期刊：
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS,2023年53(5):1571-1585 ISSN：0035-7596
通讯作者：
Li, SC
作者机构：
[Sun, Wanting] Shandong Univ, Data Sci Inst, Jinan, Peoples R China.;[Li, Shuchao; Li, SC] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan, Peoples R China.;[Li, Shuchao; Li, SC] Cent China Normal Univ, Fac Math & Stat, Wuhan, Peoples R China.;[Li, Xuechao] Univ Georgia, Div Acad Enhancement, Athens, GA USA.
通讯机构：
[Li, SC ] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan, Peoples R China.;Cent China Normal Univ, Fac Math & Stat, Wuhan, Peoples R China.
摘要：
A graph is split if its vertex set can be partitioned into a clique and an independent set. A split graph is (x,y)-bidegreed if each of its vertex degrees is equal to either x or y. Each connected split graph is of diameter at most 3. In 2017, Nikiforov proposed the A(alpha)-matrix, which is the convex combination of the adjacency matrix and the diagonal matrix of vertex degrees of the graph G. It is well-known that a connected graph of diameter l contains at least l+1 distinct A(alpha)-eigenvalues. A graph is said to be l alpha-extremal with respect to its A(alpha)-matrix if the graph is of diameter l having exactly l+1 distinct A(alpha)-eigenvalues. In this paper, using the association of split graphs with combinatorial designs, the connected 2(alpha)-extremal (resp. 3(alpha)-extremal) bidegreed split graphs are classified. Furthermore, all connected bidegreed split graphs of diameter 2 having just 4 distinct A(alpha)-eigenvalues are identified.
期刊：
Results in Mathematics,2023年78(3):1-19 ISSN：1422-6383
通讯作者：
Shuchao Li<&wdkj&>Wanting Sun
作者机构：
[Li, Shuchao; Sun, Wanting] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Shuchao Li; Wanting Sun] F;Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China<&wdkj&>Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China
关键词：
Adjacency matrix;second largest eigenvalue;outerplanar graph;\(\{K_{2, 3}, K_4\}\)-minor free graph
摘要：
Let
$$\lambda _2$$
be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2021, Liu, Chen and Stanić determined all the connected
$$\{K_{1,3}, K_5 -e\}$$
-free graphs whose second largest eigenvalue
$$\lambda _2\leqslant 1$$
. In this paper, we completely identify all the connected
$$\{K_{2,3},K_4\}$$
-minor free graphs whose second largest eigenvalue does not exceed 1. That is, we characterize all the connected outerplanar graphs satisfying
$$\lambda _2\leqslant 1$$
. Furthermore, all the maximal outerplanar graphs having the same property can be deduced by our result obtained in this paper. Our main tools include analyzing the local structure of the outerplanar graph with respect to its girth.
期刊：
Applied Mathematics and Computation,2023年438:127556 ISSN：0096-3003
通讯作者：
Sun, Wanting(wtsun2018@sina.com)
作者机构：
[Li, Shuchao; Liu, Xin; Sun, Wanting] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Yan, Lixia] Wuhan Business Univ, Sch Informat & Engn, Wuhan 430070, Peoples R China.
通讯机构：
[Wanting Sun] F;Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China
摘要：
The Steiner k-eccentricity of a vertex in a graph G is the maximum Steiner distance over all k-subsets containing the vertex. The average Steiner k-eccentricity of G is the mean value of all vertices' Steiner k-eccentricities in G. Let T-n be the set of all n-vertex trees, T-n,T-Delta be the set of n-vertex trees with maximum degree Delta, T-n,Delta(k) be the set of n-vertex trees with exactly k vertices of a given maximum degree Delta, and let MTnk be the set of n-vertex trees with exactly k vertices of maximum degree. In this paper, we first determine the sharp upper bound on the average Steiner 3-eccentricity of n-vertex trees with a given degree sequence. The corresponding extremal graphs are characterized. Consequently, together with majorization theory, all graphs among T-n,Delta(k) (resp. T-n,T-Delta, MTnk, T-n) having the maximum average Steiner 3-eccentricity are identified. Then we characterize the unique n-vertex tree with a given segment sequence having the minimum average Steiner 3-eccentricity. Finally, we determine all n-vertex trees with a given number of segments having the minimum average Steiner 3-eccentricity. (C) 2022 Elsevier Inc. All rights reserved.
