
Citation: Narges Montazeri Shahtori, Tanvir Ferdousi, Caterina Scoglio, Faryad Darabi Sahneh. Quantifying the impact of early-stage contact tracing on controlling Ebola diffusion[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1165-1180. doi: 10.3934/mbe.2018053
[1] | Margaret Jane Moore, Nele Demeyere . Neglect Dyslexia in Relation to Unilateral Visuospatial Neglect: A Review. AIMS Neuroscience, 2017, 4(4): 148-168. doi: 10.3934/Neuroscience.2017.4.148 |
[2] | Wanda M. Snow, Brenda M. Stoesz, Judy E. Anderson . The Cerebellum in Emotional Processing: Evidence from Human and Non-Human Animals. AIMS Neuroscience, 2014, 1(1): 96-119. doi: 10.3934/Neuroscience.2014.1.96 |
[3] | Chiyoko Kobayashi Frank . Reviving pragmatic theory of theory of mind. AIMS Neuroscience, 2018, 5(2): 116-131. doi: 10.3934/Neuroscience.2018.2.116 |
[4] | Erick H. Cheung, Joseph M. Pierre . The Medical Ethics of Cognitive Neuroenhancement. AIMS Neuroscience, 2015, 2(3): 105-122. doi: 10.3934/Neuroscience.2015.3.105 |
[5] | Brion Woroch, Alex Konkel, Brian D. Gonsalves . Activation of stimulus-specific processing regions at retrieval tracks the strength of relational memory. AIMS Neuroscience, 2019, 6(4): 250-265. doi: 10.3934/Neuroscience.2019.4.250 |
[6] | Md Mahbub Hossain . Alice in Wonderland syndrome (AIWS): a research overview. AIMS Neuroscience, 2020, 7(4): 389-400. doi: 10.3934/Neuroscience.2020024 |
[7] | Mani Pavuluri, Amber May . I Feel, Therefore, I am: The Insula and Its Role in Human Emotion, Cognition and the Sensory-Motor System. AIMS Neuroscience, 2015, 2(1): 18-27. doi: 10.3934/Neuroscience.2015.1.18 |
[8] | Laura Mandolesi, Noemi Passarello, Eillyn Leiva Ramirez, Deny Menghini, Patrizia Turriziani, Teresa Vallelonga, Francesco Aristei, Angela Galeotti, Stefano Vicari, Vito Crincoli, Maria Grazia Piancino . Enhancing cognition and well-being by treating the malocclusion unilateral posterior crossbite: preliminary evidence by a single case study. AIMS Neuroscience, 2025, 12(2): 95-112. doi: 10.3934/Neuroscience.2025007 |
[9] | Amory H. Danek, Virginia L. Flanagin . Cognitive conflict and restructuring: The neural basis of two core components of insight. AIMS Neuroscience, 2019, 6(2): 60-84. doi: 10.3934/Neuroscience.2019.2.60 |
[10] | Laura Serra, Sara Raimondi, Carlotta di Domenico, Silvia Maffei, Anna Lardone, Marianna Liparoti, Pierpaolo Sorrentino, Carlo Caltagirone, Laura Petrosini, Laura Mandolesi . The beneficial effects of physical exercise on visuospatial working memory in preadolescent children. AIMS Neuroscience, 2021, 8(4): 496-509. doi: 10.3934/Neuroscience.2021026 |
Population dynamics has always been an important research object of biomathematics. Various groups often have complex interspecific relationships, such as predation, competition, parasitism and mutualism [1]. Predation behavior, as a widespread interspecies relationship, has been widely studied. Lotka and Volterra were the first to propose a predator-prey system to describe the widespread interspecies relationship of predation [2]. Holling further proposed three functional responses of predators to describe the energy transfer between predators and prey [3]. These three functional responses have been applied and perfected by many scientists [4]. In 1991, Hastings and Powell proposed a food chain system with chaotic dynamics and studied the dynamics of the model [5]. In recent years, many mathematicians have also studied the development and improvement of Hastings-Powell food chain models [6,7,8,9].
Fractional calculus is a generalization of traditional calculus, and its order can be composed of integers, fractions or complex numbers[10]. Fractional calculus can better describe some systems or processes with memory and hereditary properties, and it has been widely used in many fields, such as physics, secure communication, system control, neural networks, and chaos[11,12]. The method of solving the fractional model has also been widely studied[13,14]. In [15], the Caputo fractional derivative operator is used instead of the integer first derivative to establish an effective numerical method for solving the dynamics of the reaction-diffusion model based on a new implicit finite difference scheme. In [16], a numerical approximation for the Caputo-Fabrizio derivative is used to study the dynamic complexity of a predator-prey system with a Holling-type functional response. In [17], a new fractional chaotic system described by the Caputo fractional derivative is presented, and how to use the bifurcation diagram of this chaotic system to detect chaotic regions is analyzed. In [18], the generalization of Lyapunov's direct method applying Bihari's and Bellman-Gronwall's inequalities to Caputo-type fractional-order nonlinear systems is proposed. In [19], the Fourier spectral method is introduced to explore the dynamic richness of two-dimensional and three-dimensional fractional reaction-diffusion equations. In [20], the spatial pattern formation of the predator-prey model with different functional responses was studied. [21] studied the numerical solution of the space-time fractional reaction-diffusion problem that simulates the dynamic and complex phenomena of abnormal diffusion.
Since most biological mathematical models have long-term memory, fractional differential equations can more accurately and reliably describe the actual dynamic process [1,22]. [23] proposed fractional predator-prey models and fractional rabies models and studied their equilibrium points, stability and numerical solutions. In [24], the authors studied the stability of a fractional-order system by the Lyapunov direct method, which substantially developed techniques to study the stability of fractional-order population models. In [9], the authors extended the Hastings-Powell food chain system to fractional order and analyzed its dynamic behavior.
As an important research object of biological mathematics and control theory, population model control has received extensive research and development in recent years [25,26,27]. In [28], the authors conducted random detection and contacted tracking on the HIV/AIDS epidemic model and used the Adams-type predictor-corrector method to perform fractional optimal control of the model, which significantly reduced the number of AIDS patients and HIV-infected patients. In [29], the authors applied the time-delay feedback controller to the fractional-order competitive Internet model to solve the bifurcation control problem of this model. In [30], the author considered the influence of additional predators on the Hastings-Powell food chain model and studied the control of chaos in this model.
