Parameter | Baseline values | Minimum values | Maximum values |
r | 0.17025 | 0.1374 | 0.2031 |
β | 0.6023 | 0.5682 | 0.6364 |
m1 | 0.1997 | 0.1684 | 0.2310 |
m | 0.4196 | 0.3883 | 0.4509 |
δ | 1.5024 | 1.4324 | 1.5724 |
k2 | 10.0414 | 9.6712 | 10.4116 |
Assessment data, studies of opportunity-to-learn, and the extant beliefs of some U.S. teachers about the roles and abilities of emergent bilingual students make it necessary to consider how to improve mathematics instruction. This systematic, qualitative review of 33 studies in K-12 settings used a definition of discourse informed by systemic functional linguistics and positioning theory to answer the following question: How, if at all, does the literature suggest teachers encourage emergent bilingual students' participation in mathematics discourse and discourse communities through reflexive and interactive positioning? A coding process uncovered themes from the articles, all identified through methodical elimination and published in the last 25 years. Findings suggest that teachers can position students to partake in mathematics discourse by engaging in practices that reflect these students' abilities and strategies and show the value of their current communicative practices and out-of-school contexts.
Citation: Kanushri Wadhwa. Facilitating emergent bilingual students' participation in mathematics discourse and discourse communities through positioning[J]. STEM Education, 2025, 5(2): 291-309. doi: 10.3934/steme.2025015
[1] | Fatao Wang, Ruizhi Yang, Yining Xie, Jing Zhao . Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator. AIMS Mathematics, 2023, 8(8): 17719-17743. doi: 10.3934/math.2023905 |
[2] | Kwadwo Antwi-Fordjour, Rana D. Parshad, Hannah E. Thompson, Stephanie B. Westaway . Fear-driven extinction and (de)stabilization in a predator-prey model incorporating prey herd behavior and mutual interference. AIMS Mathematics, 2023, 8(2): 3353-3377. doi: 10.3934/math.2023173 |
[3] | Weili Kong, Yuanfu Shao . The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator. AIMS Mathematics, 2023, 8(12): 29260-29289. doi: 10.3934/math.20231498 |
[4] | Yaping Wang, Yuanfu Shao, Chuanfu Chai . Dynamics of a predator-prey model with fear effects and gestation delays. AIMS Mathematics, 2023, 8(3): 7535-7559. doi: 10.3934/math.2023378 |
[5] | Binfeng Xie, Na Zhang . Influence of fear effect on a Holling type III prey-predator system with the prey refuge. AIMS Mathematics, 2022, 7(2): 1811-1830. doi: 10.3934/math.2022104 |
[6] | Xuyang Cao, Qinglong Wang, Jie Liu . Hopf bifurcation in a predator-prey model under fuzzy parameters involving prey refuge and fear effects. AIMS Mathematics, 2024, 9(9): 23945-23970. doi: 10.3934/math.20241164 |
[7] | Jing Zhang, Shengmao Fu . Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor. AIMS Mathematics, 2024, 9(11): 32514-32551. doi: 10.3934/math.20241558 |
[8] | Weili Kong, Yuanfu Shao . Bifurcations of a Leslie-Gower predator-prey model with fear, strong Allee effect and hunting cooperation. AIMS Mathematics, 2024, 9(11): 31607-31635. doi: 10.3934/math.20241520 |
[9] | Xiaoyan Zhao, Liangru Yu, Xue-Zhi Li . Dynamics analysis of a predator-prey model incorporating fear effect in prey species. AIMS Mathematics, 2025, 10(5): 12464-12492. doi: 10.3934/math.2025563 |
[10] | Reshma K P, Ankit Kumar . Stability and bifurcation in a predator-prey system with effect of fear and additional food. AIMS Mathematics, 2024, 9(2): 4211-4240. doi: 10.3934/math.2024208 |
Assessment data, studies of opportunity-to-learn, and the extant beliefs of some U.S. teachers about the roles and abilities of emergent bilingual students make it necessary to consider how to improve mathematics instruction. This systematic, qualitative review of 33 studies in K-12 settings used a definition of discourse informed by systemic functional linguistics and positioning theory to answer the following question: How, if at all, does the literature suggest teachers encourage emergent bilingual students' participation in mathematics discourse and discourse communities through reflexive and interactive positioning? A coding process uncovered themes from the articles, all identified through methodical elimination and published in the last 25 years. Findings suggest that teachers can position students to partake in mathematics discourse by engaging in practices that reflect these students' abilities and strategies and show the value of their current communicative practices and out-of-school contexts.
The relationship between predator and prey is the main field in natural ecology, and the mathematical models describing the interaction of two populations and the corresponding behaviors of those models are also topics which lots of scholars focus on, for example, the classic predator-prey model was first proposed by Lotka [1] and Volterra [2]. Due to different ecological and environmental factors, many species are affected by diseases [3]. Thus, eco-epidemiology is also one topic of concerns for many researchers, which includes ecology and epidemiology with the main purpose of studying how diseases spread. Chattopadhyay and Arino [4] first proposed a predator-prey model with disease in prey population as follows:
{˙s=r(s+i)(1−s+ik)−bsi−ηγ1(s)y,˙i=bsi−γ(i)y−ci,˙y=(εγ(i)+ηεγ1(s)−d)y, | (1.1) |
where s and i are the number of sound prey (also named the susceptible prey) and infected prey, respectively, y is the number of predator population, γ(i) and ηγ1(s) are the predator response functions. They mainly studied the persistence and extinction of the population, the existence and stability of the equilibrium and Hopf bifurcation occurring at the positive equilibrium of system (1.1). Many scholars studied the predator-prey models with disease and time delay based on system (1.1) [5,6,7,8,9,10,11,12,13].
In the natural world, in order to avoid being caught by predators, prey often likes to form a herd for life. This herd can be used as a protective shield against predators, because predators cannot easily attack the prey in the herd. For the predator population, it is obviously easy to catch the prey who does not form a herd. Let M(t) be the density of the population that forms the herd at time t and occupies area A, the population density of the individuals staying outside the herd is proportional to the length of the herd at time t, that is, its length is proportional to √A. Because M is distributed on a flat two-dimensional area, √M is an individual at the edge of the patch. At present, many scholars have studied the population with the herd behavior [14,15,16,17,18]. Recently, Saha and Samanta [19] proposed a predator-prey system with herd behavior and disease and incorporating prey refuge,
{dXS(T)dT=rXS(1−XSK)(XSK0−1)−(B+β1)XSXI−θ1√XSY1+Thθ1√XS,dXI(T)dT=β1XSXI−˜λ(1−m)XIY−δ1XI,dY(T)dT=cθ1√XSY1+Thθ1√XS−ξ˜λ(1−m)XIY−η1Y, | (1.2) |
where XS(T) and XI(T) are the density of susceptible prey and infected prey at time T, respectively, Y(T) is the density of predator at time T. K0 is the Allee threshold of susceptible prey population in absence of predator, B is the susceptible prey feels the population pressure of both susceptible and infected prey, (1−m)XI(m∈[0,1)) is the number of infected prey that can be caught by predator. The authors obtained some dynamical behaviors, such as the boundedness and positivity of the solution, the local stability of the equilibrium.
In addition, in the biological system, the fear effect is a universal phenomenon. Almost every creature can feel the crisis of being captured for prey and make a difference. For example, when the prey feel the risk of predation, they may abandon their previous high-risk predation habits, and re-choose a relatively low-risk predation habit, which deplete the energy of the prey[20]. And more and more scholars have done a lot of research on the fear effect between predator and prey [21,22,23,24,25,26]. Furthermore, due to the inherent fear of prey on predators, we need to take certain protection measures for the prey species to prevent all prey from being caught by the predators. Recently, some scholars have studied the predator-prey model with prey refuge, and showed that the use of refuge has a positive effect on the prey [27,28,29,30,31,32]. For example, Zhang et al. [30] studied a predator-prey model with fear effect and incorporating prey refuge
{dxdt=αx1+Ky−bx2−β(1−m)xy1+a(1−m)x,dydt=−γy+cβ(1−m)xy1+a(1−m)x. | (1.3) |
Where x and y are the density of prey and predator, respectively, K is the level of fear. (1−m)x(m∈[0,1)) is the number of prey that can be caught by predator. They found that the fear effect can reduce the population density of predators and change stability of system by eliminating the existence of periodic solutions.
However, some behaviors of species do not appear immediately in natural ecology, that is, there is a time delay [33,34,35,36]. Delay differential equations can reflect reality more truly and its dynamic behavior is more complicated. It is very suitable for scholars to combine time delay with stage structure, disease and other factors to study the influence of time delay [37,38,39,40]. Furthermore, the functional response of predators to prey density is essential in predator-prey system, and it can enrich the dynamics of predator-prey systems. In ecology, many factors could affect functional responses, such as prey escape ability, predator hunting ability and the structure of the prey habitat [41]. Generally, functional responses can be divided into the following types: prey-dependent (such as Holling I-III [42]) and predator-dependent (such as Beddington-DeAngelis [43], Crowley-Martin [44]). Recently, more and more scholars have studied predator-prey system with functional response functions and time delays [31,39,45].
