Time n
0
10
20
30
40
Time n
50
60
70
80
10
20
30
40
50
60
70
80
Time n
Fig. 3. Given parameters given in 2.25, we vary parameter c. As c ¼ 0, system 2.3 exhibits chaotic behavior as shown in the upper left figure. For c ¼ 0:04, an eight-cycle
appears as shown in the upper right figure. For c ¼ 0:1 and c ¼ 0:22, a four-cycle and a two-cycle appear, shown in the middle two figures. All of the cycles are stable. Notice
that there exists a positive fixed point in each of these cases, but since q > e2 for all of these cases, the positive fixed points are all unstable. As c ¼ 0:5, q ¼ 4:7551 < e2 , and
the positive fixed point ð1:0753; 14:5168Þ becomes stable as shown in the lower left figure. For c ¼ 0:8, q ¼ 0:9754, there exists no positive fixed point, and boundary fixed
point E2 is stable as shown in the lower right figure.
40
J. Li / Mathematical Biosciences 240 (2012) 35–44
point is unstable. A cycle with a large period, a four-cycle, and a
two-cycle, appear, respectively, and all are stable. As c ¼
0:5; q ¼ 4:7551 < e2 , and positive fixed point ð1:0753; 14:5168Þ becomes globally asymptotically stable. We then increase c to 0:8 such
that q ¼ 0:9754. No positive fixed point exists and boundary fixed
point E2 is stable. The dynamics are all shown in Fig. 3.
k2 x < 2;
k2
ln r 0 < 2;
k1 þ k2
i:e:
ð3:5Þ
then 1 þ det J1 < 2. It follows from
1 þ det J 1 trJ 1 ¼ ðk1 þ k2 Þx > 0
and
1 det J 1 trJ 1 ¼ ðk1 k2 Þx < 0;
3. Stage-structured models
that jtrJ 1 j < 1 þ det J1 , if k1 < k2 . Thus, we have the following
results.
The transgenic bacteria can be transmitted vertically and horizontally. They come vertically with the eggs laid by the mosquitoes
carrying the transgenic bacteria, or from the wild mosquitoes mating with the mosquitoes carrying the transgenic bacteria. They can
also be transmitted horizontally between the larvae of the two
types of mosquitoes through water during the first three metamorphic stages when they are in the same breeding sites.
We follow a line similar to the stage-structured models for
transgenic mosquitoes in [10] where the three aquatic metamorphic stages are combined as one group, call larvae. With this simplification, the wild mosquitoes are divided into two groups xn and
un , the larvae and adults, and the mosquitoes carrying transgenic
bacteria are divided into yn and v n , also the larvae and adults.
Theorem 3.1. Suppose r 0 > 1, the trivial fixed point ð0; 0Þ is unstable
and there exists a unique positive fixed point whose two components
x and u , are given in (3.3). The positive fixed point is locally
asymptotically stable if
k1 < k2 < 2
k1 þ k2
ln r 0
ð3:6Þ
and is unstable if
k1 > k2 ;
or k2 > 2
k1 þ k2
:
ln r 0
ð3:7Þ
3.1. In the absence of interaction
3.2. Interacting populations
We first study the dynamics of the mosquitoes in the absence of
interaction between the two types.
While interspecific competition and predation are rather rare
events and could be discounted as major causes of larval mortality,
intraspecific competition could represent a major density dependent source for them. Hence the effect of crowding of larvae could
be an important factor in the population dynamics of mosquitoes
[3,7,14,16], and we assume that the density-dependent larvae
mortality and emergence rate are both functions of the larvae size.