Biological models are widely studied by scientists, but many classic models study food chain models composed of herbivores and carnivores, and omnivores are rarely considered. In fact, omnivores are widespread in nature and play an important role in the food chain. In this article, the existence of omnivores is fully considered, and a food chain model in which herbivores, omnivores and carnivores coexist is studied. Based on these works, this paper proposes a fractional food chain model with a Holling type-II functional response. The main contributions of this paper are as follows. First, this paper proves the existence and uniqueness of the solution and the nonnegativity and boundedness of the solution. Second, the equilibrium point of the model is calculated, and the local stability of the equilibrium point is proven. Third, a controller is designed to prove the global asymptotic stability of the system by using the Lyapunov method.
This paper is organized as follows. In Section 2, the definitions and lemmas are given, and the food chain model is established. In Section 3, the existence, uniqueness, nonnegativity and boundedness are proven, and the local stability of the equilibrium point of the model is studied. The global stability of the model is studied through the controller. In Section 4, numerical simulations are performed to verify the theoretical results. The conclusion of this article is given in Section 5.
In this section, some basic knowledge about fractional equations and the theorems and lemmas used in this paper are given, and the fractional food chain system is introduced.
Definition 1. [10]. The Caputo fractional derivative of order α of a function f, R+→R, is defined by
Dαtf(t)=1Γ(n−α)∫t0fn(τ)(t−τ)α+1−ndτ,(n−1<α<n), n∈Z+, |
where Γ(⋅) is the Gamma function. When 0<α<1,
Dαtf(t)=1Γ(1−α)∫t0f′(τ)(t−τ)αdτ. |
Definition 2. [31]. When the order a>0, for a function f:(0,∞)→R, the Riemann-Liouville representation of the fractional integral operator is defined by
RLD−atf(t)=RLIatf(t)=1Γ(a)∫t0(t−τ)a−1f(τ)dτ,t>0, |
RLI0tg(t)=g(t), |
where a>0 and Γ(⋅) is the Gamma function.
Lemma 1. (Generalized Gronwall inequality) [32]. Assume that m≥0, γ>0, and a(t) are nonnegative, locally integrable, and nondecreasing functions defined on 0≤t≤T(T≤∞). In addition, h(t) is a nonnegative, locally integrable function defined in 0≤t≤T and satisfies
h(t)≤a(t)+m∫t0(t−s)γ−1h(t)ds, |
then,
h(t)=a(t)Eγ(mΓ(γ)tγ), |
where the Mittag-Leffler function Eγ(z)=∞∑k=0zkΓ(kγ+1).
Lemma 2. [10]. Consider the fractional-order system
{Dαtx(t)=f(t,x(t)),x(0)=x0, | (2.1) |
where f(t,x(t)) defined in R+×Rn→Rn and α∈(0,1].
The local asymptotic stability of the equilibrium point of this system can be derived from |arg(λi)|>απ2, where λi are the eigenvalues of the Jacobian matrix at the equilibrium points.
Lemma 3. [24]. Consider the system
{Dαtx(t)=f(t,x(t)),x(0)=xt0, | (2.2) |
where α∈(0,1], f:[t0,∞)×Ω→Rn, and Ω∈Rn; if f(t,x) satisfies the local Lipschitz condition about x on [t0,∞)×Ω, then there exists a unique solution of (2.2).
Lemma 4. [33]. The function x(t)∈R+ is continuous and derivable; then, for any t≥t0
Dαt[x(t)−x∗−x∗lnx(t)x∗]≤(1−x∗x(t))Dαtx(t),x∗∈R+,∀α∈(0,1). |
There are a variety of complex biological relationships in nature. Predation is the most important biological relationship, and it has received attention from and been studied by many scientists. In [5], the author proposed a three-species food chain model. The model consists of one prey ˆX and two predators ˆY and ˆZ. The top predator ˆZ feeds on the secondary predator ˆY, and the secondary predator ˆY feeds on the prey ˆX. This is the famous Hastings-Powell model:
{dˆXdT=ˆRˆX(1−ˆXˆK)−^C1^A1ˆXˆY^B1+ˆX,dˆYdT=^A1ˆXˆY^B1+ˆX−^A2ˆYˆZ^B2+ˆY−^D1ˆY,dˆZdT=^C2^A2ˆYˆZ^B2+ˆY−^D2ˆZ, | (2.3) |
where ˆR and ˆK represent the intrinsic growth rates and environmental carrying capacity, respectively. For i=1,2, parameters^Ai, ^Bi, ^Ci and ^Di are the predation coefficients, half-saturation constant, food conversion coefficients and death rates.
The Hastings-Powell model considers a food chain composed of herbivores, small carnivores and large carnivores but does not consider the existence of omnivores. We consider a food chain consisting of small herbivores X, medium omnivores Y and large carnivores Z. Among them, omnivores Y prey on herbivores X, and carnivores Z prey on omnivores Y. They all respond according to Holling II type. This system can be expressed mathematically as
{dXdT=R1X(1−XK1)−C1A1XYB1+X,dYdT=R2Y(1−YK2)+A1XYB1+X−A2YZB2+Y,dZdT=C2A2YZB2+Y−DZ, | (2.4) |
where X, Y and Z represent the densities of the prey population, primary predator population and top-predator population, respectively. For i=1,2, parameters Ri, Ki, Ai, Bi and Ci are the intrinsic growth rates, environmental carrying capacity, predation coefficients, half-saturation constant and food conversion coefficients, respectively. The parameter D is the death rates for Z.
Then, we obtain the following dimensionless version of the food chain model:
{dxdt=r1x(1−xK1)−a1xy1+b1x,dydt=r2y(1−yK2)+a1xy1+b1x−a2yz1+b2y,dzdt=a2yz1+b2y−dz, | (2.5) |
the independent variables x, y and z are dimensionless population variables; t represents a dimensionless time variable; and ai, bi(i=1,2) and d are positive.
