
A continuous-time exhaustive-limited (K = 2) two-level polling control system is proposed to address the needs of increasing network scale, service volume and network performance prediction in the Internet of Things (IoT) and the Long Short-Term Memory (LSTM) network and an attention mechanism is used for its predictive analysis. First, the central site uses the exhaustive service policy and the common site uses the Limited K = 2 service policy to establish a continuous-time exhaustive-limited (K = 2) two-level polling control system. Second, the exact expressions for the average queue length, average delay and cycle period are derived using probability generating functions and Markov chains and the MATLAB simulation experiment. Finally, the LSTM neural network and an attention mechanism model is constructed for prediction. The experimental results show that the theoretical and simulated values basically match, verifying the rationality of the theoretical analysis. Not only does it differentiate priorities to ensure that the central site receives a quality service and to ensure fairness to the common site, but it also improves performance by 7.3 and 12.2%, respectively, compared with the one-level exhaustive service and the one-level limited K = 2 service; compared with the two-level gated- exhaustive service model, the central site length and delay of this model are smaller than the length and delay of the gated- exhaustive service, indicating a higher priority for this model. Compared with the exhaustive-limited K = 1 two-level model, it increases the number of information packets sent at once and has better latency performance, providing a stable and reliable guarantee for wireless network services with high latency requirements. Following on from this, a fast evaluation method is proposed: Neural network prediction, which can accurately predict system performance as the system size increases and simplify calculations.
Citation: Zhijun Yang, Wenjie Huang, Hongwei Ding, Zheng Guan, Zongshan Wang. Performance analysis of a two-level polling control system based on LSTM and attention mechanism for wireless sensor networks[J]. Mathematical Biosciences and Engineering, 2023, 20(11): 20155-20187. doi: 10.3934/mbe.2023893
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[2] | Manoj K. Singh, Brajesh K. Singh, Poonam, Carlo Cattani . Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model. Mathematical Biosciences and Engineering, 2023, 20(6): 9625-9644. doi: 10.3934/mbe.2023422 |
[3] | Hongqiuxue Wu, Zhong Li, Mengxin He . Dynamic analysis of a Leslie-Gower predator-prey model with the fear effect and nonlinear harvesting. Mathematical Biosciences and Engineering, 2023, 20(10): 18592-18629. doi: 10.3934/mbe.2023825 |
[4] | Saheb Pal, Nikhil Pal, Sudip Samanta, Joydev Chattopadhyay . Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model. Mathematical Biosciences and Engineering, 2019, 16(5): 5146-5179. doi: 10.3934/mbe.2019258 |
[5] | Rongjie Yu, Hengguo Yu, Chuanjun Dai, Zengling Ma, Qi Wang, Min Zhao . Bifurcation analysis of Leslie-Gower predator-prey system with harvesting and fear effect. Mathematical Biosciences and Engineering, 2023, 20(10): 18267-18300. doi: 10.3934/mbe.2023812 |
[6] | Kawkab Al Amri, Qamar J. A Khan, David Greenhalgh . Combined impact of fear and Allee effect in predator-prey interaction models on their growth. Mathematical Biosciences and Engineering, 2024, 21(10): 7211-7252. doi: 10.3934/mbe.2024319 |
[7] | Shuo Yao, Jingen Yang, Sanling Yuan . Bifurcation analysis in a modified Leslie-Gower predator-prey model with fear effect and multiple delays. Mathematical Biosciences and Engineering, 2024, 21(4): 5658-5685. doi: 10.3934/mbe.2024249 |
[8] | Yuhong Huo, Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty, Renji Han . Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor. Mathematical Biosciences and Engineering, 2023, 20(10): 18820-18860. doi: 10.3934/mbe.2023834 |
[9] | Zhenzhen Shi, Huidong Cheng, Yu Liu, Yanhui Wang . Optimization of an integrated feedback control for a pest management predator-prey model. Mathematical Biosciences and Engineering, 2019, 16(6): 7963-7981. doi: 10.3934/mbe.2019401 |
[10] | Kunlun Huang, Xintian Jia, Cuiping Li . Analysis of modified Holling-Tanner model with strong Allee effect. Mathematical Biosciences and Engineering, 2023, 20(8): 15524-15543. doi: 10.3934/mbe.2023693 |
A continuous-time exhaustive-limited (K = 2) two-level polling control system is proposed to address the needs of increasing network scale, service volume and network performance prediction in the Internet of Things (IoT) and the Long Short-Term Memory (LSTM) network and an attention mechanism is used for its predictive analysis. First, the central site uses the exhaustive service policy and the common site uses the Limited K = 2 service policy to establish a continuous-time exhaustive-limited (K = 2) two-level polling control system. Second, the exact expressions for the average queue length, average delay and cycle period are derived using probability generating functions and Markov chains and the MATLAB simulation experiment. Finally, the LSTM neural network and an attention mechanism model is constructed for prediction. The experimental results show that the theoretical and simulated values basically match, verifying the rationality of the theoretical analysis. Not only does it differentiate priorities to ensure that the central site receives a quality service and to ensure fairness to the common site, but it also improves performance by 7.3 and 12.2%, respectively, compared with the one-level exhaustive service and the one-level limited K = 2 service; compared with the two-level gated- exhaustive service model, the central site length and delay of this model are smaller than the length and delay of the gated- exhaustive service, indicating a higher priority for this model. Compared with the exhaustive-limited K = 1 two-level model, it increases the number of information packets sent at once and has better latency performance, providing a stable and reliable guarantee for wireless network services with high latency requirements. Following on from this, a fast evaluation method is proposed: Neural network prediction, which can accurately predict system performance as the system size increases and simplify calculations.
Predation is a ubiquitous interaction among species and thus it has been studied extensively. Most predator-prey models are extensions and modifications of the classical Lotka-Volterra model by incorporating many other factors. The obtained theoretical results can guide us to make appropriate policies on ecological protection and ecological sustainable development [1]. One important modification is the Leslie-Gower model described by
dxdt=rx(1−xK)−P(x)y,dydt=ys(1−ynx), | (1.1) |
where x(t) and y(t) stand for the densities of the prey and predator at time t, respectively. Here both species have the logistic growths with respective intrinsic growth rates r and s and with the carrying capacities K for the prey and nx for the predator (which depends on the density of the prey). n is a measure of the quality of the prey and nx is called the Leslie-Gower term [2]. P(x) is the functional response of predators to prey. Model (1.1) has been extensively studied and promoted (see, for example, [3,4]). Recently, the influence of fear effect and Allee effect on the dynamical behaviors of predator-prey models has attracted many researchers' attention.
Fear effect of predator is a form of indirect predation impacting the prey population [5,6,7]. To some extent, the physiological states of prey are disturbed by the fear effect. As a result, in the long run, the fear effect may lead to a loss in prey population. Meanwhile, the prey suffered from scare usually forage less and then their birth rate decreases which may be not good for the survival of prey's population. There is an example that some birds make anti-predator defenses against the sound of the predator and once perceive danger, they flee from their nests. As far as prey is concerned, frequent exposure to fear and anti-predation behavior will affect their birth rate. Therefore, it is necessary to take into account of fear factor in the predator-prey interaction [8,9,10]. In [11], Wang et al. first modeled the fear effect by introducing a factor f(k,y), where k represents the intensity of anti-predator behaviors of the prey caused by fear. See the paper for more detail on the properties of f(k,y). A recent experiment designed by Elliott et al. [12] demonstrated that Allee effect [13,14,15] can be induced by the fear effect. Moreover, the strong Allee effects [16,17] should be paid more attention, because too large fear intensity will drive the species to extinction.
Allee effect mainly signifies that individual fitness is directly proportionate to population density. Due to the variety of species, the causes of Allee effect are also different. Such as sperm limitation, reproduction facilitation, cooperative breeding, anti-predator behavior and predator satiation. During the last decades, lots have been done for predator-prey models with Allee effects. For example, Merdan [18] investigated the stabilityof a Lotka-Volterra type predator-prey system involving Allee effect; Liu and Dai [19] considered an impulsive predator-prey model with double Allee effects; Wu et al. [20] tried to understand the Allee effect in a commensal symbiosis mode; studied Guan and Chen [21] studied an amensalism model with Allee effect on the second species; and Naik et al. [22] explored the effect of strong Allee effect in a discrete predator-prey model.
On the one hand, based on the experimental observations of Elliott et al. [12], Sasmal [15] proposed and investigated the following predator-prey model with prey subject to Allee effect and fear effect,
dxdt=rx(1−xK)(x−θ)11+fy−axy,dydt=αaxy−my, |
where θ and f are Allee constant and fear factor, respectively. They showed that the cost of fear has no influence on the stability of equilibria. On the other hand, González-Olivares et al. [23] studied the following Leslie-Gower model with Allee effect in prey,
dxdt=rx(1−xK)(x−m)−qxy,dydt=sy(1−ynx), |
where m represents the Allee effect threshold. Without Allee effect, there is a unique equilibrium, which is globally asymptotically stable. However, with the presence of strong Allee effect, the number of equilibria varies with respect to m. It is shown that the system exhibits local bifurcations including Hopf bifurcation and Bogdanov-Takens bifurcation.
