Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting

2020 ◽  
pp. 2050084
Author(s):  
Lakshmi Narayan Guin ◽  
Sudipta Pal ◽  
Santabrata Chakravarty ◽  
Salih Djilali
Keyword(s):  
Equilibrium Point ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Density Variation ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Homogeneous Neumann Boundary Condition ◽  
Dynamics Of Population

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge (proportion of both the species) and harvesting of prey species in this contribution. Criteria for asymptotic stability (local and global) and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point. For partial differential equation (PDE), the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated. The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species. A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space, that is, spots, stripe-spot mixtures, labyrinthine, holes, stripe-hole mixtures and stripes replication. Finally, we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.

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.

Effects of Delay and Diffusion on the Dynamics of a Leslie–Gower Type Predator–Prey Model

2014 ◽  
Vol 24 (04) ◽  
pp. 1450043
Author(s):  
Jia-Fang Zhang ◽  
Xiang-Ping Yan
Keyword(s):  
Periodic Solutions ◽  
Positive Constant ◽  
Neumann Boundary ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Constant Equilibrium ◽  
Homogeneous Neumann Boundary Condition ◽  
Delayed System ◽  
The Stability ◽  
And Diffusion

In this paper, we consider the effects of time delay and space diffusion on the dynamics of a Leslie–Gower type predator–prey system. It is shown that under homogeneous Neumann boundary condition the occurrence of space diffusion does not affect the stability of the positive constant equilibrium of the system. However, we find that the incorporation of a discrete delay representing the gestation of prey species can not only destabilize the positive constant equilibrium of the system but can also cause a Hopf bifurcation at the positive constant equilibrium as it crosses some critical values. In particular, we prove that these Hopf bifurcations' periodic solutions are all spatially homogeneous if the diffusive rates are suitably large, which has the same properties as periodic solutions of the corresponding delayed system without diffusion. However, if the diffusive rates are suitably small, then the system will generate spatially nonhomogeneous periodic solutions. The results in this work demonstrate that diffusion plays an important role in deriving complex spatiotemporal dynamics.

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.

A mathematical study of a predator–prey model with disease circulating in the both populations

2015 ◽  
Vol 08 (02) ◽  
pp. 1550015 ◽  
Author(s):  
Krishna Pada Das ◽  
J. Chattopadhyay
Keyword(s):  
Equilibrium Point ◽  
Prey Population ◽  
Global Dynamics ◽  
Predator Population ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Force Of Infection ◽  
Predator Prey Model ◽  
High Infection ◽  
Infected Prey

Disease in ecological systems plays an important role. In the present investigation we propose and analyze a predator–prey mathematical model in which both species are affected by infectious disease. The parasite is transmitted directly (by contact) within the prey population and indirectly (by consumption of infected prey) within the predator population. We derive biologically feasible and insightful quantities in terms of ecological as well as epidemiological reproduction numbers that allow us to describe the dynamics of the proposed system. Our observations indicate that predator–prey system is stable without disease but high infection rate drive the predator population toward extinction. We also observe that predation of vulnerable infected prey makes the disease to eradicate into the community composition of the model system. Local stability analysis of the interior equilibrium point near the disease-free equilibrium point is worked out. To study the global dynamics of the system, numerical simulations are performed. Our simulation results show that for higher values of the force of infection in the prey population the predator population goes to extinction. Our numerical analysis reveals that predation rates specially on susceptible prey population and recovery of infective predator play crucial role for preventing the extinction of the susceptible predator and disease propagation.

scholarly journals Stability and Hopf bifurcation of a delayed reaction–diffusion predator–prey model with anti-predator behaviour

2019 ◽  
Vol 24 (3) ◽  
pp. 387-406
Author(s):  
Jia Liu ◽  
Xuebing Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Time Delay ◽  
Predator Behaviour

In this paper, we study the dynamics of a delayed reaction–diffusion predator–prey model with anti-predator behaviour. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the model, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. By utilizing upperlower solution method, the global asymptotic stability of the interior equilibrium is studied. Finally, numerical simulation results are presented to validate the theoretical analysis.

