作者:
He, Ying;Wang, Yan;Yang, Jerry Zhijian;Yin, Hongshuang
期刊:
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS,2024年 ISSN:2079-7362
通讯作者:
Wang, Y
作者机构:
[Yang, Jerry Zhijian; Yin, Hongshuang; He, Ying] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Peoples R China.;[He, Ying] Yanqi Lake Beijing Inst Math Sci & Applicat, Beijing 101408, Peoples R China.;[Wang, Yan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Yan] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构:
[Wang, Y ] 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.
摘要:
Numerical methods for the nonlinear Dirac equation (NDE) in the massless nonrelativistic regime are considered. In this regime, the equation contains a small dimensionless parameter 0 < epsilon <= 1, and its solution is highly oscillatory in time. We present and analyze traditional numerical schemes for the NDE, including finite difference methods, time-splitting methods and exponential integrators. Error analysis indicates that all these methods require an epsilon-dependent time-step size to achieve an optimal convergence order. Utilizing an operator splitting technique, we propose a uniformly accurate (UA) scheme. The scheme enables first-order convergence in time for all epsilon is an element of (0,1] without restrictions on time-step size. Error estimates for the UA scheme are rigorously established and numerical results confirm the properties of the method.
期刊:
British Journal of Psychology,2024年 ISSN:0007-1269
通讯作者:
Zhang, Y
作者机构:
[Li, Hao; Zhang, Ying; Chen, Gaowei] Univ Hong Kong, Fac Educ, Hong Kong, Peoples R China.;[Li, Bo; Li, Na] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;[Zhou, Xin] Sichuan Univ, Middle Sch, Chengdu, Peoples R China.;[Zhang, Ying; Zhang, Y] Univ Hong Kong, Fac Educ, Pokfulam, Hong Kong, Peoples R China.
通讯机构:
[Zhang, Y ] U;Univ Hong Kong, Fac Educ, Pokfulam, Hong Kong, Peoples R China.
摘要:
Bullying victimization can undermine adolescents' well-being. However, few studies have comprehensively investigated the contributions of various victimization forms to well-being and compared which forms were more harmful than others. Evidence on whether resilience and social support moderate such associations is also limited. Using a sample of 12,058 Chinese adolescents in the Programme for International Student Assessment (PISA) 2018, this cross-sectional study aimed to (1) investigate the associations of physical, verbal and relational victimization with well-being; (2) compare the strengths of these associations; and (3) examine the moderating roles of resilience and teacher and parent support in these associations. Results showed that three victimization forms were associated with poorer well-being. Relational and physical victimization were more harmful to most studied well-being outcomes than verbal victimization. Furthermore, resilience weakened the negative effects of physical victimization on negative affect and life satisfaction but aggravated the negative effects of verbal victimization on both outcomes and the negative effect of relational victimization on school belonging. Teacher support intensified the negative effects of physical victimization on school belonging. Parent support was not effective in regulating the victimization-well-being association. The findings underscored the detrimental effects of bullying victimization on adolescents' well-being and the potentially harmful sides of resilience and social support. Implications for bullying prevention were discussed.
作者机构:
[Xia, Sijia; Zhang, Xiongjun] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Qiu, Duo] Wuhan Inst Technol, Sch Math & Phys, Wuhan 430205, Peoples R China.;[Zhang, Xiongjun] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构:
[Qiu, D ] W;Wuhan Inst Technol, Sch Math & Phys, Wuhan 430205, Peoples R China.
摘要:
In this paper, we study the problem of a batch of linearly correlated image alignment, where the observed images are deformed by some unknown domain transformations, and corrupted by additive Gaussian noise and sparse noise simultaneously. By stacking these images as the frontal slices of a third-order tensor, we propose to utilize the tensor factorization method via transformed tensor-tensor product to explore the low-rankness of the underlying tensor, which is factorized into the product of two smaller tensors via transformed tensor-tensor product under any unitary transformation. The main advantage of transformed tensor-tensor product is that its computational complexity is lower compared with the existing literature based on transformed tensor nuclear norm. Moreover, the tensor l(p) (0 < p < 1) norm is employed to characterize the sparsity of sparse noise and the tensor Frobenius norm is adopted to model additive Gaussian noise. A generalized Gauss-Newton algorithm is designed to solve the resulting model by linearizing the domain transformations, and a proximal Gauss-Seidel algorithm is developed to solve the corresponding subproblem. Furthermore, the convergence of the proximal Gauss-Seidel algorithm is established according to Kurdyka-Lojasiewicz property, whose convergence rate is also analyzed. Extensive numerical examples on real-world image datasets are carried out to demonstrate the superior performance of the proposed method as compared to several state-of-the-art methods in both accuracy and computational time.