Inspired by these two works [15,23], it is natural and interesting to investigate the combined influence of both Allee effect and fear effect on dynamics of Leslie-Gower predator-prey models. In other words, whether the qualitative structure and bifurcation phenomena will become more complex? The thoretical results will help humans manage ecosystems. Precisely, the model to be studied in this paper is
dxdt=rx(1−xK)(x−m)11+fy−qxy,dydt=ys(1−ynx), | (1.2) |
where r, K, f, q, n, s, and m are all positive constants and 0<m<K. For simplicity of analysis, we introduce new variables
ˉx=xK,ˉy=1nKy,ˉt=rKt, |
and denote
ˉm=mK,ˉq=nqr,ˉs=srk,ˉf=fnK. |
Then system (1.2) becomes
dxdt=x(1−x)(x−m)11+fy−qxy,dydt=ys(1−yx), | (1.3) |
(note that we have dropped the bars here), where m, f, q, and s are positive constants with 0<m<1.
Though not defined at x=0, the dynamical behavior of (1.3) in the absence of prey is of interest. For this end, we take the time scaling dt=(1+fy)xdτ to get a new system (still label τ as t),
dxdt=x2(1−x)(x−m)−qx2y(1+fy), | (1.4a) |
dydt=ys(x−y)(1+fy). | (1.4b) |
This new system with the time scale reaches stable equilibria much faster than the original one. As a result, in the following we only focus on (1.4). Due to the biological background, the initial conditions (x(0),y(0))∈R2+. Obviously, for such an initial condition, (1.4) has a unique global solution which stays in R2+.
In the following, we first present the results on boundedness of solutions and the attraction of the origin. Followed is the existence and local stability of equilibria. The results indicate possible occurrence of bifurcations. We thus explore the detail on saddle-node bifurcation of codimension 1, Hopf bifurcation of codimension 1, and Bogdanov-Takens bifurcation of codimension 2. A brief conclusion ends the paper.
We first show the ultimate boundedness of solutions of (1.4).
Proposition 1. Let (x(t),y(t)) be a solution of (1.4). Then
lim supt→∞x(t)≤1andlim supt→∞y(t)≤1. |
Proof. We show lim supt→∞x(t)≤1 by distinguishing two cases. Firstly, we assume x(t)≥1 for t≥0. Then it follows from (1.4a) that x is a decreasing function. Denote limt→∞x(t) by l. It is easy to see that l=1. Next, we assume that x(t0)<1 for some t0≥0. We claim that, for t≥t0, x(t)≤1. Otherwise, there exists a t∗>t0 such that x(t∗)>1. Let x(ˆt)=max{x(t)|t0≤t≤t∗}. As dx(t∗)dt<0, we have ˆt∈(t0,t∗) and hence dx(ˆt)dt=0. This is impossible as x(ˆt)>1, which gives dx(ˆt)dt<0. The claim is proved. To sum up, lim supt→∞x(t)≤1 is proved. Now, for ε0>0, there exists ˘t≥0 such that x(t)≤1+ε0 for t≥˘t. It follows that, for t≥˘t,
dy(t)dt≤ys(1+ε0−y)(1+fy). |
Then arguing similarly as for lim supt→∞x(t)≤1, we can get lim supt→∞y(t)≤1+ε0. As ε0 is arbitrary, we immediately have lim supt→∞y(t)≤1.
We start with finding the equilibria to (1.4), which are determined by solutions to
{x2(1−x)(x−m)−qx2y(1+fy)=0,ys(x−y)(1+fy)=0. |
Clearly, (1.4) always admits two positive boundary equilibria P1(m,0) and P2(1,0). Moreover, for a positive equilibrium (x,x) (or coexistence equilibrium), x satisfies
−(1+fq)x2+(m−q+1)x−m=0. | (3.1) |
If m−q+1≤0 then (3.1) has no positive roots and hence (1.4) has no positive equilibria. Now assume that m−q+1>0, which implies that q<m+1<2. Denote the discriminant of (3.1) by Δ and it can be expressed as a quadratic function of m,
Δ(m)=m2−2(2fq+q+1)m+(q−1)2. |
Note that Δ(1)=q(q−4−4f)<0 and Δ(q−1)=2(1−q)(2+fq). It follows that if q≥1 then Δ(m)<0, which again implies that (1.4) has no positive equilibrium. Finally, if 0<q<1, then Δ(m)>0 only when 0<m<m1 while Δ(m)=0 only when m=m1, where
m1=2fq+q+1−2√f2q2+fq2+fq+q. |
In the former, (3.1) has two positive roots, x3=m−q+1−√Δ(m)2(1+fq) and x4=m−q+1+√Δ(m)2(1+fq), which are in (0,1) while in the latter, ˉx=1−√fq+qfq+1∈(0,1) is the unique positive root. The above discussion is summarized as follows.
Theorem 2. (ⅰ) There are always three boundary equilibria (0,0), P1(m,0), and P2(1,0) for system (1.4).
(ⅱ) Besides the three boundary equilibria, for (1.4) to have positive equilibria it is necessary that 0<q<1. In this case,
(a) if m1<m<1, there is no positive equilibrium;
(b) if m=m1, there is a unique positive equilibrium ˉP(ˉx,ˉx), where ˉx=1−√fq+qfq+1;
(c) if 0<m<m1, there are distinct positive equilibria P3(x3,x3) and P4(x4,x4), where x3=m−q+1−√Δ(m)2(1+fq) and x4=m−q+1+√Δ(m)2(1+fq).
Remarks 3. Considering f=0, we have q+1−2√q:=m0. Obviously, m1<m0, which means that influenced by fear effect, Allee threshold value decreases. In other words, prey are more likely to die out at low density.
Next, we consider the attractivity of the origin.
Theorem 4. The origin is a non-hyperbolic attractor for (1.4).
Proof. Note that the technique of linearization is inapplicable as J(0,0) is the zero matrix for (1.4). The approach here is the blow-up method. With the horizontal blow-up
(x,y)=(x,ux)anddτ=xdt |
along the invariant line x=0, we rewrite system (1.4) as
dx(τ)dτ=−x[m−(m+1)x+qux+x2+fqu2x2],du(τ)dτ=u[m+s−su−(m+1)x+(fs+q)ux+x2−fsu2x+fqu2x2]. |
On the nonnegative u-axis, the above system admits two equilibria C1(0,0) and C2(0,m+ss). The corresponding Jacobian matrices are
JC1=(−m00m+s)andJC2=(−m0−(m+1)(m+s)s−m−s), |
respectively. Obviously, C1 is a saddle while C2 is an attractor. After blow-down, the origin is an attractor.
Theorem 5. P1 is a hyperbolic repeller node while P2 is a saddle.
Proof. At P1 and P2, the Jacobian matrices are respectively
JP1=(m(1−m)00s)andJP2=(m−100s). |
respectively. Clearly, since m<1, JP1 has two positive eigenvalues while JP2 has one positive and one negative eigenvalues. Then the results follow immediately.
The result below indicates that the stability of the positive equilibrium ˉP depends on s.
Theorem 6. Suppose that 0<q<1 and m=m1, which guarantee the existence of ˉP. Then we have the following statements.
(i) ˉP is a saddle-node with an attracting parabolic sector when s>s∗, where s∗=qˉx(1+2fˉx)fˉx+1;
(ii) ˉP is a saddle-node with a repelling parabolic sector when s<s∗;
(iii) ˉP is a degenerate equilibrium when s=s∗. Furthermore, if either (0<ˉx<2−√3 and f≠f1) or 2−√3≤ˉx<1, then ˉP is a cusp of codimension two; if 0<ˉx<2−√3 and f=f1, then ˉP is a cusp of codimension at least three, where
f1=2ˉx2−5ˉx+1+ˉx√2ˉx2−4ˉx+12ˉx2(2−ˉx). |
Fig. 1 shows the phase portraits.
Proof. Note that the Jacobian matrix of (1.4) at ˉP is
JˉP=(−ˉx2(2ˉx−m1−1)−qˉx2(1+2fˉx)ˉxs(1+fˉx)−ˉxs(1+fˉx)). |
We easily see that
det(ˉP)=sˉx31+fˉx1+fq(ˉx−1+√fq+qfq+1)=0 |
and hence JˉP has zero as an eigenvalue.
