Invasion waves for a diffusive predator–prey model with two preys and one predator

2020 ◽  
pp. 2050081
Author(s):  
Xinzhi Ren ◽  
Tianran Zhang ◽  
Xianning Liu
Keyword(s):  
Fixed Point ◽  
Wave Speed ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Traveling Fronts ◽  
Prey Model ◽  
Traveling Front ◽  
Minimal Wave Speed ◽  
Invasion Waves ◽  
Coexistence Equilibrium

In this paper, we study the existence of invasion waves of a diffusive predator–prey model with two preys and one predator. The existence of traveling semi-fronts connecting invasion-free equilibrium with wave speed [Formula: see text] is obtained by Schauder’s fixed-point theorem, where [Formula: see text] is the minimal wave speed. The boundedness of such waves is shown by rescaling method and such waves are proved to connect coexistence equilibrium by LaSalle’s invariance principle. The existence of traveling front with wave speed [Formula: see text] is got by rescaling method and limit arguments. The non-existence of traveling fronts with speed [Formula: see text] is shown by Laplace transform.

scholarly journals Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Feiying Yang ◽  
Wantong Li ◽  
Renhu Wang
Keyword(s):  
Main Concern ◽  
Invasion Process ◽  
Nonlocal Dispersal ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Minimal Wave Speed ◽  
Contracting Rectangle ◽  
Invasion Waves ◽  
The Right

<p style='text-indent:20px;'>This paper is concerned with the propagation dynamics of a nonlocal dispersal predator-prey model with two predators and one prey. Precisely, our main concern is the invasion process of the two predators into the habitat of one prey, when the two predators are weak competitors in the absence of prey. This invasion process is characterized by the spreading speed of the predators as well as the minimal wave speed of traveling waves connecting the predator-free state to the co-existence state. Particularly, the right-hand tail limit of wave profile is derived by the idea of contracting rectangle.</p>

scholarly journals Discrete-Time Predator-Prey Model with Bifurcations and Chaos

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
K. S. Al-Basyouni ◽  
A. Q. Khan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Discrete Time ◽  
Control Method ◽  
Linear Stability Theory ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Positive Fixed Point

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

scholarly journals Global Dynamics of a Predator-Prey Model with Stage Structure and Delayed Predator Response

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Lili Wang ◽  
Rui Xu
Keyword(s):  
Sufficient Conditions ◽  
Stage Structure ◽  
Global Dynamics ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Predator Prey Model ◽  
Prey Model ◽  
Coexistence Equilibrium

A Holling type II predator-prey model with time delay and stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. The existence of Hopf bifurcations at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle’s invariance principle, it is shown that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and the sufficient conditions are obtained for the global stability of the coexistence equilibrium.

scholarly journals Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Lingshu Wang ◽  
Guanghui Feng
Keyword(s):  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Comparison Arguments ◽  
Ratio Dependent ◽  
Iteration Technique ◽  
Coexistence Equilibrium

A ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, whenτ=τ0. By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system.

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.

Impact of the Fear Effect on the Stability and Bifurcation of a Leslie–Gower Predator–Prey Model

2020 ◽  
Vol 30 (14) ◽  
pp. 2050210
Author(s):  
Xiaoqin Wang ◽  
Yiping Tan ◽  
Yongli Cai ◽  
Weiming Wang
Keyword(s):  
Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Cycle Oscillation ◽  
The Stability ◽  
Coexistence Equilibrium ◽  
Predator Behavior

In this paper, we investigate analytically and numerically the dynamics of a modified Leslie–Gower predator–prey model which is characterized by the reduction of prey growth rate due to the anti-predator behavior. We prove the existence and local/global stability of equilibria of the model, and verify the existence of Hopf bifurcation. In addition, we focus on the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also destabilize the coexistence equilibrium, which are beneficial to the occurrence of limit-cycle-induced oscillation, or prevent the occurrence of limit cycle oscillation and increase the stability of the system.

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