Then, in the absence of interaction, the dynamics of the mosquito
population with or without transgenic bacteria are described by
the following system:
We then consider that the two types of mosquitoes interact. Let
xn and un be the numbers of the wild mosquito larvae and adults,
and yn and v n be the numbers of the larvae and adults of the mosquitoes carrying transgenic bacteria, respectively. We assume that
the emergence rates and the death rates of the adult wild and
transgenic mosquitoes are the same. Then the dynamics of the
interacting populations are governed by the following system:
xnþ1 ¼ run ed1 k1 xn ;
xnþ1 ¼ 1 cyn
xn þ y n
a1 un þ a2 v n
un ek1 ðxn þyn Þ ;
un þ v n
unþ1 ¼ b1 xn ek2 ðxn þyn Þ ;
b1 un þ b2 v n
cyn a1 un þ a2 v n
vn þ
un ek1 ðxn þyn Þ ;
ynþ1 ¼
un þ v n
xn þ yn un þ v n
v nþ1 ¼ b2 yn ek ðx þy Þ
2
unþ1 ¼ bxn ed2 k2 xn ;
n
n
ð3:8Þ
where r > 0 is the per capita birth rate of adults, di > 0; i ¼ 1; 2, the
death rates, ki > 0; i ¼ 1; 2, the constants describing the carrying
capacity of larvae and adults, respectively, and b > 0 the maximum
death-adjusted emergence rate from larvae to adults. For convenience, we merge ed1 and ed2 into r and b, respectively, and keep
the same r and b, without confusion, such that the system becomes
for x0 P 0; u0 P 0; y0 P 0; v 0 P 0, and ðx0 ; y0 Þ – ð0; 0Þ. If x0 P
0; y0 P 0, and ðx0 ; y0 Þ – ð0; 0Þ; xn P 0; un P 0, yn P 0; v n P 0,
and ðxn ; un ; yn ; v n Þ – ð0; 0; 0; 0Þ, for all n P 1.
We define the net reproductive numbers for the wild and transgenic mosquito populations as
xnþ1 ¼ run ek1 xn ;
r01 :¼ a1 b1 ;
unþ1 ¼ bxn e
k2 xn
ð3:1Þ
:
respectively. We assume r0i > 1; i ¼ 1; 2,
The origin ð0; 0Þ is a trivial fixed point. We define the net reproductive number of the mosquitoes as
r 0 :¼ rb
1
ln r 0 ;
k1 þ k2
u ¼
1 k1 x
¼ bx ek2 x :
xe
r
ð3:3Þ
The Jacobian matrix at this positive fixed point has the form of
J1 ¼
k1 < k2 < 2
ð3:2Þ
and assume r 0 > 1. Then the origin is unstable and there exists a positive fixed point ðx > 0; u > 0Þ, given by
x ¼
x
u
k1 x
r 02 :¼ b2 b2 ;
bek2 x bk2 ek2 x x
0
!
¼
k1 x
u
x
ð1 k2 x Þ
x
u
0
!
ð3:4Þ
and the positive fixed point is stable if jtrJ1 j < 1 þ det J 1 < 2. Now
that the trace of J1 is trJ1 ¼ k1 x , and the determinant of J1 is
det J 1 ¼ k2 x 1. If
k1 þ k2
ln r 01
ð3:9Þ
k1 þ k2
;
ln r 02
ð3:10Þ
and
k1 < k2 < 2
so that both of the origins are unstable and both of the positive fixed
points are locally asymptotically stable for the two subsystems of
(3.8) in the absence of interaction between the wild and transgenic
mosquitoes.
3.2.1. Boundary fixed points
As the interaction takes place, the coupled system (3.8) has two
boundary fixed points E01 :¼ ðx > 0; u > 0; y ¼ 0; v ¼ 0Þ and
E02 :¼ ðx ¼ 0; u ¼ 0; y > 0; v > 0Þ. Notice that if both x0 ¼ 0 and
41
J. Li / Mathematical Biosciences 240 (2012) 35–44
u0 ¼ 0, or both y0 ¼ 0 and v 0 ¼ 0, initially, the dynamics of system
(3.8) are similar to the dynamics of system (3.1). We then study the
asymptotic behavior of the two boundary fixed points for
ðx0 ; u0 Þ – ð0; 0Þ and ðy0 ; v 0 Þ – ð0; 0Þ and have the following results.
Theorem 3.2. We assume the net reproductive numbers
r 0i > 1; i ¼ 1; 2. Then system (3.8) has two boundary fixed points
E01 ¼ ðx ; u ; 0; 0Þ, where x and u are given by
x ¼
1
ln r 01 ;
k1 þ k2
u ¼ b1 x ek2 x
ð3:11Þ
and E02 ¼ ð0; 0; y ; v Þ, where y and
v are given by
1
y ¼
ln r02 ;
k1 þ k2
:
v
k2 y
¼ b2 y e
ð3:12Þ
Under condition (3.9), boundary fixed point E01 is locally asymptotically stable if
b2 a1
< ð1 cÞ
b1 b1
ð3:13Þ
b2 a1
> ð1 cÞ:
b1 b1
ð3:14Þ
Under condition (3.10), boundary fixed point E02 is locally asymptotically stable if
a2
b
ð1 cÞ < 2
b2
b1
ð3:15Þ
and is unstable if
a2
b
ð1 cÞ < 2 :
b2
b1
ð3:16Þ
!
