scholarly journals Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis

2021 ◽  
Vol 18 (5) ◽  
pp. 6672-6699
Author(s):  
Yong Luo ◽  
Keyword(s):  
Classical Solution ◽  
Linear Instability ◽  
Instability Criterion ◽  
Predator Population ◽  
Population System ◽  
Dependent Case ◽  
Density Dependent ◽  
Existence And Stability ◽  
Dimensional Distribution ◽  
Predator Model

<abstract><p>We study the existence of global unique classical solution to a density-dependent prey-predator population system with indirect prey-taxis effect. With two Lyapunov functions appropriately constructed, we then show that the solution can asymptotically approach prey-only state or coexistence state of the system under suitable conditions. Moreover, linearized analysis on the system at these two constant steady states shows their linear instability criterion. By numerical simulation we find that some density-dependent prey-taxis and predators' diffusion may either flatten the spatial one-dimensional patterns which exist in non-density-dependent case, or break the spatial two-dimensional distribution similarity which occurs in non-density-dependent case between predators and chemoattractants (released by prey).</p></abstract>

Bifurcation Analysis of a Prey–Predator Model with Beddington–DeAngelis Type Functional Response and Allee Effect in Prey

2020 ◽  
Vol 30 (16) ◽  
pp. 2050238
Author(s):  
Koushik Garain ◽  
Partha Sarathi Mandal
Keyword(s):  
Functional Response ◽  
Death Rate ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Growth Function ◽  
Global Dynamics ◽  
Density Dependent ◽  
Existence And Stability ◽  
Predator Model ◽  
The Impact

The article aims to study a prey–predator model which includes the Allee effect phenomena in prey growth function, density dependent death rate for predators and Beddington–DeAngelis type functional response. We notice the changes in the existence and stability of the equilibrium points due to the Allee effect. To investigate the complete global dynamics of the Allee model, we present here a two-parametric bifurcation diagram which describes the effect of density dependent death rate parameter of predator on dynamical changes of the system. We have also analyzed all possible local and global bifurcations that the system could go through, namely transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Finally, the impact of the Allee effect in the considered system is investigated by comparing the dynamics of both the systems with and without Allee effect.

scholarly journals Méthode d'agrégation des variables appliquée à la dynamique des populations

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Pierre Auger ◽  
Abderrahim El Abdllaoui ◽  
Rachid Mchich
Keyword(s):  
Stable Limit Cycle ◽  
Stable Limit ◽  
Dynamique Des Populations ◽  
Globally Asymptotically Stable ◽  
Density Dependent ◽  
Migration Rates ◽  
First Case ◽  
International Audience ◽  
Predator Model ◽  
Predator System

International audience We present the method of aggregation of variables in the case of ordinary differential equations. We apply the method to a prey - predator model in a multi - patchy environment. In this model, preys can go to a refuge and therefore escape to predation. The predator must return regularly to his terrier to feed his progeny. We study the effect of density-dependent migration on the global stability of the prey-predator system. We consider constant migration rates, but also density-dependent migration rates. We prove that the positif equilibrium is globally asymptotically stable in the first case, and that its stability changes in the second case. The fact that we consider density-dependent migration rates leads to the existence of a stable limit cycle via a Hopf bifurcation. Nous présentons les grandes lignes de laméthode d'agrégation des variables dans les systèmes d'équations différentielles ordinaires. Nous appliquons laméthode à un modèle proie-prédateur spatialisé. Dans ce modèle, les proies peuvent échapper à la prédation en se réfugiant sur un site. Le prédateur doit aussi retourner régulièrement dans son terrier pour nourrir sa progéniture. Nous étudions les effets de migration dépendant de la densité des populations sur la stabilité globale du système proie-prédateur. Nous considérons des taux de migration constants, puis densité-dépendants. Dans le cas de taux constants il existe un équilibre positif toujours stable alors que dans le cas de taux de migration densité-dépendants, il existe un cycle limite stable via une bifurcation de Hopf.

