
Citation: Yang Li, Jia Li. Stage-structured discrete-time models for interacting wild and sterile mosquitoes with beverton-holt survivability[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 572-602. doi: 10.3934/mbe.2019028
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To prevent mosquito-borne diseases, the sterile insect technique (SIT) has been applied to reduce or eradicate the wild mosquitoes and has shown promising results in laboratory studies [1,8,38], but predicting the impact of releasing sterile mosquitoes into the field of wild mosquito populations is still challenging. Mathematical models have proven useful in gaining insights into interactive dynamics of wild and sterile mosquito populations, and there are models in the literature for such studies [4,5,6,7,13,17,18,28,29]. However, most of them assume homogeneous mosquito populations without distinguishing the metamorphic stages of mosquitoes.
Mosquitoes undergo complete metamorphosis, going through four distinct stages of development during a lifetime, egg, pupae, larva, and adult. After a female mosquito drinking blood, she can lay from 100 to 300 eggs at a time in standing water or very slow-moving water. In her lifetime, she can produce from 1000 to 3000 eggs [34]. Within a week, the eggs hatch into larvae, which will use their tubes to breathe air by poking above the surface of the water. Larvae eat a bit of floating organic matter and each other. Larvae molt four times totally as they grow and after the fourth molt, they are called pupae. Pupae also live near the surface of water and breathe through two horn-like tubes (called siphons) on their back. But pupae do not eat. When the skin splits after a few days from a pupae, an adult mosquito emerges. The adults live for only a few weeks and a full life-cycle of a mosquito takes about a month [2,9].
To have more realistic modeling of mosquitoes, we need to include stage structure since the different stages have different responses to environment and regulating factors to the population [36]. While the interspecific competition and predation are rare events and could be discounted as major causes of larval mortality, the intraspecific competition could represent a major density dependent source. Thus the effect of crowding could be an important factor in the population dynamics of mosquitoes [15,19,35].
Moreover, since the first three stages in a mosquito's life time are aquatic and the major density dependent source comes from the larval stage, following the line in [24,26], we group the three aquatic stages of mosquitoes into one class and divide the whole mosquito population into only two classes to keep our mathematical modeling as simple as possible. We call the class consisting of the first three stages larvae and the other class adult. We assume that the density dependence is based on larvae not the adults. We still simplify our models such that no male and female individuals are distinguished.
For the density-dependent mortality, most existing works in the literature, including our previous studies, have assumed the Ricker-type nonlinearity [24,25,27,29,31]. The dynamics of the Ricker-type nonlinearity are complex, causing, e.g. period-doubling bifurcations even without any other interactions. As the sterile mosquitoes are included, the model dynamics become more complex and it is not clear whether the complexity is from the baseline model without interaction already or from the interaction. Thus, we assume that the mosquito population follow the nonlinearity of Beverton-Holt type [11,12] in this paper.
We first investigate the dynamics of the general stage-structured model with no releases of sterile mosquitoes in Section 2. We then introduce sterile mosquitoes into the model and formulate the interactive stage-structured models in Section 3. Similar to those in [4,5,13,29], we consider three strategies of releases. The case of constant releases is studied in Section 3.1. Complete mathematical analysis for the model dynamics is given. We then formulate a model for the case where the number of sterile mosquito releases is proportional to the wild mosquito population size in Section 3.2. Mathematical analysis and numerical simulations are provided to demonstrate the complexity of the model dynamics. Considering different sizes of wild mosquito population, we consider a different releasing strategy as in [30,31] in Section 3.3, where the releases of sterile mosquitoes are proportional to the wild mosquitoes size when the wild mosquitoes size is small but is saturated and approaches a constant as the wild mosquitoes size is sufficiently large. We provide complete mathematical analysis for the model dynamics. We finally provide a brief discussion on our findings, particularly on the impact of the three different strategies on the mosquito control measures in Section 4.