0
J 22
where
Jf1g
11
0
b1 x
a1 u
c
2u
0
b1 x
1
det J f1g
22
¼ c;
b1 b2
¼
:
a1 b1
f1g
Then 1 þ det det J f1g
22 < 2, and hence J 22 is stabile if
b1 b2
b1 b2
1 þ
<c<1
;
a1 b1
a1 b1
!
0
f2g
J 22
;
where
0
f2g
J11
¼ @b
0
1
v
b2 y
and
f2g
f2g
det J 11 ¼ ð1 cÞ
trJ11 ¼ 0;
a2 b1
;
b2 b2
f2g
f2g
that 1 þ det J f2g
22 < 2 and jtrJ 22 j < 1 þ det J 22 if and only if condition
(3.15) is satisfied. The proof is complete. h
3.2.2. Positive fixed points
Now we turn our attention to the positive fixed points of system
(3.8). We first have the following results for the existence.
Theorem 3.3. Define
q^ :¼ b1 1 B :¼
cb1 B
b2 þ b1 B
a1 þ a2 B
;
1þB
ð3:18Þ
b1 a1 ð1 cÞ b2 b1
b2 b2 b1 a2 ð1 cÞ
^ > 1. Then there exists a unique positive fixed point of
and assume q
system (3.8), given by
b2
^;
ln q
ðk1 þ k2 Þðb2 þ b1 BÞ
b
y ¼ 1 Bx ; v ¼ Bu ;
b2
x ¼
u ¼ b1 x ek2 ð1þpÞx ;
ð3:19Þ
if and only if
ð1cÞa2 y
b2
v
0
1
A
cy a1 u þ a2 v k1 ðx þy Þ
x ¼ 1 ue
;
x þ y
u þ v ð3:17Þ
u ¼ b1 x ek2 ðx þy Þ ;
b1 u þ b2 v cy a u þ a v v þ 1 2 u ek1 ðx þy Þ ;
y ¼
u þv
x þy
u þv
v ¼ b2 y ek
because the right inequality in (3.17) is equivalent to (3.15), and if
(3.15) holds, the left inequality in (3.17) is satisfied.
The Jacobian at E02 has the form of
J f2g
11
f21g
The stability of J f2g
22 is similar to that of J 1 in (3.4). For J 22 , it follows from
ð3:20Þ
ð3:21Þ
Proof. The components of a positive fixed point of system (3.8)
ðx ; u ; y ; v Þ need to satisfy the following equations:
A:
The stability conditions for J f1g
11 are the same as those for system
f1g
(3.1). To investigate the stability of J 22 , we have
trJ f1g
22
:
b2 b2
b b1
>1c> 2
:
b1 a2
b1 a1
has the same form as in (3.4), and
¼ @b
f1g
J22
0
or
;
f1g
v
b2 b2
b b1
<1c< 2
;
b1 a2
b1 a1
Proof. The Jacobian at E01 has the form of
f1g
k2 y
ð1 k2 y Þ vy
where
and is unstable if
J 11
J f2g
22 ¼
!