关键词:
oriented split graphs;planar digraphs;Seymour's second neighbourhood conjecture;Sullivan's second neighbourhood conjecture
摘要:
Abstract For a vertex x $x$ of a digraph, d + ( x ) ${d}^{+}(x)$ (d − ( x ) ${d}^{-}(x)$, respectively) is the number of vertices at distance 1 from (to, respectively) x $x$ and d + + ( x ) ${d}^{++}(x)$ is the number of vertices at distance 2 from x $x$. In 1995, Seymour conjectured that for any oriented graph D $D$ there exists a vertex x $x$ such that d + ( x ) ≤ d + + ( x ) ${d}^{+}(x)\le {d}^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex x $x$ in D $D$ such that d − ( x ) ≤ d + + ( x ) ${d}^{-}(x)\le {d}^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and triangle‐free graphs. An oriented graph D $D$ is an oriented split graph if the vertices of D $D$ can be partitioned into vertex sets X $X$ and Y $Y$ such that X $X$ is an independent set and Y $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when Y $Y$ induces a regular or an almost regular tournament.
摘要:
In this article, we investigate the asymptotic behavior of solutions for a non-autonomous regularized magnetohydrodynamics equations on 3D bounded domains. More precisely, the upper bounds on the number of determining modes and determining nodes for the system are established. The results show that the asymptotic behavior of the weak solution can be determined completely by its first finite number of Fourier modes and the large time behavior of the strong solution can be determined by its values on a finite number of points.
期刊:
BMC PUBLIC HEALTH,2023年23(1):1-10 ISSN:1471-2458
通讯作者:
Wang, N.;Xu, T.
作者机构:
[Li, Lin] Natl Res Inst Family Planning, Clin Med Ctr, Beijing, Peoples R China.;[Xu, Tao; Chen, Feilong] Chinese Acad Med Sci, Inst Basic Med Sci, Dept Epidemiol & Biostat, Beijing, Peoples R China.;[Xu, Tao; Chen, Feilong] Peking Union Med Coll, Sch Basic Med, Beijing, Peoples R China.;[Wang, Ning; Gao, Yiyuan; Li, Xiaoyan] Natl Res Inst Family Planning, Ctr Hlth Qual, Beijing, Peoples R China.;[Diao, Xiyezi; Zhu, Silin] Cent China Normal Univ, Dept Stat, Wuhan, Hubei, Peoples R China.
通讯机构:
[Ning Wang] C;[Tao Xu] D;Center for Health Quality, National Research Institute for Family Planning, Beijing, China<&wdkj&>Department of Epidemiology and Biostatistics Institute of Basic Medical Sciences, Chinese Academy of Medical Sciences & School of Basic Medicine, Peking Union Medical College, Beijing, China
关键词:
The elderly;Calf circumference;Incontinence;Sarcopenia;Risk factor
期刊:
Journal of Algebra,2023年635:463-485 ISSN:0021-8693
通讯作者:
Guo, HY
作者机构:
[Li, Huaimin; Guo, Hongyan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Li, Huaimin; Guo, Hongyan] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构:
[Guo, HY ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要:
In this paper, a family of infinite dimensional Lie algebras G & SIM; is introduced and investigated, called the extended Heisenberg-Virasoro algebra, denoted by G & SIM;. These Lie algebras are related to the N = 2 superconformal algebra and the Bershadsky-Polyakov algebra. We study restricted modules and associated vertex algebras of the Lie algebra G & SIM;. More precisely, we construct its associated vertex (operator) algebras VL & SIM;(e123, 0), and show that the category of vertex algebra VL & SIM;(�123, 0)-modules is equivalent to the category of restricted & SIM; of level & POUND;123. Then we give uniform constructions of simple restricted G & SIM;-modules. Also, we present several equivalent characterizations of simple restricted modules over G & SIM;.& COPY; 2023 Elsevier Inc. All rights reserved. G-modules
期刊:
Journal of Pure and Applied Algebra,2023年227(7):107322 ISSN:0022-4049
通讯作者:
Chengkang Xu
作者机构:
[Guo, Hongyan] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Xu, Chengkang] Shangrao Normal Univ, Shangrao, Jiangxi, Peoples R China.