We first consider a stage-structured model of wild mosquitoes in the absence of sterile mosquitoes. Let
xn+1=f(xn,yn)yns1(xn,yn),yn+1=g(xn,yn)xns2(xn,yn), |
where
We assume a constant birth rate and denote it as
We assume that food is abundant for mosquito adults so that the adults survival rate is constant, denoted as
xn+1=ayn1+η1xn,yn+1=γxn1+η2xn, | (2.1) |
where we merge
The origin
The
η1η2x2+(η1+η2)x+1−r0=0. | (2.2) |
Clearly, the quadratic equation (2.2) has no positive root and hence system (2.1) has no positive fixed point if
ˉx=−(η1+η2)+√Δ2η1η2,ˉy=γ(−(η1+η2)+√Δ)2η1η2+η2(−(η1+η2)+√Δ), | (2.3) |
where
The Jacobian matrix of system (2.1) at
J1:=(−η1ˉx1+η1ˉxa1+η1ˉxγ(1+η2ˉx)20). | (2.4) |
Fixed point
|trJ1|<1+detJ1<2 |
η1ˉx1+η1ˉx<1−11+η2ˉx<2. |
Thus fixed point
η1<η2, |
and is unstable if
η1>η2. |
System (2.1) may have periodic cycles of different periods. We first consider 2-cycles with
It follows from system (2.1) that
xn+2=ayn+11+η1xn+1=aγxn(1+η2xn)(1+η1xn+1),yn+2=γxn+11+η2xn+1=aγyn(1+η1xn)(1+η2xn+1). | (2.5) |
Then there exists a positive nontrivial 2-cycle if and only if
(1+η2xn)(1+η1xn+1)=(1+η1xn)(1+η2xn+1)=aγ, |
which implies
(η2−η1)(xn−xn+1)=0, |
for all
If
Synchronous 2-cycles can be found by looking for nontrivial equilibria of system (2.5) which have one component equal to zero. It follows from
xn+2=aγxn(1+η2xn)(1+η1xn+1) |
that
x∗=aγx∗(1+η2x∗)(1+η1⋅0), |
which yields
x∗=r0−1η2. | (2.6) |
Then it follows from
xn+2=ayn+11+η1xn+1, |
that
x∗=ay∗1+η1⋅0, |
which leads to
y∗=x∗a=aγ−1aη2. | (2.7) |
Thus the system will undergo a unique synchronous 2-cycle as
(x∗0)→(0y∗)→(x∗0)→(0y∗)→⋯, |
where
At the synchronous 2-cycle, the Jacobian matrix is
J2:=(aγ(1+η2x∗)2−aη1x∗1+η1x∗0aγ1+η1x∗)=(1r0−aη1x∗1+η1x∗0r01+η1x∗). |
Then, it follows from
trJ2=1r0+r01+η1x∗,detJ2=11+η1x∗, |
that the unique synchronous 2-cycle is locally asymptotically stable if
1r0+r01+η1x∗<1+11+η1x∗, |
that is,
η1>η2. |
We now assume
ˉx0=√r0−1η,ˉy0=√γ(√r0−1)√aη, | (2.8) |
whose stability is determined by the eigenvalues of the Jacobian matrix in (2.4). The eigenvalues are
λ1=−1,λ2=1√r0<1, |
which implies the possibility of bifurcated 2-cycles. We now explore the existence of such 2-cycles.