We assume that the number of susceptible prey is proportional to the number of existing susceptible prey, and the number of infected prey is proportional to the number of existing infected prey. Similarly, the number of predator is directly proportional to the number of existing predator. Motivated by the literatures [4,19,30], we will take fear effect, herd behavior, disease, Holling type II functional response, time delay and refuge for prey into model (1.2) as follows
{dXS(t)dt=rXS(t−τ)1+k2Y(t−τ)−(b+β)XSXI−α(1−m1)√XSY1+k1(1−m1)√XS−d1XS,dXI(t)dt=βXSXI−δ(1−m)XIY−d2XI,dY(t)dt=cα(1−m1)√XSY1+k1(1−m1)√XS−ξδ(1−m)XIY−d3Y, | (1.4) |
with the initial conditions
XS(θ)=ϕ1(θ)>0,XI(θ)=ϕ2(θ)>0,Y(θ)=ϕ3(θ)>0,θ∈[−τ,0),ϕi(0)>0,i=1,2,3, | (1.5) |
where XS(t) and XI(t) are the density of susceptible prey and infected prey at time t, respectively, Y(t) is the density of predator at time t. r is the intrinsic growth rate of susceptible prey, k2 is the level of fear of predators by the susceptible prey, b is the susceptible prey feels the population pressure of both susceptible and infected prey, β is the infection rate from susceptible prey to infected prey. α is the predator's capture rate for susceptible prey, k1 is the half-saturation constant, (1−m1)XS(m1∈[0,1)) is the number of susceptible prey that can be caught by predator. δ is the predator's capture rate for infected prey, (1−m)XI(m∈[0,1)) is the number of infected prey that can be caught by predator. c and ξ are the conversion rates of the predator, d1,d2 and d3 are the natural death rates of susceptible prey, infected prey and predator, respectively. τ is the reaction time of the susceptible prey when they feel the predation crisis. All parameters are positive constants.
The organization of this article is as follows: In Section 2, the positiveness and boundedness of system (1.4) without time delay, the basic reproduction number R0 and the existence and stability of the equilibria are given. In Section 3, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied, respectively. In Section 4, we give the global stability of the disease-free equilibrium and the positive equilibrium. The Lyapunov exponent of system (2.1) is calculated in Section 5. In Section 6, the direction and the stability of Hopf bifurcation are studied based on the center manifold theorem and the normal form theory. To support our theoretical results, some numerical simulations are given in last section.
Firstly, we give system (1.4) without time delay as follow:
{dXS(t)dt=rXS1+k2Y−(b+β)XSXI−α(1−m1)√XSY1+k1(1−m1)√XS−d1XS,dXI(t)dt=βXSXI−δ(1−m)XIY−d2XI,dY(t)dt=cα(1−m1)√XSY1+k1(1−m1)√XS−ξδ(1−m)XIY−d3Y, | (2.1) |
with the initial conditions
XS(0)>0,XI(0)>0,Y(0)>0. |
In natural ecology, positiveness means that the population can survive, and boundedness means that the resources of the population are limited. Therefore, we will discuss the positiveness and boundedness of system (2.1).
Lemma 2.1. All solutions of system (2.1) that start in R3+ remain positive for all time.
Proof. Because system (2.1) is continuous and satisfy the local Lipschitz condition on the continuous function space C, the solution (XS(t),XI(t),Y(t)) of system (2.1) exists and is unique with the positive initial conditions (XS(0),XI(0),Y(0)) on [0,ζ), where 0<ζ≤+∞. From the first equation of system (2.1), we can get
dXS(t)dt=(r1+k2Y−(b+β)XI−α(1−m1)Y√XS+k1(1−m1)XS−d1)XS, |
that is:
XS(t)=XS(0)exp{∫t0[r1+k2Y(s)−(b+β)XI(s)−α(1−m1)Y(s)√XS(s)+k1(1−m1)XS(s)−d1]ds}>0,forXS(0)>0. |
Similarly, we have that
XI(t)=XI(0)exp{∫t0[βXS(s)−δ(1−m)Y(s)−d2]ds}>0,forXI(0)>0.Y(t)=Y(0)exp{∫t0[cα(1−m1)XS(s)√XS(s)+k1(1−m1)XS(s)−ξδ(1−m)XI(s)−d3]ds}>0,forY(0)>0. |
Lemma 2.2. All solutions of system (2.1) which in Ω are uniformly bounded.
Proof. Let XS(t),XI(t),Y(t) be the solution of system (2.1) under the initial condition.
Case I: If XS(0)≤1, then XS(t)≤1.
Assuming that it is not true, then ∃ t1 and t2 such that XS(t1)=1 and XS(t)>1, ∀ t∈(t1,t2). So
XS(t)=XS(0)exp{∫t0ϕ(XS(s),XI(s),Y(s))ds},∀t∈(t1,t2), |
where ϕ(XS(s),XI(s),Y(s))=r1+k2Y(s)−(b+β)XI(s)−α(1−m1)Y(s)√XS(s)+k1(1−m1)XS(s)−d1.
Therefore
XS(t)=XS(0)exp{∫t10ϕ(XS(s),XI(s),Y(s))ds+∫tt1ϕ(XS(s),XI(s),Y(s))ds}=XS(t1)exp{∫tt1ϕ(XS(s),XI(s),Y(s))ds}. |
Since XS(t1)=1, ϕ(XS(s),XI(s),Y(s))<0. Hence, XS(t)<1, which is a contradiction.
Case II: If XS(0)>1, then limt→∞supXS(t)≤1.
Suppose it is not true, then XS(t)>1, ∀t>0 and ϕ(XS(s),XI(s),Y(s))<0. Therefore, we have
XS(t)=XS(0)exp{∫t0ϕ(XS(s),XI(s),Y(s))ds}<XS(0). |
Next, we construct a function W(t) as follows:
W(t)=PXS(t)+QXI(t)+RY(t), | (2.2) |
where P(b+β)=Qβ,P=cR. By differentiating (2.2) with respect to t, we get
dWdt=PdXSdt+QdXIdt+RdYdt=P(rXS1+k2Y−d1XS)+Q[−δ(1−m)XIY−d2XI]+R[−ξδ(1−m)XIY−d3Y]≤PrXS1+k2Y−Pd1XS−Qd2XI−Rd3Y≤PrXS−Pd1XS−Qd2XI−Rd3Y. |
Let E=rXS, then E attains its maximum value at Emax=1P. Hence
dWdt≤1−κW,κ=min{d1,d2,d3}, |
that is
W(t)≤1κ+W(XS(0),XI(0),Y(0))e−κt. |
That is limt→∞W(t)≤1κ. So, all solutions of system (2.1) will enter into the region:
Ω={(XS,XI,Y):0≤XS(t)≤1,0≤W(t)≤1κ+ϵ,ϵ>0}. |
In this subsection, we will discuss the existence and local stability of equilibria of system (2.1). By calculations, system (2.1) has four equilibria as follows.
Theorem 2.1. The trivial equilibrium E0 of system (2.1) is unstable.
Proof. In order to analyze the local stability of the trivial equilibrium E0(0,0,0), we redefine the variables XA(t) as XS(t)=X2A(t). Then system (2.1) transforms to the following form
{dXAdt=12[rXA1+k2Y−(b+β)XAXI−α(1−m1)Y1+k1(1−m1)XA−d1XA],dXIdt=βX2AXI−δ(1−m)XIY−d2XI,dYdt=cα(1−m1)XAY1+k1(1−m1)XA−ξδ(1−m)XIY−d3Y. | (2.3) |
The characteristic equation of system (2.3) at the trivial equilibrium E0 is
[λ−12(r−d1)](λ+d2)(λ+d3)=0. | (2.4) |
Therefore, Eq (2.4) has three eigenvalues are λ1=12(r−d1),λ2=−d2,λ3=−d3.
Thus, when r−d1<0, the trivial equilibrium E0 is locally asymptotically stable. But the trivial equilibrium E0 is unstable when r−d1>0. In fact, if r−d1<0, then susceptible prey population not exist, thus E0 always unstable.
The boundary predator-free equilibrium ˜E1(˜XS,˜XI,0) exists if r>d1, here ˜XS=d2β, ˜XI=r−d1b+β. Now we prove the stability of the boundary predator-free equilibrium ˜E1.
Theorem 2.2. The boundary predator-free equilibrium ˜E1 of system (2.1) is unstable.
Proof. The Jacobian matrix of system (2.1) is as follows:
J=(A11A12A13A21A22A23A31A32A33), | (2.5) |
where
A11=r1+k2Y−(b+β)XI−d1−α(1−m1)Y2√XS[1+k1(1−m1)√XS]2,A12=−(b+β)XS,A13=−α(1−m1)√XS1+k1(1−m1)√XS−rk2XS(1+k2Y)2,A21=βXI,A22=βXS−δ(1−m)Y−d2,A23=−δ(1−m)XI,A31=cα(1−m1)Y2√XS[1+k1(1−m1)√XS]2,A32=−ξδ(1−m)Y,A33=cα(1−m1)√XS1+k1(1−m1)√XS−ξδ(1−m)XI−d3. |
The characteristic equation of system (2.1) at the boundary predator-free equilibrium ˜E1 is
(λ−˜A33)(λ2−˜A12˜A21)=0. | (2.6) |
Where ˜A12=−(b+β)d2β, ˜A21=β(r−d1)b+β, ˜A33=cα(1−m1)d2√d2β+k1(1−m1)d2−ξδ(1−m)(r−d1)b+β−d3.
Therefore, the first eigenvalue of Eq (2.6) is λ1=˜A33, and the other two eigenvalues are determined by the following equation
λ2−˜A12˜A21=0. |
Thus, the boundary predator-free equilibrium ˜E1 is locally asymptotically stable if ˜A12˜A21>0 (r−d1<0) holds along with cα(1−m1)d2√d2β+k1(1−m1)d2<ξδ(1−m)(r−d1)b+β+d3, but ˜E1 is unstable if ˜A12˜A21>0 does not hold. In fact, if r−d1<0, then susceptible prey does not exist, thus ˜E1 is unstable.
We can obtain the disease-free equilibrium ¯E2(¯XS,0,¯Y), here ¯XS=d23(cα−k1d3)2(1−m1)2. Further ¯Y satisfies the following equation:
A1¯Y2+A2¯Y+A3=0, | (2.7) |
where A1=−k2α(1−m1),A2=−{α(1−m1)+k2d1[√¯XS+k1(1−m1)¯XS]}, A3=(r−d1)⋅[√¯XS+k1(1−m1)¯XS]. Let Δ=A22−4A1A3.
Theorem 2.3. The disease-free equilibria of system (2.1) are as follows.
(i) if A3>0, then system (2.1) has two disease-free equilibria ¯E21(¯XS,0,¯Y1) and ¯E22(¯XS,0,¯Y2), here ¯Y1=√Δ−A22A1,¯Y2=−√Δ−A22A1.