Research results show that using fractional derivatives to model real-life biological problems is more accurate than classical derivatives[15]. To better analyze the dynamics between these three populations, we studied the following fractional-order Hastings-Powell System food chain model:
{Dαtx=r1x(1−xK1)−a1xy1+b1x,Dαty=r2y(1−yK2)+a1xy1+b1x−a2yz1+b2y,Dαtz=a2yz1+b2y−dz, | (2.6) |
where α∈(0,1) is the fractional order.
Theorem 1. The fractional-order Hastings-Powell System food chain model (2.6) has a unique solution.
Proof: We will study the existence and uniqueness of the system (2.6) in [0,T]×Ω, where Ω={(x,y,z)∈R3:0≤x,y,z≤H}. Let S=(x,y,z), ˉS=(ˉx,ˉy,ˉz), F(S)=(F1(S),F2(S),F3(S)) and
{F1(S)=Dαtx=r1x(1−xK1)−a1xy1+b1x,F2(S)=Dαty=r2y(1−yK2)+a1xy1+b1x−a2yz1+b2y,F3(S)=Dαtz=a2yz1+b2y−dz, | (3.1) |
For any S,ˉS∈Ω, it follows from (3.1) that
‖F(S)−F(ˉS)‖=|F1(S)−F1(ˉS)|+|F2(S)−F2(ˉS)|+|F3(S)−F3(ˉS)|=|r1x(1−xK1)−a1xy1+b1x−(r1ˉx(1−ˉxK1)−a1ˉxˉy1+b1ˉx)|+|r2y(1−yK2)+a1xy1+b1x−a2yz1+b2y−(r2ˉy(1−ˉyK2)+a1ˉxˉy1+b1ˉx−a2ˉyˉz1+b2ˉy)|+|a2yz1+b2y−dz−(a2ˉyˉz1+b2ˉy−dˉz)|≤r1|x(1−xK1)−ˉx(1−ˉxK1)|+r2|y(1−yK2)−ˉy(1−ˉyK2)|+2a1|xy1+b1x−ˉxˉy1+b1ˉx|+2a2|yz1+b2y−ˉyˉz1+b2ˉy|+d|z−ˉz|≤r1|x−ˉx|+r2|y−ˉy|+r1K1|x2−ˉx2|+r2K2|y2−ˉy2|+d|z−ˉz|+|xy−ˉxˉy+b1ˉxxy−b1ˉxxˉy(1+b1x)(1+b1ˉx)|+|yz−ˉyˉz+b2ˉyyz−b2ˉyyˉz(1+b2y)(1+b2ˉy)|≤r1|x−ˉx|+r2|y−ˉy|+r1K1|(x+ˉx)(x−ˉx)|+r2K2|(y+ˉy)(y−ˉy)|+d|z−ˉz|+|xy−ˉxˉy+b1xˉx(y−ˉy)|+|yz−ˉyˉz+b2yˉy(z−ˉz)|≤r1|x−ˉx|+r2|y−ˉy|+r1MK1|(x−ˉx)|+r2MK2|(y−ˉy)|+d|z−ˉz|+|xy−xˉy+xˉy−ˉxˉy|+b1|xˉx||y−ˉy|+|yz−yˉz+yˉz−ˉyˉz|+b2|yˉy||z−ˉz|≤(r1+r1MK1)|x−ˉx|+(r2+r2MK2)|y−ˉy|+d|z−ˉz|+M|y−ˉy|+M|x−ˉx|+b1M2|y−ˉy|+M|z−ˉz|+M|y−ˉy|+b2M2|z−ˉz|=(r1+r1MK1+M)|x−ˉx|+(r2+r2MK2+2M+b1M2)|y−ˉy|+(d+M+b2M2)|z−ˉz|≤L‖S−ˉS‖. |
where L=max{r1+r1MK1+M,r2+r2MK2+2M+b1M2,d+M+b2M2}, Based on Lemma 3, F(S) satisfies the Lipschitz condition with respect to S in Ω. According to the Banach fixed point theorem in [34], system (2.6) has a unique solution in Ω.
Theorem 2. Set A={x,y,z)∈R3:0<x+y+z<K1(r1+v)24r1v+K2(r2+v)24r2v} as a positively invariant set of system (2.6), and the solutions are bounded.
Proof: let g(t)≜g(x(t),y(t),z(t))=x(t)+y(t)+z(t),
Dαtg(t)+vg(t)=(r1+v)x(t)−r1K1x2(t)+(r2+v)y(t)−r2K2y2(t)−(d−v)z(t)=−r1K1(x(t)−K1(r1+v)2r1)2−r2K2(y(t)−K2(r2+v)2r2)2−(d−v)z(t)+K1(r1+v)24r1+K2(r2+v)24r2, |
let
u=K1(r1+v)24r1+K2(r2+v)24r2,v=d. |
According to the positive knowledge of all parameters and the nonnegativity of the solutions,
Dαtg(t)+vg(t)≤u, |
we obtain
g(t)≤uv+[g(0)−uv]Eα(−vtα), |
Since Eα(−vtα)≥0, when g(0)≤uv, limt→∞supg(t)≤uv. According to the nonnegativity of the system (2.6), g(t)≥0(∀t≥0); hence, A={x,y,z)∈R3:0<x+y+z<K1(r1+v)24r1v+K2(r2+v)24r2v} is a positively invariant set of system (2.6), and the solutions are bounded.
Let
{r1x(1−xK1)−a1xy1+b1x=0,r2y(1−yK2)+a1xy1+b1x−a2yz1+b2y=0,a2yz1+b2y−dz=0, |
Then, the equilibrium points are E0=(0,0,0), E1=(K1,0,0), E2=(0,K2,0) and E∗=(x∗,y∗,z∗), where
x∗=b1K1−12b1+√b21K21r21+2b1K1r21−4a1b1K1r1y∗+y2∗2b1r1,y∗=da2−b2d,z∗=a1x∗(b2y∗+1)a2(b1x∗+1)−r2(b2y∗+1)(y∗−K2)a2K2. |
For system (2.6), the Jacobian matrix at the equilibrium point (x′,y′,z′) is
J(x′,y′,z′)=[r1−2r1x′K1−a1de1e23−e50e4r1+e5−a2z′e2−2dr2K2e1+a2b2dz′e1e22−d0e21z′a20], |
where e1=a2−b2d, e2=b2de1+1, e3=1+b1x′, e4=2r1x′K1+a1de1e23, e5=a1x′e3.