First of all, we use the transformation
X=x−ˉx,Y=y−ˉy |
to translate ˉP into (0,0) and system (1.4) into
dXdt=a10X+a01Y+a20X2+a11XY+a02Y2+a30X3+a21X2Y+a12XY2+a03Y3+O(|X,Y|4),dYdt=b10X+b01Y+b20X2+b11XY+b02Y2+b30X3+b21X2Y+b12XY2+b03Y3+O(|X,Y|4), | (3.2) |
where
a10=qˉx2(1+2fˉx),a01=−qˉx2(1+2fˉx),a20=−5ˉx2+(2m1+2)ˉx,a11=−2qˉx(1+2fˉx),a02=−qfˉx2,a30=m1+1−4ˉx,a21=−q(1+2fˉx),a12=−2fqˉx,a03=0,b10=s(1+fˉx),b01=−s(1+fˉx),b20=0,b11=s(1+2fˉx),b02=−s(1+2fˉx),b30=b21=0,b12=fs,b03=−fs. |
Now, assume that s≠s∗. Then the linear transformation
X=u+v,Y=u+b01a01v |
is nonsingular. This, combined with the time scale dτ=(a10+b01)dt, transforms system (3.2) into (we still denote τ as t)
dudt=a′20u2+a′11uv+a′02v2+O(|u,v|3),dvdt=v+b′20u2+b′11uv+b′02v2+O(|u,v|3), | (3.3) |
where
a′20=sˉx(1+fˉx)m1(a10+b01)2,a′11={sˉx(4ˉx4f2+2(f2s−fm1−f+4)fˉx3+2(2fm1+3fs−2m1−fq−2)fˉx2+(6fm1+5fs−q)ˉx+2m1+s)}(1+2fˉx)(a10+b01)2,a′02=s(1+fˉx)20f2qˉx5q(1+2fˉx)2(a10+b01)2+s(1+fˉx)(16f3s−10f2m1−10f2+18f)qˉx4q(1+2fˉx)2(a10+b01)2+s(1+fˉx)(−3f3s2+4f2m1q+28f2qs−9fm1q−9fq+4q)ˉx3q(1+2fˉx)2(a10+b01)2+s(1+fˉx)(−6f2s2+4fm1q+16fqs−2m1q−2q)ˉx2q(1+2fˉx)2(a10+b01)2+s(1+fˉx)(−4fs2+m1q+3qs)ˉxq(1+2fˉx)2(a10+b01)2+s(1+fˉx)−s2q(1+2fˉx)2(a10+b01)2,b′20=−s(1+fˉx)ˉxm1(a10+b01)2,b′11=12fqˉx5(a10+b01)2+(10f2qs−6fm1q+8fq2−6fq+12q)ˉx4(a10+b01)2+(−f2s2+2fm1q+6fqs−m1q−q)ˉx3(a10+b01)2+(−3fs2+2m1q+3qs)ˉx2(a10+b01)2+−s2ˉx(a10+b01)2,b′02=20f2q2ˉx6q(1+fˉx)(a10+b01)2+(8f3s−10f2m1−10f2+18f)q2ˉx5q(1+fˉx)(a10+b01)2+(5f3qs2+4f2m1q2+16f2q2s−9fm1q2−9fq2+4q2)ˉx4q(1+fˉx)(a10+b01)2+(−2f3s3+10f2qs2+4fm1q2+10fq2s−2m1q2−2q2)ˉx3q(1+fˉx)(a10+b01)2+(−5f2s3+6fqs2+m1q2+2q2s)ˉx2q(1+fˉx)(a10+b01)2+(qs2−4fs3)ˉxq(1+fˉx)(a10+b01)2−s3q(1+fˉx)(a10+b01)2. |
Since a′20>0, with the center manifold method [24], the equation near the center manifold can be approximated by
dudt=s(1+fˉx)ˉxm1(a10+b01)2u2+O(|u|3). | (3.4) |
Making use of [25,Theorem 7.1], the degenerate equilibrium ˉP is a saddle-node where the parabolic sector is located at the right half plane. From the time scaling dτ=(a10+b01)dt, we see that if s>s∗, the parabolic sector is attracting while if s<s∗ the parabolic sector is repelling.
Next, we assume s=s∗. It follows from
ˉx=m1−q+12(1+fq)andm1=2fq+q+1−2√f2q2+fq2+fq+q |
that
m1=ˉx2(f+1)1−ˉx2f+2ˉxfandq=q∗≜(ˉx−1)21−ˉx2f+2ˉxf. |
Obviously, m1∈(0,1), q∗>0, and s∗>0 still hold under the assumptions that f>0 and ˉx∈(0,1). Then (X,Y)=(u,−a01a10u+1a10v) transforms (3.2) into
dudt=v+c20u2+c11uv+c02v2+O(|u,v|3),dvdt=d20u2+d11uv+d02v2+O(|u,v|3), | (3.5) |
where
c20=ˉx2(f+1)ˉx2f−2ˉxf−1,c11=2(3ˉxf+1)ˉx(2ˉxf+1),c02=f(ˉx2f−2ˉxf−1)(2ˉxf+1)2ˉx2(ˉx−1)2,d20=−ˉx4(f+1)(ˉx−1)2(2ˉxf+1)(ˉx2f−2ˉxf−1)2,d11=−ˉx(ˉx−1)2(2ˉx2f2+4ˉxf+1)(ˉxf+1)(ˉx2f−2ˉxf−1)2,d02=3ˉx2f2+3ˉxf+1ˉx(2ˉxf+1)(ˉxf+1). |
Thus the C∞ change of coordinates in a small neighborhood of (0,0),
z1=u−c11+d122u2−c02uv,z2=v+c20u2−d02uv, |
produces the normal form of system (3.5),
dz1dt=z2+O(|z1,z2|3),dz2dt=f20z21+f11z1z2+O(|z1,z2|3), | (3.6) |
where
f20=−ˉx4(f+1)(−1+ˉx)2(2ˉxf+1)(ˉx2f−2ˉxf−1)2,f11=ˉx((2ˉx4−4ˉx3)f2+(4ˉx3−10ˉx2+2ˉx)f+ˉx2−4ˉx+1)(ˉxf+1)(1−ˉx2f+2ˉxf). |
Obviously, f20<0 and the sign of f11 relies on the sign of
φ(f)≜(2ˉx4−4ˉx3)f2+(4ˉx3−10ˉx2+2ˉx)f+ˉx2−4ˉx+1. |
Note that 2ˉx4−4ˉx3<0 since 0<ˉx<1. Denote
Δ1≜(4ˉx3−10ˉx2+2ˉx)2−4(2ˉx4−4ˉx3)(ˉx2−4ˉx+1)=4(2ˉx2−4ˉx+1)ˉx2(ˉx−1)2. |
Then
Δ1{<0 if 2−√22<ˉx<1>0 if 0<ˉx<2−√22=0 if ˉx=2−√22 |
Thus φ(f)<0 if 2−√22<ˉx<1. If ˉx=2−√22, then φ(f)=2√2−32(f+1)2<0 since f>0. Now consider the case where 0<ˉx<2−√22. Notice that
4ˉx3−10ˉx2+2ˉx{>0 if 0<ˉx<5−√174,<0 if 5−√174<ˉx<2−√22,=0 if ˉx=5−√174, |
and
ˉx2−4ˉx+1{>0if0<¯x<2−√3,=0if¯x=2−√3,<0if2−√3<ˉx<1. |
It follows that φ(f)<0 if 2−√3<ˉx<2−√22 and φ(f)=f[(90−52√3)f+(38−22√3)]<0 when ˉx=2−√3 since f>0. If 0<ˉx<2−√3, then
φ(f){>0if0<f<f1,=0iff=f1,<0iff>f1, |
where
f1=2ˉx2−5ˉx+1+(1−ˉx)√2ˉx2−4ˉx+12ˉx2(2−ˉx). |
Then the results follow and this completes the proof.
At last, we consider the positive equilibria P3 and P4.
Theorem 7. Suppose 0<q<1 and 0<m<m1. Then P3 is always a saddle point while P4 is a sink if s>x4(−2x4+m+1)fx4+1, is a source if s<x4(−2x4+m+1)fx4+1, and a center or fine focus if s=x4(−2x4+m+1)fx4+1.
Proof. After a simple calculation, we can get
det(JP3)=−sx33√Δ(m)1+fx31+fq<0anddet(JP4)=sx34√Δ(m)1+fx41+fq>0, |
where JP3 and JP4 are the Jacobian matrices of (1.4) at P3 and P4, respectively. It follows immediately that P3 is always a saddle. For the stability of P4, we need the trace of JP4, which is
Tr(JP4)=−x4[2x24+(fs−m−1)x4+s]{<0ifs>x4(−2x4+m+1)fx4+1,>0ifs<x4(−2x4+m+1)fx4+1,0ifs=x4(−2x4+m+1)fx4+1. |
Then the results follow easily.
The goal of this section is to analyse the saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation that may occur in system (1.4).