期刊：
Journal of Mathematical Biology,2023年86(2):1-3 ISSN：0303-6812
通讯作者：
Jicai Huang<&wdkj&>Hao Wang
作者机构：
[Huang, Jicai; Pan, Qin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hao] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.
通讯机构：
[Jicai Huang] S;[Hao Wang] D;School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China<&wdkj&>Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
作者机构：
[Li, Gongbao] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, GB ] C;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
Kirchhoff type equations;potential functions having non-degenerate critical points;the Lyapunov-Schmidt reduction method;multi-peak solutions;existence and local uniqueness
摘要：
In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations
$$ - \left({{\varepsilon ^2}a + \varepsilon b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}}} \right)\,\,\Delta u + V(x)u = {u^p},\,\,\,\,\,\,u > 0\,\,\,\,\,{\rm{in}}\,\,\,{\mathbb{R}^3},$$
which concentrate at non-degenerate critical points of the potential function V(x), where a, b > 0, 1 < p < 5 are constants, and ε > 0 is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at {ai}1≤i≤k, where {ai}1≤i≤k are non-degenerate critical points of V(x) as ε → 0.
期刊：
IEEE Transactions on Information Theory,2023年69(2):941-950 ISSN：0018-9448
通讯作者：
Liu, H.
作者机构：
[Chen, Bocong] South China Univ Technol, Sch Math, Guangzhou 510641, Peoples R China.;[Liu, Hongwei] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Liu, H.] C;Central China Normal University, China
关键词：
Codes;Measurement;Upper bound;Symbols;Shape;Hamming distances;Decoding;Symbol-pair read channel;symbol-pair code;coding theory;codes for storage media
作者机构：
[Yu, Yuantian; Zhou, Zihan] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Geng, Xianya] Anhui Univ Sci & Technol, Sch Math & Big Data, Huainan, Peoples R China.
通讯机构：
[Xianya Geng] S;School of Mathematics and Big Data, Anhui University of Science & Technology, Huainan, China
摘要：
This article gives some fundamental introduction to spectra of mixed graphs via its kgeneralized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from uto vis equal to the kth root of unity e(-2pi/k) (and its symmetric entry is e(-2pi/k)); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set {1, e(2pi/k), e(-2pi/k)}, which is not closed under multiplication when k >= 4. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank (H-k-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with H-k-rank 2 and 3, respectively. (C) 2022 Elsevier B.V. All rights reserved.
摘要：
A strong edge-coloring of a graph G is a proper edge-coloring such that every path of length 3 uses three different colors. The strong chromatic index of G, denoted by chi(s)'(G), is the least possible number of colors in a strong edge-coloring of G. Let G be a graph, mad(G) be the maximum average degree and delta be the maximum degree of G. In this paper, we prove that if delta >= 6 and mad(G) < 23/8 , then chi is(G) <= 3 delta - 1; if delta >= 7 and mad(G) < 26/9 , then chi is(G) <= 3 delta -1, which partially improves the result of Choi et al. (2018) who proved that if delta > 7 and mad(G) < 3, then chi is(G) <= 3 delta. (C) 2022 Elsevier B.V. All rights reserved.
期刊：
IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS,2023年27(6):3061-3071 ISSN：2168-2194
通讯作者：
Zhao, Weizhong;Shen, XJ
作者机构：
[Shen, Xianjun; Wang, Haodong; Wang, Yue; Zhao, Weizhong; Zhao, WZ; Shen, XJ; Jiang, Xingpeng; Li, Dandan] Cent China Normal Univ, Sch Comp, Hubei Prov Key Lab Artificial Intelligence & Smart, Wuhan 430079, Peoples R China.;[Sun, Han] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Shen, Xianjun; Wang, Haodong; Wang, Yue; Zhao, Weizhong; Zhao, WZ; Shen, XJ; Jiang, Xingpeng; Li, Dandan] Cent China Normal Univ, Natl Language Resources Monitoring & Res Ctr Netwo, Wuhan 430079, Peoples R China.
通讯机构：
[Zhao, WZ; Shen, XJ ] C;Cent China Normal Univ, Sch Comp, Hubei Prov Key Lab Artificial Intelligence & Smart, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Natl Language Resources Monitoring & Res Ctr Netwo, Wuhan 430079, Peoples R China.