It follows from system (2.1) that
xn+2=aγxn(1+ηxn)(1+ηxn+1)=r0xn1+ηxn+aηyn,yn+2=aγyn(1+ηxn)(1+ηxn+1)=r0yn1+ηxn+aηyn. | (2.9) |
For a positive 2-cycle with initial values
1+ηx0+aηy0=r0. | (2.10) |
Thus, if we let
Any positive 2-cycle has the form
(x1y1)→(x2y2)→(x1y1)→(x2y2)→⋯ |
with
If the initial point
dk=ηxk+aηyk+1−r0√η2+(aη)2. |
Thus
dk+1−dk=η(xk+1−xk)+aη(yk+1−yk)√η2+(aη)2, |
where
xk+1−xk=ayk−xk(1+ηxk)1+ηxk,yk+1−yk=γxk−yk(1+ηxk)1+ηxk. |
Therefore,
dk+1−dk=ηxk(1+ηxk)√η2+(aη)2(r0−(1+ηxk+aηyk)), | (2.11) |
where
We use mathematical induction method to prove that
1+ηxm+1+aηym+1−r0=1+ηaym1+ηxm+aηγxm1+ηxm−r0=1+ηxm+aηym−r01+ηxm>0, |
which implies that the point
Then we have
dk+1−dk<0, |
which indicates that the distance
Similarly, if the initial point is below the line, we can show that the distance
Example 1. Given parameters
a=5,γ=0.4,η2=0.3, | (2.12) |
there exists a continuum of positive 2-cycles of system (2.1) if
We would like to point out that if we define parameter
We further prove, in Appendix 4, that other
Theorem 2.1. The trivial fixed point
Now suppose sterile mosquitoes are released into the field of wild mosquitoes. Since sterile mosquitoes do not reproduce, the birth input term will be their releases rate. Let
bn=C(Nn)aynyn+Bn=C(Nn)aynNn, |
where
xn+1=C(Nn)aynyn+Bnyn1+η1xn,yn+1=γxn1+η2xn. | (3.1) |
We first consider the case where
xn+1=aynyn+byn1+η1xn=ay2n(yn+b)(1+η1xn),yn+1=γxn1+η2xn. | (3.2) |
Clearly, the origin
x=ay2(y+b)(1+η1x),y=γx1+η2x, |
which lead to
ay(y+b)(1+η1x)γ1+η2x=1. |
Solving for
b=aγy(1+η1x)(1+η2x)−y=γx(1+η1x)(1+η2x)2(r0−(1+η1x)(1+η2x)):=γH(x), | (3.3) |
for
Clearly, there exists no positive fixed point if
Ω:={x:0<x<ˉx}. | (3.4) |
Since
H′(x)=1(1+η1x)(1+η2x)3(r0(1−η2x−η1x(1+η2x)1+η1x)−(1+η1x)(1+η2x)), | (3.5) |
we define
Since
L′(x)=r0(−η2−η1+2η1η2x+η21η2x2(1+η1x)2)<0, |
With this unique
{H′(x)>00<x<xc,H′(x)<0xc<x<ˉx. |
We define
bc:=γH(xc). | (3.6) |
Then
To investigate the existence of synchronous 2-cycles, we have such a form
(x∗0)→(0y∗)→(x∗0)→(0y∗)→⋯, |
where
x∗=ay∗2b+y∗,y∗=γx∗1+η2x∗. | (3.7) |
To solve for
aη2y∗2+(1−r0)y∗+b=0. | (3.8) |
Since
b1:=(r0−1)24aη2. | (3.9) |
Then there exists no, one synchronous 2-cycle with
y∗=r0−12aη2, |
or two synchronous 2-cycles with
y{1}∗=r0−1−√(r0−1)2−4aη2b2aη2,y{2}∗=r0−1+√(r0−1)2−4aη2b2aη2, | (3.10) |
if
Now we claim
P(x):=γH(x)−b1=Q(x)4aη2(1+η1x)(1+η2x)2, |
where
Q(x):=−Ax3−Bx2+Cx−(r0−1)2 |
with
A=4r0η1η22+η1η22(r0−1)2>0,B=4r0η2(η1+η2)+η22(r0−1)2+2η1η2(r0−1)2>0,C=4r0η2(r0−1)−2η2(r0−1)2−η1(r0−1)2. | (3.11) |
Then the function
Assume
Q′(x)=−3Ax2−2Bx+C |
that
Q(x0)=−Ax30−Bx20+Cx0−(r0−1)2, |
at
x0=√B2+3AC−B3A>0, |
where,
Using
Q(x0):=q(η1)=−A(η1)⋅x30(η1)−B(η1)⋅x20(η1)+C(η1)⋅x0(η1)−(r0−1)2, |
where
Taking the derivative of
q′(η1)=(−3A(η1)x20(η1)−2B(η1)x0(η1)+C(η1))⋅x′0(η1)+(−A′(η1)x30(η1)−B′(η1)x20(η1)+C′(η1)x0(η1))=−A′(η1)x30(η1)−B′(η1)x20(η1)+C′(η1)x0(η1) |
since
−3Ax20−2Bx0+C=Q′(x0)=0. |
Notice that
A′(η1)=4r0η22+η22(r0−1)2>0,B′(η1)=4r0η2+2η2(r0−1)2>0,C′(η1)=−(r0−1)2<0. |
Thus
Moreover, it follows from
x0(η1)|η1→0=limη1→0x0(η1)=C(0)2B(0), |
and thus
limη1→0q(η1)=−B(0)x20(0)+C(0)x0(0)−(r0−1)2=−B(0)C2(0)4B2(0)+C(0)C(0)2B(0)−(r0−1)2=C2(0)4B(0)−(r0−1)2=14B(0)(C2(0)−4B(0)(r0−1)2)=0. |
Hence
We next investigate the stability of the positive fixed points and the synchronous 2-cycles.