From Theorems 6 and 7, it is not difficult to obtain the saddle-node surface,
SN={(m,q,f,s):m=m1,s≠s∗,f>0,s>0,q>0,0<m<1}. |
The phase of the saddle-node bifurcation corresponding to the ecology system has obvious critical Allee value m1. When m>m1, the prey will face the risk of extinction; when m<m1, the dynamic behavior of system (1.4) becomes complex since the saddle and anti-saddle come out in the phase, which means that the density of prey must be large enough for survival.
From Theorem 7, the local stability of P4 will change as the parameter s varies. Besides, there exists an sH=x4(−2x4+m+1)fx4+1 which satisfies Tr(JP4)|sH=0 and then P4 becomes a non-hyperbolic equilibrium which loses its stability. Meanwhile, it is easy to verify the transversality condition,
ddsTr(JP4)|sH=−x4(fx4+1)<0. |
Therefore, there exists a Hopf bifurcation in a small neighbourhood of P4. This is summarized as follows.
Theorem 8. Let s be the Hopf bifurcation parameter. When s crosses the threshold value sH=x4(−2x4+m+1)fx4+1, system (1.4) undergoes a non-degenerate Hopf bifurcation around P4.
There is at least one limit cycle which appears around P4 due to the occurrence of Hopf bifurcation. The stability of the limit cycle is very significant for ecology systems and we can use the sign of the Lyapunov number to determine it. In order to simplify the computation, we translate P4(x4,x4) to (1,1) with the following state variable scaling and time rescaling:
ˉx=xx4,ˉy=yx4,τ=x24t,ˉK=1x4,ˉm=mx4,ˉq=q,ˉf=fx4,ˉs=sx4. | (4.1) |
After dropping the bars and relabelling τ as t, we transform system (1.4) into
dxdt=x2(1−xK)(x−m)−qx2y(1+fy),dydt=ys(x−y)(1+fy). | (4.2) |
Because system (4.2) has an equilibrium ¯P4(1,1) (i.e., P4(x4,x4) of (1.4)), we have q=(K−1)(1−m)K(1+f)>0. Then we can get another positive equilibrium ¯P3(ˉx3,ˉx3), where ˉx3 and ˉx4 are positive distinct roots of the equation
(1−m)(K−1)+(1+f)K(1+f)x2+(m+K)(1+f)−(1−m)(K−1)K(1+f)x−m=0. |
By Vieta theorem [26], ˉx3ˉx4=Km(1+f)−Kfm+Kf+fm+1<1. From the scalings (4.1), we see that the parameters in system (4.2) satisfy
K>1,m<1,f>0,s>0,Km(1+f)−Kfm+Kf+fm+1<1. | (4.3) |
With q being replaced by (K−1)(1−m)K(1+f), system (4.2) becomes
dxdt=x2(1−xK)(x−m)−(K−1)(1−m)K(1+f)x2y(1+fy),dydt=ys(x−y)(1+fy), | (4.4) |
where the parameters K, m, f, and s satisfy (4.3). Therefore, we study the stability of the weak focus ¯P4(1,1) for system (4.4) to acquire the stability of the weak focus P4(x4,y4) in system (1.4).
The Jacobian matrix of system (4.4) at ¯E4(1,1) is
J¯E4=(−2+m+KK(K−1)(−1+m)(1+2f)K(1+f)s(1+f)−s(1+f)). |
It follows that
det(J¯P4)=−(2Kfm−Kf+Km−fm−1)sK |
and
Tr(J¯P4)=−Kfs+Ks−K−m+2K. |
With the help of (4.3), we see that det(J¯P4)>0. Thus the necessary conditions on parameters for Hopf bifurcation occurring around ¯E4 are
m+K−2K(1+f)>0,K>1,m<1,f>0,s>0,Km(1+f)−Kfm+Kf+fm+1<1. | (4.5) |
Note that at s=shf≜m+K−2K(1+f), J¯P4 has a pair of conjugate pure imaginary eigenvalues. Moreover,
ddsTr(J¯P4)|shf=−(f+1)<0, |
that is, the transversality condition holds.
Now translating ¯P4(1,1) to the origin (0,0) by the change (ˆx,ˆy)=(x−1,y−1) when s=shf, we rewrite (4.4) as
{dˆxdt=^a10ˆx+^a01ˆy+^a20ˆx2+^a11ˆxˆy+^a02ˆy2+^a30ˆx3+^a21ˆx2ˆy+^a12ˆxˆy2+^a03ˆy3+O(|ˆx,ˆy|4),dˆydt=^b10ˆx+^b01ˆy+^b20ˆx2+^b11ˆxˆy+^b02ˆu2+^b30ˆx3+^b21ˆx2ˆy+^b12ˆxˆy2+^b03ˆy3+O(|ˆx,ˆy|4), | (4.6) |
where
^a10=−2+m+KK,^a01=(K−1)(−1+m)(1+2f)K(1+f),^a20=2K+2m−5K,^a11=2(K−1)(−1+m)(1+2f)K(1+f),^a02=2(K−1)(−1+m)(1+2f)K(1+f),^a30=−4+K+mK,^a21=(K−1)(−1+m)(1+2f)K(1+f),^a12=2(K−1)(−1+m)fK(1+f),^a03=0,^b10=−2+m+KK,^b01=−−2+m+KK,^b20=0,^b11=(−2+m+K)(1+2f)K(1+f),^b02=−(−2+m+K)(1+2f)K(1+f),^b30=0,^b21=0,^b12=(−2+m+K)fK(1+f),^b03=−(−2+m+K)fK(1+f). |
We now make another transformation ˆu=−ˆx, ˆv=1√D(^a10ˆx+^a01ˆy), and dτ=√Ddt to change system (4.6) into
{dˆudt=−ˆv+f(ˆu,ˆv),dˆvdt=ˆu+g(ˆu,ˆv), |
where
D=(2−m−K)(2Kfm−Kf+Km−fm−1)K2(1+f),f(ˆu,ˆv)=^c20ˆu2+^c11ˆuˆv+^c02ˆv2+^c30ˆu3+^c21ˆu2ˆv+^c12ˆuˆv2+^c03ˆv3+O(|ˆu,ˆv|4),g(ˆu,ˆv)=^d20ˆu2+^d11ˆuˆv+^d02ˆv2+^d30ˆu3+^d21ˆu2ˆv+^d12ˆuˆv2+^d03ˆv3+O(|ˆu,ˆv|4). |
Here we have omitted the expressions of ^cij and ^dij.
Applying the formal series method in [25] and calculating the first Lyapunov number with the help of MAPLE, we get
l1=18p1f2+p2f+p3√Dp4, |
where
p1=(2K−1)2m3+(4K3−32K2+33K−8)m2−(4K−1)(K2−8K+6)m+K3−8K2+6K,p2=(4K2−3K)m3+(4K3−30K2+24K−1)m2+(−3K3+24K2−12K−6)m−K2−6K+6,p3=(K2−K)m3+K(K−1)(K−7)m2−(K−1)(K2−6K−2)m−2K+2,p4=(1+2f)(−1+m)(K−1)K((2Km−K−m)f+Km−1). |
Under condition (4.5), we have p4>0. Thus the sign of l1 depends on that of
l11=p1f2+p2f+p3. |
Theorem 9. If l11>0 then system (4.4) exhibits a subcritical Hopf bifurcation around ¯P4 at s=shf (see Fig. 2(a)) while if l11<0 then it exhibits a supercritical Hopf bifurcation around ¯P4 at s=shf (see Fig. 2(b)).
When l11=0, we should consider the second Lyapunov number l2 to determine the stability of the order-two weak focus. The expression of l2 is too complicated, so we just provide a numerical example with (K,f,m,s)=(2,3628121430,25,428657711). For such a set of parameter values, we have l1=0 and l2=0.4025314467>0, which means that system (4.4) undergoes a multiple Hopf bifurcation of codimension two and there are two limit cycles (the inner one is stable and the outer one is unstable) bifurcated from ¯P4(1,1) when the bifurcation parameters (f,s)=(3628121430,428657711) is disturbed (see Fig. 2(c)).
From Theorem 6, system (1.4) admits that ˉP is a cusp of codimension 2 when m=m1, s=s∗, q=q∗, and φ(f)≠0. In the following, we show that Bogdanov-Takens bifurcation around ˉP can occur.
Theorem 10. Under the condition that ˉP is a cusp of codimension 2, we choose the parameters m and s as bifurcation parameters. As the parameters (m,s) vary around (m1,s∗), system (1.4) experiences Bogdanov-Takens bifurcation in a small neighborhood of\ ˉP. Furthermore,
(i) Given φ(f)>0, system (1.4) exhibits a supercritical Bogdanov -Takens bifurcation, which accompanies with the appearance of a stable limit cycle originated from the supercritical Hopf bifurcation followed by a stable homoclinic loop;
(ii) Given φ(f)<0, system (1.4) exhibits a subcritical Bogdanov -Takens bifurcation, which accompanies with the appearance of an unstable limit cycle originated from the subcritical Hopf bifurcation followed by an unstable homoclinic loop.
