scholarly journals Mathematical analysis of harvested predator-prey system with prey refuge and intraspecific competition

2021 ◽  
Vol 47 (2) ◽  
pp. 728-737
Author(s):  
Alanus Mapunda ◽  
Thadei Sagamiko
Keyword(s):  
Mathematical Analysis ◽  
Intraspecific Competition ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behaviour ◽  
Predator Population ◽  
Bifurcation Parameter ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System

In this paper, a predator-prey relationship in the presence of prey refuge was studied. The analysis of the dependence of locally stable equilibrium points on the parameters of the problem was carried out. Bifurcation and limit cycles for the model were analyzed to show the dynamical behaviour of the system. The results showed that the system is stable at a constant prey refuge m = 0.3 and prey harvesting rate H = 0.3. However, increasing m and decreasing H or vice versa, the predator-prey system remains stable. It was further observed that for a constant prey refuge m ≥ 0.78, the predator population undergoes extinction. Therefore, m was found to be a bifurcation parameter and m = 0.78 is a bifurcation value. Keywords: Prey refuge, bifurcation, harvesting, intraspecific competition, phase portrait

OCCURRENCE OF CHAOS AND ITS POSSIBLE CONTROL IN A PREDATOR-PREY MODEL WITH DENSITY DEPENDENT DISEASE-INDUCED MORTALITY ON PREDATOR POPULATION

2010 ◽  
Vol 18 (02) ◽  
pp. 399-435 ◽  
Author(s):  
KRISHNA PADA DAS ◽  
SAMRAT CHATTERJEE ◽  
J. CHATTOPADHYAY
Keyword(s):  
Equilibrium Points ◽  
Dynamical Behaviour ◽  
Predator Population ◽  
Epidemiological Models ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Chaotic Nature ◽  
Mortality Function ◽  
Infected Prey

Eco-epidemiological models are now receiving much attention to the researchers. In the present article we re-visit the model of Holling-Tanner which is recently modified by Haque and Venturino1 with the introduction of disease in prey population. Density dependent disease-induced predator mortality function is an important consideration of such systems. We extend the model of Haque and Venturino1 with density dependent disease-induced predator mortality function. The existence and local stability of the equilibrium points and the conditions for the permanence and impermanence of the system are worked out. The system shows different dynamical behaviour including chaos for different values of the rate of infection. The model considered by Haque and Venturino1 also exhibits chaotic nature but they did not shed any light in this direction. Our analysis reveals that by controlling disease-induced mortality of predator due to ingested infected prey may prevent the occurrence of chaos.

scholarly journals Dynamics of a Predator-Prey Population in the Presence of Resource Subsidy under the Influence of Nonlinear Prey Refuge and Fear Effect

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-38
Author(s):  
Sudeshna Mondal ◽  
G. P. Samanta ◽  
Juan J. Nieto
Keyword(s):  
Equilibrium Points ◽  
Time Lag ◽  
Sufficient Conditions ◽  
Food Sources ◽  
Prey Population ◽  
Predator Population ◽  
Input Rate ◽  
Prey Refuge ◽  
Predator Prey ◽  
The Impact

In this work, our aim is to investigate the impact of a non-Kolmogorov predator-prey-subsidy model incorporating nonlinear prey refuge and the effect of fear with Holling type II functional response. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). The positivity and asymptotically uniform boundedness of the solutions of the system have been derived. Analytically, we have studied the criteria for the feasibility and stability of different equilibrium points. In addition, we have derived sufficient conditions for the existence of local bifurcations of codimension 1 (transcritical and Hopf bifurcation). It is also observed that there is some time lag between the time of perceiving predator signals through vocal cues and the reduction of prey’s birth rate. So, we have analyzed the dynamical behaviour of the delayed predator-prey-subsidy model. Numerical computations have been performed using MATLAB to validate all the analytical findings. Numerically, it has been observed that the predator, prey, and subsidy can always exist at a nonzero subsidy input rate. But, at a high subsidy input rate, the prey population cannot persist and the predator population has a huge growth due to the availability of food sources.