通讯机构:
[Chengkang Xu] S;Shangrao Normal University, Shangrao, Jiangxi, China
摘要:
This paper studies restricted modules of gap-p Virasoro algebra g(p) and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted g(p)-modules of level (l) under bar and the category of twisted modules of vertex algebra V-Np((l) under bar, 0), where N-p is a new Lie algebra, (l) under bar :=(l(0), 0, center dot center dot center dot, 0) is an element of C[p/2]+1, l(0) is an element of C is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted g(p)-modules of level (l) under bar. More explicitly, we give a uniform construction of simple restricted g(p)-modules as induced modules. We present several equivalent characterizations of simple restricted g(p)-modules, as locally nilpotent (equivalently, locally finite) modules with respect to certain positive part of g(p). Moreover, simple restricted g(p)-modules of level (l) under bar are classified. They are either highest weight modules or simple induced modules. At the end, we exhibit several concrete examples of simple restricted g(p)-modules of level (l) under bar (including Whittaker modules). (c) 2023 Elsevier B.V. All rights reserved.
期刊:
Journal of Mathematical Analysis and Applications,2023年527(1):127340 ISSN:0022-247X
通讯作者:
Zhang, MM
作者机构:
[Zhang, Min-Min; Zhang, MM; Lu, Jian-Feng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Min-Min; Zhang, MM; Lu, Jian-Feng] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Wang, Shi] Hu Bei Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Shi] Hu Bei Univ, Hubei Key Lab Appl Math, Wuhan 430079, Peoples R China.
通讯机构:
[Zhang, MM ] 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.
摘要:
Let t >= 2 be an integer and p, q be two prime numbers, D =E-p circle plus (pq)l-1p Eq be a product-form digit set with l >= 1. Let mu t,D be the self-similar measure generated by the pair (t, D). It is known that Fu et al. [4,17] characterized the spectrality of mu t,D. In this paper, given a spectrum ? of mu t,D, we study some integers b such that b is also a spectrum of mu t,D. (c) 2023 Elsevier Inc. All rights reserved.
期刊:
Journal of Functional Analysis,2023年284(6):109816 ISSN:0022-1236
通讯作者:
Xiaohua Yao
作者机构:
[Li, Ping] Yangtze Univ, Sch Informat & Math, Jingzhou 434000, Peoples R China.;[Soffer, Avy] Rutgers State Univ, Math Dept, New Brunswick, NJ 08903 USA.;[Yao, Xiaohua] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;[Yao, Xiaohua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Peoples R China.
通讯机构:
[Xiaohua Yao] D;Department of Mathematics and Hubei Province Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan, 430079, China
关键词:
Decay estimates;Fourth-order Schrödinger operators;Asymptotic expansion of resolvent;Dimension two
摘要:
In this paper we study the decay estimates of the fourth order Schrodinger operator H = Delta 2 + V(x) on R2 with a bounded decaying potential V(x). We first deduce the asymptotic expansions of resolvent of H near zero threshold in the presence of resonances or eigenvalue, and then use them to establish the L1 - L infinity decay estimates of e-itH generated by the fourth order Schrodinger operator H. Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we also classify these zero resonances as the distributional solutions of H phi = 0 in suitable weighted spaces. Due to the degeneracy of Delta 2 at zero threshold, we remark that the asymptotic expansions of resolvent RV (lambda 4) and the classifications of resonances are much more involved than Schrodinger operator -Delta + V in dimension two.(c) 2022 Elsevier Inc. All rights reserved.
