scholarly journals Discrete-Time Predator-Prey Interaction with Selective Harvesting and Predator Self-Limitation

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Zhihua Chen ◽  
Qamar Din ◽  
Muhammad Rafaqat ◽  
Umer Saeed ◽  
Muhammad Bilal Ajaz
Keyword(s):  
Discrete Time ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
State Feedback Control ◽  
Time Model ◽  
Predator Prey ◽  
Selective Harvesting ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Interaction

Selective harvesting plays an important role on the dynamics of predator-prey interaction. On the other hand, the effect of predator self-limitation contributes remarkably to the stabilization of exploitative interactions. Keeping in view the selective harvesting and predator self-limitation, a discrete-time predator-prey model is discussed. Existence of fixed points and their local dynamics is explored for the proposed discrete-time model. Explicit principles of Neimark–Sacker bifurcation and period-doubling bifurcation are used for discussion related to bifurcation analysis in the discrete-time predator-prey system. The control of chaotic behavior is discussed with the help of methods related to state feedback control and parameter perturbation. At the end, some numerical examples are presented for verification and illustration of theoretical findings.

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.

Fear Induced Multistability in a Predator-Prey Model

2021 ◽  
Vol 31 (10) ◽  
pp. 2150150
Author(s):  
N. C. Pati ◽  
Shilpa Garai ◽  
Mainul Hossain ◽  
G. C. Layek ◽  
Nikhil Pal
Keyword(s):  
Parameter Space ◽  
Periodic Structures ◽  
Chaotic Oscillations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Prey Interaction ◽  
Type Ii Functional Response

In ecology, the predator’s impact goes beyond just killing the prey. In the present work, we explore the role of fear in the dynamics of a discrete-time predator-prey model where the predator-prey interaction obeys Holling type-II functional response. Owing to the increasing strength of fear, the system becomes stable from chaotic oscillations via inverse Neimark–Sacker bifurcation. Extensive numerical simulations are carried out to investigate the intricate dynamics for the organization of periodic structures in the bi-parameter space of the system. We observe fear induced multistability between different pairs of coexisting heterogeneous attractors due to the overlapping of multiple periodic domains in the bi-parameter space. The basin sets of the coexisting attractors are obtained and discussed at length. Multistability in the predator-prey system is important because the dynamics of the predator and prey populations in the critical parameter zone becomes uncertain.

Mutual interference and its effects on searching efficiency between predator–prey interaction with bifurcation analysis and chaos control

2021 ◽  
pp. 107754632110564
Author(s):  
Waqas Ishaque ◽  
Qamar Din ◽  
Muhammad Taj
Keyword(s):  
Chaos Control ◽  
Chaotic Behavior ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Mutual Interference ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Existence And Stability ◽  
Prey Interaction

In this paper, we study the dynamic of the predator–prey model based on mutual interference and its effects on searching efficiency. The parametric conditions, existence, and stability for trivial and boundary equilibrium points are studied. Also, it has shown that by applying the center manifold theorem and bifurcation theory, system undergoes Neimark–Sacker bifurcation across the neighborhood of a positive fixed point. Moreover, due to the bifurcation and chaos which objectively exist in a system, three chaos control strategies are designed and used. Moreover, to validate our theoretical and analytical discussions, numerical simulations are applied to show complex and chaotic behavior. Finally, theoretical discussions are validated with experimental field data.

Spatiotemporal Model of a Predator–Prey System with Herd Behavior and Quadratic Mortality

2019 ◽  
Vol 29 (04) ◽  
pp. 1950049 ◽  
Author(s):  
Teekam Singh ◽  
Sandip Banerjee
Keyword(s):  
Herd Behavior ◽  
Spatiotemporal Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Prey System ◽  
Prey Model ◽  
Prey Interaction ◽  
Mixed Pattern ◽  
Spotted Pattern

In this paper, we have investigated a diffusive predator–prey model with herd behavior. Also, we considered that the mortality of predators is linear as well as quadratic. Using linear stability analysis, we obtain the condition for diffusive instability and identify the corresponding domain in the space of control parameters. Using extensive numerical simulations, we obtain non-Turing spatiotemporal patterns in the model with linear mortality of predators, and Turing pattern formation, namely, spotted pattern and mixed pattern (spots-stripes) in model with quadratic mortality of predators. The results focus on the effect of the changing mortality rates of predator in pattern dynamics of a diffusive predator–prey model and help us in the better understanding of the dynamics of the predator–prey interaction in real environment.

