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 The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoli Wang ◽  
Zhihao Ge
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

Dynamical Analysis of Prey Refuge in a Predator-Prey System with Rosenzweig Functional Response

2011 ◽  
Vol 271-273 ◽  
pp. 577-580
Author(s):  
Zhi Hui Ma ◽  
Shu Fan Wang ◽  
Wen Ting Wang
Keyword(s):  
Functional Response ◽  
Stability Property ◽  
Dynamical Analysis ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Stabilizing Effect ◽  
The Stability ◽  
Prey Refuges

In this paper, we proposed a predator-prey system incorporating Rosenzweig functional response and prey refuges. We will consider the stability property of the equilibria. Our results show that refuges using by prey have stabilizing effect on the considered system.

scholarly journals Scrounging by foragers can resolve the paradox of enrichment

2016 ◽  
Author(s):  
Wataru Toyokawa
Keyword(s):  
Group Living ◽  
Theoretical Models ◽  
Social Foraging ◽  
Predator Population ◽  
Ecological Communities ◽  
Predator Prey ◽  
Paradox Of Enrichment ◽  
Prey System ◽  
The Stability ◽  
Predator System

AbstractTheoretical models of predator-prey system predict that sufficient enrichment of prey can generate large amplitude limit cycles, paradoxically causing a high risk of extinction (the paradox of enrichment). While real ecological communities contain many gregarious species whose foraging behaviour should be influenced by socially transmitted information, few theoretical studies have examined the possibility that social foraging might be a resolution of the paradox. I considered a predator population in which individuals play the producer-scrounger foraging game both in a one-prey-one-predator system and a two-prey-one-predator system. I analysed the stability of a coexisting equilibrium point in the former one-prey system and that of non-equilibrium dynamics of the latter two-prey system. The result showed that social foraging can stabilise both systems and thereby resolves the paradox of enrichment when scrounging behaviour is prevalent in predators. This suggests a previously neglected mechanism underlying a powerful effect of group-living animals on sustainability of ecological communities.

scholarly journals Stability analysis of an eco-epidemiological model incorporating a prey refuge

2010 ◽  
Vol 15 (4) ◽  
pp. 473-491 ◽  
Author(s):  
A. K. Pal ◽  
G. P. Samanta
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Computer Simulations ◽  
Discrete Time ◽  
Predator Population ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Discrete Time Delay ◽  
The Stability

The present paper deals with the problem of a predator-prey model incorporating a prey refuge with disease in the prey-population. We assume the predator population will prefer only infected population for their diet as those are more vulnerable. Dynamical behaviours such as boundedness, permanence, local and global stabilities are addressed. We have also studied the effect of discrete time delay on the model. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.

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

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.

scholarly journals STABILITY AND BIFURCATION IN A DELAYED RATIO-DEPENDENT PREDATOR–PREY SYSTEM

2002 ◽  
Vol 46 (1) ◽  
pp. 205-220 ◽  
Author(s):  
Dongmei Xiao ◽  
Wenxia Li
Keyword(s):  
Periodic Solutions ◽  
Time Delays ◽  
Analytical Study ◽  
Real Life ◽  
Primary 34C25 ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Prey System ◽  
Ratio Dependent ◽  
The Stability

AbstractRecently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay $\tau$ as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions.AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14

scholarly journals Bifurcation and control of a predator-prey system with unfixed functional responses

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lizhi Fei ◽  
Xingwu Chen
Keyword(s):  
Fixed Points ◽  
Hybrid Control ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Functional Responses ◽  
Predator Prey ◽  
Prey System ◽  
The Stability ◽  
Necessary And Sufficient ◽  
And Control

<p style='text-indent:20px;'>In this paper we investigate a discrete-time predator-prey system with not only some constant parameters but also unfixed functional responses including growth rate function of prey, conversion factor function and predation probability function. We prove that the maximal number of fixed points is <inline-formula><tex-math id="M1">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> and give necessary and sufficient conditions of exactly <inline-formula><tex-math id="M2">\begin{document}$ j $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M3">\begin{document}$ j = 1,2,3 $\end{document}</tex-math></inline-formula>) fixed points, respectively. For transcritical bifurcation and Neimark-Sacker bifurcation, we provide bifurcation conditions depending on these unfixed functional responses. In order to regulate the stability of this biological system, a hybrid control strategy is used to control the Neimark-Sacker bifurcation. Finally, we apply our main results to some examples and carry out numerical simulations for each example to verify the correctness of our theoretical analysis.</p>

scholarly journals Scrounging by foragers can resolve the paradox of enrichment

2017 ◽  
Vol 4 (3) ◽  
pp. 160830 ◽  
Author(s):  
Wataru Toyokawa
Keyword(s):  
Group Living ◽  
Theoretical Models ◽  
Social Foraging ◽  
Predator Population ◽  
Ecological Communities ◽  
Predator Prey ◽  
Paradox Of Enrichment ◽  
Prey System ◽  
The Stability ◽  
The One

Theoretical models of predator–prey systems predict that sufficient enrichment of prey can generate large amplitude limit cycles, paradoxically causing a high risk of extinction (the paradox of enrichment). Although real ecological communities contain many gregarious species, whose foraging behaviour should be influenced by socially transmitted information, few theoretical studies have examined the possibility that social foraging might resolve this paradox. I considered a predator population in which individuals play the producer–scrounger foraging game in one-prey-one-predator and two-prey-one-predator systems. I analysed the stability of a coexisting equilibrium point in the one-prey system and that of non-equilibrium dynamics in the two-prey system. The results revealed that social foraging could stabilize both systems, and thereby resolve the paradox of enrichment when scrounging behaviour (i.e. kleptoparasitism) is prevalent in predators. This suggests a previously neglected mechanism underlying a powerful effect of group-living animals on the sustainability of ecological communities.

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