The Jacobian matrix evaluated at a positive fixed point has the form
J:=(−η1x1+η1xxyy+2by+byx11+η2x0). |
Since
trJ=−η1x1+η1x,detJ=−y+2b(y+b)(1+η2x), |
a positive fixed point
η1x1+η1x+y+2b(y+b)(1+η2x)<1, |
which is equivalent to
b(1+2η1x−η2x)−(η2−η1)xy<0. | (3.12) |
If
We then assume
γH(x)⋅(1+2η1x−η2x)−(η2−η1)xγx1+η2x=γx(1+η1x)(1+η2x)(r0(1+2η1x−η2x)−(1+η1x)2(1+η2x))<0. |
Define the following function
h(x):=r0(1+2η1x−η2x)−(1+η1x)2(1+η2x). |
Thus, the positive fixed point
For
(1+η1xc)(1+η2xc)=r0(1−η2xc−η1xc(1+η2xc)1+η1xc). |
Then we have
h(xc)=r0(1+2η1xc−η2xc)−(1+η1xc)r0(1−η2xc−η1xc(1+η2xc)1+η1xc)=r0(1+2η1xc−η2xc)−r0((1+η1xc)−η2xc(1+η1xc)−η1xc(1+η2xc))=r0(2η1xc+2η1η2x2c)=2r0η1xc(1+η2xc)>0, |
and thus this unique positive fixed point
For
We first consider the positive fixed point
(1+η1x1)(1+η2x1)<r0(1−η2x1−η1x1(1+η2x1)1+η1x1). |
Then we have
h(x1)>r0(1+2η1x1−η2x1)−(1+η1x1)r0(1−η2x1−η1x1(1+η2x1)1+η1x1)=r0(1+2η1x1−η2x1)−r0((1+η1x1)−η2x1(1+η1x1)−η1x1(1+η2x1))=r0(2η1x1+2η1η2x21)=2r0η1x1(1+η2x1)>0, |
and thus fixed point
Next we consider the positive fixed point
h(xs)=−η21η2x3s−(η21+2η1η2)x2s+(r0(2η1−η2)−2η1−η2)xs+(r0−1)=0. |
Notice that
h(ˉx)=r0(η1−η2)ˉx<0 |
and
Then, the positive fixed point
bs:=γH(xs), | (3.13) |
where
For the stability of the synchronous 2-cycles with components
(λ000), |
where
λ=2r0y∗(1+η2x∗)(b+y∗)−2aη2y∗2(1+η2x∗)(b+y∗)−r0y∗2(1+η2x∗)2(b+y∗)2. |
Since
λ=2r0y∗r0y∗−2aη2y∗2r0y∗−r0y∗2r20y∗2=2−2η2γy∗−1r0. |
If
λ{1}=2−r0−√(r0−1)2−4baη2r0=r0+√(r0−1)2−4baη2r0>1, |
at
λ{2}=2−r0+√(r0−1)2−4baη2r0=r0−√(r0−1)2−4baη2r0<1, |
at
We summarize our results in the following theorem and Table 1.