y
2
ðx þy Þ
:
ð3:22aÞ
ð3:22bÞ
ð3:22cÞ
ð3:22dÞ
It follows from (3.22b) and (3.22d) that
u
v
¼
b1 x
:
b2 y
Letting y ¼ p x and v ¼ q u , and substituting them into
(3.22a) and (3.22c), we obtain
cp a1 þ a2 q b1 þ b2 q cp a1 þ a2 q
p 1 ¼
q þ
;
1 þ p
1 þ q
1 þ q
1 þ p 1 þ q
b
q ¼ 2 p :
b1
ð3:23Þ
Solving (3.23), we have
p ¼
b1
B;
b2
q ¼ B:
ð3:24Þ
42
J. Li / Mathematical Biosciences 240 (2012) 35–44
^ > 1 and condition (3.20) or (3.21), solving for x ; u ; y ,
With q
and v , yields (3.19). h
To study stability of the positive fixed point, we first let
pn :¼ yn =xn and qn :¼ v n =un , and consider the system
cyn a1 un þa2 v n
b1 un þb2 v n
un
n þ x þy
u þv
u þv
pnþ1 ¼ n xnnþð1cÞy an un nþa v nn n
n 1
2
u
n
xn þyn
un þv n
v
qnþ1 ¼
ðb1 þ b2 qn Þð1 þ pn Þqn þ cpn ða1 þ a2 qn Þ
¼
;
ð1 þ ð1 cÞpn Þða1 þ a2 qn Þ
b2 yn eðk3 un þk4 v n Þ b2
¼ p :
b1 xn eðk3 un þk4 v n Þ b1 n
cp a1 þ a2 q k1 ð1þp Þxn
xnþ1 ¼ 1 un e
;
1 þ p
1 þ q
unþ1 ¼ b1 xn ek2 ð1þp
ð3:25Þ
Theorem 3.4. The unique positive fixed point ðp ; q Þ ¼ ðb1 =b2 B; BÞ of
system (3.25) is globally asymptotically stable if condition (3.20) is
satisfied, and is unstable if condition (3.21) is satisfied.
ð3:27Þ
n
and
1
b1 þ b2 q
cp a1 þ a2 q
þ
v n ek1 1þp yn ;
1þq
1 þ p ð1 þ q Þq
1
b
¼ 2 yn ek2 1þp yn :
p
ynþ1 ¼
v nþ1
We have the following results for system (3.25).
Þx
ð3:28Þ
Systems (3.27) and (3.28) are the same type as system (3.1). Directly from Theorem 3.1, the positive fixed point of system (3.27) is
asymptotically stable if
k1 < k2 < 2
k1 þ k2
;
lnðr 1 Þ
where
cp a1 þ a2 q
r1 ¼ b1 1 ;
1þp
1 þ q
Proof. Define functions
1 þ pn
f1 ðpn Þ :¼
;
1 þ ð1 cÞpn
ðb1 þ b2 qn Þqn
:
gðqn Þ :¼
a1 þ a2 qn
f 2 ðpn Þ :¼
cpn
1 þ ð1 cÞpn
;
^ . From (3.23), we also have
which is indeed equal to q
Then they are increasing functions of pn and qn , respectively. It follows from
pnþ1 p ¼ f 1 ðpn Þgðqn Þ f1 ðp Þgðq Þ þ f2 ðpn Þ f2 ðp Þ
> 0; if p0 > p and q0 > q ;
< 0; if p0 < p and q0 < q ;
that pn > p and qn > q , if p0 > p and q0 > q , and pn < p and
qn < q , if p0 < p and q0 < q , for all n P 1.
Similarly as in Section 2.2, we have
1 þ pn
b1 þ b2 qn
qn ð1 cÞpn
1 þ ð1 cÞpn a1 þ a2 qn
ð1 þ pn Þqn
ðb ðb1 þ b2 qn Þ
¼
b2 ð1 þ ð1 cÞpn Þða1 þ a2 qn Þ 2
b1 ð1 cÞða1 þ a2 qn ÞÞ
ð1 þ pn Þqn
ðb b2 b1 ð1 cÞa2 Þ
¼
b2
ð1 þ ð1 cÞpn Þða1 þ a2 qn
Þ 2
b a1 ð1 cÞ b2 b1
qn 1
b2 b2 b1 a2 ð1 cÞ b1 a2 ð1 þ pn Þqn
b2 b2
ð1 cÞ ðqn q Þ
¼
b2 ð1 þ ð1 cÞpn Þða1 þ a2 qn Þ b1 a2
pnþ1 pn ¼
ð3:26Þ
Hence we have the following stability results for the positive
fixed point of system (3.8).
Theorem 3.5. Suppose the unique positive fixed point ðx ; u ; y ; v Þ
of system (3.8) exists. Then it is locally asymptotically stable if
condition (3.20) holds and
k1 < k2 < 2
k1 þ k2
;
^Þ
lnðq
ð3:29Þ
^ is defined in (3.18). The positive fixed point is unstable if
where q
k1 > k2 ;
k2 > 2
k1 þ k2
;
^Þ
lnðq
ð3:30Þ
or condition (3.21) holds.