Proof. We give the unfolding system of system (1.4),
dxdt=x2(1−x)(x−m1−λ1)−qx2y(1+fy),dydt=y(s∗+λ2)(x−y)(1+fy), | (4.7) |
where λ1 and λ2 are disturbed in a small neighbourhood of the origin. For convenience, we should find the versal unfolding of system (4.7), which is helpful to study the change of the topological structure. We make some transformations as follows to achieve it.
Firstly, use the transformation u1=x−ˉx and u2=y−ˉx to translate the positive equilibrium ˉP to (0,0) and transform system (4.7) into
du1dt=˜a00+˜a10u1+˜a01u2+˜a20u21+˜a11u1z2+˜a02u22+O(|u1,u2|3),du2dt=˜b10u1+˜b01u2+˜b20u21+˜b11u1u2+˜b02u22+O(|u1,u2|3), | (4.8) |
where
˜a00=ˉx3λ1−ˉx2λ1,˜a10=ˉx(2ˉx4f+(−3fλ1−4f+1)ˉx3+(8fλ1+2f−2)ˉx2+(−4fλ1+3λ1+1)ˉx−2λ1)2ˉxf+1−ˉx2f,˜a01=(ˉx2−2ˉx+1)ˉx2(2ˉxf+1)ˉx2f−2ˉxf−1,˜a20=5ˉx4f+(−3fλ1−10f+2)ˉx3+(7fλ1+4f−5)ˉx2+(−2fλ1+3λ1+2)ˉx−λ12ˉxf+1−ˉx2f,˜a11=2ˉx(ˉx2−2ˉx+1)(2ˉxf+1)ˉx2f−2ˉxf−1,˜a02=(ˉx2−2ˉx+1)ˉx2fˉx2f−2ˉxf−1,˜b10=(2ˉx4f+(−f2λ2−4f+1)ˉx3+(2f2λ2−fλ2+2f−2)ˉx2+(3fλ2+1)ˉx+λ2)ˉx2ˉxf+1−ˉx2f,˜b01=−˜b10,˜b20=0,˜b11=(2ˉx4f+(−f2λ2−4f+1)ˉx3+(2f2λ2−fλ2+2f−2)ˉx2(3fλ2+1)ˉx+λ2)(2ˉxf+1)(ˉxf+1)(ˉx2f−2ˉxf−1),˜b02=−˜b11. |
Next, letting u3=u2, u4=du2dt, we transform system (4.8) into
du3dt=u4,du4dt=˜c00+˜c10u3+˜c01u4+˜c20u23+˜c11u3u4+˜c02u24+O(|u3,u4|3), | (4.9) |
where
˜c00=˜a00˜b10,˜c10=˜b11˜a00−˜a10˜b01+˜a01˜b10,˜c01=˜a10+˜b01,˜c20=−˜a10˜b10˜b02−˜a01˜b10˜b11−˜a20˜b201+˜a11˜b01˜b10−˜a02˜b210˜b10,˜c11=−2˜a20~b01−˜a11˜b10−2˜b02˜b10+˜b11˜b01˜b10,˜c02=˜a20+˜b11˜b10. |
In order to remove the ˜c02u24 term in system (4.9), we make a new time variable τ by dt=(1−˜c02u3)dτ and let u5=u3, u6=u4(1−˜c02u3). We still denote τ as t to get from (4.9) that
du5dt=u6,du6dt=˜d00+˜d10u5+˜d01u6+˜d20u25+˜d11u5u6+O(|u5,u6|3), | (4.10) |
where
˜d00=˜c00,˜d10=˜c10−2˜c02˜c00,˜d01=˜c01,˜d20=˜c20−2˜c10˜c02+˜c00˜c202,˜d11=˜c11−˜c01˜c02. |
Notice that ˜d20→−(ˉx−1)2(2ˉxf+1)ˉx4(f+1)(ˉx2f−2ˉxf−1)2<0 when (λ1,λ2)→(0,0). Then we employ the transformation
u7=u5,u8=u6√−˜d20,τ=√−˜d20t |
to change ˜d20 into −1. Meanwhile with this transform and relabelling τ as t, system (4.10) becomes
du7dt=u8,du8dt=˜e00+˜e10u7+˜e01u8−u27+˜e11u7u8+O(|u7,u8|3), | (4.11) |
where
˜e00=−˜d00˜d20,˜e10=−˜d10˜d20,˜e01=˜d01√−˜d20,˜e11=˜d11√−˜d20. |
Now eliminate u7 by the transformation u9=u7−˜e102, u10=u8 to change system (4.11) into
du9dt=u10,du10dt=˜f00+˜f01u10−u29+˜f11u9u10+O(|u9,u10|3), | (4.12) |
where
˜f00=˜e00+14˜e210,˜f01=˜e01+12˜e10˜e11,˜f11=˜e11. |
Lastly, to obtain the versal unfolding of (4.7), we need to make ~f11 be 1. Since ˜f11→ˉxφ(f)(ˉxf+1)(1−ˉx2f+2ˉxf)≠0 as (λ1,λ2)→(0,0), we can perform the transformation
x=−~f112u9,y=˜f311u10,τ=−t˜f11. | (4.13) |
Relabelling τ as t, we can rewrite (4.12) as
dxdt=y,dydt=˜g00+˜g01y+x2+xy+O(|x,y|3), | (4.14) |
where
˜g00=−~f00~f114,˜g01=−~f01~f11. |
With the assistance of MAPLE,
∂(˜g00,˜g01)∂(λ1,λ2)|λ1=λ2=0=φ5(f)(ˉx2f−2ˉxf−1)2(f+1)4(ˉxf+1)4(ˉx−1)5(2ˉxf+1)3ˉx6≠0, | (4.15) |
which implies that ˜g00 and ˜g10 are independent parameters. Therefore, system (4.14) is the versal unfolding of system (4.7) such that both have the same topological structure. As ˜g00 and ˜g10 vary, the topological structure of system (4.14) will change and system (1.4) will undergo Bogdanov-Takens bifurcation. Noticing that there exists time scale t=−~f11τ in the transformation (4.13), we have the following conclusions:
(i) if φ(f)>0, then −˜f11<0. Thus system (1.4) undergoes a supercritical Bogdanov-Takens bifurcation of codimension 2, which consists of saddle-node bifurcation of codimension 1, supercritical Hopf bifurcation of codimension 1, and homoclinic bifurcation of codimension 1.
(ii) if φ(f)<0, then −˜f11>0. Thus system (1.4) undergoes a subcritical Bogdanov-Takens bifurcation of codimension 2, which consists of saddle-node bifurcation of codimension 1, subcritical Hopf bifurcation of codimension 1, and homoclinic bifurcation of codimension 1.
Before drawing the phase portraits, we need to acquire the expression of the bifurcation curves within Bogdanov-Takens bifurcation in the (λ1,λ2)-plane.
By the conclusion in Bogdanov [27] and Takens [28] (see also Chow et al. [29] and Perko [24]), we can obtain the following local representations of the bifurcation curves.
(i) The saddle-node bifurcation curve SN={(˜g00,˜g01):˜g00=0,˜g01≠0};
(ii) the Hopf bifurcation curve H={(˜g00,˜g01):˜g01=√−˜g00,˜g00<0};
(iii) the homoclinic bifurcation curve HL={(˜g00,˜g01):˜g01=57√−˜g00,˜g00<0}.