(ii) if Δ>0, A3<0, then system (2.1) has one disease-free equilibrium ¯E21(¯XS,0,¯Y1), here ¯Y1=√Δ−A22A1. \label{Theorem2.2.4}
In addition, the basic reproduction number is given by R0=β¯XSd2. The R0 mainly represents the number of new infective generated from a single infected individual [46]. If R0<1, then the disease dies out from the system, but it remains the endemic in the host population if R0>1. Now we prove the stability of the disease-free equilibrium ¯E21(¯XS,0,¯Y1), and can use the similar methods to obtain the local stability of other disease-free equilibria.
Theorem 2.4. The disease-free equilibrium ¯E21 is locally asymptotically stable if (H1) and (H2) hold along with β¯XS−δ(1−m)¯Y−d2<0, but ¯E21 is unstable if (H1) and (H2) does not hold.
Proof. The characteristic equation of system (2.1) at the disease-free equilibrium ¯E21 is
(λ−J22)[λ2−(J11+J33)λ+J11J33−J13J31]=0, | (2.8) |
where
J11=r1+k2¯Y−d1−α(1−m1)¯Y2√¯XS[1+k1(1−m1)√¯XS]2,J13=−α(1−m1)√¯XS1+k1(1−m1)√¯XS−rk2¯XS(1+k2¯Y)2,J22=β¯XS−δ(1−m)¯Y−d2,J31=cα(1−m1)¯Y2√¯XS[1+k1(1−m1)√¯XS]2,J33=cα(1−m1)√¯XS1+k1(1−m1)√¯XS−d3. |
Therefore, the first eigenvalue of Eq (2.8) is λ1=J22=β¯XS−δ(1−m)¯Y−d2, and the other two eigenvalues are determined by the following equation
λ2−(J11+J33)λ+J11J33−J13J31=0. | (2.9) |
All eigenvalues of Eq (2.9) have negative real parts if
(H1): r1+k2¯Y<d1+(1−m1)2(cα−d3k1)3¯Y2d3c2α,
(H2): cα(1−m1)¯Y2√¯XS[1+k1(1−m1)√¯XS]2(−d3c−rk2¯XS(1+k2¯Y)2)<0
holds. Thus the disease-free equilibrium ¯E21 is locally asymptotically stable if assumptions (H1) and (H2) hold along with β¯XS−δ(1−m)¯Y−d2<0, but ¯E21 is unstable if assumptions (H1) and (H2) does not hold.
Theorem 2.5. The positive equilibrium E∗(X∗S,X∗I,Y∗) of system (2.1) exists if the assumption (H3) is true.
Proof. We assume that E∗(X∗S,X∗I,Y∗) is a positive equilibrium of system (2.1), then X∗S,X∗I and Y∗ satisfy the following equations
{r1+k2Y∗−(b+β)X∗I−α(1−m1)Y∗√X∗S+k1(1−m1)X∗S−d1=0,βX∗S−δ(1−m)Y∗−d2=0,cα(1−m1)X∗S√X∗S+k1(1−m1)X∗S−ξδ(1−m)X∗I−d3=0. | (2.10) |
According to Eq (2.10), we can obtain that X∗I=1ξδ(1−m)[cα(1−m1)X∗S√X∗S+k1(1−m1)X∗S−d3],Y∗=βX∗S−d2δ(1−m), and X∗S(s=√X∗S) is the positive root of the equation:
Z1s4+Z2s3+Z3s2+Z4s+Z5=0, | (2.11) |
where
Z1=−βk2(1−m1)[(b+β)(cα+k1d3)+ξβα+d1k1ξδ(1−m)],Z2=−βk2[(b+β)d3+d1ξδ(1−m)],Z3=δ(1−m1)(1−m)[d1d2ξk1k2−rξk1δ(1−m)−(cα+k1d3)(b+β)−βαξ−d1k1ξδ(1−m)]+d2(1−m)[(b+β)(cαk2+k1d3)−2αβξd2k2],Z4=ξδ2(1−m)2(r−d1)+d1ξk2d2δ(1−m)+(b+β)d3[k2d2−δ(1−m)],Z5=ξd2α(1−m1)[δ(1−m)−k2d2]. |
Thus, the positive equilibrium E∗(X∗S,X∗I,Y∗) of system (2.1) exists if the assumption
(H3):cα(1−m1)X∗S√X∗S+k1(1−m1)X∗S−d3>0,βX∗S−d2>0. |
is true.
Next, we discuss the dynamical behavior of the positive equilibrium E∗(X∗S,X∗I,Y∗) of system (2.1) with time delay.
In this section, we discuss the local stability of the system at the positive equilibrium and the existence of Hopf bifurcation of system (1.4). For convenience, let ˉXS(t)=XS(t)−X∗S,ˉXI(t)=XI(t)−X∗I,ˉY(t)=Y(t)−Y∗, then we have the following linearized system
{ˉXS(t)=a11ˉXS(t)+a12ˉXI(t)+a13ˉY(t)+b11ˉXS(t−τ)+b13ˉY(t−τ),ˉXI(t)=a21ˉXS(t)+a22ˉXI(t)+a23ˉY(t),ˉY(t)=a31ˉXS(t)+a32ˉXI(t)+a33ˉY(t), | (3.1) |
where
a11=−(b+β)X∗I−d1−α(1−m1)Y∗2√X∗S[1+k1(1−m1)√X∗S]2,a12=−(b+β)X∗S,a13=−α(1−m1)√X∗S1+k1(1−m1)√X∗S,b11=r1+k2Y∗,b13=−rk2X∗S(1+k2Y∗)2,a21=βX∗I,a22=βX∗S−δ(1−m)Y∗−d2,a23=−δ(1−m)X∗I,a31=cα(1−m1)Y∗2√X∗S[1+k1(1−m1)√X∗S]2,a32=−ξδ(1−m)Y∗,a33=cα(1−m1)√X∗S1+k1(1−m1)√X∗S−ξδ(1−m)X∗I−d3. |
Therefore, the characteristic equation corresponding to system (3.1) can be given
λ3+p2λ2+p1λ+p0+(s2λ2+s1λ+s0)e−λτ=0, | (3.2) |
where
p2=−(a11+a22+a33),p1=a22a33+a11a33+a11a22−a12a21−a13a31−a23a32,p0=a11a32a23+a12a21a33+a13a22a31−a11a22a33−a12a23a31−a21a32a13,s2=−b11,s1=a33b11+a22b11−a13b13,s0=a32a23b11+a22a31b13−a21a32b13−a22a33b11. |
In order to study the distribution of the root of Eq (3.2), we will discuss it in the following cases.
Case I: τ=0
The Eq (3.2) is reduced to
λ3+p12λ2+p11λ+p10=0, | (3.3) |
where p12=p2+s2,p11=p1+s1,p10=p0+s0.
So we know that all roots of Eq (3.3) have negative real parts if the following assumption
(H4):p12>0,p10>0,p12p11>p10,
holds. That is, the positive equilibrium E∗(X∗S,X∗I,Y∗) of system (3.1) is locally asymptotically stable if the condition (H4) is satisfied.
Case II: τ≠0
Let iω(ω>0) be a root of Eq (3.2). By separating real and imaginary parts, it follows that
{s1ωsinωτ+(s0−s2ω2)cosωτ=p2ω2−p0,s1ωcosωτ−(s0−s2ω2)sinωτ=ω3−p1ω. | (3.4) |
Adding squares of Eq (3.4), we can get
ω6+e12ω4+e11ω2+e10=0, | (3.5) |
where
e12=p22−2p1−s22,e11=p21+2(s0s2−p0p2)−s21,e10=p20−s20. |
Let ω2=v, then Eq (3.5) can be written as
v3+e12v2+e11v+e10=0. | (3.6) |
Denote f(v)=v3+e12v2+e11v+e10, then f(0)=e10,limv→+∞f(v)=+∞, f′(v)=3v2+2e12v+e11.
After discussion about the roots of Eq (3.6) by the method in [47], there are the following conditions.
(H5):e10≥0,△=e212−3e11≤0,
(H6):e10≥0,△=e212−3e11>0,v∗=−e12+√△3>0,f1(v∗)≤0,
(H7):e10<0.
Then we have the following lemma.
Lemma 3.1. For the polynomial Eq (3.6), we have the following results.
(1) If the assumption (H5) holds, then Eq (3.6) has no positive root.
(2) If the assumption (H6) or the assumption (H7) holds, then Eq (3.6) has at least one positive root.
Without loss of generality, we assume that Eq (3.6) has three positive roots defined as v1,v2 and v3. Then Eq (3.5) has three positive roots ωk=√vk, k=1,2,3. According to (3.4), if vk>0, the corresponding critical value of time delay is
τ(j)k=1ωkarccos{A40ω4k+A20ω2k+A10B40ω4k+B20ω2k+B10}+2πjωk,k=1,2,3;j=0,1,2,…, | (3.7) |
where A40=s1−p2s2,A20=p2s0+p0s2−p1s1,A10=−p0s0,B40=s22,B20=s21−2s0s2,B10=s20. Therefore, ±iωk is a pair of purely imaginary roots of Eq (3.2) with τ=τ(j)k. And let τ0=mink∈{1,2,3}{τ(0)k},ω=ωk0.
Lemma 3.2. Suppose that (H8):f′(ω2)≠0, then the following transversality condition d(Reλ)dτ|λ=iω≠0 holds.
Proof. Differentiating Eq (3.2) with respect to τ, and noticing that λ is a function of τ, then we obtain
(dλdτ)−1=−3λ2+2p2λ+p1λ(λ3+p2λ2+p1λ+p0)+2λs2+s1λ(s2λ2+s1λ+s0)−τλ. | (3.8) |
Thus, we have
Re(dλdτ)−1=Re(−3λ2+2p2λ+p1λ(λ3+p2λ2+p1λ+p0))λ=iω+Re(2λs2+s1λ(s2λ2+s1λ+s0))λ=iω=3ω4+2(p22−2p1)ω2+p21−2p0p2(ω3−p1ω2)+(p0−p2ω2)2−2s22ω2+s21−2s0s2(s2ω2−s0)2+s21ω2. | (3.9) |
From Eq (3.4) and Eq (3.9),
sign{d(Reλ)dτ}λ=iω=sign{Re(dλdτ)−1}λ=iω=3(ω2)2+2e12ω2+e11s21ω2+(s0−s2ω2)2=f′(ω2)s21ω2+(s0−s2ω2)2≠0. |
It follows that d(Reλ)dτ|λ=iω≠0 and the proof is complete.