It is clear that if b1 ¼ b2 , the dynamics of system (3.8) are the
same as those of system (2.3). We then notice, from Theorems
3.2, 3.3, and 3.5, that if b2 < b1 , increasing b1 is stabilizing the
boundary fixed point E1 , and if b1 < b2 , increasing b2 is stabilizing
the boundary fixed point E2 . We provide two examples to verify the
fact as follows.
Example 4. We use the following parameters:
a1 ¼ 5; a2 ¼ 11; b1 ¼ 1; b2 ¼ 3; c ¼ 0:65; k1 ¼ 0:2 k2 ¼ 0:4;
and
qnþ1 qn ¼
b2 b1 þ b2 q
cp a1 þ a2 q
^:
¼ r1 ¼ q
þ
p
1 þ q
1 þ p ð1 þ q Þq
b2 p
pn :
b1 nþ1
Suppose condition (3.20) holds. Then, sequences fpn g and fqn g are
both increasing if p0 < p and q0 < q , and are both decreasing if
p0 > p and q0 > q . Hence, ðpn ; qn Þ approaches ðp ; q Þ globally, as
n ! 1.
On the other hand, if condition (3.21) holds, then fpn g and fqn g
are both decreasing, provided p0 < p and q0 < q , and increasing,
provided p0 > p and q0 > q . Hence the positive fixed point is
unstable. h
We then investigate the stability of the positive fixed point of
system (3.8) as follows.
As in Section 2.2, we let the positive fixed point ðp ; q Þ of system (3.25) be globally asymptotically stable. Then the x-limit set
of solutions of system (3.8) with x0 > 0; u0 > 0; y0 > 0, and
v 0 > 0, lies in ðy; v Þ ¼ ðp x; q uÞ. By setting yn ¼ p xn and
v n ¼ q un , system (3.25) becomes the following two uncoupled
systems:
ð3:31Þ
which are the same as in Example 1 while no stage structure is included. As b1 ¼ 0:6 and b2 ¼ 0:7, both net reproductive numbers
r0i > 1; i ¼ 1; 2. Conditions (3.9), (3.13), and (3.16) are all satisfied.
Boundary fixed point E1 ¼ ð1:8310; 0:5282; 0; 0Þ is asymptotically
stable and E2 ¼ ð0; 0; 1:2366; 0:4524Þ is unstable. No positive foxed
point exists. The dynamics are similar as those in Example 1. Since
b1 < b2 , increasing b2 is stabilizing E2 . With the same b1 ¼ 0:6 but b2
is increased to 0:8, conditions (3.9), (3.13), and (3.15) are satisfied.
Now both E1 and E2 ¼ ð0; 0; 1:2366; 0:4524Þ are locally asymptotically stable. A positive fixed point exists, but is unstable. Solutions
approach either E1 or E2 , depending on their initial values, as shown
in Fig. 4.
Example 5. We use the following parameters:
a1 ¼ 5; a2 ¼ 8; b1 ¼ 4; b2 ¼ 3; c ¼ 0:65; k1 ¼ 0:2 k2 ¼ 0:4
ð3:32Þ
43
J. Li / Mathematical Biosciences 240 (2012) 35–44
3
2.5
wild larvae
transgenic larvae
wild larvae
transgenic larvae
2.5
Mosquito Populations
Mosquito Populations
2
2
1.5
1
1
0.5
0.5
0
1.5
10
20
30
40
50
60
70
0
80
10
20
30
Time n
40
50
60
70
80
Time n
Fig. 4. Use the parameters in 3.31. For b1 ¼ 0:6 and b2 ¼ 0:8, both boundary fixed points E1 and E2 are locally asymptotically stable. The solution with initial value
ð1; 0:5; 0:1; 0:1Þ approaches E1 , as shown in the left figure. The solution with initial value ð0:2; 0:01; 1:5; 0:5Þ approaches E2 , as shown in the right figure.
