scholarly journals Spatial pattern formation and delay induced destabilization in predator-prey model with fear effect

2021 ◽  
Author(s):  
Swati Mishra ◽  
RANJIT UPADHYY
Keyword(s):  
Hopf Bifurcation ◽  
Field Experiments ◽  
Turing Instability ◽  
Predator Prey ◽  
Gestation Delay ◽  
Interior Equilibrium ◽  
Defensive Strategies ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Cost

Recent field experiments showed that predators influence the prey population not only by direct consumption but also by stimulating various defensive strategies. The cost of these defensive strategies can include energetic investment in defensive structures, reduced energy income, lower mating success, and emigration which ultimately reduces the reproduction of prey. To explore the effect of these defensive strategies (anti-predator behaviors), a modified Leslie-Gower predator-prey model with the cost of fear has been considered. Gestation delay is also incorporated in the system for a more realistic formulation. Boundedness, equilibria and stability analysis of the temporal model are studied. By considering gestation delay as a bifurcation parameter, the existence of Hopf-bifurcation around the interior equilibrium point is discussed together with the direction, stability and period of bifurcating solutions arising through Hopf-bifurcation. The spatial extension of the proposed model incorporating density-dependent cross-diffusion is also investigated and the conditions for diffusion-driven instability are obtained. To illustrate the analytical findings, detailed numerical simulations are performed. Biologically realistic Turing patterns as hexagonal spots, spots and stripes mixture, and labyrinthine type patterns are identified. It is found that the fear level has a stabilizing impact on delay induced destabilization and both stabilizing and destabilizing effects on Turing instability.

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.

Stability and Hopf bifurcation analysis of a diffusive predator–prey model with Smith growth

2015 ◽  
Vol 08 (01) ◽  
pp. 1550013 ◽  
Author(s):  
M. Sivakumar ◽  
M. Sambath ◽  
K. Balachandran
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Neumann Boundary Condition ◽  
Local Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

In this paper, we consider a diffusive Holling–Tanner predator–prey model with Smith growth subject to Neumann boundary condition. We analyze the local stability, existence of a Hopf bifurcation at the co-existence of the equilibrium and stability of bifurcating periodic solutions of the system in the absence of diffusion. Furthermore the Turing instability and Hopf bifurcation analysis of the system with diffusion are studied. Finally numerical simulations are given to demonstrate the effectiveness of the theoretical analysis.

scholarly journals Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bo Yang
Keyword(s):  
Boundary Condition ◽  
Hopf Bifurcation ◽  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Flux Boundary Condition ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

The spatiotemporal dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey model with Smith growth subject to zero-flux boundary condition are investigated analytically and numerically. The asymptotic stability of the positive equilibrium and the existence of Hopf bifurcation around the positive equilibrium are shown; the conditions of Turing instability are obtained. And with the help of numerical simulations, it is found that the model exhibits complex pattern replication: stripes, spots-stripes mixtures, and spots Turing patterns.

scholarly journals Drivers of pattern formation in a predator–prey model with defense in fearful prey

2021 ◽  
Author(s):  
Purnedu Mishra ◽  
Barkha Tiwari
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Direct Method ◽  
Turing Instability ◽  
Steady States ◽  
Predator Prey ◽  
Random Movement ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fear And Anxiety

AbstractExistence of predator is routinely used to induce fear and anxiety in prey which is well known for shaping entire ecosystem. Fear of predation restricts the development of prey and promotes inducible defense in prey communities for the survival. Motivated by this fact, we investigate the dynamics of a Leslie–Gower predator prey model with group defense in a fearful prey. We obtain conditions under which system possess unique global-in-time solutions and determine all the biological feasible states of the system. Local stability is analyzed by linearization technique and Lyapunov direct method has been applied for global stability analysis of steady states. We show the occurrence of Hopf bifurcation and its direction at the vicinity of coexisting equilibrium point for temporal model. We consider random movement in species and establish conditions for the stability of the system in the presence of diffusion. We derive conditions for existence of non-constant steady states and Turing instability at coexisting population state of diffusive system. Incorporating indirect prey taxis with the assumption that the predator moves toward the smell of prey rather than random movement gives rise to taxis-driven inhomogeneous Hopf bifurcation in predator–prey model. Numerical simulations are intended to demonstrate the role of biological as well as physical drivers on pattern formation that go beyond analytical conclusions.

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.

Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response

2014 ◽  
Vol 07 (05) ◽  
pp. 1450047 ◽  
Author(s):  
Lakshmi Narayan Guin ◽  
Prashanta Kumar Mandal
Keyword(s):  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Turing Instability ◽  
Spatial Pattern Analysis ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper, spatial patterns of a diffusive predator–prey model with sigmoid (Holling type III) ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The (local and global) asymptotic stability behavior of the corresponding non-spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pattern analysis through numerical simulations are depicted and analyzed. Furthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.

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