In the case of repelling Bogdanov-Takens bifurcation (φ(f)>0), we will use λ1 and λ2 to describe the bifurcation curves SN, H, and HL. By (4.15) and the Implicit Function Theorem, we get the expressions of λ1 and λ2 in terms of ~g00 and ~g10,
λ1=a1˜g00+a2˜g10+O(∣˜g00,˜g10∣),λ2=b1˜g00+b2˜g10+O(∣˜g00,˜g10∣), | (4.16) |
where
a1=ˉx6(ˉx3f2+ˉx3f+ˉx2f+ˉx2)(ˉxf+1)3(f+1)2(2ˉx3f−4ˉx2f+ˉx2+2ˉxf−2ˉx+1)2(ˉx2f−2ˉxf−1)(ˉx−1)(2ˉx5f2−4ˉx4f2+4ˉx4f−10ˉx3f+ˉx3+2ˉx2f−4ˉx2+ˉx)4,a2=0,b1=¯b1ˉx3(f+1)2(fˉx+1)(ˉx−1)2(2fˉx+1)φ4(f)(1−fˉx2+2fˉx),b2=ˉx2(ˉx−1)2(f+1)(2ˉxf+1)(1−ˉx2f+2ˉxf)φ(f)>0,¯b1=16ˉx8f4−(68f4−50f3)ˉx7+(76f4−246f3+48f2)ˉx6−(12f4−330f3+284f2−17f)ˉx5−(8f4+114f3−436f2+131f−2)ˉx4−(4f3+188f2−231f+21)ˉx3+(12f2−113f+43)ˉx2+(12f−23)ˉx+3. |
For the saddle-node bifurcation curve SN, let Λ1≜~g00(λ1,λ2)=0. Due to
∂Λ1∂λ1|λ=0=φ4(f)(ˉx2f−2ˉxf−1)ˉx4(ˉx−1)3(2ˉxf+1)2(f+1)3(ˉxf+1)4≠0, |
turning to the Implicit Function Theorem, there exists a unique function λ1(λ2), which satisfies λ1(0)=0 and Λ1(λ1(λ2),λ2)=0. It follows that
λ1(λ2)=0. |
Furthermore, on the curve Λ1=0, from (4.16), we know λ2=b2~g10+O(∣~g10∣2), which implies that λ2>0 if ~g10>0 and λ2<0 if ~g10<0. Therefore, we acquire
SN+={(λ1,λ2)|λ1=0,λ2<0}andSN−={(λ1,λ2)|λ1=0,λ2>0}. |
For the Hopf bifurcation curve H, let Λ2≜~g00(λ1,λ2)+~g102(λ1,λ2)=0. Due to
∂Λ2∂λ1|λ=0=φ4(f)(ˉx2f−2ˉxf−1)ˉx4(ˉx−1)3(2ˉxf+1)2(f+1)3(ˉxf+1)4≠0, |
there exists a unique function λ1(λ2), which satisfies λ1(0)=0 and Λ2(λ1(λ2),λ2)=0. Then
λ1(λ2)=(fˉx2−2fˉx−1)(f+1)(fˉx+1)4φ2(f)(1−ˉx)λ22+O(∣λ2∣2). |
Furthermore, on the curve Λ2=0, from (4.16), we know λ2=b2~g10+O(∣~g10∣2), which implies that λ2<0 if ~g10<0. Therefore, we get
H={(λ1,λ2)∣λ1=(fˉx2−2fˉx−1)(f+1)(fˉx+1)4φ2(f)(1−ˉx)λ22+O(|λ2|2)}. |
For the Homoclinic bifurcation curve HL, let Λ3≜2549~g00(λ1,λ2)+~g102(λ1,λ2)=0. Due to
∂Λ3∂λ1|λ=0=2549φ4(f)(ˉx2f−2ˉxf−1)ˉx4(ˉx−1)3(2ˉxf+1)2(f+1)3(ˉxf+1)4≠0, |
there exists a unique function λ1(λ2), which satisfies λ1(0)=0 and Λ3(λ1(λ2),λ2)=0. Then
λ1(λ2)=4925(fˉx2−2fˉx−1)(f+1)(fˉx+1)4φ2(f)(1−ˉx)λ22+O(∣λ2∣2). |
Moreover, on the curve Λ3=0, from (4.16), we know λ2=b2~g10+O(∣~g10∣2), which implies that λ2<0 if ~g10<0. Therefore,
HL={(λ1,λ2)|λ1=4925(fˉx2−2fˉx−1)(f+1)(fˉx+1)4φ2(f)(1−ˉx)λ22+O(∣λ2∣2)}. |
Analogously, the Bogdanov-Takens bifurcation curve in terms of λ1 and λ2 can be described.
To end this section, we present two sets of numerical simulations to verify the theoretical results with the help of MATLAB.
First, let ˉx=0.5 and f=3. Then m1=413, q∗=113, s∗=465, and φ(f)=−578<0. Therefore, system (1.4) undergoes a repelling Bogdanov-Takens bifurcation. With the disturbance of (λ1,λ2) in a small neighborhood of (0,0), the repelling Bogdanov-Takens bifurcation diagram and its local phase portraits are shown in Fig 3.
Next, let ˉx=0.1 and f=8. Then m1=128, q∗=928, s∗=13280, and φ(f)=1.1988>0. Therefore, system (1.4) undergoes an attracting Bogdanov-Takens bifurcation. With the disturbance of (λ1,λ2) in a small neighborhood of (0,0), the attracting Bogdanov-Takens bifurcation diagram and its local phase portraits are illustrated by Fig. 4.
In the present study, the dynamical behavior of a Leslie-Gower model with linear (Holling Type I) functional response and with both Allee effect and fear effect in prey is investigated. Although the model is not defined when the density of the prey is zero, the origin is an attractor with the help of the blow-up method and the technique of time scaling. Correspondingly, prey will die out at low densities. Moreover, solutions are bounded. Based on the work of Korobeinikov [30] that system (1.4) admits a unique globally asymptotically stable positive equilibrium when m=0 and f=0, qualitative analysis reveals that Allee effect results in complex dynamics. The number and stability of equilibria in system (1.4) depend on the Allee constant. The model has two boundary equilibria, one an unstable node and the other a saddle which means that the prey population will not fully reach the environmental capacity. There is a cusp coexistence equilibrium of codimension 2 or 3 if Theorem 6 holds. Furthermore, with the help of the formal series method in [25], we calculated the order of the weak focus to determine the stability of limit cycles bifurcated from Hopf bifurcation, which means that there is the possibility of periodic distributions of prey and predator populations. Choosing the Allee constant and intrinsic growth rate of the predator as bifurcation parameters and by means of a series of near-identity transformations, it was demonstrated that system (1.4) undergoes Bogdanov-Takens bifurcation of codimension 2 through calculating the versal unfolding.
Compared to the work of González-Olivares et al. [23], we focus on the cost of the anti-predation response in the prey reproduction. Our investigation indicates that the cost of fear still does not chang the stability of equilibria. This is similar to the result that the stability of positive equilibrium will not be influenced and the system still undergoes analogous bifurcation.
Different from the work of Sasmal [15], we considered a Leslie-Gower model where the carrying capacity of the predator relies on the abundance of prey. The model has abundant bifurcation phenomena. Through rigorous mathematical analysis, as the Allee constant varies, the model experiences saddle-node bifurcation and there are at most two equilibria in the phase diagram. Meanwhile, the model can undergo more complex bifurcation, Bogdanov-Takens bifurcation, than the traditional predator-prey model. Further, the model undergoes subcritical or supercritical Hopf bifurcation, which means that there is a stable or unstable limit cycle around a positive equilibrium. The model can even exhibit multiple Hopf bifurcations by a set of numerical simulations when the first Lyapunov number is zero.
This work was supported partially by the Natural Science Foundation of Fujian Province (Nos. 2020J01499, 2021J01613) and the National Natural Science Foundation of China (No. 12071418).
The authors declare there is no conflict of interest.