Theorem 3.3. For system (2.6), the equilibrium points E0 and E1 are saddle points.
Proof: The Jacobian matrices evaluated at E0 and E1 are
J(0,0,0)=[r1000r2000−d],J(K1,0,0)=[−r1−a1K1b1K1+100r2+a1K1b1K1+1000−d], |
According to Lemma 2, when the eigenvalues are all real numbers and all negative, the equilibrium points are locally asymptotically stable.
The eigenvalues of J(E0) are λ01=r1, λ02=r2 and λ03=−d. The eigenvalues of J(E1) are λ11=−r1, λ12=r2+a1K1b1K1+1 and λ13=−d.
Then, we have λ01,λ02>0, λ03,λ11<0, and λ13<0, so the equilibrium points E0 and E1 are saddle points.
Theorem 4. For system (2.6), if r1<a1K2 and a2K2(b2K2+1)<d, then the equilibrium point E2=(0,K2,0) is locally asymptotically stable.
Proof: The Jacobian matrix evaluated at E2 is
J(0,K2,0)=[r1−a1K200a1K2−r2−a2K2b2K2+100a2K2b2K2+1−d], |
The eigenvalues of J(E2) are λ1=r1−a1K2, λ2=−r2 and λ3=a2K2b2K2+1−d.
Therefore, if r1<a1K2 and a2K2(b2K2+1)<d, the equilibrium point E2 is locally asymptotically stable.
The characteristic equation of equilibrium points E∗=(x∗,y∗,z∗) is given as
P(λ)=λ3+Aλ2+Bλ+C=0, | (3.2) |
where
A=2f4−2r1−e5+f2z∗+f1de23+f5−f2b2dz∗e1e2,
B=(B1+B2),
B1=e4e5+e5r1+r21−2r1f4−f2r1z∗−2e5f4+e21dz∗a2−f6e5−f6r1−2r1f5,
B2=2e23f1f5+2f2f4z∗+a1f3e23z∗+4dr2f4e1k1+r1f7−f1f7de3−2f4f7,
C=dz∗(a1dK1e23−e1K1r1+2e1r1x∗)f1K1,
where f1=a1e1, f2=a2e2, f3=f2de1, f4=r1x∗K1, f5=r2de1K2, f6=a1de23e1, f7=b2f3z∗e2.
For Eq. (3.2), define the discriminant as
D(P)=18ABC+(AB)2−4CA2−4B3−27C2, |
With reference to the results of [35] and [36], we obtain the following fractional Routh-Hurwitz conditions:
1. If D(P)>0, A>0, C>0, and AB−C>0, E∗ is locally asymptotically stable.
2. If D(P)<0 and A≥0, B≥0, and C>0, when α<23, E∗ is locally asymptotically stable.
3. If D(P)<0, A>0, B>0, and AB=C, then for all α∈(0,1), E∗ is locally asymptotically stable.
Theorem 5. If D(P)<0, C>0 and AB≠C, then α∗∈(0,1) exists; when α∈(0,α∗), E∗ is locally asymptotically stable; when α∈(α∗,1), E∗ is unstable. The system diverges at the critical value E∗.
Proof: If D(P)<0, then the eigenvalues of Eq. (3.2) have one real root λ1=a and two complex conjugate roots λ2,3=b±ci. Then, Eq. (3.2) can be written as
P(λ)=(λ−a)[λ−(b+ci)][λ−(b−ci)]=0, | (3.3) |
where A=−a−2b, B=b2+c2+2ab, C=−a(b2+c2), c>0, a,b,c∈R.
From C>0, then a<0, and then |arg(λ1)|=π>απ2.
From AB≠C, then −a2b+b(b2+c2)≠−2ab2 ⟹ −2b[(a+b)2+c2]≠0 ⟹ b≠0 and (a+b)2+c2≠0.
Thus, we can obtain |arg(λ2,3)|=|arctan(cb)|=arctan|cb|∈(0,π2).
Then, α∗∈(0,1) exists; when α∈(0,α∗), απ2<arctan|cb|, according to Lemma 2, E∗ is locally asymptotically stable, and when α∈(α∗,1), απ2>arctan|cb|, E∗ is unstable.
To study the asymptotic stability of system (2.6), three controllers will be added. The controller is proposed as follows: μ1=m1x(x−x∗), μ2=m2y(y−y∗), and μ3=m3z(z−z∗). where m1, m2 and m3 represent negative feedback gains, which are defined as real numbers. Clearly, if mi=0(i=1,2,3) or x=x∗(y=y∗,z=z∗), then μi=0(i=1,2,3), so it will not change the equilibrium point of system (2.6).