1.6
1.6
wild larvae
transgenic larvae
1.4
1.2
Mosquito Populations
Mosquito Populations
1.2
1
0.8
0.6
1
0.8
0.6
0.4
0.4
0.2
0.2
0
wild larvae
transgenic larvae
1.4
10
20
30
40
50
60
70
80
Time n
0
10
20
30
40
50
60
70
80
Time n
Fig. 5. Use the parameters in 3.32 and fix b2 ¼ 0:5. For b2 ¼ 0:51, boundary fixed point E1 is unstable and E2 is asymptotically stable. No positive foxed point exists. The
dynamics are shown in the left figure. By increasing b2 to b2 ¼ 0:7, both boundary fixed points become unstable and a locally asymptotically stable positive fixed point
appears. The dynamics are shown in the right figure.
and fix b2 ¼ 0:5 As b1 ¼ 0:51, conditions (3.9), (3.14), and (3.15) are
all satisfied. Boundary fixed point E1 ¼ ð1:5602; 0:4263; 0; 0Þ is
unstable and E2 ¼ ð0; 0; 0:6758; 0:2630Þ is asymptotically stable.
No positive foxed point exists. Since b2 < b1 , increasing b1 is destabilizing E2 . Fix b2 ¼ 0:5 but increase b1 to b1 ¼ 0:7. Both E1 and E2
are unstable and a locally asymptotically stable positive fixed point
ð0:4779; 0:1761; 1:1267; 0:2965Þ appears. The dynamics are shown
in Fig. 5.
4. Concluding remarks
We have formulated two simple discrete-time models to study
the dynamics of the interacting wild mosquitoes and mosquitoes
carrying genetically-modified bacteria in this preliminary research.
We first assume that the both mosquito populations are homogeneous without distinguishing their metamorphic stages and formulate model system (2.3) in Section 2. We then include the
metamorphic stages of mosquitoes but combine the three aquatic
metamorphic stages as one group to formulate stage-structured
model system (3.8). For both of the model systems, we fully investigated their dynamics and obtained conditions for the existence of
all possible fixed points and conditions for the stability of the fixed
points. We assume that the net reproductive numbers for the two
mosquito populations are greater than one so that the boundary
fixed points, E1 and E2 for (2.3) and E01 and E02 for (3.8), always exist. Under such a hypothesis, we showed that a unique positive
fixed point exists if and only if the two boundary fixed points are
both stable or unstable. If both of the boundary fixed points are locally asymptotically stable, the positive fixed point may exist but
must be unstable. If both of the boundary fixed points are unstable,
the positive fixed point is asymptotically stable provided such
additional conditions as q < e2 for model (2.3), and (3.29) for model (3.8), are satisfied. Keep the both boundary fixed points unstable.
A period-doubling bifurcation may appear as parameters vary.
Example 3 demonstrates such dynamical features by using c as a
bifurcation parameter.
The transgenic bacteria can be transmitted vertically and horizontally. The vertical transmission is determined by the matings
between the wild and transgenic mosquitoes, described by ai and
bi ; i ¼ 1; 2, and the horizontal transmission is described by c. The
mixing of these parameters determines the dynamics of the model
^,
systems. In particular, q, defined in (2.17), for model (2.3) and q
defined in (3.18), for model (3.8), combine all of these parameters,
characterize the interaction, and establish bases for the asymptotic
dynamics of the model systems. It affects the two boundary fixed
points, but determines more the existence and stability of the positive fixed point – the coexistence of the two types of mosquitoes.
44
J. Li / Mathematical Biosciences 240 (2012) 35–44
While the vertical transmission plays an important role in changing the wild mosquitoes to carrying transgenic bacteria, it seems
that the horizontal transmission makes a clear picture that it can
drive the wild mosquitoes to extinct as c becomes sufficiently
large, as shown in Example 3.
While the qualitative behavior of the stage-structured model
(3.8) is similar to that of the homogeneous model (2.3), the characters of the adults can also change the dynamics of the interactive
populations. As is shown in Examples 4 and 5, the death-adjusted
emergence rates, bi ; i ¼ 1; 2, can reverse the stability of the boundary and positive fixed points. In addition, the carrying capacity
parameter for adults, k2 can also change the stability of the fixed
points.
We realize that we have simplified our model formulation by
assuming that the mosquitoes carrying transgenic bacteria do not
have fitness difference from the wild mosquitoes, which makes
our mathematical analysis more tractable. Notice that the paratransgenic approach employed to controlling malaria transmission
is still in its early stages [17,18] and new developments are ongoing.