[1] |
X. M. Wang, L. J. Du, Y. Zhang, X. Z. Zhao, X. Z. Cheng, Y. L. Tao, Priority queue-based polling mechanism on seismic equipment cluster monitoring, Cluster Comput., 20 (2017), 611–619. https://doi.org/10.1007/s10586-017-0726-6 doi: 10.1007/s10586-017-0726-6
![]() |
[2] |
G. Sudha, C. Tharini, Trust-based clustering and best route selection strategy for energy efficient wireless sensor networks, Automatika, 64 (2023), 634–641. https://doi.org/10.1080/00051144.2023.2208462 doi: 10.1080/00051144.2023.2208462
![]() |
[3] |
O. V. Semenova, D. T. Bui, The software package and its application to study the polling systems, Vestn. Tomsk. Gos. Univ. Upr. Vychislitelnaja Teh. Inform., 50 (2020), 106–113. https://doi.org/10.17223/19988605/50/13 doi: 10.17223/19988605/50/13
![]() |
[4] |
J. Y. Cao, W. X. Xie, Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time limited services disciplines, Queueing Syst., 85 (2017), 117–147. https://doi.org/10.1007/s11134-016-9504-z doi: 10.1007/s11134-016-9504-z
![]() |
[5] |
Z. Yang, Y. Sun, J. Gan, New polling scheme based on busy/idle queues mechanism, Int. J. Perform. Eng., 14 (2018), 2522–2531. https://doi.org/10.23940/ijpe.18.10.p28.25222531 doi: 10.23940/ijpe.18.10.p28.25222531
![]() |
[6] |
Y. X. Lv, Z. X. Liu, M. H. Bi, C. Hao, Y. R. Zhai, Selective polling and controlled contention based WLAN MAC scheme for low-latency applications, IEEE Commun. Lett., 27 (2023), 1050–1054. https://doi.org/10.1109/LCOMM.2023.3240191 doi: 10.1109/LCOMM.2023.3240191
![]() |
[7] |
B. R. Sathishkumar, An effectual spectrum sharing and battery utilization strategy using fuzzy enabled dynamic relay polling transmission in WSN, J. Intell. Fuzzy Syst., 44 (2023), 4907–4930. https://doi.org/10.3233/JIFS-223001 doi: 10.3233/JIFS-223001
![]() |
[8] |
M. F. Li, C. C. Fang, H. W. Ferng, On-demand energy transfer and energy-aware polling-based MAC for wireless powered sensor networks, Sensors, 22 (2022), 2476. https://doi.org/10.3390/s22072476 doi: 10.3390/s22072476
![]() |
[9] |
S. Siddiqui, A. A. khan, S. Ghani, Achieving energy efficiency in wireless sensor networks using dynamic channel polling and packet concatenation, China Commun., 18 (2021), 249–270. https://doi.org/10.23919/JCC.2021.08.018 doi: 10.23919/JCC.2021.08.018
![]() |
[10] | Z. Yang, Y. Su, H. W. Ding, Analysis of two-level polling system characteristics of exhaustive service and asymmetrically gated service, Acta Automatica Sin., 44 (2018), 2228–2237. |
[11] |
Z. J. Han, H. W. Ding, L. Y. Bao, Z. J. Yang, Q. L. Liu, Analysis of new multi-priority P-CSMA of non-persistent type random multiple access ad hoc network in MAC protocol analysis, Int. J. Commun. Netw. Distr. Syst., 25 (2020), 57–77. https://doi.org/10.1504/IJCNDS.2020.108163 doi: 10.1504/IJCNDS.2020.108163
![]() |
[12] |
Z. Yang, L. Zhu, H. Ding, Z. Guan, A Priority-based parallel schedule polling MAC for wireless sensor networks, J. Commun., 11 (2016), 792–797. https://doi.org/10.12720/jcm.11.8.792-797 doi: 10.12720/jcm.11.8.792-797
![]() |
[13] |
A. Mercian, E. I. Gurrola, F. Aurzada, M. P. McGarry, M. Reisslein, Upstream polling protocols for flow control in PON/xDSL hybrid access networks, IEEE Trans. Commun., 64 (2016), 2971–2984. https://doi.org/10.1109/TCOMM.2016.2576450 doi: 10.1109/TCOMM.2016.2576450
![]() |
[14] |
H. Ding, C. Li, L. Bao, Z. Yang, L. Li, Q. Liu, Research on multi-level priority polling MAC protocol in FPGA tactical data chain, IEEE Access, 7 (2019), 33506–33516. https://doi.org/10.1109/ACCESS.2019.2902488 doi: 10.1109/ACCESS.2019.2902488
![]() |
[15] |
Z. Guan, Z. J. Yang, M. He, W. H. Qian, Analysis of time-delay characteristics of two-level polling control system relying on station states, J. Autom., 42 (2016), 1207–1214. https://doi.org/10.16383/j.aas.2016.c150226 doi: 10.16383/j.aas.2016.c150226
![]() |
[16] |
T. Jiang, X. Lu, L. Liu, J. Lv, X. Chai, Strategic behavior of customers and optimal control for batch service polling systems with priorities, Complexity, 2020 (2020). https://doi.org/10.1155/2020/6015372 doi: 10.1155/2020/6015372
![]() |
[17] |
W. H. Mu, L. Y. Bao, H. W. Ding, Y. F. Zhao, An exact analysis of discrete time two-level priority polling system based on multi-times gated service policy, Acta Electron. Sinica, 46 (2018), 276–280. https://doi.org/10.3969/j.issn.0372-2112.2018.02.003 doi: 10.3969/j.issn.0372-2112.2018.02.003
![]() |
[18] | Z. J. Yang, L. Mao, H. W. Ding, Q. L. Kou, Research on two-level priority polling access control protocol based on continuous time, in 2020 IEEE 6th International Conference on Computer and Communications, (2020), 136–140. https://doi.org/10.1109/ICCC51575.2020.9345093 |
[19] |
Z. J. Yang, Z. Liu, H. W. Ding, Research of continuous-time two-level polling system performance of exhaustive service and gated service, J. Comput. Appl., 39 (2019), 2019–2023. https://doi.org/10.11772/j.issn.1001-9081.2019010063 doi: 10.11772/j.issn.1001-9081.2019010063
![]() |
[20] |
R. Gupta, J. Gupta, Federated learning using game strategies: state-of-the art and future trends, Comput. Netw., 225 (2023), 109650. https://doi.org/10.1016/j.comnet.2023.109650 doi: 10.1016/j.comnet.2023.109650
![]() |
[21] |
P. Boobalan, S. P. Ramu, Q. V. Pham, K. Dev, S. Pandya, P. K. R. Maddikunta, et al., Fusion of federated learning and industrial Internet of Tings: A survey, Comput. Netw., 212 (2022), 109048. https://doi.org/10.1016/j.comnet.2022.109048 doi: 10.1016/j.comnet.2022.109048
![]() |
[22] |
I. Guarino, G. Aceto, D. Ciuonzo, A. Montieri, V. Persico, A. Pescapè, Contextual counters and multimodal Deep Learning for activity-level traffic classification of mobile communication apps during COVID-19 pandemic, Comput. Netw., 219 (2022), 109452. https://doi.org/10.1016/j.comnet.2022.109452 doi: 10.1016/j.comnet.2022.109452
![]() |
[23] |
J. Liu, Q. Wang, Y. Xu, AR-GAIL: adaptive routing protocol for FANETs using generative adversarial imitation learning, Comput. Netw., 218 (2022), 109382. https://doi.org/10.1016/j.comnet.2022.109382 doi: 10.1016/j.comnet.2022.109382
![]() |
[24] |
Y. H. Liu, X. Y. Zhang, W. Y. Liu, Y. Lin, F. Su, J. Cui, et al., Seismic vulnerability and risk assessment at the urban scale using support vector machine and GI science technology: a case study of the Lixia District in Jinan City, China, Geomat. Nat. Haz. Risk, 14 (2023), 1947–5705. https://doi.org/10.1080/19475705.2023.2173663 doi: 10.1080/19475705.2023.2173663
![]() |
[25] |
P. Bhatt, A. L. Maclean, Comparison of high-resolution NAIP and unmanned aerial vehicle (UAV) imagery for natural vegetation communities classification using machine learning approaches, Gisci. Remote Sens., 60 (2023), 2177448. https://doi.org/10.1080/15481603.2023.2177448 doi: 10.1080/15481603.2023.2177448
![]() |
[26] |
M. B. Haile, A. O. Salau, B. Enyew, A. J. Belay, Detection and classification of gastrointestinal disease using convolutional neural network and SVM, Cogent Eng., 9 (2022), 2084878. https://doi.org/10.1080/23311916.2022.2084878 doi: 10.1080/23311916.2022.2084878
![]() |
[27] |
Wang J., Zhao Y. L., Y. C. Fu, L. L. Xia, J. S. Chen, Improving LSMA for impervious surface estimation in an urban area, Eur. J. Remote Sens., 55 (2022), 37–51. https://doi.org/10.1080/22797254.2021.2018666 doi: 10.1080/22797254.2021.2018666
![]() |
[28] |
N. Ali, Z. Halim, S. F. Hussain, An artificial intelligence-based framework for data-driven categorization of computer scientists: a case study of world's Top 10 computing departments, Scientometrics, 128 (2022), 1513–1545. https://doi.org/10.1007/s11192-022-04627-9 doi: 10.1007/s11192-022-04627-9
![]() |
[29] |
B. Akyuz, S. Karatay, F. Erken, Comparison of the performance of the regression models in GPS-Total electron content prediction, Politeknik Dergisi, 26 (2022), 321–328. https://doi.org/10.2339/politeknik.1137658 doi: 10.2339/politeknik.1137658