Controllers added into system (2.6) as follows
{Dαtx=r1x(1−xK1)−a1xy1+b1x−m1x(x−x∗),Dαty=r2y(1−yK2)+a1xy1+b1x−a2yz1+b2y−m2y(y−y∗),Dαtz=a2yz1+b2y−dz−m3z(z−z∗), | (3.4) |
One gives a Lyapunov function as:
V(x,y,z)=x−x∗−x∗lnxx∗+y−y∗−y∗lnyy∗+z−z∗−z∗lnzz∗. |
then,
DαtV≤x−x∗xDαx+y−y∗yDαy+z−z∗zDαz=(x−x∗)(r1−r1xK1−a1y1+b1x)−m1(x−x∗)2+(y−y∗)(r2−r2yK2+a1x1+b1x−a2z1+b2y)−m2(y−y∗)2+(z−z∗)(a2y1+b2y−d)−m3(z−z∗)2. |
Consider E∗ to be the equilibrium point:
{r1−r1x∗K1−a1y∗1+b1x∗=0,r2−r2y∗K2+a1x∗1+b1x∗−a2z∗1+b2y∗=0,a2y∗1+b2y∗−d=0, |
According to Lemma 4, we can obtain
DαtV≤(x−x∗)(r1x∗K1+a1y∗1+b1x∗−r1xK1−a1y1+b1x)−m1(x−x∗)2+(y−y∗)(r2y∗K2−a1x∗1+b1x∗+a2z∗1+b2y∗−r2yK2+a1x1+b1x−a2z1+b2y)−m2(y−y∗)2+(z−z∗)(a2y1+b2y−a2y∗1+b2y∗)−m3(z−z∗)2=a1(x−x∗)(y∗1+b1x∗−y1+b1x)+a1(y−y∗)(x1+b1x−x∗1+b1x∗)+a2(y−y∗)(z∗1+b2y∗−z1+b2y)+a2(z−z∗)(y1+b2y−y∗1+b2y∗)−(m1+r1K1)(x−x∗)2−(m2+r2K2)(y−y∗)2−m3(z−z∗)2=a1(x−x∗)(y∗1+b1x∗−y∗1+b1x+y∗1+b1x−y1+b1x)+a1(y−y∗)(x1+b1x−x∗1+b1x+x∗1+b1x−x∗1+b1x∗)+a2(y−y∗)(z∗1+b2y∗−z∗1+b2y+z∗1+b2y−z1+b2y)+a2(z−z∗)(y1+b2y−y∗1+b2y+y∗1+b2y−y∗1+b2y∗)−(m1+r1K1)(x−x∗)2−(m2+r2K2)(y−y∗)2−m3(z−z∗)2=a1(x−x∗)(b1y∗(x−x∗)(1+b1x∗)(1+b1x)+y∗−y1+b1x)+a1(y−y∗)(x−x∗1+b1x+b1x∗(x∗−x)(1+b1x∗)(1+b1x))+a2(y−y∗)(b2z∗(y−y∗)(1+b2y∗)(1+b2y)+z∗−z1+b2y)+a2(z−z∗)(y−y∗1+b2y+b2y∗(y∗−y)(1+b2y∗)(1+b2y))−(m1+r1K1)(x−x∗)2−(m2+r2K2)(y−y∗)2−m3(z−z∗)2≤a1b1y∗1+b1x∗(x−x∗)2+a2b2z∗1+b2y∗(y−y∗)2+a1b1x∗1+b1x∗(x−x∗)(y∗−y)+a2b2y∗1+b2y∗(y−y∗)(z∗−z)−(m1+r1K1)(x−x∗)2−(m2+r2K2)(y−y∗)2−m3(z−z∗)2≤a1b1x∗2(1+b1x∗)((x−x∗)2+(y−y∗)2)+a2b2y∗2(1+b2y∗)((y−y∗)2+(z−z∗)2)+(a1b1y∗1+b1x∗−m1−r1K1)(x−x∗)2+(a2b2z∗1+b2y∗−m2−r2K2)(y−y∗)2−m3(z−z∗)2=(a1b1(2y∗+x∗)2(1+b1x∗)−m1−r1K1)(x−x∗)2+(a2b2y∗2(1+b2y∗)−m3)(z−z∗)2+(a2b2(2z∗+y∗)2(1+b2y∗)+a1b1x∗2(1+b1x∗)−m2−r2K2)(y−y∗)2. |
When m1≥a1b1(2y∗+x∗)2(1+b1x∗)−r1K1, m2≥a2b2(2z∗+y∗)2(1+b2y∗)+a1b1x∗2(1+b1x∗)−r2K2, and m3≥a2b2y∗2(1+b2y∗), it follows that DαV≤0. We can show that the equilibrium point E∗ is uniformly asymptotically stable.
In this section, we use the Adams-Bashforth-Molton predictor-corrector algorithm numerical simulation. This method is described in detail in [37] and [38].
Example 1. In system (2.6), let r1=1, r2=0.6, K1=50, K2=10, a1=1, a2=0.6, b1=5, b2=0.01 and d=0.5. System (2.6) has a positive equilibrium point E∗=(49.8479,0.8403,1.2597). According to Theorem 3.5, when α=1, α=0.9, α=0.8, and E∗ is locally asymptotically stable, it can be seen from Figure 1 that the order α will affect the speed at which the system converges to the equilibrium point. The relevant results are shown in Figure 1.
Example 2. In system (2.6), let r1=1, r2=0.6, K1=50, K2=30, a1=1, a2=0.6, b1=5, b2=0.2 and d=0.2. System (2.6) has a positive equilibrium point E∗=(49.9382,0.3571,1.4158). It follows from Theorem 3.5 that system (2.6) has a bifurcation at α∗. When α=0.95 and α=0.8, E∗ is locally asymptotically stable, and when α=0.98, E∗ is unstable. The relevant results are shown in Figure 2.
Example 3. To verify the sensitivity of the system (2.6) to initial conditions and other parameters, according to the method in [39], apply the positive Euler format to transform the differential model into the following discrete form:
{xt+1=xt+δ(r1xt(1−xtK1)−a1xtyt1+b1xt),yt+1=yt+δ(r2yt(1−ytK2)+a1xtyt1+b1xt−a2ytzt1+b2yt),zt+1=zt+δ(a2ytzt1+b2yt−dzt), | (4.1) |
where δ is the time step size. We use the parameters of Example 1 to study Lyapunov exponents. Figure 3 shows that system (2.6) is in a stable state and is less sensitive to initial conditions.
This paper studies a new fractional-order food chain model with a Holling type-II functional response. First, the existence, uniqueness, nonnegativity and boundedness of the solution of the model are discussed. Second, the local stability of each equilibrium point is discussed. Third, controllers μ1=m1x(x−x∗), μ2=m2y(y−y∗) and μ3=m3z(z−z∗) are proposed and added to the system. Using the Lyapunov method, sufficient conditions for the positive equilibrium point to reach the global uniformly asymptotically stable state are obtained. Finally, we use numerical simulations to verify the theoretical results.
This work was supported by the Shandong University of Science and Technology Research Fund (2018 TDJH101).
The author declares no conflicts of interest in this paper.