scholarly journals Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zhenzhen Shi ◽  
Qingjian Li ◽  
Weiming Li ◽  
Huidong Cheng
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Poincaré Map ◽  
Global Dynamics ◽  
Poincare Map ◽  
Existence And Stability ◽  
Sufficient And Necessary Conditions ◽  
Predator Model ◽  
Order One Periodic Solution

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained when ΦyA<yA, and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when ΦyA>yA. Furthermore, we prove the existence of the order-kk≥2 periodic solution under certain conditions. Finally, we verify the main results by numerical simulation.

EFFECT OF TIME-DELAY ON A PREY–PREDATOR MODEL WITH MICROPARASITE INFECTION IN THE PREDATOR

2011 ◽  
Vol 19 (02) ◽  
pp. 365-387 ◽  
Author(s):  
SWETA PATHAK ◽  
ALAKES MAITI ◽  
SHYAM PADA BERA
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Predator Population ◽  
Dynamical Characteristics ◽  
Dynamical Behaviors ◽  
Discrete Time Delay ◽  
Mathematical Analyses ◽  
Predator Model ◽  
Living Organisms

To increase a prey population that is attacked by a predator it is more convenient and economical to choose the living organisms to control the predator. In this paper, the dynamical behaviors of a prey–predator model with microparasitic infection in the predator have been discussed. In this epidemiological model the microparasite is horizontally transmitted and attacks the predator population only. The infected population does not recover or become immune. The dynamical characteristics of the system are studied through mathematical analyses. The role of discrete time-delay has been discussed to show that time-delay can induce instability and oscillation. Numerical simulations are carried out. Biological implications have been discussed.

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 Global Stability Analysis and Optimal Control of a Harvested Ecoepidemiological Prey Predator Model with Vaccination and Taxation

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Chao Liu ◽  
Qingling Zhang ◽  
Jinna Li
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Global Stability ◽  
Interaction Mechanism ◽  
Prey Population ◽  
Predator Population ◽  
Optimal Control Strategy ◽  
Global Stability Analysis ◽  
Selective Harvest ◽  
Predator Model

We propose an ecoepidemiological prey predator model, where selective harvest effort on predator population is considered. Vaccination and taxation are introduced as control instruments, which are utilized to control number of susceptible prey population and protect predator population from overexploitation, respectively. Conditions which influence nonnegativity and boundedness of solutions are studied. Global stability analysis around disease-free equilibrium is discussed based on robust Bendixson criterion, which is theoretically beneficial to studying coexistence and interaction mechanism of population within harvested ecoepidemiological system. By using Pontryagin’s maximum principle, an optimal control strategy is derived to maximize the total discounted net economic revenue to society as well as protect prey population from infectious disease. Numerical simulations are carried out to show the consistency with theoretical analysis.

An Imprecise Eco-Epidemic Model with Pesticide in Relevance to Agricultural Pest Control

2018 ◽  
Vol 13 (03) ◽  
pp. 109-131
Author(s):  
Anjana Das ◽  
M. Pal
Keyword(s):  
Pest Control ◽  
Dynamical Behavior ◽  
Predator Population ◽  
Agricultural Pest ◽  
Stability Criteria ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Proposed Model ◽  
Existence And Stability ◽  
Imprecise Parameters

In this paper, we have proposed and analyzed an agricultural pest control system. For this purpose, an eco-epidemiological type predator–prey model has been proposed with the consideration of a sound predator population and two classes of pest populations namely susceptible pest and infected pest. Further to consider uncertainty, we modify our model and transform it into a fuzzy system with incorporation of imprecise parameters. The dynamical behavior of the proposed model has been investigated by examining the existence and stability criteria of all feasible equilibria. An optimal control problem is formed by considering the pesticide control as the control parameter and then the problem is solved both theoretically and numerically with the help of some computer simulation works.

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