By Lemmas 3.1-3.2 and the Hopf bifurcation theorem [48,49], we have the following results.
Theorem 3.1. For system (1.4) with τ≠0, the following results are true.
(1) If the assumption (H5) holds, then the positive equilibrium E∗(X∗S,X∗I,Y∗) is locally asymptotically stable for all τ≥0.
(2) If the assumption (H6) or the assumption (H7) and (H8) hold, then the positive equilibrium E∗(X∗S,X∗I,Y∗) is locally asymptotically stable for all τ∈[0,τ0) and is unstable for τ>τ0. Furthermore, system (1.4) undergoes a Hopf bifurcation at the positive equilibrium E∗(X∗S,X∗I,Y∗) when τ=τ0.
Remark 3.1. The biological meaning of delay τ is the reaction time of the susceptible prey when they feel the predation crisis in our system (1.4). More detailed discussions on similar time delay can be found in [45]. According to our analysis of system (1.4), we find the minimum critical value of τ is τ0. According to Theorem 3.1, system (1.4) can keep its stability if the minimum reaction time of susceptible prey does not exceed τ0, otherwise system (1.4) will change its stability.
In this section, we will discuss the global stability of equilibria which are locally asymptotically stable under some conditions.
Theorem 4.1. The disease-free equilibrium ¯E2(¯XS,0,¯Y) is globally asymptotically stable if the assumptions (H1), (H2) and (H9) are true, here
(H9):1−m1≥0,1−m≥0,(cα−k1d3)(1−m1)≤√2d3
Proof. We construct a suitable Lyapunov function
V1(XS,XI,Y)=βb+β[XS−¯XS−¯XSln(XS¯XS)]+XI+[Y−¯Y−¯Yln(Y¯Y)]. | (4.1) |
Here, V1(XS,XI,Y) is a positive definite function for all (XS,XI,Y). Then differentiating (4.1) with respect to t, we get
dV1dt=βb+β(1−¯XSXS)dXSdt+dXIdt+(1−¯YY)dYdt=βb+β(1−¯XSXS)[rXS1+k2Y−(b+β)XSXI−α(1−m1)XSY√XS+k1(1−m1)XS−d1XS]+[βXSXI−δ(1−m)XIY−d2XI]+(1−¯YY)[cα(1−m1)XS√XS+k1(1−m1)XS−ξδ(1−m)XI−d3]Y.=βb+β(XS−¯XS)XS[r(XS−¯XS)1+k2Y−d1(XS−¯XS)]−βb+β(XS−¯XS)[α(1−m1)Y√XS+k1(1−m1)XS]+[−δ(1−m)Y−d2]XI+(Y−¯Y)Y[cα(1−m1)XS(Y−¯Y)√XS+k1(1−m1)XS−ξδ(1−m)XI(Y−¯Y)−d3(Y−¯Y)] |
=−βb+β[kr(1+k2Y)(1+k2¯Y)+d1](XS−¯XS)2XS−βb+β[α(1−m1)[1+2k1(1−m1)√XS]¯Y2√XS[√XS+k1(1−m1)XS][√¯XS+k1(1−m1)¯XS]](XS−¯XS)2−[[δ(1−m)]3Y¯Y+d2]XI−[[ξδ(1−m)]3XI¯XI+d3](Y−¯Y)2Y+cα(1−m1)(12√¯XS−√¯XS)[√XS+k1(1−m1)XS][√¯XS+k1(1−m1)¯XS](Y−¯Y)2Y. |
Therefore, for all t≥T(T>0) if (H9) holds. That is, d3≥1√2(cα−k1d3)(1−m1), then dV1dt≤0. Let M1 be the largest invariant set in {(XS,XI,Y)|dV1dt=0}, we obtained that dV1dt=0 if and only if XS=¯XS,XI=¯XI=0,Y=¯Y. Thus, M1={¯E2}. By the LaSalle's invariance principle [50], the disease-free equilibrium ¯E2(¯XS,0,¯Y) is globally asymptotically stable.
Theorem 4.2. The positive equilibrium E∗(X∗S,X∗I,Y∗) is globally asymptotically stable if (H6) or (H7) and (H8), (H10) hold, here
(H10): 1−m1≥0,1−m≥0,12√X∗S−√X∗S≤0 and XS(t−τ)[1+k2Y]XS[1+k2Y(t−τ)]=1.
Proof. We construct a suitable Lyapunov function:
V2(XS,XI,Y)=βb+β[XS−X∗S−X∗Sln(XSX∗S)]+[XI−X∗I−X∗Iln(XIX∗I)]+[Y−Y∗−Y∗ln(YY∗)]. | (4.2) |
Then differentiating (4.2) with respect to t, we get
dV2dt=βb+β(1−X∗SXS)dXSdt+(1−X∗IXI)dXIdt+(1−Y∗Y)dYdt=βb+β(1−X∗SXS)[rXS(t−τ)1+k2Y(t−τ)−(b+β)XSXI−α(1−m1)XSY√XS+k1(1−m1)XS−d1XS]+(1−X∗IXI)[βXSXI−δ(1−m)XIY−d2XI]+(1−Y∗Y)[cα(1−m1)√XSY1+k1(1−m1)√XS−ξδ(1−m)XIY−d3Y].=βb+β(XS−X∗S)XS[rXS1+k2Y−α(1−m1)XSY√XS+k1(1−m1)XS−d1XS]+(XI−X∗I)XI[−δ(1−m)XIY−d2XI]+(Y−Y∗)Y[cα(1−m1)XSY√XS+k1(1−m1)XS−ξδ(1−m)XIY−d3Y].+rXS(t−τ)1+k2Y(t−τ)−X∗SXSrXS(t−τ)1+k2Y(t−τ)−rXS1+k2Y+X∗SXSrXS1+k2Y=βb+β(XS−X∗S)XS[r(XS−X∗S)1+k2Y−d1(XS−X∗S)]−βb+β(XS−X∗S)[α(1−m1)Y√XS+k1(1−m1)XS]+(XI−X∗I)XI[−δ(1−m)(XI−X∗I)Y−d2(XI−X∗I)]+(Y−Y∗)Y[cα(1−m1)√XS(Y−Y∗)1+k1(1−m1)√XS−ξδ(1−m)XI(Y−Y∗)−d3(Y−Y∗)]+rXS(t−τ)1+k2Y(t−τ)−X∗SXSrXS(t−τ)1+k2Y(t−τ)−rXS1+k2Y+X∗SXSrXS1+k2Y=−βb+β[kr(1+k2Y)(1+k2Y∗)+d1](XS−X∗S)2XS−βb+β[α(1−m1)[1+2k1(1−m1)√XS]Y∗2√XS[√XS+k1(1−m1)XS][√X∗S+k1(1−m1)X∗S]](XS−X∗S)2−[[δ(1−m)]3YY∗+d2](XI−X∗I)2XI−[[ξδ(1−m)]3XIX∗I+d3](Y−Y∗)2Y+[cα(1−m1)(12√X∗S−√X∗S)[√XS+k1(1−m1)XS][√X∗S+k1(1−m1)X∗S]](Y−Y∗)2Y+rXS(t−τ)1+k2Y(t−τ)−rXS(t−τ)1+k2Y(t−τ)X∗SXS−rXS1+k2Y+rXS1+k2YX∗SXS. |
Now let
V3=V2+r∫tt−τ{XS(s)1+k2Y(s)−X∗S1+k2Y∗−X∗S1+k2Y∗lnXS(s)[1+k2Y]XS[1+k2Y(s)]}ds. | (4.3) |
Then differentiating (4.3) with respect to t, we get
dV3dt=dV2dt+rXS1+k2Y−rXS(t−τ)1+k2Y(t−τ)−rX∗S1+k2Y∗lnXS[1+k2Y(t−τ)]XS(t−τ)[1+k2Y]=−βb+β[kr(1+k2Y)(1+k2Y∗)+d1](XS−X∗S)2XS−βb+β[α(1−m1)[1+2k1(1−m1)√XS]Y∗2√XS[√XS+k1(1−m1)XS][√X∗S+k1(1−m1)X∗S]](XS−X∗S)2−[[δ(1−m)]3YY∗+d2](XI−X∗I)2XI−[[ξδ(1−m)]3XIX∗I+d3](Y−Y∗)2Y+[cα(1−m1)(12√X∗S−√X∗S)[√XS+k1(1−m1)XS][√X∗S+k1(1−m1)X∗S]](Y−Y∗)2Y+rXS1+k2YX∗SXS−rXS(t−τ)1+k2Y(t−τ)X∗SXS−rX∗S1+k2Y∗lnXS[1+k2Y(t−τ)]XS(t−τ)[1+k2Y]. | (4.4) |
Define
dV12dt=−βb+β[kr(1+k2Y)(1+k2Y∗)+d1](XS−X∗S)2XS−βb+β[α(1−m1)[1+2k1(1−m1)√XS]Y∗2√XS[√XS+k1(1−m1)XS][√X∗S+k1(1−m1)X∗S]](XS−X∗S)2−[[δ(1−m)]3YY∗+d2](XI−X∗I)2XI−[[ξδ(1−m)]3XIX∗I+d3](Y−Y∗)2Y+[cα(1−m1)(12√X∗S−√X∗S)[√XS+k1(1−m1)XS][√X∗S+k1(1−m1)X∗S]](Y−Y∗)2Y. | (4.5) |
dV22dt=rXS1+k2YX∗SXS−rXS(t−τ)1+k2Y(t−τ)X∗SXS−rX∗S1+k2Y∗lnXS[1+k2Y(t−τ)]XS(t−τ)[1+k2Y]. | (4.6) |
For Eq (4.5), for all t≥T(T>0) and 1−m1≥0,1−m≥0,12√X∗S−√X∗S≤0, then dV12dt≤0. Then
dV3dt=dV12dt+dV22dt=dV12dt−(rXS1+k2Y)X∗SXS{XS(t−τ)[1+k2Y]XS[1+k2Y(t−τ)]−1+X∗S(1+k2Y)XS(1+k2Y∗)lnXS[1+k2Y(t−τ)]XS(t−τ)[1+k2Y]}. |
Noting that XS1+k2Y=X∗S1+k2Y∗ at E∗, we have
dV3dt=dV12dt−(rX∗S1+k2Y∗)X∗SXS{XS(t−τ)[1+k2Y]XS[1+k2Y(t−τ)]−1−ln[XS(t−τ)[1+k2Y]XS[1+k2Y(t−τ)]]}. |
When n=1 in [51], if
XS(t−τ)[1+k2Y]XS[1+k2Y(t−τ)]=1, | (4.7) |
then dV3dt≤0. Let M2 be the largest invariant set in {(XS,XI,Y)|dV3dt=0}. We obtained that dV3dt=0 if and only if XS=X∗S,XI=X∗I,Y=Y∗,XS(t−τ)[1+k2Y]XS[1+k2Y(t−τ)]=1. Thus, M2={E∗}. By the LaSalle's invariance principle [50], the positive equilibrium E∗(X∗S,X∗I,Y∗) is globally asymptotically stable.