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

2016 ◽  
Vol 24 (02n03) ◽  
pp. 345-365 ◽  
Author(s):  
SUDIP SAMANTA ◽  
RIKHIYA DHAR ◽  
IBRAHIM M. ELMOJTABA ◽  
JOYDEV CHATTOPADHYAY
Keyword(s):  
Stable Equilibrium ◽  
Predator Population ◽  
Stable Limit ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

NONCONSTANT PREY HARVESTING IN RATIO-DEPENDENT PREDATOR–PREY SYSTEM INCORPORATING A CONSTANT PREY REFUGE

2012 ◽  
Vol 05 (02) ◽  
pp. 1250021 ◽  
Author(s):  
SAPNA DEVI
Keyword(s):  
Rate Function ◽  
Predator Population ◽  
Catch Per Unit Effort ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Optimal Harvest ◽  
Ratio Dependent ◽  
The Stability ◽  
Optimal Harvest Policy

This paper deals with the problem of nonconstant harvesting of prey in a ratio-dependent predator–prey system incorporating a constant prey refuge. Here we use the reasonable catch-rate function instead of usual catch-per-unit-effort hypothesis. The existence, as well as the stability of possible equilibria, is carried out. Bionomic equilibrium of the system is determined and optimal harvest policy is studied with the help of Pontryagin's maximum principle. The key results developed in this paper are illustrated using numerical simulations. Our results indicate that dynamic behavior of the system very much depends on the prey refuge parameter and increasing amount of refuge could increase prey density and may lead to the extinction of predator population density.

scholarly journals On a predator-prey system interaction under fluctuating water level with nonselective harvesting

2020 ◽  
Vol 18 (1) ◽  
pp. 458-475
Author(s):  
Na Zhang ◽  
Yonggui Kao ◽  
Fengde Chen ◽  
Binfeng Xie ◽  
Shiyu Li
Keyword(s):  
Water Level ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Optimal Harvesting ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Model Interaction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fluctuating Water Level

Abstract A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global stability of the positive equilibrium are obtained. The bionomic equilibrium and the optimal harvesting policy are also presented. Numerical simulations are carried out to show the feasibility of the main results.

scholarly journals Complex Dynamical Behavior of a Predator-Prey System with Group Defense

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jianglin Zhao ◽  
Min Zhao ◽  
Hengguo Yu
Keyword(s):  
Numerical Simulation ◽  
Theoretical Basis ◽  
Dynamical Behavior ◽  
Turing Instability ◽  
Equilibrium Density ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Group Defense ◽  
Complex Dynamical Behavior

A diffusive predator-prey system with prey refuge is studied analytically and numerically. The Turing bifurcation is analyzed in detail, which in turn provides a theoretical basis for the numerical simulation. The influence of prey refuge and group defense on the equilibrium density and patterns of species under the condition of Turing instability is explored by numerical simulations, and this shows that the prey refuge and group defense have an important effect on the equilibrium density and patterns of species. Moreover, it can be obtained that the distributions of species are more sensitive to group defense than prey refuge. These results are expected to be of significance in exploration for the spatiotemporal dynamics of ecosystems.

scholarly journals Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Caifeng Du
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

AbstractIn this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

scholarly journals Dynamical Behaviour in Two Prey-Predator System with Persistence

2016 ◽  
Vol 16 ◽  
pp. 20-37
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Limit Cycle ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Behaviour ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Interior Equilibrium ◽  
Trivial Equilibrium ◽  
Necessary And Sufficient

In this work, the dynamical behavior of the system with two preys and one predator population is investigated. The predator exhibits a Holling type II response to one prey which is harvested and a Beddington-DeAngelis functional response to the other prey. The boundedness of the system is analyzed. We examine the occurrence of positive equilibrium points and stability of the system at those points. At trivial equilibrium E0and axial equilibrium (E1); the system is found to be unstable. Also we obtain the necessary and sufficient conditions for existence of interior equilibrium point (E6) and local and global stability of the system at the interior equilibrium (E6): Depending upon the existence of limit cycle, the persistence condition is established for the system. The numerical simulation infer that varying the parameters such as e and λ1it is possible to change the dynamical behavior of the system from limit cycle to stable spiral. It is also observed that the harvesting rate plays a crucial role in stabilizing the system.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

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