scholarly journals Existence and Uniqueness of Caputo Fractional Predator-Prey Model of Holling-Type II with Numerical Simulations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
A. Al Themairi ◽  
Manar A. Alqudah
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Real Life ◽  
Population Based ◽  
Type Ii ◽  
Predator Species ◽  
Susceptible Population ◽  
Euler’S Method ◽  
Predator Prey Model ◽  
Caputo Fractional Order Derivative

We suggested a new mathematical model for three prey-predator species, predator is considered to be divided into two compartments, infected and susceptible predators, as well as the prey and susceptible population based on Holling-type II with harvesting. We considered the model in Caputo fractional order derivative to have significant consequences in real life since the population of prey create memory and learn from their experience of escaping and resisting any threat. The existence, uniqueness, and boundedness of the solution and the equilibrium points for the considered model are studied. Numerical simulations using Euler’s method are discussed to interpret the applicability of the considered model.

scholarly journals Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Wanlin Chen ◽  
Yuhua Lin
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.

Predator–Prey System: Bachelor Herding of the Prey Imposes Ecological Constraints on Predation

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Rajat Kaushik ◽  
Sandip Banerjee
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Herd Behavior ◽  
Ecological Constraints ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Schooling Behavior ◽  
Prey System ◽  
Parent Groups

Bachelor herd behavior is very common among juvenile animals who have not become sexually matured but have left their parent groups. The complex grouping or schooling behavior provides vulnerable juveniles refuge from predation and opportunities for foraging, especially when their parents are not within the area to protect them. In spite of this, juvenile/immature prey may easily become victims because of their greenness while on the other hand, adult prey may be invulnerable to attack due to their tricky manoeuvring abilities to escape from the predators. In this study, we propose a stage-structured predator–prey model, in which predators attack only the bachelor herds of juvenile prey while adult prey save themselves due to small predator–prey size ratio and their fleeing capability, enabling them to avoid confrontation with the predators. Local and global stability analysis on the equilibrium points of the model are performed. Sufficient conditions for uniform permanence and the impermanence are derived. The model exhibits both transcritical as well as Hopf bifurcations and the corresponding numerical simulations are carried out to support the analytical results. Bachelor herding of juvenile prey as well as inaccessibility of adult prey restricts the uncontrolled predation so that prey abundance and predation remain balanced. This investigation on bachelor group defence brings out some unpredictable results, especially close to the zero steady state. Altogether, bachelor herding of the juvenile prey, which causes unconventional behavior near the origin, plays a significant role in establishing uniform permanence conditions, also increases richness of the dynamics in numerical simulations using the bifurcation theory and thereby, shapes ecosystem properties tremendously and may have a large influence on the ecosystem functioning.

External Source of Infection and Nutritional Efficiency Control Chaos in a Predator–Prey Model with Disease in the Predator

2017 ◽  
Vol 12 (02) ◽  
pp. 87-115 ◽  
Author(s):  
Krishna pada Das ◽  
Prodip Roy ◽  
Subhabrata Ghosh ◽  
Somnath Maiti
Keyword(s):  
Basic Reproduction Number ◽  
Chaotic Dynamics ◽  
Nutritional Value ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
External Source ◽  
Maximum Lyapunov Exponent ◽  
Predator Species ◽  
Predator Prey Model ◽  
Source Of Infection

This paper deals with an eco-epidemiological approach with disease circulating through the predator species. Disease circulation in the predator species can be possible by contact as well as by external sources. Here, we try to discuss the role of external source of infection along with nutritional value on system dynamics. To establish our findings, we have worked out the local and global stability analysis of the equilibrium points with Hopf bifurcation analysis associated with interior equilibrium point. The ecological consequence by ecological basic reproduction number as well as the disease basic reproduction number or basic reproductive ratio are obtained and we have analyzed the community structure of the particular system with the help of ecological and disease basic reproduction numbers. Further we pay attention to the chaotic dynamics which is produced by disease circulating in predator species by contact. Our numerical simulations reveal that eco-epidemiological system without external source of infection induced chaotic dynamics for increasing force of infection due to contact, whereas in the presence of external source of infection, it exhibits stable solution. It is also observed that nutritional value can prevent chaotic dynamics. We conclude that chaotic dynamics can be controlled by the external source of infection as well as nutritional value. We apply basic tools of nonlinear dynamics such as Poincare section and maximum Lyapunov exponent to investigate chaotic behavior of the system.