关键词：
graph representation learning;heterogeneous information network;multi-head attention mechanism;Phage-host interactions prediction
摘要：
In the treatment of bacterial infectious diseases, overuse of antibiotics may lead to not only bacterial resistance to antibiotics but also dysbiosis of beneficial bacteria which are essential for maintaining normal human life activities. Instead, phage therapy, which invades and lyses specific pathogenic bacteria without affecting beneficial bacteria, becomes more and more popular to treat bacterial infectious diseases. For the effective phage therapy, it requires to accurately predict potential phage-host interactions from heterogeneous information network consisting of bacteria and phages. Although many models have been proposed for predicting phage-host interactions, most methods fail to consider fully the sparsity and unconnectedness of phage-host heterogeneous information network, deriving the undesirable performance on phage-host interactions prediction. To address the challenge, we propose an effective model called GERMAN-PHI for predicting Phage-Host Interactions via Graph Embedding Representation learning with Multi-head Attention mechaNism. In GERMAN-PHI, the multi-head attention mechanism is utilized to learn representations of phages and hosts from multiple perspectives of phage-host associations, addressing the sparsity and unconnectedness in phage-host heterogeneous information network. More specifically, a module of GAT with talking-heads is employed to learn representations of phages and bacteria, on which neural induction matrix completion is conducted to reconstruct the phage-host association matrix. Results of comprehensive experiments demonstrate that GERMAN-PHI performs better than the state-of-the-art methods on phage-host interactions prediction. In addition, results of case study for two high-risk human pathogens show that GERMAN-PHI can predict validated phages with high accuracy, and some potential or new associated phages are provided as well.
摘要：
It has been established that the local mass of blow-up solutions to Toda systems associated with the simple Lie algebras |$\textbf{A}_{n}$|, |$\textbf{B}_{n}$|, |$\textbf{C}_{n}$|, and |$\textbf{G}_{2}$| can be represented by a finite Weyl group. In particular, at each blow-up point, after a sequence of bubbling steps (via scaling) is performed, the transformation of the local mass at each step corresponds to the action of an element in the Weyl group. In this article, we present the results in the same spirit for the affine |$\textbf{B}_{2}^{(1)}$| Toda system with singularities. Compared with the Toda system with simple Lie algebras, the computation of local masses is more challenging due to the infinite number of elements of the affine Weyl group of type |$\textbf{B}_{2}^{(1)}$|. In order to give an explicit expression for the local mass formula, we introduce two free integers and write down all the possibilities into |$8$| types. This shows a striking difference to previous results on Toda systems with simple Lie algebras. The main result of this article seems to provide the first major advance in understanding the relation between the blow-up analysis of affine Toda system and the affine Weyl group of the associated Lie algebras.
期刊：
MATHEMATICAL BIOSCIENCES AND ENGINEERING,2023年20(2):1938-1959 ISSN：1547-1063
作者机构：
[Huang, Gang; Yang, Yu] China Univ Geosci, Sch Math & Phys, Wuhan 430074, Peoples R China.;[Dong, Yueping] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
HIV infection model;intracellular delay;immune response delay;stability analysis;Hopf bifurcation
摘要：
This work focuses on an HIV infection model with intracellular delay and immune response delay, in which the former delay refers to the time it takes for healthy cells to become infectious after infection, and the latter delay refers to the time when immune cells are activated and induced by infected cells. By investigating the properties of the associated characteristic equation, we derive sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model. Based on normal form theory and center manifold theorem, the stability and the direction of the Hopf bifurcating periodic solutions are studied. The results reveal that the intracellular delay cannot affect the stability of the immunity-present equilibrium, but the immune response delay can destabilize the stable immunity-present equilibrium through the Hopf bifurcation. Numerical simulations are provided to support the theoretical results.
作者机构：
[Lim, MF; Lim, Meng Fai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Lim, MF; Lim, Meng Fai] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Lim, MF ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
even K-groups;class groups;Kummer's criterion
期刊：
Calculus of Variations and Partial Differential Equations,2023年62(7):1-31 ISSN：0944-2669
通讯作者：
Guo, YJ
作者机构：
[Guo, Yujin] Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, POB 71010, Wuhan 430079, Peoples R China.;[Li, Yan; Liu, Qiang] Cent China Normal Univ, Sch Math & Stat, POB 71010, Wuhan 430079, Peoples R China.;[Lu, Lu] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430073, Peoples R China.