More data will be available and may propose possible fitness differences for mosquitoes carrying transgenic bacteria. Further analysis
for the models with difference death rates and carrying capacity
parameters therefore is needed. Moreover, we have assumed the
Ricker-type nonlinearity in the models in our study. It is possible
that different types of nonlinearity would be more appropriate for
such modeling. These projects will be in our future research.
References
[1] Iliano V. Coutinho-Abreu, Kun Yan Zhu, Marcelo Ramalho-Ortigao,
Transgenesis and paratransgenesis to control insect-borne diseases: current
status and future challenges, Parasitol. Int. 59 (2010) 1.
[2] J.M. Cushing, An Introduction to Structured Population Dynamics, SIAM,
Philadelphia, 1998.
[3] C. Dye, Intraspecific competition amongst larval aedes aegypti: food
exploitation or chemical interference, Ecol. Entom. 7 (1982) 39.
[4] Guido Favia et al., Facing malaria parasite with mosquito symbionts, in:
Omolade O. Okwa (Ed.), Malaria Parasites, InTechOpen, 2012, pp. 57–70.
[5] O. Galor, Discrete Dynamical Systems, Springer-Verlag, Berlin, 2007.
[6] A.K. Ghosh, P.E.M. Ribolla, M. Jacobs-Lorena, Targeting plasmodium ligands on
mosquito salivary glands and midgut with a phage display peptide library,
Proc. Natl. Acad. Sci. USA 98 (2001) 13278.
[7] R.M. Gleiser, J. Urrutia, D.E. Gorla, Effects of crowding on populations of aedes
albifasciatus larvae under laboratory conditions, Entomologia Experimentalis
et Applicata 95 (2000) 135.
[8] J. Ito, A. Ghosh, L.A. Moreira, E.A. Wilmmer, M. Jacobs-Lorena, Transgenic
anopheline mosquitoes impaired in transmission of a malaria parasite, Nature
417 (2002) 452.
[9] Jia Li, Simple mathematical models for interacting wild and transgenic
mosquito populations, Math. Biosci. 189 (2004) 39.
[10] Jia Li, Malaria model with stage-structured mosquitoes, Math. Biol. Eng. 8
(2011) 753.
[11] Jia Li, Simple discrete-time malarial models, J. Diff. Equat. Appl. (2012) 1. iFirst
article.
[12] G.J. Lycett, F.C. Kafatos, Anti-malarial mosquitoes?, Nature 417 (2002) 387.
[13] Omolade O. Okwa (Ed.), Malaria Parasites, InTechOpen, 2012.
http://
www.intechopen.com/books/malaria-parasites.
[14] M. Otero, H.G. Solari, N. Schweigmann, A stochastic population dynamics
model for Aedes aegypti: formulation and application to a city with temperate
climate, Bull. Math. Biol. 68 (2006) 1945.
[15] Paratransgenesis,
Wikipedia,
2010.
<http://en.wikipedia.org/wiki/
Paratransgenesis>.
[16] M.H. Reiskind, L.P. Lounibos, Effects of intraspecific larval competition on adult
longevity in the mosquitoes Aedes aegypti and Aedes albopictus, Med. Vet.
Entomol. 23 (2009) 62.
[17] M.A. Riehle, M. Jacobs-Lorena, Using bacteria to express and display antiparasite molecules in mosquitoes: current and future strategies, Insect
Biochem. Mol. Biol. 35 (2005) 699.
[18] M.A. Riehle, C.K. Moreira, D. Lampe, C. Lauzon, M. Jacobs-Lorena, Using
bacteria to express and display anti-plasmodium molecules in the mosquitoes
midgut, Inter. J. Parasitol. 37 (2007) 595.
[19] R.J. Sacker, H. von Bremen, Global asymptotic stability in the Jia Li model for
genetically altered mosquitos, in: Proceedings of 9th International Conference
on Difference Equations and Applications, World Scientific, Los Angeles, 2005,
pp. 87–100.
[20] G.S. Sayler, S. Ripp, Field applications of genetically engineered
microorganisms for bioremediation processes, Curr. Opin. Biotechnol. 11
(2000) 286.
[21] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,
second ed., Springer, New York, 2003.