[1] | [ C. Browne,H. Gulbudak,G. Webb, Modeling contact tracing in outbreaks with application to Ebola, Journal of Theoretical Biology, 384 (2015): 33-49. |
[2] | [ G. Chowell,C. Viboud, Is it growing exponentially fast?-Impact of assuming exponential growth for characterizing and forecasting epidemics with initial near-exponential growth dynamics, Infectious Disease Modelling, 1 (2016): 71-78. |
[3] | [ O. Diekmann,J. A. P. Heesterbeek,J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990): 365-382. |
[4] | [ M. G. Dixon,I. J. Schafer, Ebola Viral Disease Outbreak - West Africa, 2014, MMWR Morb Mortal Wkly Rep, 63 (2014): 548-551. |
[5] | [ K. T. Eames,M. J. Keeling, Contact tracing and disease control, Proceedings of the Royal Society of London B: Biological Sciences, 270 (2003): 2565-2571. |
[6] | [ F. O. Fasina, A. Shittu, D. Lazarus, O. Tomori, L. Simonsen, C. Viboud and G. Chowell, Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014, Eurosurveillance, 19 (2014), 20920. |
[7] | [ M. Greiner,D. Pfeiffer,R. D. Smith, Principles and practical application of the receiver-operating characteristic analysis for diagnostic tests, Preventive Veterinary Medicine, 45 (2000): 23-41. |
[8] | [ J. M. Heffernan,R. J. Smith,L. M. Wahl, Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2 (2005): 281-293. |
[9] | [ M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2008. |
[10] | [ I. Z. Kiss,D. M. Green,R. R. Kao, Disease contact tracing in random and clustered networks, Proceedings of the Royal Society of London B: Biological Sciences, 272 (2005): 1407-1414. |
[11] | [ D. Klinkenberg,C. Fraser,H. Heesterbeek, The effectiveness of contact tracing in emerging epidemics, PloS One, 1 (2006): e12. |
[12] | [ S. Liu,N. Perra,M. Karsai,A. Vespignani, Controlling contagion processes in activity driven networks, Physical review letters, 112 (2014): 118702. |
[13] | [ A. S. da Mata and R. Pastor-Satorras, Slow relaxation dynamics and aging in random walks on activity driven temporal networks, The European Physical Journal B, 88 (2015), Art. 38, 8 pp. |
[14] | [ N. Perra, B. Gonçalves, R. Pastor-Satorras and A. Vespignani, Activity Driven Modeling of Time Varying Networks, Scientific Reports, 2012. |
[15] | [ N. Perra,A. Baronchelli,D. Mocanu,B. Gonc'calves,R. Pastor-Satorras,A. Vespignani, Random walks and search in time-varying networks, Physical review letters, 109 (2012): 238701. |
[16] | [ A. Rizzo,B. Pedalino,M. Porfiri, A network model for Ebola spreading, Journal of theoretical biology, 394 (2016): 212-222. |
[17] | [ A. Rizzo,M. Frasca,M. Porfiri, Effect of individual behavior on epidemic spreading in activity-driven networks, Physical Review E, 90 (2014): 042801. |
[18] | [ A. Rizzo and M. Porfiri, Toward a realistic modeling of epidemic spreading with activity driven networks, in Temporal Network Epidemiology (N. Masuda and P. Holme), Springer, (2017), 317-319. |
[19] | [ N. M. Shahtori,C. Scoglio,A. Pourhabib,F. D. Sahneh, Sequential Monte Carlo filtering estimation of Ebola progression in West Africa, American Control Conference(ACC), null (2016): 1277-1282. |
[20] | [ M. D. Shirley,S. P. Rushton, The impacts of network topology on disease spread, Ecological Complexity, 2 (2005): 287-299. |
[21] | [ F. Shuaib,R. Gunnala,E. O. Musa,F. J. Mahoney,O. Oguntimehin,P. M. Nguku,S. B. Nyanti,N. Knight,N. S. Gwarzo,O. Idigbe,A. Nasidi,J. F. Vertefeuille, Ebola virus disease outbreak-Nigeria, July-September 2014, MMWR Morb Mortal Wkly Rep, 63 (2014): 867-872. |
[22] | [ M. Starnini,R. Pastor-Satorras, Temporal percolation in activity-driven networks, Physical Review E, 89 (2014): 032807. |
[23] | [ M. Starnini,R. Pastor-Satorras, Topological properties of a time-integrated activity-driven network, Physical Review E, 87 (2013): 062807. |
[24] | [ K. Sun, A. Baronchelli and N. Perra, Contrasting effects of strong ties on SIR and SIS processes in temporal networks, The European Physical Journal B, 88 (2015), Art. 326, 8 pp. |
[25] | [ L. Zino,A. Rizzo,M. Porfiri, Continuous-Time Discrete-Distribution Theory for Activity-Driven Networks, Physical Review Letters, 117 (2016): 228302. |
[26] | [ Cases of Ebola Diagnosed in the United States, 2014. Available from: https://www.cdc.gov/vhf/ebola/outbreaks/2014-west-africa/united-states-imported-case.html. |
[27] | [ Implementation and Management of Contact Tracing for Ebola Virus Disease, Emergency Guideline by the World Health Organization, 2015. Available from: https://www.cdc.gov/vhf/ebola/pdf/contact-tracing-guidelines.pdf. |
1. | Emily Frith, Paul D. Loprinzi, Stephanie E. Miller, Role of Embodied Movement in Assessing Creative Behavior in Early Childhood: A Focused Review, 2019, 126, 0031-5125, 1058, 10.1177/0031512519868622 | |
2. | Ignatius G.P. Gous, The quest for context-relevant online education, 2019, 75, 2072-8050, 10.4102/hts.v75i1.5346 | |
3. | Dean D'Souza, Annette Karmiloff-Smith, Why a developmental perspective is critical for understanding human cognition, 2016, 39, 0140-525X, 10.1017/S0140525X15001569 | |
