Pattern Formation Controlled by External Forcing in a Spatial Harvesting Predator-Prey Model

2011 ◽  
pp. 214-221
Author(s):  
Feng Rao
Keyword(s):  
Reaction Diffusion ◽  
Euler Method ◽  
External Forcing ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
Predator Prey Models ◽  
The Stability

Predator–prey models in ecology serve a variety of purposes, which range from illustrating a scientific concept to representing a complex natural phenomenon. Due to the complexity and variability of the environment, the dynamic behavior obtained from existing predator–prey models often deviates from reality. Many factors remain to be considered, such as external forcing, harvesting and so on. In this chapter, we study a spatial version of the Ivlev-type predator-prey model that includes reaction-diffusion, external periodic forcing, and constant harvesting rate on prey. Using this model, we study how external periodic forcing affects the stability of predator-prey coexistence equilibrium. The results of spatial pattern analysis of the Ivlev-type predator-prey model with zero-flux boundary conditions, based on the Euler method and via numerical simulations in MATLAB, show that the model generates rich dynamics. Our results reveal that modeling by reaction-diffusion equations with external periodic forcing and nonzero constant prey harvesting could be used to make general predictions regarding predator-prey equilibrium,which may be used to guide management practice, and to provide a basis for the development of statistical tools and testable hypotheses.

scholarly journals Spatial Nonhomogeneous Periodic Solutions Induced by Nonlocal Prey Competition in a Diffusive Predator–Prey Model

2019 ◽  
Vol 29 (04) ◽  
pp. 1950043 ◽  
Author(s):  
Shanshan Chen ◽  
Junjie Wei ◽  
Kaiqi Yang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Flux Boundary Conditions ◽  
The Stability ◽  
The Given

The diffusive Holling–Tanner predator–prey model with no-flux boundary conditions and nonlocal prey competition is considered in this paper. We show the existence of spatially nonhomogeneous periodic solutions, which is induced by nonlocal prey competition. In particular, the constant positive steady state may lose the stability through Hopf bifurcation when the given parameter passes through some critical values, and the bifurcating periodic solutions near such values could be spatially nonhomogeneous and orbitally asymptotically stable.

scholarly journals An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Behrooz Basirat ◽  
Hamid Reza Elahi
Keyword(s):  
Time Lag ◽  
Euler Polynomials ◽  
Algebraic Equations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Collocation Points ◽  
Prey Model ◽  
Collocation Approach ◽  
Predator Prey Models ◽  
The Stability

This paper deals with an approach to obtaining the numerical solution of the Lotka–Volterra predator-prey models with discrete delay using Euler polynomials connected with Bernoulli ones. By using the Euler polynomials connected with Bernoulli ones and collocation points, this method transforms the predator-prey model into a matrix equation. The main characteristic of this approach is that it reduces the predator-prey model to a system of algebraic equations, which greatly simplifies the problem. For these models, the explicit formula determining the stability and the direction is given. Numerical examples illustrate the reliability and efficiency of the proposed scheme.

scholarly journals Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Heping Jiang ◽  
Huiping Fang ◽  
Yongfeng Wu
Keyword(s):  
Hopf Bifurcation ◽  
Neumann Boundary ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Abstract This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

scholarly journals Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Shengmao Fu ◽  
Lina Zhang
Keyword(s):  
Reaction Diffusion ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Sex Structure ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Ode System ◽  
Prey Model ◽  
Weakly Coupled ◽  
Coupled Reaction

In this paper, we consider a cross-diffusion predator-prey model with sex structure. We prove that cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system. As a result, we find that stationary patterns arise solely from the effect of cross-diffusion.

A Semi-Analytical Method for Solving Problems on the Role of Prey Taxis in a Biological Control-Mathematical Model

2019 ◽  
Vol 10 (02) ◽  
pp. 1850009
Author(s):  
OPhir Nave ◽  
Yifat Baron ◽  
Manju Sharma
Keyword(s):  
Biological Control ◽  
Analytical Method ◽  
Reaction Diffusion ◽  
Prey Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Homotopy Analysis ◽  
The Stability ◽  
Pest Density

In this paper, we applied the well-known homotopy analysis methods (HAM), which is a semi-analytical method, perturbation method, to study a reaction–diffusion–advection model for the dynamics of populations under biological control. According to the predator–prey model, the advection expression represents the predator density movement in which the acceleration is proportional to the prey density gradient. The prey population reproduces logistically, and the interactions of prey population obey the Holling’s prey-dependent Type II functional response. The predation process splits into the following subdivided processes: random movement which is represented by diffusion, direct movement which is described by prey taxis, local prey interactions, and consumptions which are represented by the trophic function. In order to ensure a successful biological control, one should make the predator-pest population to stabilize at a very low level of pest density. One reason for this effect is the intermediate taxis activity. However, when the system loses stability, for example very intensive prey taxis destroys the stability, it leads to chaotic dynamics with pronounced outbreaks of pest density.

scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

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