期刊:
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES,2023年170:1-32 ISSN:0021-7824
通讯作者:
Guo, YJ
作者机构:
[Guo, Yujin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Guo, Yujin] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构:
[Guo, YJ ] 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:
Bose-Einstein condensate;Rotational velocity;Nonexistence of vortices;Limit profiles
摘要:
As a continuation of [34], we consider ground states of rotating Bose-Einstein condensates with attractive interactions in non-radially harmonic traps V(x)=x12+Λ2x22 , where 0<Λ≠1 and x=(x1,x2)∈R2 . For any fixed rotational velocity ⁎ 0≤Ω<Ω⁎:=2min{1,Λ} , it is known that ground states exist if and only if ⁎ a<a⁎ for some critical constant ⁎ 0<a⁎<∞ , where a>0 denotes the product of the number of particles and the absolute value of the scattering length. We analyze the asymptotic expansions of ground states as ⁎ a↗a⁎ , which display the visible effect of Ω on ground states. As a consequence, we further prove that ground states do not have any vortex in the region ⁎ R(a):={x∈R2:|x|≤C(a⁎−a)−112} as ⁎ a↗a⁎ for some constant C>0 , which is independent of ⁎ 0<a<a⁎ .
En continuant le travail [34], nous considérons les états fondamentaux des condensats de Bose-Einstein en rotation avec des interactions attractives dans des pièges non radialement harmoniques V(x)=x12+Λ2x22 , où 0<Λ≠1 et x=(x1,x2)∈R2 . Pour toute vitesse de rotation fixe ⁎ 0≤Ω<Ω⁎:=2min{1,Λ} , il est connu que les états fondamentaux existent si et seulement si ⁎ a<a⁎ pour certaine constante critique ⁎ 0<a⁎<∞ , où a>0 désigne le produit du nombre de particules et la valeur absolue de la longueur de dispersion. Nous analysons les développements asymptotiques des états fondamentaux quand ⁎ a↗a⁎ , ce qui montrent l'effet visible de Ω sur les états fondamentaux. Comme conséquence, nous prouvons que les états fondamentaux ne présentent aucun vortex dans la région ⁎ R(a):={x∈R2:|x|≤C(a⁎−a)−112} quand ⁎ a↗a⁎ pour une certaine constante C>0 , qui est indépendante de ⁎ 0<a<a⁎ .
期刊:
Mathematical Methods in the Applied Sciences,2023年46(5):5099-5118 ISSN:0170-4214
通讯作者:
Hangzhou Hu<&wdkj&>Hangzhou Hu Hangzhou Hu Hangzhou Hu
作者机构:
[Hu, Hangzhou; Zhao, Dun] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Gansu, Peoples R China.;[Li, Yuan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.
通讯机构:
[Hangzhou Hu; Hangzhou Hu Hangzhou Hu Hangzhou Hu] S;School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000 Gansu, China
关键词:
asymptotic behavior;ground states;NLS equations;periodic magnetic field;variational methods for second-order elliptic equations;XFEL Schrodinger equation
摘要:
We consider the following X‐ray free electron lasers Schrödinger equation (i∇−A)2u+V(x)u−μ|x|u=1|x|∗|u|2u−K(x)|u|q−2u,x∈ℝ3,$$ {\left(i\nabla -A\right)}&#x0005E;2u&#x0002B;V(x)u-\frac{\mu }{\mid x\mid }u&#x0003D;\left(\frac{1}{\mid x\mid}\ast {\left&#x0007C;u\right&#x0007C;}&#x0005E;2\right)u-K(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{q-2}u,x\in {\mathbb{R}}&#x0005E;3, $$ where A∈Lloc2(ℝ3,ℝ3)$$ A\in {L}_{loc}&#x0005E;2\left({\mathbb{R}}&#x0005E;3,{\mathbb{R}}&#x0005E;3\right) $$ denotes the magnetic potential such that the magnetic field B=curlA$$ B&#x0003D;\operatorname{curl}\kern0.4em A $$ is ℤ3$$ {\mathbb{Z}}&#x0005E;3 $$‐periodic, μ∈ℝ,K∈L∞ℝ3$$ \mu \in \mathbb{R},K\in {L}&#x0005E;{\infty}\left({\mathbb{R}}&#x0005E;3\right) $$ is ℤ3$$ {\mathbb{Z}}&#x0005E;3 $$ periodic and non‐negative, q∈(2,4)$$ q\in \left(2,4\right) $$. Using the variational method, based on a profile decomposition of the Cerami sequence in HA1ℝ3$$ {H}_A&#x0005E;1\left({\mathbb{R}}&#x0005E;3\right) $$, we obtain the existence of the ground state solution for suitable μ≥0$$ \mu \ge 0 $$. When μ<0$$ \mu &lt;0 $$ is small, we also obtain the non‐existence. Furthermore, we give a description of the asymptotic behavior of the ground states as μ→0+$$ \mu \to {0}&#x0005E;{&#x0002B;} $$.