![]() |
[30] |
X. Y. Li, X. S. Han, M. Yang, Day-Ahead optimal dispatch strategy for active distribution network based on improved deep reinforcement learning, IEEE Access, 10 (2022), 9357–9370. https://doi.org/10.1109/ACCESS.2022.3141824 doi: 10.1109/ACCESS.2022.3141824
![]() |
[31] |
S. Dudey, F. Olimov, M. A. Rafique, J. Kim, M. Jeon, Label-attention transformer with geometrically coherent objects for image captioning, Inform. Sci., 623 (2022), 812–831. https://doi.org/10.1016/j.ins.2022.12.018 doi: 10.1016/j.ins.2022.12.018
![]() |
[32] |
R. Inokuchi, M. Iwagami, Y. Sun, A. Sakamoto, N. Tamiya, Machine learning models predicting under-triage in telephone triage, Ann. Med., 54 (2022), 2990–2997. https://doi.org/10.1080/07853890.2022.2136402 doi: 10.1080/07853890.2022.2136402
![]() |
[33] |
S. Janizadeh, S. M. Bateni, C. Jun, J. Im, H. T. Pai, S. S. Band, et al., Combination four different ensemble algorithms with the generalized linear model (GLM) for predicting forest fire susceptibility, Geomat. Nat. Haz. Risk, 14 (2023), 2206512. https://doi.org/10.1080/19475705.2023.2206512 doi: 10.1080/19475705.2023.2206512
![]() |
[34] |
S. Latif, X. W. Fang, K. Arshid, A. Almuhaimeed, A. Imran, M. Alghamdi, Analysis of birth data using ensemble modeling techniques, Appl. Artif. Intell., 37 (2023), 2158273. https://doi.org/10.1080/08839514.2022.2158273 doi: 10.1080/08839514.2022.2158273
![]() |
[35] |
Y. W. Tang, F. Qiu, B. J. Wang, D. Wu, L. H. Jing, Z. C. Sun, A deep relearning method based on the recurrent neural network for land cover classification, Gisci. Remote Sens., 59 (2022), 1344–1366. https://doi.org/10.1080/15481603.2022.2115589 doi: 10.1080/15481603.2022.2115589
![]() |
[36] |
G. C. Habek, M. A. Tocoglu, A. Onan, Bi-directional CNN-RNN architecture with group-wise enhancement and attention mechanisms for cryptocurrency sentiment analysis, Appl. Artif. Intell., 36 (2022), 2145641. https://doi.org/10.1080/08839514.2022.2145641 doi: 10.1080/08839514.2022.2145641
![]() |
[37] |
L. Jin, S. Li, B. Hu, RNN models for dynamic matrix inversion: A control-theoretical perspective, IEEE Trans. Ind. Inform., 14 (2018), 189–199. https://doi.org/10.1109/TII.2017.2717079 doi: 10.1109/TII.2017.2717079
![]() |
[38] |
Y. S. Zhang, J. Zhang, Y. R. Jiang, G. J. Huang, R. Y. Chen, A text sentiment classification modeling method based on coordinated CNN-LSTM-Attention model, Chinese J. Electr., 28 (2019), 120–126. https://doi.org/10.1049/cje.2018.11.004 doi: 10.1049/cje.2018.11.004
![]() |
[39] |
D. Zhang, G. Lindholm, H. Ratnaweera, Use long short-term memory to enhance Internet of Things for combined sewer overflow monitoring, J. Hydrol., 556 (2018), 409–418. https://doi.org/10.1016/j.jhydrol.2017.11.018 doi: 10.1016/j.jhydrol.2017.11.018
![]() |
[40] |
Y. N. Zhou, S. Y. Wang, T. J. Wu, L. Feng, W. Wu, J. C. Luo, et al., For-backward LSTM-based missing data reconstruction for time-series Landsat images, Gisci. Remote Sens., 59 (2022), 410–430. https://doi.org/10.1080/15481603.2022.2031549 doi: 10.1080/15481603.2022.2031549
![]() |
[41] |
J. Sridhar, R. Gobinath, M. S. Kirgiz, Evaluation of artificial neural network predicted mechanical properties of jute and bamboo fiber reinforced concrete ALONG with silica fume, J. Nat. Fibers, 20 (2023), 2162186. https://doi.org/10.1080/15440478.2022.2162186 doi: 10.1080/15440478.2022.2162186
![]() |
[42] |
W. H. AlAlaween, O. A. Abueed, A. H. AlAlawin, O. H. Abdallah, N. T. Albashabsheh, E. S. AbdelAll, et al., Artificial neural networks for predicting the demand and price of the hybrid electric vehicle spare parts, Cogent Eng., 9 (2022), 2075075. https://doi.org/10.1080/23311916.2022.2075075 doi: 10.1080/23311916.2022.2075075
![]() |
[43] |
T. A. H. Alghamdi, O. T. E. Abdusalam, F. Anayi, M. Packianather, An artificial neural network based harmonic distortions estimator for grid-connected power converter-based applications, Ain Shams Eng. J., 14 (2022), 101916. https://doi.org/10.1016/j.asej.2022.101916 doi: 10.1016/j.asej.2022.101916
![]() |
[44] |
M. K. Wei, X. B. Hu, H. X. Yuan, Residual displacement estimation of the bilinear SDOF systems under the near-fault ground motions using the BP neural network, Adv. Struct. Eng., 25 (2021), 552–571. https://doi.org/10.1177/13694332211058530 doi: 10.1177/13694332211058530
![]() |
[45] |
H. X. Zhou, A. L. Che, X. H. Shuai, Y. Zhang, A spatial evaluation method for earthquake disaster using optimized BP neural network model, Geomat. Nat. Haz. Risk, 14 (2023), 1–26. https://doi.org/10.1080/19475705.2022.2160664 doi: 10.1080/19475705.2022.2160664
![]() |
[46] |
Z. B. Qiu, Z. J. Wu, Y. Song, Sphere gap breakdown voltage prediction based on ISSA optimized BP neural network and effective electric field feature set, IEE J. Trans. Electr., 18 (2022), 506–514. https://doi.org/10.1002/tee.23750 doi: 10.1002/tee.23750
![]() |
[47] |
J. W. Hou, Y. J. Wang, B. Hou, J. Zhou, Q. Tian, Spatial simulation and prediction of air temperature based on CNN-LSTM, Appl. Artif. Intell., 37 (2023), 2166235. https://doi.org/10.1080/08839514.2023.2166235 doi: 10.1080/08839514.2023.2166235
![]() |
[48] |
W. S. Zhang, W. W. Guo, X. Liu, Y. Liu, J. Zhou, B. Li, et al. LSTM-based analysis of industrial IoT equipment. IEEE Access, 6 (2018), 23551–23560. https://doi.org/10.1109/access.2018.2825538 doi: 10.1109/ACCESS.2018.2825538
![]() |
[49] |
X. B. Shu, L. Y. Zhang, Y. L. Sun, J. H. Tang, Host-parasite: Graph LSTM-in-LSTM for group activity recognition, IEEE Trans. Neural Netw. Learning Syst., 32 (2020), 663–674. https://doi.org/10.1109/TNNLS.2020.2978942 doi: 10.1109/TNNLS.2020.2978942
![]() |
[50] |
T. X. Shu, J. H. Chen, V. K. Bhargava, C. W. de Silva, An energy-efficient dual prediction scheme using LMS filter and LSTM in wireless sensor networks for environment monitoring, IEEE Int. Things J., 6 (2019), 6736–6747. https://doi.org/10.1109/JIOT.2019.2911295 doi: 10.1109/JIOT.2019.2911295
![]() |
[51] |
R. Okumura, K. Mizutani, H. Harada, Efficient polling communications for multi-hop networks based on receiver-initiated MAC protocol, Ieice Trans. Commun., E104-B (2021), 550–562. https://doi.org/10.1587/transcom.2020EBP3095 doi: 10.1587/transcom.2020EBP3095
![]() |
[52] |
J. L. Guo, F. F. Li, T. Wang, S. B. Zhang, Y. Q. Zhao, Parameter analysis and optimization of polling-based medium access control protocol for multi-sensor communication, Int. J. Distrib. Sens. Netw., 17 (2021). https://doi.org/10.1177/15501477211007412 doi: 10.1177/15501477211007412
![]() |
[53] |
J. K. van Ommeren, A. Al Hanbali, R. J. Boucherie, Analysis of polling models with a self-ruling server, Queueing Syst., 94 (2020), 77–107. https://doi.org/10.1007/s11134-019-09639-6 doi: 10.1007/s11134-019-09639-6
![]() |
[54] | Y. Y. Sun, Z. J. Yang, Analysis and research on polling system of wireless sensor network, Electr. Meas. Tech., 41 (2018), 100–104. |
[55] |
Z. J. Yang, Q. L. Kou, H. W. Ding, BSCP-MAC: A blockchain-based synchronous control polling MAC protocol for wireless sensor networks, J. Electr. Eng. Technol., 18 (2023), 3799–3810. https://doi.org/10.1007/s42835-023-01440-z doi: 10.1007/s42835-023-01440-z
![]() |
[56] | J. Y. Ge, L. Y. Bao, H. W. Ding, X. Y. Ding, Performance analysis of the first-order characteristics of two-level priority polling system based on parallel gated and exhaustive services mode, in 2021 IEEE 4th International Conference on Electronic Information and Communication Technology (ICEICT), (2021), 10–13. https://doi.org/10.1109/ICEICT53123.2021.9531122 |
[57] | Z. H. Liu, Y. J. Li, J. Q. Yao, Z. N. Cai, G. B. Han, X. Y. Xie, Ultra-short-term forecasting method of wind power based on W-BiLSTM, in 2021 IEEE 4th International Electrical and Energy Conference (CIEEC), (2021), 1–6. https://doi.org/10.1109/CIEEC50170.2021.9511041 |