Remark 4.1. According to Eq (4.5), if 12√X∗S−√X∗S≤0, then dV12dt≤0, here X∗S is the positive root of Eq (2.11). That is, X∗ has a certain mathematical expression, which is greater than or equal to 0.5. Furthermore, if Eq (4.7) is also true, then dV3dt≤0, which is consistent with the numerical simulation in Section 7 (see Figure 6), thus the positive equilibrium E∗ is globally asymptotically stable.
In a dynamic system, the Lyapunov exponent is an important indicator to measure the dynamics of the system [52]. It indicates the average exponential rate of convergence or divergence of the system between adjacent orbits in phase space.
Theorem 5.1. If the Lyapunov exponents are all negative, then system (2.1) is globally asymptotically stable, but if one is positive, then system (2.1) is unstable.
Proof. The Jacobian matrix corresponding of system (2.1) is Eq (2.5), on the basis of given some parameters values, initialize the Jacobian matrix (2.5), and initialize the three Lyapunov exponents to be calculated, we define a starting time:
t1=t∗+tss,I1=ws, | (5.1) |
where s is the number of steps in each evolution, ts is the time step, I1 is the number of evolutions and w is the total number of cycles. In Eq (2.5), we let:
J1=(A11A21A31),J2=(A12A22A32),J3=(A13A23A33). | (5.2) |
Next, use the Schmidt orthogonalization method to orthogonalize Eq (5.2), we can obtain that
a1=J1,a2=J2−(a′1,J2)(a′1,a1)a1,a3=J3−(a′1,J3)(a′1,a1)a1−(a′2,J3)(a′2,a2)a2. |
By calculation, we make the orthogonalized matrix A=[a1,a2,a3], and assume that the matrix after orthogonalization is:
a1=(a01a02a03),a2=(a04a05a06),a3=(a07a08a09). | (5.3) |
Then take the modulus length of the three vectors as:
mod(i)=√a′iai(i=1,2,3). | (5.4) |
Finally, the three Lyapunov exponents of system (1.4) are given:
Li=log|mod(i)|t1(i=1,2,3). | (5.5) |
Therefore, if the Lyapunov exponents are all negative, then system (2.1) is globally asymptotically stable, but if one is positive, then system (2.1) is unstable.
In this part, we will study the direction of Hopf bifurcation and the stability of bifurcating periodic solutions of system (1.4). The theoretical approach is the normal form theory and center manifold theorem [48]. Throughout this section, we assume that system (1.4) undergoes a Hopf bifurcation at τ=τ0.
Without loss of generality, let μ=τ−τ0,μ∈R,t=sτ,XS(sτ)=ˆXS(s),XI(sτ)=ˆXI(s),Y(sτ)=ˆY(s). We denote u1(t)=XS(t)−X∗S,u2(t)=XI(t)−X∗I,u3(t)=Y(t)−Y∗, then system (1.4) can be written as a functional differential equation (FDE) in C=C([−1,0],R3):
˙u(t)=Lμ(ut)+F(μ,ut), | (6.1) |
where u(t)=(u1(t),u2(t),u3(t))T∈C, ut(θ)=u(t+θ)=(u1(t+θ),u2(t+θ),u3(t+θ))T∈C and Lμ:C→R3, F:R×C→R3 are given by
Lμ(ϕ)=(τ0+μ){˜Aϕ(0)+˜Bϕ(−1)}, | (6.2) |
and
F(μ,ϕ)=(τ0+μ)(F1,F2,F3)T, |
where
ϕ(θ)=(ϕ1(θ),ϕ2(θ),ϕ3(θ))T∈C, |
˜A=(a11a12a13a21a22a23a31a32a33),˜B=(b110b13000000), |
{F1=k11ϕ21(0)+k12ϕ1(0)ϕ2(0)+k13ϕ1(0)ϕ3(0)+k14ϕ23(−τ),F2=k21ϕ1(0)ϕ2(0)+k22ϕ2(0)ϕ3(0),F3=k31ϕ1(0)ϕ3(0)+k32ϕ2(0)ϕ3(0). |
Here
k11=α(1−m1)[1√XS+4k1(1−m1)+3k21(1−m1)2√XS]4XS[1+k1(1−m1)√XS]4,k12=−(b+β),k13=−α(1−m1)2√XS[1+k1(1−m1)√XS]2,k14=−k2r[1+k2Y(t−τ)],k21=β,k22=−δ(1−m),k31=cα(1−m1)2√XS[1+k1(1−m1)√XS]2,k32=−ξδ(1−m). |
By the Riesz representation theorem [53], there exists a 3×3 matrix function η(θ,μ) for θ∈[−1,0) such that
Lμ(ϕ)=∫0−1dη(θ,μ)ϕ(θ),ϕ∈C([−1,0],R3). | (6.3) |
In fact, we can choose
η(θ,μ)={(τ0+μ)(˜A+˜B),θ=0(τ0+μ)˜B,θ∈(−τ0,0)−(τ0+μ)˜B,θ∈(−1,−τ0)0,θ=−1 | (6.4) |
for ϕ∈C1([−1,0],R3), we define
A(μ)ϕ={dϕ(θ)dθ,−1≤θ<0∫0−1dη(μ,s)ϕ(s),θ=0 |
and
Rμ(ϕ)={0,−1≤θ<0F(μ,ϕ).θ=0 |
Then, Eq (6.1) can be rewritten as
˙ut=A(μ)ut+R(μ)ut. | (6.5) |
For φ∈C1([−1,0],(R3)∗), where (R3)∗ is the three-dimensional space of row vectors, we further define the adjoint operator A∗ of A(0):
A∗φ(s)={−dφ(s)ds,0<s≤1∫0−1dηT(t,0)φ(−t),s=0 |
for ϕ∈C1([−1,0],R3) and φ∈C1([−1,0],(R3)∗), we define the bilinear form
⟨φ(s),ϕ(s)⟩=ˉφ(0)ϕ(0)−∫0−1∫θξ=0ˉφ(ξ−θ)dη(θ)ϕ(ξ)dξ, | (6.6) |
where η(θ)=η(θ,0),A=A(0) and A∗ are adjoint operators. By the discussion in Section 3, we know that ±iωτ0 are eigenvalues of A(0). Thus, they are also the eigenvalues of A∗.
We suppose that q(θ)=(1,q2,q3)Teiωτ0θ is the eigenvector of A(0) corresponding to the eigenvalue iωτ0, and q∗(s)=D(1,q∗2,q∗3)e−iωτ0s is the eigenvector of A∗ corresponding to the eigenvalue −iωτ0. By computation, we obtain
q2=(iω−a33)q3−a31a32,q3=(iω−a22)a31+a32a21(iω−a22)(iω−a33)−a23,q∗2=(−iω−a11−b11eiωτ0)−a13q∗3a21,q∗3=a21iω+a23(−iω−a11−b11eiωτ0)−a21(−a13−b13eiωτ0)a23a13−a21a33. |
Then, from Eq (6.6), we get
⟨q∗(s),q(θ)⟩=ˉq∗(0)q(0)−∫0−1∫θξ=0ˉq∗dη(θ)q(ξ)dξ=ˉD[1+ˉq∗2q2+ˉq∗3q3+τ0e−iωτ0(b13q3+b11)]. | (6.7) |
Therefore, we choose ˉD=[1+ˉq∗2q2+ˉq∗3q3+τ0e−iωτ0(b13q3+b11)]−1, such that ⟨q∗(s),q(θ)⟩=1,⟨q∗(s),ˉq(θ)⟩=0.