通讯机构：
[Guo, YJ ] C;Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, POB 71010, Wuhan 430079, Peoples R China.
摘要：
We study ground states of attractive Bose gases, which are confined in a harmonic trap
$$V(x)=x_1^2+\Lambda x_2^2$$
(
$$\Lambda \ge 1$$
) rotating at the velocity
$$\Omega $$
. For any
$$0\le \Omega <\Omega ^*:=2$$
, where
$$\Omega ^*$$
is called a critical rotational velocity, it is well known that ground states exist if and only if
$$ a<a^*$$
for some critical constant
$$0<a^*<\infty $$
, where
$$a>0$$
denotes the product for the number of particles times the absolute value of the scattering length. In this paper, we consider the critical rotating case, where the rotational velocity
$$\Omega =\Omega ^*$$
, to study the existence and non-existence of ground states with respect to
$$a>0$$
. As imposed in Remark 2.2 of Lewin et al. (Blow-up profile of rotating 2D focusing bose gases. macroscopic limits of quantum systems, Springer, Berlin, 2018), we also analyze the limiting behavior of ground states as
$$a\nearrow a^*$$
for the case where
$$\Omega =\Omega _{a}:=\Omega ^*\sqrt{1-C_0(a^*-a)^m}\nearrow \Omega ^*$$
,
$$0\le m<\frac{1}{2}$$
and
$$0<C_0<1$$
.
期刊：
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY,2023年31(05):2350037 ISSN：0218-348X
通讯作者：
Yi, SF
作者机构：
[Pan, Yuewei] GuiZhou Normal Univ, Sch Math Sci, Guiyang 550001, Guizhou, Peoples R China.;[Yi, Shanfeng; Yi, SF] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构：
[Yi, SF ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
摘要：
Let Sigma(infinity)(i=1) c(i)R(i)(x) be the Rademacher series, where {R-i(x)}(i=1)(infinity) is the classical Rademacher function system and {c(i)}(i=1)(infinity) is an arbitrary real number sequence. In this paper, we first show that the value range of the Rademacher series at any subinterval of [0, 1] is R boolean OR {+/- 8} when {c(i)}(1)(infinity) is an element of l(2)\l(1). This result provides us with the basic facts that when {c(i)}(1)(infinity) is an element of l(2)\l(1), the Rademacher series cannot converge to an approximate continuous function, and there is no approximate limit at any point of [0,1]. Further, when {c(i)}(1)(infinity) is an element of l(2)\l(1), we show various dimensions of the level set of Rademacher series on any subinterval of [0, 1]. Finally, we give the relationship between the box dimension and the coefficient of Rademacher series when {ci}(1)(infinity) is an element of l(1), and the exact values of box dimension, packing dimension and Hausdorff dimension are obtained in some special cases.
期刊：
Journal of Fixed Point Theory and Applications,2023年25(1):1-22 ISSN：1661-7738
通讯作者：
Tao Yang
作者机构：
[Luo, Xiao] Hefei Univ Technol, Sch Math, Hefei 230009, Peoples R China.;[Yang, Tao] Zhejiang Normal Univ, Dept Math, Jinhua 321004, Zhejiang, Peoples R China.;[Yang, Tao] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Tao Yang] D;Department of Mathematics, Zhejiang Normal University, Jinhua, People’s Republic of China<&wdkj&>School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China
摘要：
We consider ground states of three-dimensional dipolar Bose–Einstein condensate, which can be described equivalently by positive
$$L^2$$
-constraint critical points of the Gross–Pitaevskii energy functional involving mass-subcritical perturbation. When the physical parameters describing the strength of nonlinearities lie in a defined regime, the corresponding energy functional is unbounded on the
$$L^2$$
-spheres, so we turn to study a suitable local minimization problem and prove the existence of ground states. Furthermore, we show that any ground state is a local minimizer and it is stable under the Cauchy flow, which indicates that the mass-subcritical perturbation stabilizes the system that is initially unstable. Finally, by refining the upper bound of the ground state energy, we provide a precise description of the asymptotic behavior of the ground states as the mass vanishes.
