On periodic solutions of 2-periodic Lyness difference equations

We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears.Comment: 27 pages; 1 figur

Invariant Manifolds for Competitive Systems in the Plane

Let $T$ be a competitive map on a rectangular region $\mathcal{R}\subset \mathbb{R}^2$, and assume $T$ is $C^1$ in a neighborhood of a fixed point $\bar{\rm x}\in \mathcal{R}$. The main results of this paper give conditions on $T$ that guarantee the existence of an invariant curve emanating from $\bar{\rm x}$ when both eigenvalues of the Jacobian of $T$ at $\bar{\rm x}$ are nonzero and at least one of them has absolute value less than one, and establish that $\mathcal{C}$ is an increasing curve that separates $\mathcal{R}$ into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. Several applications to planar systems of difference equations with non-hyperbolic equilibria are given.Comment: 20 pages, 2 figure

Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

By using the KAMtheory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: xn+1 = (2axn)/(1 + x2n) – xn-1, n = 0, 1, …, where x-1, x0, ∈ (−∞,∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions

Enhancing Discovery of Genetic Variants for Posttraumatic Stress Disorder Through Integration of Quantitative Phenotypes and Trauma Exposure Information

Funding Information: This work was supported by the National Institute of Mental Health / U.S. Army Medical Research and Development Command (Grant No. R01MH106595 [to CMN, IL, MBS, KJRe, and KCK], National Institutes of Health (Grant No. 5U01MH109539 to the Psychiatric Genomics Consortium ), and Brain & Behavior Research Foundation (Young Investigator Grant [to KWC]). Genotyping of samples was provided in part through the Stanley Center for Psychiatric Genetics at the Broad Institute supported by Cohen Veterans Bioscience . Statistical analyses were carried out on the LISA/Genetic Cluster Computer ( https://userinfo.surfsara.nl/systems/lisa ) hosted by SURFsara. This research has been conducted using the UK Biobank resource (Application No. 41209). This work would have not been possible without the financial support provided by Cohen Veterans Bioscience, the Stanley Center for Psychiatric Genetics at the Broad Institute, and One Mind. Funding Information: MBS has in the past 3 years received consulting income from Actelion, Acadia Pharmaceuticals, Aptinyx, Bionomics, BioXcel Therapeutics, Clexio, EmpowerPharm, GW Pharmaceuticals, Janssen, Jazz Pharmaceuticals, and Roche/Genentech and has stock options in Oxeia Biopharmaceuticals and Epivario. In the past 3 years, NPD has held a part-time paid position at Cohen Veterans Bioscience, has been a consultant for Sunovion Pharmaceuticals, and is on the scientific advisory board for Sentio Solutions for unrelated work. In the past 3 years, KJRe has been a consultant for Datastat, Inc., RallyPoint Networks, Inc., Sage Pharmaceuticals, and Takeda. JLM-K has received funding and a speaking fee from COMPASS Pathways. MU has been a consultant for System Analytic. HRK is a member of the Dicerna scientific advisory board and a member of the American Society of Clinical Psychopharmacology Alcohol Clinical Trials Initiative, which during the past 3 years was supported by Alkermes, Amygdala Neurosciences, Arbor Pharmaceuticals, Dicerna, Ethypharm, Indivior, Lundbeck, Mitsubishi, and Otsuka. HRK and JG are named as inventors on Patent Cooperative Treaty patent application number 15/878,640, entitled “Genotype-guided dosing of opioid agonists,” filed January 24, 2018. RP and JG are paid for their editorial work on the journal Complex Psychiatry. OAA is a consultant to HealthLytix. All other authors report no biomedical financial interests or potential conflicts of interest. Funding Information: This work was supported by the National Institute of Mental Health/ U.S. Army Medical Research and Development Command (Grant No. R01MH106595 [to CMN, IL, MBS, KJRe, and KCK], National Institutes of Health (Grant No. 5U01MH109539 to the Psychiatric Genomics Consortium), and Brain & Behavior Research Foundation (Young Investigator Grant [to KWC]). Genotyping of samples was provided in part through the Stanley Center for Psychiatric Genetics at the Broad Institute supported by Cohen Veterans Bioscience. Statistical analyses were carried out on the LISA/Genetic Cluster Computer (https://userinfo.surfsara.nl/systems/lisa) hosted by SURFsara. This research has been conducted using the UK Biobank resource (Application No. 41209). This work would have not been possible without the financial support provided by Cohen Veterans Bioscience, the Stanley Center for Psychiatric Genetics at the Broad Institute, and One Mind. This material has been reviewed by the Walter Reed Army Institute of Research. There is no objection to its presentation and/or publication. The opinions or assertions contained herein are the private views of the authors and are not to be construed as official or as reflecting true views of the U.S. Department of the Army or the Department of Defense. We thank the investigators who comprise the PGC-PTSD working group and especially the more than 206,000 research participants worldwide who shared their life experiences and biological samples with PGC-PTSD investigators. We thank Mark Zervas for his critical input. Full acknowledgments are in Supplement 1. MBS has in the past 3 years received consulting income from Actelion, Acadia Pharmaceuticals, Aptinyx, Bionomics, BioXcel Therapeutics, Clexio, EmpowerPharm, GW Pharmaceuticals, Janssen, Jazz Pharmaceuticals, and Roche/Genentech and has stock options in Oxeia Biopharmaceuticals and Epivario. In the past 3 years, NPD has held a part-time paid position at Cohen Veterans Bioscience, has been a consultant for Sunovion Pharmaceuticals, and is on the scientific advisory board for Sentio Solutions for unrelated work. In the past 3 years, KJRe has been a consultant for Datastat, Inc. RallyPoint Networks, Inc. Sage Pharmaceuticals, and Takeda. JLM-K has received funding and a speaking fee from COMPASS Pathways. MU has been a consultant for System Analytic. HRK is a member of the Dicerna scientific advisory board and a member of the American Society of Clinical Psychopharmacology Alcohol Clinical Trials Initiative, which during the past 3 years was supported by Alkermes, Amygdala Neurosciences, Arbor Pharmaceuticals, Dicerna, Ethypharm, Indivior, Lundbeck, Mitsubishi, and Otsuka. HRK and JG are named as inventors on Patent Cooperative Treaty patent application number 15/878,640, entitled ?Genotype-guided dosing of opioid agonists,? filed January 24, 2018. RP and JG are paid for their editorial work on the journal Complex Psychiatry. OAA is a consultant to HealthLytix. All other authors report no biomedical financial interests or potential conflicts of interest. Publisher Copyright: © 2021 Society of Biological PsychiatryBackground: Posttraumatic stress disorder (PTSD) is heritable and a potential consequence of exposure to traumatic stress. Evidence suggests that a quantitative approach to PTSD phenotype measurement and incorporation of lifetime trauma exposure (LTE) information could enhance the discovery power of PTSD genome-wide association studies (GWASs). Methods: A GWAS on PTSD symptoms was performed in 51 cohorts followed by a fixed-effects meta-analysis (N = 182,199 European ancestry participants). A GWAS of LTE burden was performed in the UK Biobank cohort (N = 132,988). Genetic correlations were evaluated with linkage disequilibrium score regression. Multivariate analysis was performed using Multi-Trait Analysis of GWAS. Functional mapping and annotation of leading loci was performed with FUMA. Replication was evaluated using the Million Veteran Program GWAS of PTSD total symptoms. Results: GWASs of PTSD symptoms and LTE burden identified 5 and 6 independent genome-wide significant loci, respectively. There was a 72% genetic correlation between PTSD and LTE. PTSD and LTE showed largely similar patterns of genetic correlation with other traits, albeit with some distinctions. Adjusting PTSD for LTE reduced PTSD heritability by 31%. Multivariate analysis of PTSD and LTE increased the effective sample size of the PTSD GWAS by 20% and identified 4 additional loci. Four of these 9 PTSD loci were independently replicated in the Million Veteran Program. Conclusions: Through using a quantitative trait measure of PTSD, we identified novel risk loci not previously identified using prior case-control analyses. PTSD and LTE have a high genetic overlap that can be leveraged to increase discovery power through multivariate methods.publishersversionpublishe