4. | Darren J. Yeo, Eric D. Wilkey, Gavin R. Price, The search for the number form area: A functional neuroimaging meta-analysis, 2017, 78, 01497634, 145, 10.1016/j.neubiorev.2017.04.027 | |
5. | Richard J. Binney, Richard Ramsey, Social Semantics: The role of conceptual knowledge and cognitive control in a neurobiological model of the social brain, 2020, 112, 01497634, 28, 10.1016/j.neubiorev.2020.01.030 | |
6. | Ruth Campos, If you want to get ahead, get a good master. Annette Karmiloff-Smith: the developmental perspective / Si quieres avanzar, ten una buena maestra. Annette Karmiloff-Smith: la mirada desde el desarrollo, 2018, 41, 0210-3702, 90, 10.1080/02103702.2017.1401318 | |
7. | Ruth Campos, Like mermaids or centaurs: psychological functions that are constructed through interaction (Como sirenas o centauros: funciones psicológicas que se construyen en interacción), 2020, 43, 0210-3702, 713, 10.1080/02103702.2020.1810942 | |
8. | Mauricio de Jesus Dias Martins, Laura Desirèe Di Paolo, Hierarchy, multidomain modules, and the evolution of intelligence, 2017, 40, 0140-525X, 10.1017/S0140525X16001710 | |
9. | André Knops, Hans-Christoph Nuerk, Silke M. Göbel, Domain-general factors influencing numerical and arithmetic processing, 2017, 3, 2363-8761, 112, 10.5964/jnc.v3i2.159 | |
10. | Rebecca J. Landa, Efficacy of early interventions for infants and young children with, and at risk for, autism spectrum disorders, 2018, 30, 0954-0261, 25, 10.1080/09540261.2018.1432574 | |
11. | Julia Siemann, Franz Petermann, Innate or Acquired? – Disentangling Number Sense and Early Number Competencies, 2018, 9, 1664-1078, 10.3389/fpsyg.2018.00571 | |
12. | Manja Attig, Sabine Weinert, What Impacts Early Language Skills? Effects of Social Disparities and Different Process Characteristics of the Home Learning Environment in the First 2 Years, 2020, 11, 1664-1078, 10.3389/fpsyg.2020.557751 | |
13. | Janna M. Gottwald, Sheila Achermann, Carin Marciszko, Marcus Lindskog, Gustaf Gredebäck, An Embodied Account of Early Executive-Function Development, 2016, 27, 0956-7976, 1600, 10.1177/0956797616667447 | |
14. | Chiara Colliva, Marta Ferrari, Cristina Benatti, Azzurra Guerra, Fabio Tascedda, Joan M.C. Blom, Executive functioning in children with epilepsy: Genes matter, 2019, 95, 15255050, 137, 10.1016/j.yebeh.2019.02.019 | |
15. | Ruth Campos, Carmen Nieto, María Núñez, Research domain criteria from neuroconstructivism: A developmental view on mental disorders, 2019, 10, 1939-5078, e1491, 10.1002/wcs.1491 | |
16. | Arturo E. Hernandez, Hannah L. Claussenius-Kalman, Juliana Ronderos, Kelly A. Vaughn, Symbiosis, Parasitism and Bilingual Cognitive Control: A Neuroemergentist Perspective, 2018, 9, 1664-1078, 10.3389/fpsyg.2018.02171 | |
17. | Luca Rinaldi, Annette Karmiloff-Smith, Intelligence as a Developing Function: A Neuroconstructivist Approach, 2017, 5, 2079-3200, 18, 10.3390/jintelligence5020018 | |
18. | David C. Geary, Daniel B. Berch, Kathleen Mann Koepke, 2019, 9780128159521, 1, 10.1016/B978-0-12-815952-1.00001-3 | |
19. | Giacomo Vivanti, Taralee Hamner, Nancy Raitano Lee, Neurodevelopmental Disorders Affecting Sociability: Recent Research Advances and Future Directions in Autism Spectrum Disorder and Williams Syndrome, 2018, 18, 1528-4042, 10.1007/s11910-018-0902-y | |
20. | Stephen Ramanoël, Elena Hoyau, Louise Kauffmann, Félix Renard, Cédric Pichat, Naïla Boudiaf, Alexandre Krainik, Assia Jaillard, Monica Baciu, Gray Matter Volume and Cognitive Performance During Normal Aging. A Voxel-Based Morphometry Study, 2018, 10, 1663-4365, 10.3389/fnagi.2018.00235 | |
21. | Renata Sarmento-Henrique, Laura Quintanilla, Beatriz Lucas-Molina, Patricia Recio, Marta Giménez-Dasí, The longitudinal interplay of emotion understanding, theory of mind, and language in the preschool years, 2020, 44, 0165-0254, 236, 10.1177/0165025419866907 | |
22. | Tuukka Kaidesoja, Matti Sarkia, Mikko Hyyryläinen, Arguments for the cognitive social sciences, 2019, 49, 0021-8308, 480, 10.1111/jtsb.12226 | |
23. | Phillip Hamrick, Jarrad A. G. Lum, Michael T. Ullman, Child first language and adult second language are both tied to general-purpose learning systems, 2018, 115, 0027-8424, 1487, 10.1073/pnas.1713975115 | |
24. | Jeremy I.M. Carpendale, Stuart I. Hammond, The development of moral sense and moral thinking, 2016, 28, 1040-8703, 743, 10.1097/MOP.0000000000000412 | |
25. | David M. G. Lewis, Laith Al-Shawaf, Mike Anderson, Contemporary evolutionary psychology and the evolution of intelligence, 2017, 40, 0140-525X, 10.1017/S0140525X16001692 | |
26. | Antonella Nonnis, Nick Bryan-Kinns, 2019, Mazi, 9781450366908, 672, 10.1145/3311927.3325340 | |
27. | Hana D'Souza, Annette Karmiloff‐Smith, Neurodevelopmental disorders, 2017, 8, 1939-5078, e1398, 10.1002/wcs.1398 | |
28. | Chiyoko Kobayashi Frank, Reviving pragmatic theory of theory of mind, 2018, 5, 2373-7972, 116, 10.3934/Neuroscience.2018.2.116 | |
29. | Karen L Campbell, Lorraine K Tyler, Language-related domain-specific and domain-general systems in the human brain, 2018, 21, 23521546, 132, 10.1016/j.cobeha.2018.04.008 | |