A Michaelis–Menten Predator–Prey Model with Strong Allee Effect and Disease in Prey Incorporating Prey Refuge

2018 ◽  
Vol 28 (06) ◽  
pp. 1850073 ◽  
Author(s):  
Sangeeta Saha ◽  
Alakes Maiti ◽  
G. P. Samanta
Keyword(s):  
Allee Effect ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Susceptible Population ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Lasalle Theorem ◽  
Transcritical Bifurcations

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

Application of Mathematical Models Two Predators and Infected Prey by Pesticide Control in Nilaparvata Lugens Spreading in Bantul Regency

2020 ◽  
Vol 2 (1) ◽  
pp. 41-50
Author(s):  
Irham Taufiq ◽  
Denik Agustito
Keyword(s):  
Equilibrium Points ◽  
Nilaparvata Lugens ◽  
Logistic Function ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Using Data ◽  
Infected Prey ◽  
Model Predator

AbstractIn this paper, we develop a mathematical model to analyze interactions between planthopper pests as prey and menochilus sexmaculatus and mirid ladybug as two predators where prey is controlled by pesticides. The interaction between predator and prey is modeled using the Holling type II response function. The predator and prey growth are modeled using a logistic function. From this model, we obtain eight equilibrium points. The three of these equilibrium points are analyzed using linearization and locally asymptotically stable. We simulate this model using data to predict the dynamics of planthopper population and its predators. Simulation result shows that all of these populations will survive because they are influenced by pesticide control and predation rates.Keywords: control of pest; predator-prey model; the Holling type II; the logistic function.                                                                                     AbstrakPada penelitian ini, kami membangun model matematika untuk menganalisis interaksi antara hama wereng sebagai mangsa (prey) dan menochilus sexmaculatus dan mirid ladybug sebagai dua pemangsa (predator) dimana mangsa dikontrol oleh pestisida. Interaksi antara predator dan prey dimodelkan menggunakan fungsi respon Holling tipe II sedangkan pertumbuhan predator dan prey dimodelkan menggunakan fungsi logistik. Dari model tersebut diperoleh delapan titik ekuilibrium. Tiga titik ekuilibrium dari titik-titik equilibrium tersebut dianalisis menggunakan metode linierisasi dan bersifat stabil asimtotik lokal. Kemudian model ini diaplikasikan pada data.  Untuk memudahkan interpretasi antara mangsa dan dua pemangsa dilakukan simulasi numerik untuk memprediksikan dinamika populasi wereng dan predatornya. Hasil simulasi menunjukkan bahwa semua populasi tersebut akan bertahan hidup karena dipengaruhi oleh kontrol pestisida dan tingkat pemangsaan.Kata Kunci: kontrol pestisida; model predator-prey; Holling tipe II; fungsi logistik.

scholarly journals Global Dynamics of an Exploited Prey-Predator Model with Constant Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Uttam Das ◽  
T. K. Kar ◽  
U. K. Pahari
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Global Dynamics ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Predator Model ◽  
Theoretical Results ◽  
Type Ii Functional Response

This paper describes a prey-predator model with Holling type II functional response incorporating constant prey refuge and harvesting to both prey and predator species. We have analyzed the boundedness of the system and existence of all possible feasible equilibria and discussed local as well as global stabilities at interior equilibrium of the system. The occurrence of Hopf bifurcation of the system is examined, and it was observed that the bifurcation is either supercritical or subcritical. Influences of prey refuge and harvesting efforts are also discussed. Some numerical simulations are carried out for the validity of theoretical results.

scholarly journals A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect

2021 ◽  
Vol 5 (3) ◽  
pp. 84
Author(s):  
Emli Rahmi ◽  
Isnani Darti ◽  
Agus Suryanto ◽  
Trisilowati
Keyword(s):  
Functional Response ◽  
Memory Effect ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Predator Population ◽  
Predator Prey Model ◽  
Proposed Model ◽  
Caputo Fractional Order Derivative ◽  
Extinction Point

In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation.

scholarly journals Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator

2020 ◽  
Vol 1 (1) ◽  
pp. 16-24
Author(s):  
Hasan S. Panigoro ◽  
Dian Savitri
Keyword(s):  
Hopf Bifurcation ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Biological Condition ◽  
Existence Of Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Caputo Fractional Order Derivative ◽  
Theoretical Results ◽  
Extinction Point

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results.

scholarly journals Predator-Prey Model with Refuge, Fear and Z-Control

2021 ◽  
Vol 26 (1) ◽  
pp. 40-57
Author(s):  
Ibrahim M. Elmojtaba ◽  
Kawkab Al-Amri ◽  
Qamar J.A. Khan
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Predator Population ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Long Run ◽  
Constant State

In this paper, we consider a predator-prey model incorporating fear and refuge.  Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium.  We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity.  Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium.  However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.

Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

2019 ◽  
Vol 20 (1) ◽  
pp. 89-104 ◽  
Author(s):  
Hafizul Molla ◽  
Md. Sabiar Rahman ◽  
Sahabuddin Sarwardi
Keyword(s):  
Hopf Bifurcation ◽  
Uniform Boundedness ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Predator Model ◽  
The Stability ◽  
Manifold Theory

AbstractWe propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.

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