Turing Pattern Induced by Cross-Diffusion in a Predator–Prey Model with Pack Predation-Herd Behavior

2020 ◽  
Vol 30 (07) ◽  
pp. 2050103
Author(s):  
Jingen Yang ◽  
Tonghua Zhang ◽  
Sanling Yuan
Keyword(s):  
Neumann Boundary ◽  
Multiple Scale ◽  
Turing Instability ◽  
Average Density ◽  
Field Analysis ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition

In this paper, we propose a diffusive predator–prey model with herd behavior in prey and pack predation behavior in predator under the homogeneous Neumann boundary condition. Due to the pack predation-herd behavior, the predator–prey interaction occurs only at the outer edge of each group. First, we analyze the existence and local stability of the equilibria of temporal model using vector field analysis and characteristic equations. Second, we deduce the conditions under which Turing instability occurs with the help of linear stability analysis. Then, using standard multiple-scale analysis, we derive the amplitude equations for the excited modes. By numerical simulations, we find that the model exhibits complex pattern replication by varying the value of parameters [Formula: see text] and [Formula: see text], and study the relationships between the average density of both prey and predator and these two parameters.

Existence of spatiotemporal patterns in the reaction–diffusion predator–prey model incorporating prey refuge

2016 ◽  
Vol 09 (06) ◽  
pp. 1650085 ◽  
Author(s):  
Lakshmi Narayan Guin ◽  
Benukar Mondal ◽  
Santabrata Chakravarty
Keyword(s):  
Pattern Formation ◽  
Spatial Dynamics ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Model System ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

The pattern formation in reaction–diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal existence and importance. The present investigation deals with a spatial dynamics of the Beddington–DeAngelis predator–prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique positive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion-driven instability of the spatiotemporal model are investigated. Based on the appropriate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes replication. The results obtained appear to enrich the findings of the model system under consideration.

BIFURCATION ANALYSIS OF A TWO-DIMENSIONAL PREDATOR–PREY MODEL WITH HOLLING TYPE IV FUNCTIONAL RESPONSE AND NONLINEAR PREDATOR HARVESTING

2020 ◽  
Vol 28 (04) ◽  
pp. 839-864
Author(s):  
UTTAM GHOSH ◽  
PRAHLAD MAJUMDAR ◽  
JAYANTA KUMAR GHOSH
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Conversion Efficiency ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
Trivial Equilibrium

The aim of this paper is to investigate the dynamical behavior of a two-species predator–prey model with Holling type IV functional response and nonlinear predator harvesting. The positivity and boundedness of the solutions of the model have been established. The considered system contains three kinds of equilibrium points. Those are the trivial equilibrium point, axial equilibrium point and the interior equilibrium points. The trivial equilibrium point is always saddle and stability of the axial equilibrium point depends on critical value of the conversion efficiency. The interior equilibrium point changes its stability through various parametric conditions. The considered system experiences different types of bifurcations such as Saddle-node bifurcation, Hopf bifurcation, Transcritical bifurcation and Bogdanov–Taken bifurcation. It is clear from the numerical analysis that the predator harvesting rate and the conversion efficiency play an important role in stability of the system.

scholarly journals Global existence and boundedness of solutions in a Lotka-Volterra reaction-diffusion system of predator-prey model with nonlinear prey-taxis

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 5023-5035
Author(s):  
Demou Luo
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Boundedness Of Solutions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Global Classical Solutions ◽  
Prey Model

In this paper, we investigate a diffusive Lotka-Volterra predator-prey model with nonlinear prey-taxis under Neumann boundary conditions. This system describes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which lead to the collection of u). We apply some methods to overcome the substantial difficulty of the existence of nonlinear prey-taxis term and prove that the unique global classical solutions of Lotka-Volterra predator-prey model are globally bounded.

scholarly journals Dynamics and Pattern Formation of a Diffusive Predator-prey Model in the Subdiffusive Regime in Presence of Toxicity

2020 ◽  
Author(s):  
Daniel Cassidi Bitang A Ziem ◽  
Carlos Lawrence Gninzanlong ◽  
Conrad Bertrand Tabi ◽  
Timoléon Crépin Kofane
Keyword(s):  
Neumann Boundary ◽  
Turing Instability ◽  
Stability Domain ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Diffusive Regime ◽  
Ecological Literature ◽  
Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
The Stability

Motivated by the fact that the restrictive conditions for a Turing instability are relaxed in sub- diffusive regime, we investigate the effects of subdiffu- sion in the predator−prey model with toxins under the homogeneous Neumann boundary condition. First, the stability analysis of the corresponding ordinary differ- ential equation is carried out. From this analysis, it fol- lows that stability is closely related to the coefficient of toxicity. In addition, the temporal fractional derivative does not systematically widen the range of parameters to maintain a point in the stability domain. Further- more, we derive the condition which links the Turing instability to the coefficient of toxicity in the subdif- fusive regime. System parameters are varied in order to test our mathematical predictions while comparing them to ecological literature. It turns out that the mem- ory effects, linked to the transport process can, depend- ing on the parameters, either stabilize an ecosystem or make a completely different configuration.

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