30. | Thiago T. Varella, Asif A. Ghazanfar, Cooperative care and the evolution of the prelinguistic vocal learning, 2021, 0012-1630, 10.1002/dev.22108 | |
31. | Mauricio Martínez, José Manuel Igoa, 2022, Chapter 5, 978-3-031-08922-0, 123, 10.1007/978-3-031-08923-7_5 | |
32. | Selma Dündar-Coecke, To What Extent Is General Intelligence Relevant to Causal Reasoning? A Developmental Study, 2022, 13, 1664-1078, 10.3389/fpsyg.2022.692552 | |
33. | Lisa S. Scott, Michael J. Arcaro, A domain-relevant framework for the development of face processing, 2023, 2731-0574, 10.1038/s44159-023-00152-5 | |
34. | Dilara Berkay, Adrianna C. Jenkins, A Role for Uncertainty in the Neural Distinction Between Social and Nonsocial Thought, 2022, 1745-6916, 174569162211120, 10.1177/17456916221112077 | |
35. | Miriam D. Lense, Eniko Ladányi, Tal-Chen Rabinowitch, Laurel Trainor, Reyna Gordon, Rhythm and timing as vulnerabilities in neurodevelopmental disorders, 2021, 376, 0962-8436, 10.1098/rstb.2020.0327 | |
36. | Raamy Majeed, Does the Problem of Variability Justify Barrett’s Emotion Revolution?, 2022, 1878-5158, 10.1007/s13164-022-00650-0 | |
37. | Ulrike Malmendier, Experience Effects in Finance: Foundations, Applications, and Future Directions, 2021, 25, 1572-3097, 1339, 10.1093/rof/rfab020 | |
38. | Santeri Yrttiaho, Anneli Kylliäinen, Tiina Parviainen, Mikko J. Peltola, Neural specialization to human faces at the age of 7 months, 2022, 12, 2045-2322, 10.1038/s41598-022-16691-5 | |
39. | Giulia Calignano, Eloisa Valenza, Francesco Vespignani, Sofia Russo, Simone Sulpizio, The unique role of novel linguistic labels on the disengagement of visual attention, 2021, 74, 1747-0218, 1755, 10.1177/17470218211014147 | |
40. | Raamy Majeed, The “puzzle” of emotional plasticity, 2022, 35, 0951-5089, 546, 10.1080/09515089.2021.2002293 | |
41. | Christina M. Vanden Bosch der Nederlanden, Xin Qi, Sarah Sequeira, Prakhar Seth, Jessica A. Grahn, Marc F. Joanisse, Erin E. Hannon, Developmental changes in the categorization of speech and song, 2022, 1363-755X, 10.1111/desc.13346 | |
42. | Claudia Karwath, Manja Attig, Jutta von Maurice, Sabine Weinert, Does Poverty Affect Early Language in 2-year-old Children in Germany?, 2022, 1062-1024, 10.1007/s10826-022-02500-0 | |
43. | Rebecca Merkley, Eric D. Wilkey, Anna A. Matejko, Exploring the Origins and Development of the Visual Number Form Area: A Functionally Specialized and Domain-Specific Region for the Processing of Number Symbols?, 2016, 36, 0270-6474, 4659, 10.1523/JNEUROSCI.0710-16.2016 | |
44. | Ulrike Malmendier, FBBVA Lecture 2020 Exposure, Experience, and Expertise: Why Personal Histories Matter in Economics, 2021, 19, 1542-4766, 2857, 10.1093/jeea/jvab045 | |
45. | Lorijn Zaadnoordijk, Tarek R. Besold, Rhodri Cusack, Lessons from infant learning for unsupervised machine learning, 2022, 4, 2522-5839, 510, 10.1038/s42256-022-00488-2 | |
46. | Ulrike Malmendier, Experience Effects in Finance: Foundations, Applications, and Future Directions, 2021, 1556-5068, 10.2139/ssrn.3893355 | |
47. | Zhian Lv, Chong Liu, Liya Liu, Qi Zeng, Past Exposure, Risk Perception, and Risk-Taking During a Local Covid-19 Shock, 2022, 1556-5068, 10.2139/ssrn.4113772 | |
48. | Raamy Majeed, On How to Develop Emotion Taxonomies, 2024, 1754-0739, 10.1177/17540739241259566 | |
49. | Andreas Demetriou, George Spanoudis, Timothy C. Papadopoulos, The typical and atypical developing mind: a common model, 2024, 0954-5794, 1, 10.1017/S0954579424000944 | |
50. | K. J. Hiersche, E. Schettini, J. Li, Z. M. Saygin, Functional dissociation of the language network and other cognition in early childhood, 2024, 45, 1065-9471, 10.1002/hbm.26757 | |
51. | Maeve R. Boylan, Bailey Garner, Ethan Kutlu, Jessica Sanches Braga Figueira, Ryan Barry‐Anwar, Zoe Pestana, Andreas Keil, Lisa S. Scott, How labels shape visuocortical processing in infants, 2024, 1525-0008, 10.1111/infa.12621 | |
52. | Itziar Lozano, Ruth Campos, Mercedes Belinchón, Sensitivity to temporal synchrony in audiovisual speech and language development in infants with an elevated likelihood of autism: A developmental review, 2025, 78, 01636383, 102026, 10.1016/j.infbeh.2024.102026 | |
53. | Emily E. Meyer, Marcelina Martynek, Sabine Kastner, Margaret S. Livingstone, Michael J. Arcaro, Expansion of a conserved architecture drives the evolution of the primate visual cortex, 2025, 122, 0027-8424, 10.1073/pnas.2421585122 | |
54. | Valeria Costanzo, Fabio Apicella, Lucia Billeci, Alice Mancini, Raffaella Tancredi, Carolina Beretta, Filippo Muratori, Giacomo Vivanti, Sara Calderoni, Altered Visual Attention at 12 Months Predicts Joint Attention Ability and Socio-Communicative Development at 24 Months: A Single-Center Eye-Tracking Study on Infants at Elevated Likelihood to Develop Autism, 2025, 15, 2076-3417, 3288, 10.3390/app15063288 | |
55. | Ioannis Xenakis, Argyris Arnellos, Relating creativity to aesthetics through learning and development: an Interactivist-Constructivist framework, 2025, 1568-7759, 10.1007/s11097-025-10084-5 |