(PFP stands for positive fixed point and STC stands for synchronous 2-cycle.) | ||||
Two PFP | Two PFP | No PFP | ||
One stable | both unstable | |||
One unstable | ||||
Two STC | No STC | |||
One stable | ||||
One unstable | ||||
Two PFP | No PFP | |||
both unstable | ||||
Two STC | No STC | |||
One stable | ||||
One unstable |
Theorem 3.1. The trivial fixed point
We then give the following example to demonstrate the results in Theorem 3.1, but only address the case of
Example 2. Choosing the following parameters
a=25,γ=0.8,η1=0.2,η2=0.7, | (3.14) |
we have
Instead of constant releases of sterile mosquitoes, we assume that the releases are proportional to the population size of the wild mosquitoes such that the number of releases is
We assume that there is no mating difficulty for mosquitoes. Then the model dynamics are governed by the following system:
xn+1=aynyn+byn⋅yn1+η1xn=ayn(1+b)(1+η1xn)=ˉayn1+η1xn,yn+1=γxn1+η2xn, | (3.15) |
where
Mathematically, the system is the same as system (2.1). It follows from Theorem 2.1 that if
bc:=aγ−1=r0−1. |
Then the trivial fixed point is globally asymptotically stable if
If
x∗=−(η1+η2)+√Δ2η1η2,y∗=γx∗1+η2x∗=γ(−(η1+η2)+√Δ)2η1η2+η2(−(η1+η2)+√Δ), | (3.16) |
where
(x∗0)→(0y∗)→(x∗0)→(0y∗)→⋯ | (3.17) |
with
Theorem 3.2. The trivial fixed point
It follows from Theorem 3.2 that if
Example 3. Given the parameters
a=5,γ=0.8, | (3.18) |
we have threshold value
Compared to the case of constant releases, the proportional releases may be a good strategy when the size of the wild mosquito population is small since the size of releases is also small. However, if the wild mosquito population size is significantly large, the release size would be large as well, which may exceed our affordability. Then, as in [13], we consider a different strategy where the number of releases is proportional to the wild adult mosquito population size when it is small, but it is saturated and approaches a constant when the wild adult mosquito population size is sufficiently large. To this end, we let the releases be of Holling-Ⅱ type [21] such that
xn+1=aynyn+byn1+yn⋅yn1+η1xn=ayn(1+yn)(1+b+yn)(1+η1xn),yn+1=γxn1+η2xn. | (3.19) |
We assume
A positive fixed point
a(1+y)(1+b+y)(1+η1x)⋅γ1+η2x=1, |
that is,
b=(1+γx1+η2x)(r0(1+η1x)(1+η2x)−1)=:G(x). | (3.20) |
Let
G′(x)=A1x3+A2x2+A3x+A4(1+η1x)2(1+η2x)3, | (3.21) |
where
A1=−η21η2γ<0,A2=−2η1η22r0−2η1η2γr0−η21γ−2η1η2γ<0,A3=−3η1η2r0−η22r0−η2γr0−2η1γ−η2γ<0,A4=γ((aγ−aη2)−(1+aη1)), |
that if
Next we assume
The existence of the positive fixed points, based on the release value of the sterile mosquitoes
The system may also have positive cycles of different periods. We only consider synchronous 2-cycles with components
(x∗0)→(0y∗)→(x∗0)→(0y∗)→⋯ |
that
x∗=ay∗(1+y∗)1+b+y∗,y∗=γx∗1+η2x∗, |
where
aη2y∗2−(b0−aη2)y∗+b−b0=0. | (3.22) |
If
b1:=(b0+aη2)24aη2≥b0. |
Then there exists no synchronous 2-cycle if
We illustrate our results in Table 2.
