scholarly journals Analysis of a mutualism model with stochastic perturbations

2015 ◽  
Vol 08 (06) ◽  
pp. 1550072 ◽  
Author(s):  
Mei Li ◽  
Hongjun Gao ◽  
Chenfeng Sun ◽  
Yuezheng Gong
Keyword(s):  
Ecological Model ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Stochastic Perturbations ◽  
Stochastic Permanence ◽  
Local Existence And Uniqueness ◽  
Mutualism Model ◽  
Stochastically Bounded ◽  
Uniformly Continuous

This paper is concerned with a mutualism ecological model with stochastic perturbations. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded, uniformly continuous and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistent are also established. At last, some figures are presented to illustrate our main results.

Strong solutions to the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system

2019 ◽  
Vol 16 (04) ◽  
pp. 701-742 ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Stokes System ◽  
Initial Density ◽  
Local Existence ◽  
Strong Solutions ◽  
Incompressible Fluids ◽  
Navier Stokes ◽  
Initial Value ◽  
Two Phase ◽  
Density Dependent ◽  
Local Existence And Uniqueness

We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local existence and uniqueness of strong solutions to the initial value problem in a bounded domain, when the initial density function enjoys a positive lower bound.

scholarly journals Local existence and uniqueness of increasing positive solutions for non-singular and singular beam equation with a parameter

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hui Wang ◽  
Lingling Zhang
Keyword(s):  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Beam Equation ◽  
Mixed Monotone Operator ◽  
Local Existence And Uniqueness ◽  
Beam Equations ◽  
Dependent Parameter

AbstractThis paper is concerned with a class of beam equations with a parameter. By using the fixed point theorems of mixed monotone operator and the properties of cone, we study the non-singular and singular case, respectively, and obtain the sufficient conditions which guarantee the local existence and uniqueness of increasing positive solutions. Also, we present an iterative algorithm that converges to the solution. Moreover, we get some pleasant properties of the solutions for the boundary value problem dependent parameter. At last, two examples are given to illustrate the main results.

scholarly journals An abstract inverse problem

1991 ◽  
Vol 4 (2) ◽  
pp. 117-128 ◽  
Author(s):  
M. Choulli
Keyword(s):  
Inverse Problem ◽  
Uniqueness Theorem ◽  
Integrodifferential Equation ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Existence And Uniqueness Theorem ◽  
Local Existence And Uniqueness ◽  
Abstract Integrodifferential Equation

In this paper we consider an inverse problem that corresponds to an abstract integrodifferential equation. First, we prove a local existence and uniqueness theorem. We also show that every continuous solution can be locally extended in a unique way. Finally, we give sufficient conditions for the existence and a stability of the global solution.

scholarly journals Global Existence and Blowup Analysis to Single-Species Bacillus System with Free Boundary

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Zhi Ling ◽  
Zhigui Lin
Keyword(s):  
Free Boundary ◽  
Death Rate ◽  
Local Existence ◽  
Energy Condition ◽  
Global Solutions ◽  
Reaction Diffusion Equation ◽  
Time Behavior ◽  
Initial Value ◽  
Contraction Mapping Theorem ◽  
Local Existence And Uniqueness

This paper is concerned with a reaction-diffusion equation which describes the dynamics of single bacillus population with free boundary. The local existence and uniqueness of the solution are first obtained by using the contraction mapping theorem. Then we exhibit an energy condition, involving the initial data, under which the solution blows up in finite time. Finally we examine the long time behavior of global solutions; the global fast solution and slow solution are given. Our results show that blowup occurs if the death rate is small and the initial value is large enough. If the initial value is small the solution is global and fast, which decays at an exponential rate while there is a global slow solution provided that the death rate is small and the initial value is suitably large.

scholarly journals A Singular Initial-Value Problem for Second-Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Second Order ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Local Existence And Uniqueness ◽  
Singular Initial Value Problem

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

A generalized stochastic competitive system with Ornstein–Uhlenbeck process

2020 ◽  
pp. 2150001
Author(s):  
Baodan Tian ◽  
Liu Yang ◽  
Xingzhi Chen ◽  
Yong Zhang
Keyword(s):  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Competitive System ◽  
Uhlenbeck Process ◽  
Stochastic Perturbations ◽  
Stochastic Permanence ◽  
Persistence In The Mean ◽  
Ornstein Uhlenbeck Process ◽  
Global Positive Solution ◽  
The Mean

A generalized competitive system with stochastic perturbations is proposed in this paper, in which the stochastic disturbances are described by the famous Ornstein–Uhlenbeck process. By theories of stochastic differential equations, such as comparison theorem, Itô’s integration formula, Chebyshev’s inequality, martingale’s properties, etc., the existence and the uniqueness of global positive solution of the system are obtained. Then sufficient conditions for the extinction of the species almost surely, persistence in the mean and the stochastic permanence for the system are derived, respectively. Finally, by a series of numerical examples, the feasibility and correctness of the theoretical analysis results are verified intuitively. Moreover, the effects of the intensity of the stochastic perturbations and the speed of the reverse in the Ornstein–Uhlenbeck process to the dynamical behavior of the system are also discussed.

scholarly journals Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev space

1997 ◽  
Vol 40 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Nakao Hayashi ◽  
Hitoshi Hirata
Keyword(s):  
Sobolev Space ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Initial Value ◽  
Hyperbolic Case ◽  
Local Existence And Uniqueness ◽  
Usual Sobolev Space

We study the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness H5/2 with a condition such that the L2 norm of the data is sufficiently small.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere

2017 ◽  
Vol 147 (6) ◽  
pp. 1243-1277 ◽  
Author(s):  
Eiji Onodera
Keyword(s):  
Fourth Order ◽  
Energy Method ◽  
Initial Value Problem ◽  
Local Existence ◽  
Three Dimensional ◽  
Closed Curve ◽  
Initial Value ◽  
Curve Flow ◽  
Local Existence And Uniqueness ◽  
Loss Of Derivatives

A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

scholarly journals The spreading fronts in a mutualistic model with delay

2016 ◽  
Vol 09 (06) ◽  
pp. 1650080
Author(s):  
Mei Li
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Local Existence ◽  
Free Boundaries ◽  
Local Existence And Uniqueness ◽  
Fast Solution ◽  
Global Fast Solution ◽  
Semilinear Parabolic

This paper is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite limits at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the intra-specific competitions are strong, while if the intra-specific competitions are weak, there exist the blowup solution and global fast solution.

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