摘要：
This paper investigates the sparse beta model with l(1) penalty in the field of network data models, which is a hot topic in both statistical and social network research. We present a refined algorithm designed for parameter estimation in the proposed model. Its effectiveness is highlighted through its alignment with the proximal gradient descent method, stemming from the convexity of the loss function. We study the estimation consistency and establish an optimal bound for the proposed estimator. Empirical validations facilitated through meticulously designed simulation studies corroborate the efficacy of our methodology. These assessments highlight the prospective contributions of our methodology to the advanced field of network data analysis.
作者机构：
[Fan, Aihua] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Fan, Aihua] Univ Picardie, LAMFA, UMR 7352, CNRS, F-80039 Amiens, France.;[Meyer, Yves] Univ Paris Saclay, CNRS, ENS Cachan, CMLA, F-91190 Paris, France.
通讯机构：
[Aihua Fan] S;School of Mathematics and Statistics, Central China Normal University, Wuhan, China<&wdkj&>LAMFA, CNRS UMR 7352, University of Picardie, Amiens, France
摘要：
The random trigonometric series
$$\sum\nolimits_{n = 1}^\infty {{\rho _n}\cos \left( {nt + {\omega _n}} \right)} $$
on the circle
$$\mathbb{T}$$
are studied under the conditions ∑∣ρn∣2 = ∞ and ρn → 0, where {ωn} are independent and uniformly distributed random variables on
$$\mathbb{T}$$
. They are almost surely not Fourier-Stieltjes series but determine pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed. The behavior of the partial sums of the above series is proved to be multifractal. Our theory holds on the torus
$${\mathbb{T}^d}$$
of dimension d ⩾ 1.
期刊：
Journal of Functional Analysis,2023年284(10):109882 ISSN：0022-1236
作者机构：
[Chen, Haixia] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Medina, Maria] Univ Autonoma Madrid, Dept Matemat, Ciudad Univ Cantoblanco, Madrid 28049, Spain.;[Pistoia, Angela] Sapienza Univ Roma, Dipartimento Sci Base & Applicate Ingn, Via Scarpa 16, I-00161 Rome, Italy.
摘要：
We consider the elliptic system -Delta u(i) = u(i)(3)+ Sigma(q+1)(j=1 j not equal i) beta(ij)u(i)u(j)(2) in R-4, i = 1, ..., q + 1, when alpha := beta(ij) and beta := beta(i(q+1)) = beta((q+1)j) for any i, j = 1, ... , q. If beta < 0 and |beta| is small enough we build solutions such that each component u(1), . . . , u(q) blows-up at the vertices of q polygons placed in different great circles which are linked to each other, and the last component u(q+1) looks like the radial positive solution of the single equation. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org /licenses/by-nc-nd /4 .0/).
期刊：
Mathematical Methods in the Applied Sciences,2023年46(4):3429-3444 ISSN：0170-4214
通讯作者：
Shuang Li<&wdkj&>Shuang Li Shuang Li Shuang Li
作者机构：
[Li, Shuang] Henan Normal Univ, Coll Math & Informat Sci, Xinxiang 453007, Henan, Peoples R China.;[Li, Yong] Yangtze Univ, Sch Informat & Math, Jingzhou, Peoples R China.;[Zhang, Xinan] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Shuang Li; Shuang Li Shuang Li Shuang Li] C;College of Mathematics and Information Science, Henan Normal University, Xinxiang, China
关键词：
Lévy noise;persistence in the mean;predator–prey;stochastic perturbation
摘要：
An eco-epidemiological system with Levy noise and white noise is proposed. In this prey-predator system, assume that the predator population is infected by disease and divided into two classes: susceptible predator and infected predator. The existence of global positive solution is derived. Threshold conditions of extinction and persistence in the mean for prey and predator population are given by Ito's formula and comparison theorem for stochastic model. The long-time behavior of this random system is also discussed by constructing suitable Lyapunov function, which shows the relationship between the solution of stochastic system and the equilibria of corresponding deterministic system. Finally, these results are verified by numerical simulation. Our results illustrate whether Levy noise and white noise are both unfavorable to the population size of species.
