scholarly journals Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency

2021 ◽  
Vol 11 (1) ◽  
pp. 212-224
Author(s):  
Bo Zheng ◽  
Jianshe Yu
Keyword(s):  
Periodic Solution ◽  
Periodic Orbits ◽  
Discrete Model ◽  
Wolbachia Infection ◽  
Discrete Models ◽  
Infection Frequency ◽  
Globally Asymptotically Stable ◽  
Release Ratio ◽  
Impulsive Release ◽  
Special Case

Abstract In this paper, we study a discrete model on Wolbachia infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first N generations with the release ratio α, and the release is terminated from (N + 1)-th generation to T-th generation. We find a release ratio threshold denoted by α *(N, T), and prove the existence of a T-periodic solution for the model when α ∈ (0, α *(N, T)). For the special case when N = 1 and T = 2, we prove that the model has a unique T-periodic solution which is unstable when α ∈ (0, α *(N, T)). While α ≥ α *(N, T), no periodic phenomenon occurs and the Wolbachia fixation equilibrium is globally asymptotically stable. Numerical simulations are also provided to illustrate our theoretical results. One main contribution of this work is to offer a new method to determine the exact number of periodic orbits to discrete models.

Wolbachia and Cytoplasmic Incompatibility in the CaliforniaCulex pipiensMosquito Species Complex: Parameter Estimates and Infection Dynamics in Natural Populations

Genetics ◽  
2003 ◽  
Vol 165 (4) ◽  
pp. 2029-2038 ◽  
Author(s):  
Jason L Rasgon ◽  
Thomas W Scott
Keyword(s):  
Species Complex ◽  
Cytoplasmic Incompatibility ◽  
Natural Populations ◽  
Wolbachia Infection ◽  
Field Conditions ◽  
Parameter Estimates ◽  
Infection Prevalence ◽  
Infection Frequency ◽  
Specific Pcr ◽  
Vector Borne

AbstractBefore maternally inherited bacterial symbionts like Wolbachia, which cause cytoplasmic incompatibility (CI; reduced hatch rate) when infected males mate with uninfected females, can be used in a program to control vector-borne diseases it is essential to understand their dynamics of infection in natural arthropod vector populations. Our study had four goals: (1) quantify the number of Wolbachia strains circulating in the California Culex pipiens species complex, (2) investigate Wolbachia infection frequencies and distribution in natural California populations, (3) estimate the parameters that govern Wolbachia spread among Cx. pipiens under laboratory and field conditions, and (4) use these values to estimate equilibrium levels and compare predicted infection prevalence levels to those observed in nature. Strain-specific PCR, wsp gene sequencing, and crossing experiments indicated that a single Wolbachia strain infects Californian Cx. pipiens. Infection frequency was near or at fixation in all populations sampled for 2 years along a >1000-km north-south transect. The combined statewide infection frequency was 99.4%. Incompatible crosses were 100% sterile under laboratory and field conditions. Sterility decreased negligibly with male age in the laboratory. Infection had no significant effect on female fecundity under laboratory or field conditions. Vertical transmission was >99% in the laboratory and ∼98.6% in the field. Using field data, models predicted that Wolbachia will spread to fixation if infection exceeds an unstable equilibrium point above 1.4%. Our estimates accurately predicted infection frequencies in natural populations. If certain technical hurdles can be overcome, our data indicate that Wolbachia can invade vector populations as part of an applied transgenic strategy for vector-borne disease reduction.

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

Asymptotic Behavior of Periodic Cohen-Grossberg Neural Networks with Delays

2009 ◽  
Vol 21 (12) ◽  
pp. 3444-3459 ◽  
Author(s):  
Wei Lin
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Lyapunov Method ◽  
Autonomous Systems ◽  
Neural Systems ◽  
Stability Criteria ◽  
Volterra System ◽  
Numerical Examples ◽  
The Stability ◽  
Special Case

Without assuming the positivity of the amplification functions, we prove some M-matrix criteria for the [Formula: see text]-global asymptotic stability of periodic Cohen-Grossberg neural networks with delays. By an extension of the Lyapunov method, we are able to include neural systems with multiple nonnegative periodic solutions and nonexponential convergence rate in our model and also include the Lotka-Volterra system, an important prototype of competitive neural networks, as a special case. The stability criteria for autonomous systems then follow as a corollary. Two numerical examples are provided to show that the limiting equilibrium or periodic solution need not be positive.

scholarly journals Stability Analysis of an Alcoholism Model with Public Health Education and NSFD Scheme

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Zhaofeng An ◽  
Suxia Zhang ◽  
Jinhu Xu
Keyword(s):  
Public Health ◽  
Health Education ◽  
Differential Equations ◽  
Discrete Model ◽  
Public Health Education ◽  
Continuous Model ◽  
Basic Reproductive Number ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nsfd Scheme