Next, let ut be the solution of Eq (6.5) when μ=0. We define
z(t)=⟨q∗,ut⟩,W(t,θ)=ut−zq−ˉzˉq=ut−2Rez(t)q(θ). | (6.8) |
On the center manifold C0, it comes to the conclusion that
W(t,θ)=W(z(t),ˉz(t),θ)=W20(θ)z22+W11(θ)zˉz+W02ˉz22+⋯, | (6.9) |
where z and ˉz are local coordinates for C0 in the direction of q∗ and ˉq∗. Note that W is real if ut is real, we only consider real solutions. From Eq (6.8), we get
⟨q∗,W⟩=⟨q∗,ut−zq−ˉzˉq⟩=⟨q∗,ut⟩−⟨q∗,q⟩z−⟨q∗,ˉq⟩ˉz. | (6.10) |
For a solution ut∈C0 of Eqs (6.3), (6.5), (6.6) and μ=0, we have
˙z(t)=⟨q∗,˙u(t)⟩=⟨q∗,A(0)ut+R(0)ut⟩=⟨A∗(0)q∗,ut⟩+ˉq∗(θ)F(0,ut):=iq0z(t)+ˉq∗F0(z,ˉz). | (6.11) |
What's more, Eq (6.11) can be rewritten as follows,
˙z(t)=iωz(t)+g(z,ˉz), | (6.12) |
where
g(z,ˉz)=g20z22+g11zˉz+g02ˉz22+g21z2ˉz2+⋯. | (6.13) |
It follows from Eqs (6.8) and (6.9) that
ut(θ)=W20(θ)z22+W11(θ)zˉz+W02(θ)ˉz22+qTeiωθz+q∗Teiωθˉz+⋯. | (6.14) |
By Eqs (6.13) and (6.14), we obtain the following relevant parameters, which determine the direction and stability of Hopf bifurcation:
g20=2ˉDτ0[k11+k12q2+k13q3+k14e−2iωτ0q23+ˉq∗2(k21q2+k22q2q3)+ˉq∗3(k31q3+k32q2q3)],g11=ˉDτ0{2k11+k12(ˉq2+q2)+k13(ˉq3+q3)+2k14ˉq3q3e−iωτ0+ˉq∗2[k21(ˉq2+q2)+k22(q2ˉq3+ˉq2q3)]+ˉq∗3[k31(ˉq3+q3)+k32(q2ˉq3+ˉq2q3)]},g02=2ˉDτ0[k11+k12ˉq2+k13ˉq3+k14ˉq23e−2iωτ0+ˉq∗2(k21ˉq2+k22ˉq2ˉq3)+ˉq∗3(k31ˉq3+k32ˉq2ˉq3)],g21=2ˉDτ0{k11(12W(1)20(0)+W(1)11(0)+12W(1)20(0)+W(1)11(0))+k12(12W(1)20(0)ˉq2+W(1)11(0)q2+12W(2)20(0)+W(2)11(0))+k13(12W(1)20(0)ˉq3+W(1)11(0)q3+12W(3)20(0)+W(3)11(0))+k14[e−iωτ0(12W(3)20(−τ)ˉq3+W(3)11(−τ)q3+12W(3)20(−τ)ˉq3+W(3)11(−τ)q3)]+ˉq∗2[k21(12W(1)20(0)ˉq2+W(1)11(0)q2+12W(2)20(0)+W(2)11(0))+k22(12W(2)20(0)ˉq3+W(2)11(0)q3+12W(3)20(0)ˉq2+W(3)11(0)q2)]+ˉq∗3[k31(12W(1)20(0)ˉq3+W(1)11(0)q3+12W(3)20(0)+W(3)11(0))+k32(12W(2)20(0)ˉq3+W(2)11(0)q3+12W(3)20(0)ˉq2+W(3)11(0)q2)]}, |
and
W20(θ)=ig20ωτ0q(0)eiωτ0θ+iˉg023ωτ0ˉq(0)e−iωτ0θ+E1e2iωτ0θ,W11(θ)=−ig11ωτ0q(0)eiωτ0θ+iˉg11ωτ0ˉq(0)e−iωτ0θ+E2, | (6.15) |
where E1=(E(1)1,E(2)1,E(3)1)T∈R3 and E2=(E(1)2,E(2)2,E(3)2)T∈R3 are also constant vectors and can be determined by the following equations, respectively,
(2iω−a11−b11e−2iωτ0−a12−a13−b13e−2iωτ0−a212iω−a22−a23−a31−a322iω−a33)E1=2(H1H2H3), | (6.16) |
(−a11−b11−a12−a13−b13−a21−a22−a23−a31−a32−a33)E2=(P1P2P3), | (6.17) |
where
H1=k11+k12q2+k13q3+k14e−2iωτ0q23,H2=k21q2+k22q2q3,H3=k31q3+k32q2q3,P1=2k11+k12(ˉq2+q2)+k13(ˉq3+q3)+2k14ˉq3q3e−iωτ0,P2=k21(ˉq2+q2)+k22(q2ˉq3+ˉq2q3),P3=k31(ˉq3+q3)+k32(q2ˉq3+ˉq2q3). |
Therefore, we can calculate g21 and the following values:
C1(0)=i2ωτ0(g20g11−2|g11|2−|g02|23)+g212,μ2=−Re{C1(0)}Re{λ′(τ0)},β2=2Re{C1(0)},T2=−Im{C1(0)}+μ2Im{λ′(τ0)}ωτ0, | (6.18) |
which determine the properties of bifurcating periodic solutions at τ=τ0. From the discussion above, we have the following results.
Theorem 6.1. For system (1.4), the direction of Hopf bifurcation is determined by the sign of μ2: if μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical). The stability of the bifurcating periodic solutions is determined by the sign of β2: if β2<0(β2>0), then the bifurcating periodic solutions are stable (unstable). The period of the bifurcating periodic solutions is determined by the sign of T2: if T2>0(T2<0), then the bifurcating periodic solutions increase (decrease).
In this section, we present some numerical simulations of system (1.4) and system (2.1) to support our theoretical results. First, we will conduct the sensitivity analysis of some parameters of system (2.1).
To determine the parameters that have significant impact on output variables of system (2.1), we conduct global sensitivity analysis of some parameters, many scholars have studied the sensitivity analysis of the parameters in system [54,55]. We calculate partial rank correlation coefficients (PRCCs) between the parameters r,δ,β,m1,m and k2 from system (2.1). Nonlinear and monotone relationships are observed with the input parameters of system (2.1), which is a prerequisite for computing PRCCs. Then, a total of 1000 simulations of the model per Latin hypercube sampling (LHS) were carried out using the baseline values tabulated in Table 1. According to the parameter values in Table 1, we analyze the influence of some parameters in the system on the correlation of infected prey. By sampling these parameters for 1000 times and a scatter plot with a fixed time point of 80, we obtain the sampling results in Figure 1 and scatter plot in Figure 2. Monotonical increasing (decreasing) indicates a positive (negative) correlation of the parameter with the model output. It is known from Figure 2 that several selected parameters exhibit periodic correlation, we can know that the parameters r,β and m1 show a positive correlation with the output of system, the parameters m,k2 show a little negative correlation with the output of system, and the parameter δ has no correlation with the output of system.
Parameter | Baseline values | Minimum values | Maximum values |
r | 0.17025 | 0.1374 | 0.2031 |
β | 0.6023 | 0.5682 | 0.6364 |
m1 | 0.1997 | 0.1684 | 0.2310 |
m | 0.4196 | 0.3883 | 0.4509 |
δ | 1.5024 | 1.4324 | 1.5724 |
k2 | 10.0414 | 9.6712 | 10.4116 |
In this subsection, we will analyze the stability of system (2.1) and assume that the initial values of system (2.1) are XS(0)=0.6,XI(0)=0.1,Y(0)=0.2. First, we choose a set of parameter values: r=0.17,k2=10,b=0.5,β=0.6,α=0.6,k1=1,d1=0.01,d2=0.9,δ=1.5,c=0.89,ξ=0.14,d3=0.2,m1=0.2,m=0.42. According to the Theorem 2.5, there is a positive equilibrium E∗(0.6002,0.0355,0.0971). We find that the positive equilibrium E∗ is locally asymptotically stable under this set of parameter values, the susceptible prey, infected prey and predator population have stable dynamics behavior(see Figure 3). Furthermore, according to the discussion in Section 2, we can get R0=β¯XSd2=0.37<1(because ¯XS=0.56 from Section 2.2.3), which indicates that the disease dies out from the system.
Next, for the given parameters, we get ω=0.1486 and τ0=1.059 when system (1.4) with time delay. According to the Theorem 3.1, we can obtain that the positive equilibrium E∗(0.6002,0.0355,0.0971) of the system (1.4) is locally asymptotically stable when τ=0.5<τ0=1.059, the susceptible prey, infected prey and predator population have stable dynamics behavior (see Figure 4). Then, as the value of τ gradually increases, we choose the value of τ as τ=2>τ0=1.059 and obtain that C1(0)=−4.5406−1.7383i,μ2=5.1018>0,β2=−9.0812<0,T2=−2.0684<0 by Eq (6.18). Moreover, we know that system (1.4) undergoes a supercritical Hopf bifurcation when the value of τ exceeds its threshold value τ0. In addition, the bifurcating periodic solutions of system (1.4) are stable, and the period of bifurcating periodic solutions decreases. We clearly see from Figure 5 that the positive equilibrium E∗(0.6002,0.0355,0.0971) is destabilized through a Hopf bifurcation in τ0=1.059 and a stable limit cycle appeared in phase portrait.
Then, we choose r=0.6,d2=0.5,δ=3, the initial conditions of the variables and other parameter values are the same as those given above. According to the Theorem 4.2, system (1.4) has a positive equilibrium E∗(0.9023,0.1255,0.1943). By numerical simulations, we find that the positive equilibrium E∗ is global asymptotically stable under this set of parameter values (see Figure 6). Figure 6(b) shows that the trajectory of the system (1.4) tends to the positive equilibrium E∗ for different initial values at any time.
Finally, we give some parameter values: t∗=0,ts=1×10−3,s=10,w=1×105, then the Lyapunov exponents have been derived numerically of system (2.1) for different species (see Figure 7 (a)). According to Theorem 5.1, all Lyapunov exponents are negative (L1=−0.0871,L2=−1.0740,L3=−0.8451), thus system (2.1) is globally asymptotically stable. We draw the maximum Lyapunov exponent of system (1.4) for τ=2 (see Figure 7 (b)). In the figure, positive values of the maximum Lyapunov exponent indicates that system (1.4) is unstable. Figure 7 (b) corresponding to Figure 6, the value of time delay τ can also determine the global asymptotic stability of system (1.4).
In this subsection, we will discuss how the susceptible refuge parameter m1 affects on each of the population when the positive equilibrium exists and is locally asymptotically stable. We study the influence of different parameter values of m1 on the population of prey and predator, which can be seen in Figure 8 and Figure 9.
