scholarly journals Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Kenan Shi ◽  
Jianhui Tian ◽  
Tongqian Zhang
Keyword(s):  
Fixed Point ◽  
Functional Response ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Point Index ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fixed Point Index Theory

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when b and d are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when k>0 is sufficiently large.

scholarly journals Positive Solutions for a General Gause-Type Predator-Prey Model with Monotonic Functional Response

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Guohong Zhang ◽  
Xiaoli Wang
Keyword(s):  
Functional Response ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Index Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fixed Point Index Theory ◽  
Local And Global Bifurcations ◽  
Necessary And Sufficient

We study a general Gause-type predator-prey model with monotonic functional response under Dirichlet boundary condition. Necessary and sufficient conditions for the existence and nonexistence of positive solutions for this system are obtained by means of the fixed point index theory. In addition, the local and global bifurcations from a semitrivial state are also investigated on the basis of bifurcation theory. The results indicate diffusion, and functional response does help to create stationary pattern.

scholarly journals Steady state for a predator–prey cross-diffusion system with the Beddington–DeAngelis and Tanner functional response

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Demou Luo
Keyword(s):  
Fixed Point ◽  
Functional Response ◽  
Fixed Point Index ◽  
Index Theory ◽  
Diffusion System ◽  
Point Index ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Fixed Point Index Theory ◽  
Positive Steady States

AbstractThe main goal of this paper is investigating the existence of nonconstant positive steady states of a linear prey–predator cross-diffusion system with Beddington–DeAngelis and Tanner functional response. An analytical method and fixed point index theory plays a significant role in our main proofs.

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.

scholarly journals Multiple Positive Solutions for a System of Nonlinear Caputo-Type Fractional Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hongyu Li ◽  
Yang Chen
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fixed Point Index ◽  
Index Theory ◽  
Multiple Positive Solutions ◽  
Point Index ◽  
Fixed Point Index Theory ◽  
Fractional Differential

By using fixed-point index theory, we consider the existence of multiple positive solutions for a system of nonlinear Caputo-type fractional differential equations with the Riemann-Stieltjes boundary conditions.

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.

Existence of positive solutions to multi-point third order problems with sign changing nonlinearities

2020 ◽  
Vol 24 (1) ◽  
pp. 109-129
Author(s):  
Abdulkadir Dogan ◽  
John R. Graef
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Index Theory ◽  
Point Index ◽  
Third Order ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results.

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