scholarly journals Discretization, Bifurcation and Control for a Class of Predator–Prey Interactions

2022 ◽  
Vol 6 (1) ◽  
pp. 31
Author(s):  
Asifa Tassaddiq ◽  
Muhammad Sajjad Shabbir ◽  
Qamar Din ◽  
Humera Naaz
Keyword(s):  
Chaotic Behavior ◽  
Period Doubling ◽  
Fractal Dimensions ◽  
Phase Portraits ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Piecewise Constant Argument ◽  
Feedback Control Strategy

The present study focuses on the dynamical aspects of a discrete-time Leslie–Gower predator–prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark–Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings.

scholarly journals Stoichiometric knife-edge model on discrete time scale

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ming Chen ◽  
Lale Asik ◽  
Angela Peace
Keyword(s):  
Discrete Time ◽  
Time Scale ◽  
Ecological Stoichiometry ◽  
Discrete Model ◽  
Nutrient Content ◽  
Continuous Model ◽  
Knife Edge ◽  
Time Model ◽  
Predator Prey ◽  
Predator Prey Model

AbstractEcological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.

Role of Alternative Food in Controlling Chaotic Dynamics in a Predator–Prey Model with Disease in the Predator

2016 ◽  
Vol 26 (09) ◽  
pp. 1650147 ◽  
Author(s):  
Krishna Pada Das ◽  
Nandadulal Bairagi ◽  
Prabir Sen
Keyword(s):  
Chaotic Dynamics ◽  
Period Doubling ◽  
Host Population ◽  
Alternative Food ◽  
Predator Population ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Oscillatory State ◽  
Prey Interaction

It is generally, but not always, accepted that alternative food plays a stabilizing role in predator–prey interaction. Parasites, on the other hand, have the ability to change both the qualitative and quantitative dynamics of its host population. In recent times, researchers are showing growing interest in formulating models that integrate both the ecological and epidemiological aspects. The present paper deals with the effect of alternative food on a predator–prey system with disease in the predator population. We show that the system, in the absence of alternative food, exhibits different dynamics viz. stable coexistence, limit cycle oscillations, period-doubling bifurcation and chaos when infection rate is gradually increased. However, when predator consumes alternative food coupled with its focal prey, the system returns to regular oscillatory state from chaotic state through period-halving bifurcations. Our study shows that alternative food may have larger impact on the community structure and may increase population persistence.

Hunting Cooperation in a Discrete-Time Predator–Prey System

2018 ◽  
Vol 28 (07) ◽  
pp. 1850083 ◽  
Author(s):  
Saheb Pal ◽  
Nikhil Pal ◽  
Joydev Chattopadhyay
Keyword(s):  
Numerical Simulations ◽  
Discrete Time ◽  
Period Doubling ◽  
Phase Portraits ◽  
Predator Population ◽  
Predator Prey ◽  
Prey System ◽  
Cooperation Rate ◽  
Route To Chaos ◽  
The Impact

The present paper mainly investigates the impact of hunting cooperation in a discrete-time predator–prey system through numerical simulations. We show that without hunting cooperation, an increase in the growth rate of prey population produces chaotic dynamics. We also show that hunting cooperation has the potential to modify the well-known period-doubling route to chaos by reverse period-halving bifurcations and makes the system stable. However, very high hunting cooperation can be detrimental and populations go to extinction. We observe that hunting cooperation induces strong demographic Allee effect in the system, where predator population persists due to hunting cooperation and would go to extinction without hunting cooperation. We perform extensive numerical simulations of the system and draw phase portraits, bifurcation diagrams, maximum Lyapunov exponents, two-parameter stability regions. We also observe the occurrence of flip and Neimark–Sacker bifurcations by taking the hunting cooperation rate as a bifurcation parameter.

CHAOTIC BEHAVIOR OF A PERIODICALLY FORCED PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE AND IMPULSIVE PERTURBATIONS

2006 ◽  
Vol 09 (03) ◽  
pp. 209-222 ◽  
Author(s):  
SHUWEN ZHANG ◽  
DEJUN TAN ◽  
LANSUN CHEN
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Periodic Variation ◽  
Periodic Forcing ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Impulsive Perturbations

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

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