作者机构:
[Yaping WU] School of Artificial Intelligence,Jianghan University,Hubei 430056,P.R.China;[Huiqing LIU] School of Mathematics and Statistics,Hubei University,Hubei 430062,P.R.China;[Qiong FAN] School of Mathematics and Statistics,Central China Normal University,Hubei 430079,P.R.China
摘要:
Given a simple graph G, the oriented graph Gσ is obtained from G by orienting each edge and G is called the underlying graph of Gσ. The skew-symmetric adjacency matrix S(Gσ)of Gσ, where the(u, v)-entry is 1 if there is an arc from u to v, and -1 if there is an arc from v to u(and 0 otherwise), has eigenvalues of 0 or pure imaginary. The k-th-skew spectral moment of Gσis the sum of power k of all eigenvalues of S(Gσ), where k is a non-negative integer. The skew spectral moments can be used to produce graph catalogues. In this paper, we researched the skew spectral moments of some oriented trees and oriented unicyclic graphs and produced their catalogues in lexicographical order. We determined the last 2■ oriented trees with underlying graph of diameter d and the last 2■ + 1 oriented unicyclic graphs with underlying graph of girth g, respectively.
期刊:
Communications in Statistics - Theory and Methods,2023年52(15):5470-5482 ISSN:0361-0926
通讯作者:
Zhao, Hui
作者机构:
[Luo, Lin] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;[Zhao, Hui] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430073, Peoples R China.
通讯机构:
[Zhao, Hui] Z;Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan 430073, Peoples R China.
关键词:
Estimating equation;propensity score;linear transformation model;interval-censored data
摘要:
Interval-censored failure time data often occur in medical follow-up studies among other areas. Regression analysis of linear transformation models with interval-censored data has been investigated by several authors under different contexts, but most of the existing methods assume that the covariates are discrete because these methods rely on the estimation of conditional survival distribution function. Without this assumption, this paper constructs a new generalized estimating equation using the propensity score. The proposed inference procedure does not need to estimate the conditional survival distribution any more and then can be used not only in the discrete but also in the continuous covariate situation. The asymptotic properties of the resulting estimates are given, and an extensive simulation study is performed. Finally, the application to two real datasets is also provided. Key words: Estimating equation; Interval-censored data; Propensity score; Linear transformation model.
期刊:
Journal of Mathematical Biology,2023年86(6):1-45 ISSN:0303-6812
通讯作者:
Huang, JC;Wang, H
作者机构:
[Zhang, Yuyue; Huang, Jicai] Cent China Normal Univ, Sch Math & Stat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;[Zhang, Yuyue; Huang, Jicai] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;[Wang, Hao] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.;[Wang, Hao] Univ Alberta, Interdisciplinary Lab Math Ecol & Epidemiol, Edmonton, AB T6G 2G1, Canada.
通讯机构:
[Wang, H ] U;[Huang, JC ] C;Cent China Normal Univ, Sch Math & Stat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.
关键词:
Degenerate Bogdanov–Takens bifurcation;Generalist predator;Nilpotent cusp of codimension 4;Nilpotent focus of codimension 3;Predator–prey model;Specialist predator
摘要:
In this paper, we revisit a predator-prey model with specialist and generalist predators proposed by Hanski et al. (J Anim Ecol 60:353-367, 1991) , where the density of generalist predators is assumed to be a constant. It is shown that the model admits a nilpotent cusp of codimension 4 or a nilpotent focus of codimension 3 for different parameter values. As the parameters vary, the model can undergo cusp type (or focus type) degenerate Bogdanov-Takens bifurcations of codimension 4 (or 3). Our results indicate that generalist predation can induce more complex dynamical behaviors and bifurcation phenomena, such as three small-amplitude limit cycles enclosing one equilibrium, one or two large-amplitude limit cycles enclosing one or three equilibria, three limit cycles appearing in a Hopf bifurcation of codimension 3 and dying in a homoclinic bifurcation of codimension 3. In addition, we show that generalist predation stabilizes the limit cycle driven by specialist predators to a stable equilibrium, which clearly explains the famous Fennoscandia phenomenon.