(PFP stands for positive fixed point and STC stands for synchronous 2-cycle.) | ||||
One PFP | No PFP | |||
One STC | No STC | |||
One PFP | No PFP | |||
One STC | Two STC | No STC | ||
One PFP | Two PFP | No PFP | ||
One STC | Two STC | No STC |
We next investigate the stability of the positive fixed points. The Jacobian matrix at a positive fixed point
ˉJ:=(−η1x1+η1x(b(1+2y)+(y+1)2)(1+η2x)γ(1+y)(1+y+b)γ(1+η2x)20). |
Since
trˉJ=−η1x1+η1x,detˉJ=−b(1+2y)+(y+1)2(1+η2x)(1+y+b)(1+y), |
b(1+2y)+(y+1)2(1+η2x)(1+y+b)(1+y)<1−η1x1+η1x, |
that is,
[(1+η1x)(1+2y)−(1+η2x)(1+y)]b<(η2−η1)x(1+y)2. | (3.23) |
Substituting
Φ(x):=[(1+η1x)(1+2y)−(1+η2x)(1+y)]b−(η2−η1)x(1+y)2=γx(η2x+γx+1)(1+η1x)(1+η2x)3⋅ϕ(x), |
where
ϕ(x)=−η21η2x3−(η21+2η1η2)x2+(η1(2b0+aη2)−η2(r0+1+aη2))x+b0−aη2+aη1. |
Thus the positive fixed point is locally asymptotically stable if
It is clear from (3.23) that if
η1(2b0+aη2)−η2(r0+1+aη2)<η1(2b0+aη2)−η2(r0+1+b0+aη1)=2b0η1−2(b0+1)η2=2b0(η1−η2)−2η2<0, |
which implies that
If
bs:=G(xs), |
where
For the case of
For the case of
For the case of
To prove
dG(x):=A1x3+A2x2+A3x+A4, |
where
γϕ(x)−dG(x)=(2η1η22r0+2η1η2γr0)x2+(4η1η2r0+2η1γr0)x+2r0η1>0, |
for all
We use Table 3 to summarize our results, and give Example 4 to demonstrate the existence and stability results for the positive fixed point of system (3.19).
(PFP stands for positive fixed point and L.A.S stands for locally asymptotically stable.) | ||||
One PFP | No PFP | |||
L.A.S. | - | |||
One PFP | One PFP | No PFP | ||
L.A.S. | unstable | - | ||
One PFP | One PFP | Two PFP | No PFP | |
L.A.S. | unstable | both unstable | - | |
|
||||
One PFP | Two PFP | Two PFP | No PFP | |
L.A.S. | larger one L.A.S. | both unstable | - |
Example 4. With the following parameters,
a=2.25,γ=0.8,η1=0.1,η2=0.3, | (3.24) |
we have
Example 5. Given the following parameters,
a=2.25,γ=0.8,η1=0.1,η2=0.2, | (3.25) |
we have the threshold values
Example 6. With the following parameters,
a=2.25,γ=0.8,η1=0.02,η2=0.1, | (3.26) |
we have the threshold values
In this paper, we first formulated a discrete-time stage-structured mosquito model where the mosquito population is divided into two groups, the larvae and the adults. We assume that the survivability and progression of larvae are both of Beverton-Holt type nonlinearity. We determined the existence and stability for the positive fixed points and the synchronous 2-cycles, respectively. When the intrinsic growth rate of the population
We then introduced sterile mosquitoes in the stage-structured wild mosquito population and considered three different strategies for the releases of sterile mosquitoes in model system (3.2) where the sterile mosquitoes are released constantly, (3.15) where the releases are proportional to the size of the wild mosquitoes, and (3.19) where the releases are of Holling-Ⅱ type, respectively. We established threshold value
We note that, in the absence of sterile mosquitoes, if the density-dependent death has less effect than the density-dependent progression from the larvae, that is,
Such dynamical features are similarly carried out when the sterile mosquitoes are released constantly or proportionally. More specifically, with the constant release rate and in the case of
For any of the three release strategies, it is not surprising that the amount of releases changes the model dynamics. As small amounts of sterile mosquitoes are released, there exist stable positive fixed points or synchronous 2-cycles. When the release amount is gradually increasing greater than the stability thresholds, first positive fixed points or synchronous 2-cycles become unstable, and then all disappear leading to the extinction of wild mosquitoes.