Global Dynamics of Three Anticompetitive Systems of Difference Equations in the Plane

We investigate the global dynamics of several anticompetitive systems of rational difference equations which are special cases of general linear fractional system of the forms ., where all parameters and the initial conditions are arbitrary nonnegative numbers, such that both denominators are positive. We find the basins of attraction of all attractors of these systems

Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect

We consider the following system of difference equations: xn+1=xn2/B1xn2+C1yn2, yn+1=yn2/A2+B2xn2+C2yn2,  n=0, 1, …,   where B1, C1, A2, B2, C2 are positive constants and x0, y0≥0 are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at (0,0), which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at (0,0) and thus describe the global dynamics of this system. Since the singular point at (0,0) always possesses a basin of attraction this system exhibits Allee’s effect

Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations

Global dynamic scenarios for competitive maps in the plane

Abstract In this paper we present some global dynamic scenarios for general competitive maps in the plane. We apply these results to the class of second-order autonomous difference equations whose transition functions are decreasing in the variable xn $x_{n}$ and increasing in the variable xn−1 $x_{n-1}$. We illustrate our results with the application to the difference equation xn+1=Cxn−12+Exn−1axn2+dxn+f,n=0,1,…, $$x_{n+1}=\frac{Cx_{n-1}^{2}+Ex_{n-1}}{a x_{n}^{2}+d x_{n}+f},\quad n=0,1,\ldots,$$ where the initial conditions x−1 $x_{-1}$ and x0 $x_{0}$ are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C,E,a,d,f≥0 $C,E,a,d,f\geq0$, C+E>0 $C+E>0$, a+C>0 $a+C>0$, and a+d>0 $a+d>0$. We characterize the global dynamics of this equation with the basins of attraction of its equilibria and periodic solutions