The dynamic response diagram of system (2.1) under different parameter values m1 is shown in Figure 8, which corresponds to Figure 9. Figure 9 shows that the solution curve of state variables of system (2.1) under different values of m1. It can be seen from Figure 9 that as the value of the m1 increases, the number of susceptible prey and infected prey gradually increases, which results in an immediate reduction in the number of predators, but system (2.1) is locally asymptotically stable all time. This shows that taking refuge measures has a positive effect on the prey population, but has a negative effect on the growth of the predator population. This conclusion is consistent with the laws of nature.
In this subsection, we will discuss how the infected refuge parameter m affects on each of the population when the positive equilibrium exists and is locally asymptotically stable. We study the influence of different parameter values of m on the population of prey and predator, whose results are shown in Figure 10 and Figure 12.
The dynamic response diagram of system (2.1) under different parameter values m is shown in Figure 10, which corresponds to Figure 12. Figure 12 shows that the solution curve of state variables of system (2.1) under different values of m. It can be obtained from Figure 12 that as the value of the m increases from 0 to 0.8, the number of susceptible prey and predator decreases, but the number of infected prey does not change significantly. This is mainly impact due to disease factors.
In this subsection, we will discuss how the fear effect parameter k2 affects on each of the population when the positive equilibrium exists and is locally asymptotically stable. We study the influence of different parameter values of k2 on the population of prey and predator (seen Figure 11 and Figure 13).
Figure 11 shows that the dynamic response diagram of system (2.1) corresponding to the fear effect parameter k2 in the time interval [0,100]. It can be seen that the influence of the fear effect k2 on the population is negative for prey and predator. That is, when the value of the fear parameter k2 increases from 0 to 40, the number of prey and predator population decreases (see Figure 13), but the level of fear effect k2 does not affect the stability of the system (2.1). This conclusion is consistent with the conclusions obtained from system (2.1) state response diagrams in Figure 13.
Finally, we give the effect of the predator's catch rate δ for diseased prey on the number of species in system (see Figure 14). It can be seen from Figure 14 (a) that the number of susceptible prey gradually increase as δ increases, which indicates that the capture rate has a positive effect on the number of susceptible prey. However, we can obtain that the population of infected prey and predator first increases and then decreases when δ increases from Figure 14 (b) and Figure 14 (c). When it reached its critical value δC=28, the number of infected prey and predator begins to decrease. At this time, the number of predators tends to become extinct, because disease factors inhibit the growth of the number of predators and infected prey.
In this paper, a predator-prey model with fear effect and disease in prey incorporating prey refuge is studied. To make such system be more realistic, the effect of delay denoting the reaction time of the susceptible prey who feels the predation crisis on the system is considered, and the herd behavior of prey is also considered. We assume that the predator's capture rate on prey population is in line with Holling-II type functional response function. According to calculation, the basic reproduction number is given that R0=0.37<1, which indicates the disease dies out from system. And system (2.1) has a trivial equilibrium E0, a boundary predator-free equilibrium ˜E1, a disease-free equilibrium ¯E2 and a unique positive equilibrium E∗. By discussing the distribution of the roots of the characteristic equation of system (2.1) and taking the time delay τ as bifurcation parameters, the stability of this system is discussed. If time delay τ is greater than its corresponding critical values, the Hopf bifurcation occurs at the positive equilibrium E∗. Next, the global stability of the equilibrium ¯E2 and E∗ is proved. Furthermore, we discuss the direction and stability of Hopf bifurcation by using the center manifold theorem and the normal form theory. Finally, the stability and bifurcations affecting the prey and predator population are supported by numerical simulations.
In the absence of time delay, we select m1, m, k2 and δ as the analysis parameters, and discuss the influence of these parameters on stability of system (2.1). We found that the parameter m1 has a positive effect on the increase of the prey population, and vice versa for the predator population (see Figure 8). The parameter m has a negative effect on the change of the prey population, but does not change significantly for the predator population (see Figure 10). In the literature [19], they studied that if the prey is infected with the disease, then the three species do not coexist. Our paper mainly introduces the influence of the fear effect k2 on system (2.1). For the parameter k2 of the level of fear of prey on the predator, as the level of fear increases, the number of prey and predator population decreases (see Figure 11). But the level of fear effect k2 does not affect the stability of system (2.1). The catch rate δ of predators to infected prey has a positive effect on the increase in the number of susceptible prey. The catch rate δ inhibits the increase in the number of predators and infected prey due to disease factors (see Figure 13). In addition, the stability of system (1.4) with time delay is discussed. The study shows that when τ=0.5<τ0=1.059, system (1.4) is locally asymptotically stable at the positive equilibrium E∗ (see Figure 4), but system (1.4) is unstable when τ=2>τ0=1.059. Thus, Hopf bifurcation takes place when τ=τ0 (see Figure 5). In other words, the minimum reaction time of susceptible prey cannot exceed τ0, otherwise the stability of the system (1.4) will change. At the same time, the direction of Hopf bifurcation and the stability of periodic solutions are discussed by the center manifold theorem and normal form theory.
In future, in order to obtain the greatest economic benefits, we will consider the effective harvest of the susceptible prey and predator. In addition, by considering the intrapsychic competition in the susceptible prey, we have the following model
{dXS(t)dt=rXS(t−τ)1+k2Y(t−τ)−(b+β)XSXI−α(1−m1)√XS√Y1+k1(1−m1)√XS−d1X2S−q1EXS,dXI(t)dt=βXSXI−δ(1−m)XIY−d2XI,dY(t)dt=cα(1−m1)√XS√Y1+k1(1−m1)√XS−ξδ(1−m)XIY−d3Y−q2EY. |
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11661050 and 11861044), and the HongLiu First-class Disciplines Development Program of Lanzhou University of Technology. We also would like to thank two anonymous reviewers for their valuable comments and suggestions.
All authors declare no conflicts of interest in this paper.
[1] | Kaje, E.S., More than just good teaching: Teachers engaging culturally and linguistically diverse learners with content and language in mainstream classrooms. Doctoral dissertation, University of Washington, 2009, 1‒191. |
[2] | García, O. and Kleifgen, J., Introduction, in Educating emergent bilinguals: Policies, programs, and practices for English language learners (Eds. Author 1 and Author 2), 2010, Teachers College Press. |
[3] |
Brenner, M., Development of mathematical communication in problem solving groups by language minority students. Bilingual Research Journal, 1998, 22(2/3/4): 103–128. https://doi.org/10.1080/15235882.1998.10162720 doi: 10.1080/15235882.1998.10162720
![]() |
[4] | Willey, C., A case study of two teachers attempting to create active mathematics discourse communities with Latinos. Unpublished doctoral dissertation. University of Illinois, 2013, 1‒166. |
[5] | National Center for Education Statistics. National assessment of educational progress 2019, 2019. Available from: https://www.nationsreportcard.gov/mathematics/nation/achievement/?grade = 4 |
[6] |
Aguirre-Muñoz, Z. and Boscardin, C., Opportunity to learn and English learner achievement: Is increased content exposure beneficial? Journal of Latinos and Education, 2008, 7(3): 186‒205. https://doi.org/10.1080/15348430802100089 doi: 10.1080/15348430802100089
![]() |
[7] |
Bailey, A., Blackstock-Bernstein, A. and Heritage, M., At the intersection of mathematics and language: Examining mathematical strategies and explanations by grade and English learner status. The Journal of Mathematical Behavior, 2015, 40: 6‒28. https://doi.org/10.1016/j.jmathb.2015.03.007 doi: 10.1016/j.jmathb.2015.03.007
![]() |
[8] |
Aguirre-Muñoz, Z. and Amabisca, A., Defining opportunity to learn for English language learners: Linguistic and cultural dimensions of ELLs' instructional contexts. Journal of Education for Students Placed at Risk, 2010, 15(3): 259‒278. https://doi.org/10.1080/10824669.2010.495691 doi: 10.1080/10824669.2010.495691
![]() |
[9] |
de Araujo, Z., Roberts, S., Willey, C., and Zahner, W., English learners in K–12 mathematics education: A Review of the literature. Review of Educational Research, 2018, 88(6): 879‒919. https://doi.org/10.3102/0034654318798093 doi: 10.3102/0034654318798093
![]() |
[10] |
Drageset, O.G. and Ell, F., Using positioning theory to think about mathematics classroom talk. Educational Studies in Mathematics, 2023,115(1): 353‒385. https://doi.org/10.1007/s10649-023-10295-0 doi: 10.1007/s10649-023-10295-0
![]() |
[11] |
Schleppegrell, M., The Linguistic challenges of mathematics teaching and learning: A research review. Reading & Writing Quarterly: Overcoming Learning Difficulties, 2007, 23(2): 139‒159. http://dx.doi.org/10.1080/10573560601158461 doi: 10.1080/10573560601158461
![]() |
[12] |
Walshaw, M. and Anthony, G., The Teacher's role in classroom discourse: A review of recent literature into mathematics classrooms. Review of Educational Research, 2017, 78(3): 516‒551. https://doi.org/10.3102/0034654308320292 doi: 10.3102/0034654308320292