In this paper, an alcoholism model of SEAR type with different susceptibilities due to public health education is investigated, with the form of continuous differential equations as well as discrete differential equations by applying the Mickens nonstandard finite difference (NSFD) scheme to the continuous equations. Threshold dynamics of the continuous model are performed by constructing Lyapunov functions. The analysis of a discrete model indicates that the alcohol-free equilibrium is globally asymptotically stable if the basic reproductive number R0<1, and conversely, the alcohol-present equilibrium is globally asymptotically stable if R0>1, revealing the consistency and efficiency of the discrete model to preserve the dynamical properties of the corresponding continuous model. In addition, stability preserving and the impact of the parameters related with public health education are conducted by numerical simulations.

scholarly journals Impulsive Biological Pest Control Strategies of the Sugarcane Borer

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Marat Rafikov ◽  
Alfredo Del Sole Lordelo ◽  
Elvira Rafikova
Keyword(s):  
Pest Control ◽  
Floquet Theory ◽  
Control Strategies ◽  
Egg Parasitoid ◽  
Asymptotically Stable ◽  
Biological Pest Control ◽  
Globally Asymptotically Stable ◽  
Larval Stages ◽  
Sugarcane Borer ◽  
Impulsive Release

We propose an impulsive biological pest control of the sugarcane borer (Diatraea saccharalis) by its egg parasitoidTrichogramma galloibased on a mathematical model in which the sugarcane borer is represented by the egg and larval stages, and the parasitoid is considered in terms of the parasitized eggs. By using the Floquet theory and the small amplitude perturbation method, we show that there exists a globally asymptotically stable pest-eradication periodic solution when some conditions hold. The numerical simulations show that the impulsive release of parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.

scholarly journals Stability of Periodic Planetary-Type Orbits of the General Planar N-Body Problem

1979 ◽  
Vol 81 ◽  
pp. 23-28
Author(s):  
John D. Hadjidemetriou
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
Frame Of Reference ◽  
Rotating Frame ◽  
N Body Problem ◽  
Families Of Periodic Orbits ◽  
Special Case

It is known that families of periodic orbits in the general N-body problem (N≥3) exist, in a rotating frame of reference (Hadjidemetriou 1975, 1977). A special case of the above families of periodic orbits are the periodic orbits of the planetary type. In this latter case only one body, which we shall call sun, is the more massive one and the rest N-1 bodies, which we shall call planets, have small but not negligible masses. The aim of this paper is to study the properties of the families of periodic planetary-type orbits, with particular attention to stability. To make the presentation clearer, we shall start first with the case N=3 and we shall extend the results to N>3. We shall discuss planar orbits only.

scholarly journals Global asymptotic stability in a periodic Lotka-Volterra system

1985 ◽  
Vol 27 (1) ◽  
pp. 66-72 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Volterra System ◽  
Periodic Coefficients ◽  
Globally Asymptotically Stable ◽  
Stable Periodic

AbstractA set of easily verifiable sufficient conditions are obtained for the existence of a globally asymptotically stable periodic solution in a Lotka-Volterra system with periodic coefficients.

Modeling Flow and Transport in Fractured Crystalline Rock using the Discrete Fracture Network Concept

1988 ◽  
Vol 127 ◽  
Author(s):  
Björn Dverstorp ◽  
Wille Nordqvist ◽  
Johan Andersson
Keyword(s):  
Discrete Model ◽  
Continuum Model ◽  
Large Scale ◽  
Flow Analysis ◽  
Flow Distribution ◽  
Crystalline Rock ◽  
Discrete Models ◽  
Large Variability ◽  
Rock Heterogeneity ◽  
Modeling Flow

ABSTRACTThe conductive properties of fractured crystalline rock vary considerably in space, which implies that the flow is very unevenly distributed in space. The large variability raises doubts on modeling the flow with a large scale continuum model. Modeling flow in fractured crystalline rock in a network of discrete fractures provides an increased understanding of the character of the rock heterogeneity. Compared to a continuum model discrete models introduce new parameters such as statistical distributions for fracture orientation, radii, density and transmissivity that need to be estimated. By analyzing the migration experiment in the Stripa research mine in Sweden it is demonstrated how to calibrate and eventually validate a discrete model on field data. The flow analysis shows that the flow distribution on the drift roof and in two out of three vertical boreholes can be modelled with the same discrete model. The properties of the third borehole differ substantially. Initial attempts of analyzing the tracer experiment are also described.

THE PERIODIC PREDATOR-PREY LOTKA–VOLTERRA MODEL WITH IMPULSIVE EFFECT

2002 ◽  
Vol 02 (03n04) ◽  
pp. 267-296 ◽  
Author(s):  
SANYI TANG ◽  
LANSUN CHEN
Keyword(s):  
Periodic Solution ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Volterra Model ◽  
Impulsive Effect ◽  
Biological Pest Control ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Special Cases

In this paper, a classical periodic Lotka–Volterra predator-prey system with impulsive effect is investigated. We analyze the dynamics of positive solutions of such models. Among other results we show that if some trivial or semi-trivial positive solution is linearly stable, then it is globally asymptotically stable with respect to the positive solutions. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution. We use bifurcation theorem to show the existence of coexistence states which arise near the sem-trivial periodic solution. As an application, we also examine some special cases of the system which can be used in the biological pest control.

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