scholarly journals Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response

2018 ◽  
Vol 16 (1) ◽  
pp. 623-635 ◽  
Author(s):  
Xiaozhou Feng ◽  
Yi Song ◽  
Xiaomin An
Keyword(s):  
Fixed Point ◽  
Behavior Analysis ◽  
Dynamic Behavior ◽  
Fixed Point Index ◽  
Global Bifurcation ◽  
Index Theory ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Coexistence States ◽  
Predator Model

AbstractThis paper investigates the dynamic behavior analysis on the prey-predator model with ratio-dependent Monod-Haldane response function under the homogeneous Dirichlet boundary conditions, which is used to simulate a class of biological system. Firstly, the sufficient and necessary conditions on existence and non-existence of coexistence states of this model are discussed by comparison principle and fixed point index theory. Secondly, taking a as a main bifurcation parameter, the structure of global bifurcation curve on positive solutions is established by using global bifurcation theorem and properties of principal eigenvalue. Finally, the stability of coexistence states is obtained by the eigenvalue perturbation theory; the multiplicity of coexistence states is investigated when a satisfies some condition by the fixed point index theory.

scholarly journals Uniqueness and Multiplicity of a Prey-Predator Model with Predator Saturation and Competition

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Xiaozhou Feng ◽  
Lifeng Li
Keyword(s):  
Fixed Point ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Dirichlet Boundary Conditions ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Sufficient And Necessary Conditions ◽  
Predator Model ◽  
Homogeneous Dirichlet Boundary Conditions

We investigate positive solutions of a prey-predator model with predator saturation and competition under homogeneous Dirichlet boundary conditions. First, the existence of positive solutions and some sufficient and necessary conditions is established by using the standard fixed point index theory in cones. Second, the changes of solution branches, multiplicity, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory. Finally, the exact number and type of positive solutions are proved whenkormconverges to infinity.

scholarly journals Eigenvalue of Coupled Systems for Hammerstein Integral Equation with Two Parameters

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Wanjun Li
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Coupled Systems ◽  
Point Index ◽  
Hammerstein Integral Equation ◽  
Fixed Point Index Theory ◽  
Two Parameters

By using the fixed-point index theory, we discuss the existence, multiplicity, and nonexistence of positive solutions for the coupled systems of Hammerstein integral equation with parameters.

scholarly journals Positive Solutions for a Class of Fourth-Order Boundary Value Problems in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Jingjing Cai ◽  
Guilong Liu
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Positive Solutions ◽  
Fourth Order ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Existence And Nonexistence

Using a specially constructed cone and the fixed point index theory, this work shows existence and nonexistence results of positive solutions for fourth-order boundary value problem with two different parameters in Banach spaces.

Positive Radial Solutions for Elliptic Systems with Nonlocal Conditions

2012 ◽  
Vol 204-208 ◽  
pp. 4800-4808
Author(s):  
Sheng Li Xie
Keyword(s):  
Fixed Point ◽  
Elliptic System ◽  
Fixed Point Index ◽  
Nonlocal Conditions ◽  
Elliptic Systems ◽  
Index Theory ◽  
Point Index ◽  
Radial Solutions ◽  
Fixed Point Index Theory ◽  
Positive Radial Solutions

By using fixed point index theory,we study the existence of positive radial solutions and multiple positive radial solutions for the elliptic system with nonlocal conditions. Our results extend and improve some existing ones.

scholarly journals Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Li Wu ◽  
Chuanzhi Bai
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Fractional Boundary Value Problem ◽  
Point Index ◽  
Advanced Argument ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

scholarly journals On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs

2021 ◽  
Vol 22 (2) ◽  
pp. 259
Author(s):  
Svetlin Georgiev Georgiev ◽  
Karima Mebarki
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
First Order ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions ◽  
Theoretical Results ◽  
Positive Half

The aim of this work is two fold: first  we  extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction  obtained in \cite{DjebaMeb, Svet-Meb}, to  the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction.  Secondly, as  illustration of some our theoretical results,  we study  the existence of positive solutions  for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as  a class of  partial differential equations (PDEs for short).

scholarly journals On Six Solutions form-Point Differential Equations System with Two Coupled Parallel Sub-Super Solutions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Jian Liu ◽  
Hua Su ◽  
Shuli Wang
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Index ◽  
Index Theory ◽  
Second Order ◽  
Point Index ◽  
Fixed Point Index Theory

Under the assumption of two coupled parallel subsuper solutions, the existence of at least six solutions for a kind of second-orderm-point differential equations system is obtained using the fixed point index theory. As an application, an example to demonstrate our result is given.

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