scholarly journals Analytical approach to a steady-state predator-prey system of Lotka-Volterra model

2020 ◽  
Author(s):  
M. Nivethitha ◽  
R. Senthamarai
Keyword(s):  
Steady State ◽  
Analytical Approach ◽  
Volterra Model ◽  
Predator Prey ◽  
Prey System

Stability and Bifurcation in a Predator–Prey System with Prey-Taxis

2020 ◽  
Vol 30 (02) ◽  
pp. 2050022 ◽  
Author(s):  
Huanhuan Qiu ◽  
Shangjiang Guo ◽  
Shangzhi Li
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Steady States ◽  
Mode Interaction ◽  
Small Range ◽  
Steady State Mode ◽  
Predator Prey ◽  
Prey System

In this paper, we consider a generalized predator–prey system with prey-taxis under Neumann boundary condition, that is, the predators can survive even in the absence of the prey species. It is proved that for an arbitrary spatial dimension, the corresponding initial boundary value problem possesses a unique global bounded classical solution when the prey-taxis is restricted to a small range. Moreover, the local stabilities of constant steady states (including trivial, semi-trivial and positive constant steady states) are investigated. A further study on the coexistence steady state implies that the prey-taxis term suppresses the global asymptotical stability and influences the steady-state/Hopf bifurcations (if they exist). Analyses of steady-state bifurcation, Hopf bifurcation, and even Hopf/steady-state mode interaction are carried out in detail by means of the Lyapunov–Schmidt procedure. In particular, we obtain stable or unstable steady states, time-periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found in numerical simulations.

scholarly journals Effects of Behavioral Tactics of Predators on Dynamics of a Predator-Prey System

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Hui Zhang ◽  
Zhihui Ma ◽  
Gongnan Xie ◽  
Lukun Jia
Keyword(s):  
Time Scale ◽  
Stable State ◽  
Volterra Model ◽  
Fast Time ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Interaction ◽  
Game Dynamic

A predator-prey model incorporating individual behavior is presented, where the predator-prey interaction is described by a classical Lotka-Volterra model with self-limiting prey; predators can use the behavioral tactics of rock-paper-scissors to dispute a prey when they meet. The predator behavioral change is described by replicator equations, a game dynamic model at the fast time scale, whereas predator-prey interactions are assumed acting at a relatively slow time scale. Aggregation approach is applied to combine the two time scales into a single one. The analytical results show that predators have an equal probability to adopt three strategies at the stable state of the predator-prey interaction system. The diversification tactics taking by predator population benefits the survival of the predator population itself, more importantly, it also maintains the stability of the predator-prey system. Explicitly, immediate contest behavior of predators can promote density of the predator population and keep the preys at a lower density. However, a large cost of fighting will cause not only the density of predators to be lower but also preys to be higher, which may even lead to extinction of the predator populations.

scholarly journals Investigating Lotka-Volterra model using computer simulation

2020 ◽  
Vol 3 (10) ◽  
Author(s):  
F. Kunis ◽  
M. Dimitrov
Keyword(s):  
Computer Simulation ◽  
Population Dynamics ◽  
Numerical Solution ◽  
Fourth Order ◽  
Real Data ◽  
Volterra Model ◽  
Predator Prey ◽  
Prey System ◽  
Two Populations ◽  
Over Time

In this project we study the Lotka-Volterra model, also known as the model describing the population dynamics in the Predator-prey system. This model describes the interaction of the two species and also the development of their populations over time. We simulate this model using the fourth-order Runge-Kutta algorithm. This is the most widely used method for numerical solution of ordinary differential equations. Based on the obtained program, we simulated two populations and traced their behavior over time. We optimized the parameters and managed to obtain results that are very close to real data for such populations.

scholarly journals Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Changtong Li ◽  
Hao Sun ◽  
Yuzhen Wang
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Bifurcation Theory ◽  
A Priori Estimates ◽  
Global Bifurcation ◽  
Local Bifurcation ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
Steady State Solutions

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

Stability in predator-prey system Lotka-Volterra model incorporating a prey refuge like the law of mass action

2018 ◽  
Vol 11 (42) ◽  
pp. 2049-2057 ◽  
Author(s):  
Edilber Almanza-Vasquez ◽  
Ruben-Dario Ortiz-Ortiz ◽  
Ana-Magnolia Marin-Ramirez
Keyword(s):  
Mass Action ◽  
Volterra Model ◽  
Law Of Mass Action ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
The Law

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

2021 ◽  
Author(s):  
Haixia Li ◽  
Wenbin Yang ◽  
Meihua Wei ◽  
Aili Wang
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
A Priori ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
A Priori Bound ◽  
Steady State Solutions

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

2019 ◽  
Vol 29 (11) ◽  
pp. 1950145 ◽  
Author(s):  
Yu-Xia Wang ◽  
Wan-Tong Li
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Spatial Patterns ◽  
Bifurcation Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Positive Steady State ◽  
Prey System

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

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