scholarly journals Pattern Formation and Bistability in a Generalist Predator-Prey Model

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 20
Author(s):  
Vagner Weide Rodrigues ◽  
Diomar Cristina Mistro ◽  
Luiz Alberto Díaz Rodrigues
Keyword(s):  
Pattern Formation ◽  
Temporal Dynamics ◽  
Complex Dynamics ◽  
Species Conservation ◽  
Turing Instability ◽  
Food Sources ◽  
Generalist Predators ◽  
Generalist Predator ◽  
Predator Prey ◽  
Predator Prey Model

Generalist predators have several food sources and do not depend on one prey species to survive. There has been considerable attention paid by modellers to generalist predator-prey interactions in recent years. Erbach and collaborators in 2013 found a complex dynamics with bistability, limit-cycles and bifurcations in a generalist predator-prey system. In this paper we explore the spatio-temporal dynamics of a reaction-diffusion PDE model for the generalist predator-prey dynamics analyzed by Erbach and colleagues. In particular, we study the Turing and Turing-Hopf pattern formation with special attention to the regime of bistability exhibited by the local model. We derive the conditions for Turing instability and find the region of parameters for which Turing and/or Turing-Hopf instability are possible. By means of numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. In the Turing-Hopf range of the parameters, we observed either stable patterns or homogeneous periodic distributions. Our findings reveal that movement can break the effect of hysteresis observed in the local dynamics, what can have important implication in pest management and species conservation.

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.

scholarly journals Predator-prey interactions under fear effect and multiple foraging strategies

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Susmita Halder ◽  
Joydeb Bhattacharyya ◽  
Samares Pal
Keyword(s):  
Complex Dynamics ◽  
Model Simulation ◽  
Prey Population ◽  
Foraging Strategies ◽  
Generalist Predator ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Simulation Results

<p style='text-indent:20px;'>We propose and analyze the effects of a generalist predator-driven fear effect on a prey population by considering a modified Leslie-Gower predator-prey model. We assume that the prey population suffers from reduced fecundity due to the fear of predators. We investigate the predator-prey dynamics by incorporating linear, Holling type Ⅱ and Holling type Ⅲ foraging strategies of the generalist predator. As a control strategy, we have considered density-dependent harvesting of the organisms in the system. We show that the systems with linear and Holling type Ⅲ foraging exhibit transcritical bifurcation, whereas the system with Holling type Ⅱ foraging has a much more complex dynamics with transcritical, saddle-node, and Hopf bifurcations. It is observed that the prey population in the system with Holling type Ⅲ foraging of the predator gets severely affected by the predation-driven fear effect in comparison with the same with linear and Holling type Ⅱ foraging rates of the predator. Our model simulation results show that an increase in the harvesting rate of the predator is a viable strategy in recovering the prey population.</p>

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 Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Zongmin Yue ◽  
Xiaoqin Wang ◽  
Haifeng Liu
Keyword(s):  
Allee Effect ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Turing Instability ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Highly Sensitive ◽  
The Impact

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

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.

scholarly journals Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Hua Liu ◽  
Yong Ye ◽  
Yumei Wei ◽  
Weiyuan Ma ◽  
Ming Ma ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Pattern Formation ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Point Of View ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Pattern Formations

In this paper, we establish a reaction-diffusion predator-prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.

scholarly journals Spatiotemporal pattern formation in a prey-predator model with generalist predator

2021 ◽  
Author(s):  
Kalyan Manna ◽  
Malay Banerjee
Keyword(s):  
Prey Species ◽  
Landau Equation ◽  
Traveling Wave Solutions ◽  
Turing Instability ◽  
Prey Consumption ◽  
Food Sources ◽  
Spatiotemporal Pattern ◽  
Generalist Predators ◽  
Generalist Predator ◽  
Density Level

Generalist predators exploit multiple food sources and it is economical for them to reduce predation pressure on a particular prey species when their density level becomes comparatively less. As a result, a prey-predator system tends to become more stable in the presence of a generalist predator. In this article, we investigate the roles of both the diffusion and nonlocal prey consumption in shaping the population distributions for interacting generalist predator and its focal prey species. In this regard, we first derive the conditions associated with Turing instability through linear analysis. Then, we perform a weakly nonlinear analysis and derive a cubic Stuart-Landau equation governing amplitude of the resulting patterns near Turing bifurcation boundary. Further, we present a wide variety of numerical simulations to corroborate our analytical findings as well as to illustrate some other complex spatiotemporal dynamics. Interestingly, our study reveals the existence of traveling wave solutions connecting two spatially homogeneous coexistence steady states in Turing domain under the influence of temporal bistability phenomenon. Also, our investigation shows that nonlocal prey consumption acts as a stabilizing force for the system dynamics.

scholarly journals Pattern Formation in a Cross-Diffusive Holling-Tanner Model

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Weiming Wang ◽  
Zhengguang Guo ◽  
R. K. Upadhyay ◽  
Yezhi Lin
Keyword(s):  
Pattern Formation ◽  
Sufficient Conditions ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Diffusion Controlled ◽  
Model Dynamics ◽  
Flux Boundary Conditions ◽  
Spatially Distributed

We present a theoretical analysis of the processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; Hopf and Turing bifurcation in a spatial domain is presented, too. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replication. And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive model.

Spatial Pattern of Ratio-Dependent Predator–Prey Model with Prey Harvesting and Cross-Diffusion

2019 ◽  
Vol 29 (03) ◽  
pp. 1950036 ◽  
Author(s):  
R. Sivasamy ◽  
M. Sivakumar ◽  
K. Balachandran ◽  
K. Sathiyanathan
Keyword(s):  
Temporal Dynamics ◽  
Intake Rate ◽  
Diffusion Term ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonlinear Harvesting ◽  
Ratio Dependent ◽  
The Cross

This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.

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