![]() |
[13] | Smith, E.M., Facilitating emergent bilinguals' participation in mathematics: An examination of a teacher's positioning acts. Unpublished doctoral dissertation. University of Missouri, 2018, 1‒166. |
[14] |
Celedón-Pattichis, S., Peters, S., Borden, L., Males, J., Pape, S., Chapman, O., et al., Asset-based approaches to equitable mathematics education research and practice. Journal for Research in Mathematics Education, 2018, 49(4): 373‒389. https://doi.org/10.5951/jresematheduc.49.4.0373 doi: 10.5951/jresematheduc.49.4.0373
![]() |
[15] | Gee, J., Social Linguistics and Literacies: Ideology in Discourses. Falmer Press, 1996. |
[16] | Halliday, M.A.K., Language as social semiotic: The Social interpretation of languages and meaning. Hodder & Stoughton Educational, 1976. |
[17] | Wenger, E., Communities of practice: Learning, meaning, and identity. Cambridge University Press, 1998. |
[18] |
Wagner, D. and Herbel-Eisenmann, B., Re-mythologizing mathematics through attention to classroom positioning. Educational Studies in Mathematics, 2009, 72(1): 1‒15. https://doi.org/10.1007/s10649-008-9178-5 doi: 10.1007/s10649-008-9178-5
![]() |
[19] |
Khisty, L.L. and Chval, K.B., Pedagogic discourse and equity in mathematics: When teachers' talk matters. Mathematics Education Research Journal, 2002, 14(3): 154–168. https://doi.org/10.1007/BF03217360 doi: 10.1007/BF03217360
![]() |
[20] |
Celedón-Pattichis, S., Kussainova, G., López-Leiva, C.A. and Pattichis, M.S., "Fake it until you make it": Participation and positioning of a Bilingual Latina student in mathematics and computing. Teachers College Record, 2022,124(5): 3‒228. https://doi.org/10.1177/01614681221103929 doi: 10.1177/01614681221103929
![]() |
[21] |
Banse, H., Palacios, N., Merritt, E. and Rimm-Kaufman, S., Scaffolding English language learners' mathematical talk in the context of Calendar Math. The Journal of Educational Research, 2017,110(2): 199‒208. http://dx.doi.org/10.1080/00220671.2015.1075187 doi: 10.1080/00220671.2015.1075187
![]() |
[22] |
Celedón-Pattichis, S. and Turner, E., "Explicame tu respuesta": Supporting the development of mathematical discourse in emergent bilingual kindergarten students. Bilingual Research Journal: The Journal of the National Association for Bilingual Education, 2012, 35(2): 197–216. https://doi.org/10.1080/15235882.2012.703635 doi: 10.1080/15235882.2012.703635
![]() |
[23] |
Chval, K., Pinnow, R. and Thomas, A., Learning how to focus on language while teaching mathematics to English language learners: A case study of Courtney. Mathematics Education Research Journal, 2014, 27(1): 103–127. https://doi.org/10.1007/s13394-013-0101-8 doi: 10.1007/s13394-013-0101-8
![]() |
[24] |
Dominguez, H., Bilingual students' articulation and gesticulation of mathematical knowledge during problem solving. Bilingual Research Journal, 2005, 29(2): 269–293. https://doi.org/10.1080/15235882.2005.10162836 doi: 10.1080/15235882.2005.10162836
![]() |
[25] |
Dominguez, H., Using what matters to students in bilingual mathematics problems, Educational Studies in Mathematics, 2011, 76(1): 305‒328. https://doi.org/10.1007/s10649-010-9284-z doi: 10.1007/s10649-010-9284-z
![]() |
[26] |
Dominguez, H., Lopez-Leiva, C. and Khisty, L., Relational engagement: Proportional reasoning with bilingual Latino/a students. Educational Studies in Mathematics, 2014, 85(1): 143‒160. https://doi.org/10.1007/S10649-013-9501-7 doi: 10.1007/S10649-013-9501-7
![]() |
[27] | Dominguez, H., Social risk takers: understanding bilingualism in mathematical discussions. Issues in Teacher Education, 2017, 26(2): 35‒47. |
[28] |
Enyedy, N., Rubel, L., Castellón, V., Mukhopadhyay, S., Esmonde, I. and Secada, W., Revoicing in a multilingual classroom. Mathematical Thinking and Learning, 2008, 10(2): 134–162. https://doi.org/10.1080/10986060701854458 doi: 10.1080/10986060701854458
![]() |
[29] | Hansen-Thomas, H., Learning to use math discourse in a standards-based, reform-oriented middle school classroom: How Latino/a ELLs becomes socialized into the math community of practice. Unpublished doctoral dissertation. University of Texas at San Antonio, 2005. |
[30] |
Hansen-Thomas, H., Reform-oriented mathematics in three 6th grade classes: How teachers draw in ELLs to academic discourse. Journal of Language, Identity and Education, 2009, 8(2-3): 88‒106. https://doi.org/10.1080/15348450902848411 doi: 10.1080/15348450902848411
![]() |
[31] |
Hufferd-Ackles, K., Fuson, K. and Sherin, M., Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 2004, 35(2): 81‒116. https://doi.org/10.2307/30034933 doi: 10.2307/30034933
![]() |
[32] |
McGraw, R. and Rubinstein-Ávila, E., Middle school immigrant students developing mathematical reasoning in Spanish and English. Bilingual Research Journal, 2008, 31(1-2): 147‒173. https://doi.org/10.1080/15235880802640656 doi: 10.1080/15235880802640656
![]() |
[33] |
Merritt, E., Palacios, N., Banse, H., Rimm-Kaufman, S. and Leis, M., Teaching practices in Grade 5 mathematics class with high-achieving English learner students. The Journal of Education Research, 2014,110(1): 17‒31. https://doi.org/10.1080/00220671.2015.1034352 doi: 10.1080/00220671.2015.1034352
![]() |
[34] |
Moschkovich, J., Academic literacy in mathematics for English Learners. The Journal of Mathematical Behavior, 2015, 40(A): 43–62. https://doi.org/10.1016/j.jmathb.2015.01.005 doi: 10.1016/j.jmathb.2015.01.005
![]() |
[35] | Musanti, S. and Celedón-Pattichis, S., Promising pedagogical practices for emergent bilinguals in kindergarten: Towards a mathematics discourse community. Journal of Multilingual Education Research, 2013, 4(1): 41‒62. |
[36] |
Musanti, S.I., Celedón-Pattichis, S. and Marshall, M.E., Reflections on language and mathematics problem-solving: A Study of a bilingual first grade teacher. Bilingual Research Journal, 2009, 32(1): 25‒41.https://doi.org/10.1080/15235880902965763 doi: 10.1080/15235880902965763
![]() |
[37] | Petkova, M., Classroom discourse and teacher talk influences on English language learner students' mathematics experience. Doctoral dissertation, University of South Florida. Proquest, 2009. |
[38] |
Rubinstein-Ávila, E., Sox, A.A., Kaplan, S. and McGraw, R., Does biliteracy + mathematical discourse = binumerate development? Language use in a middle-school dual-language mathematics classroom? Urban Education, 2014, 50(8): 899‒937. https://doi.org/10.1177/0042085914536997 doi: 10.1177/0042085914536997
![]() |
[39] |
Shein, P., Seeing with two eyes: A Teacher's use of gestures in questioning and revoicing to engage English language learners in the repair of mathematical errors. Journal for Research in Mathematics Education, 2012, 43(2): 182–222. https://doi.org/10.5951/jresematheduc.43.2.0182 doi: 10.5951/jresematheduc.43.2.0182
![]() |
[40] | Smith, E., Facilitating emergent bilinguals' participation in mathematics: An examination of a teacher's positioning acts. Electronic Journal for Research in Science and Mathematics Education, 2023, 27(4): 77‒106. |
[41] |
Turner, E. and Celedón-Pattichis, S., Mathematical problem solving among Latina/o kindergarteners: An Analysis of opportunities to learn. Journal of Latinos and Education, 2011, 10(2): 146‒169. https://doi.org/10.1080/15348431.2011556524 doi: 10.1080/15348431.2011556524
![]() |
[42] |
Turner, E., Dominguez, H., Empson, S. and Maldonado, L., Latino/a bilinguals and their teachers developing a shared communicative space. Educational Studies in Mathematics, 2013, 84(1): 349‒370. https://doi.org/10.1007/s10649-013-9486 doi: 10.1007/s10649-013-9486
![]() |
[43] |
Turner, E., Dominguez, H., Maldonado, L., and Empson, S., English learners' participation in mathematical discussion: Shifting positionings and dynamic identities. Journal for Research in Mathematics Education, 2013, 44(1): 199–234. https://doi.org/10.5951/jresematheduc.44.1.0199 doi: 10.5951/jresematheduc.44.1.0199
![]() |
[44] |
Yang, J., Ozbek, G., Liang, S. and Cho, S., Effective teaching strategies for teaching mathematics to young gifted English learners. Gifted Education International, 2023, 39(2): 226‒246. https://doi.org/10.1177/02614294231165121 doi: 10.1177/02614294231165121
![]() |
[45] | Zahner, W., How to do math with words: Learning algebra through peer discussions. Unpublished doctoral dissertation. University of California, 2011. |
[46] |
Zahner, W., The rise and run of a computational understanding of slope in a conceptually focused bilingual algebra class. Educational Studies in Mathematics, 2015, 88(1): 19–41. https://doi.org/10.1007/s10649-014-9575-x doi: 10.1007/s10649-014-9575-x
![]() |
[47] |
Zahner, W., Calleros, E.D. and Pelaez, K., Designing learning environments to promote academic literacy in mathematics in multilingual secondary mathematics classrooms. ZDM- Mathematics Education, 2021, 53(1): 359‒373. https://doi.org/10.1007/s11858-021-01239-0 doi: 10.1007/s11858-021-01239-0
![]() |
[48] |
Zahner, W., Velazquez, G., Moschkovich, J., Vahey, P. and Lara-Meloy, T., Mathematics teacher practices with technology that support conceptual understanding for Latino/a students. The Journal of Mathematical Behavior, 2012, 31(4): 431‒446. https://doi.org/10.1016/j.jmathb.2012.06.002 doi: 10.1016/j.jmathb.2012.06.002
![]() |
[49] | Echevarria, J. and Short, D., Programs and practices for effective sheltered content instruction in Improving Education for English Learners: Research-Based Approaches (eds. Author 1 and Author 2), CDE Press, 2010. |
[50] | Wadhwa, K., Positioning emergent bilingual and racially minoritized students to participate in mathematics discourse and discourse communities, Unpublished doctoral dissertation, University of Miami, 2023. |
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Parameter | Baseline values | Minimum values | Maximum values |
r | 0.17025 | 0.1374 | 0.2031 |
β | 0.6023 | 0.5682 | 0.6364 |
m1 | 0.1997 | 0.1684 | 0.2310 |
m | 0.4196 | 0.3883 | 0.4509 |
δ | 1.5024 | 1.4324 | 1.5724 |
k2 | 10.0414 | 9.6712 | 10.4116 |