We would also like to point out that the outcomes from the models studied in this paper seem to be similar to those with the Ricker-type nonlinearity in [27]. However, it is well known that the Beverton-Holt nonlinearity excludes the possibility of the period doubling bifurcation and chaotic feature for the models without stage structure. When stage structure is included, the relatively simple dynamics with no period doubling bifurcation and chaotic feature are carried out, which makes the analysis more tractable. Such model structure has been applied to the discrete-time malaria transmission models incorporating releases of sterile mosquitoes in [32].
The authors thanks the two anonymous reviewers for their careful reading and valuable comments and suggestions.
All authors declare no conflicts of interest in this paper.
Proof. To prove that there exist no positive
xn+3=a2γyn1+aηyn+η(r0+1)xn,yn+3=aγ2xn1+aηyn+η(r0+1)xn. | (4.1) |
Any point
1+aηy+η(r0+1)x=a2γyx. |
Plugging
x=√r0−1η, |
which is exactly the fixed point from (2.8). Thus 3-cycles do not exist and as a consequence, any
We next check for 4-cycles. If
xn+4=r20xn1+η(r0+1)xn+aη(r0+1)yn,yn+4=r20yn1+η(r0+1)xn+aη(r0+1)yn. | (4.2) |
For a point
r20=1+η(r0+1)x+aη(r0+1)y. |
Equivalently, we can write it in this form
r20−1=(r0+1)(ηx+aηy), |
that is,
r0=1+ηx+aηy. |
It is exactly the positive 2-cycle from (2.10). Thus there exist no 4-cycles, and as a consequence, there exist no
Therefore, there exist no positive
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(PFP stands for positive fixed point and STC stands for synchronous 2-cycle.) | ||||
Two PFP | Two PFP | No PFP | ||
One stable | both unstable | |||
One unstable | ||||
Two STC | No STC | |||
One stable | ||||
One unstable | ||||
Two PFP | No PFP | |||
both unstable | ||||
Two STC | No STC | |||
One stable | ||||
One unstable |
(PFP stands for positive fixed point and STC stands for synchronous 2-cycle.) | ||||
One PFP | No PFP | |||
One STC | No STC | |||
One PFP | No PFP | |||
One STC | Two STC | No STC | ||
One PFP | Two PFP | No PFP | ||
One STC | Two STC | No STC |
(PFP stands for positive fixed point and L.A.S stands for locally asymptotically stable.) | ||||
One PFP | No PFP | |||
L.A.S. | - | |||
One PFP | One PFP | No PFP | ||
L.A.S. | unstable | - | ||
One PFP | One PFP | Two PFP | No PFP | |
L.A.S. | unstable | both unstable | - | |
|
||||
One PFP | Two PFP | Two PFP | No PFP | |
L.A.S. | larger one L.A.S. | both unstable | - |
(PFP stands for positive fixed point and STC stands for synchronous 2-cycle.) | ||||
Two PFP | Two PFP | No PFP | ||
One stable | both unstable | |||
One unstable | ||||
Two STC | No STC | |||
One stable | ||||
One unstable | ||||
Two PFP | No PFP | |||
both unstable | ||||
Two STC | No STC | |||
One stable | ||||
One unstable |
(PFP stands for positive fixed point and STC stands for synchronous 2-cycle.) | ||||
One PFP | No PFP | |||
One STC | No STC | |||
One PFP | No PFP | |||
One STC | Two STC | No STC | ||
One PFP | Two PFP | No PFP | ||
One STC | Two STC | No STC |
(PFP stands for positive fixed point and L.A.S stands for locally asymptotically stable.) | ||||
One PFP | No PFP | |||
L.A.S. | - | |||
One PFP | One PFP | No PFP | ||
L.A.S. | unstable | - | ||
One PFP | One PFP | Two PFP | No PFP | |
L.A.S. | unstable | both unstable | - | |
|
||||
One PFP | Two PFP | Two PFP | No PFP | |
L.A.S. | larger one L.A.S. | both unstable | - |