This document analyzes a prey-predator model with a harvested predator species using a Holling type IV functional response. It contains the following key points:
1) A prey-predator differential equation model is presented that includes harvesting of the predator species.
2) Three possible equilibrium points are identified: the trivial equilibrium of (0,0), the axial equilibrium of (1/b, 0), and a possible interior equilibrium.
3) A stability analysis is conducted of the equilibrium points by analyzing the eigenvalues of the Jacobian matrix evaluated at each point. Conditions for stability and instability are determined.
4) Numerical simulations and bifurcation diagrams supporting the theoretical stability results are generated using MATLAB.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document reports on a study that tested unidimensional unfolding theory, which posits that people's preferences between stimuli can be represented by their ideal point on an underlying dimension. Specifically, the study:
1) Used 8 attitude statements about interpersonal distance toward homosexual people and had participants complete pairwise comparisons. This allowed the study to identify which pairs were unilateral vs. bilateral to participants' ideal points.
2) Tested the ordinal and quantitative components of unfolding theory using a multinomial Dirichlet model. It found 98% agreement with the quantitative pairwise axiom for bilateral pairs and 97% agreement with the ordinal pairwise axiom for unilateral pairs.
3) Concluded that the data provided strong support for both the
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
This document summarizes a research paper that proposes a new algorithm for influence maximization in social networks. The algorithm draws inspiration from previous works on community detection and a data-based credit distribution model. It first assigns credits to users based on their past actions to determine probabilistic influence between users. It then uses a community detection approach to identify groups of similar users before applying an influence maximization algorithm based on the independent cascade model. The proposed approach aims to better learn mutual influence from user data and improve time complexity by leveraging the relationship between community detection and viral marketing.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document summarizes a research paper that proposes the SPARROW (SPAtial ClusteRing AlgoRithm thrOugh SWarm Intelligence) algorithm for clustering software components. SPARROW is a multi-agent algorithm that uses principles from flocking behavior to identify clusters of components in parallel. It begins by randomly positioning agents within component data and identifying core points to label neighboring components. Agents then move according to flocking rules to explore the data space and continuously update component clusters. The algorithm is extended to multidimensional data and can optimize clustered test cases using other algorithms. The paper argues SPARROW provides an efficient way to cluster components early in development for component-based software architecture.
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
This document reports on a study that tested unidimensional unfolding theory, which posits that people's preferences between stimuli can be represented by their ideal point on an underlying dimension. Specifically, the study:
1) Used 8 attitude statements about interpersonal distance toward homosexual people and had participants complete pairwise comparisons. This allowed the study to identify which pairs were unilateral vs. bilateral to participants' ideal points.
2) Tested the ordinal and quantitative components of unfolding theory using a multinomial Dirichlet model. It found 98% agreement with the quantitative pairwise axiom for bilateral pairs and 97% agreement with the ordinal pairwise axiom for unilateral pairs.
3) Concluded that the data provided strong support for both the
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
This document summarizes a research paper that proposes a new algorithm for influence maximization in social networks. The algorithm draws inspiration from previous works on community detection and a data-based credit distribution model. It first assigns credits to users based on their past actions to determine probabilistic influence between users. It then uses a community detection approach to identify groups of similar users before applying an influence maximization algorithm based on the independent cascade model. The proposed approach aims to better learn mutual influence from user data and improve time complexity by leveraging the relationship between community detection and viral marketing.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document summarizes a research paper that proposes the SPARROW (SPAtial ClusteRing AlgoRithm thrOugh SWarm Intelligence) algorithm for clustering software components. SPARROW is a multi-agent algorithm that uses principles from flocking behavior to identify clusters of components in parallel. It begins by randomly positioning agents within component data and identifying core points to label neighboring components. Agents then move according to flocking rules to explore the data space and continuously update component clusters. The algorithm is extended to multidimensional data and can optimize clustered test cases using other algorithms. The paper argues SPARROW provides an efficient way to cluster components early in development for component-based software architecture.
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Typical Measures on Discrete Time Prey-Predator Model with HarvestedPrey MangaiK4
Abstract - Prey-predator model has received much attention during the last few decades due to its wide range of applications. There are many different kinds of prey-predator models in mathematical ecology. The discrete time models governed by difference equations are more appropriate than the continuous time models to describe the prey-predator relations. Thispaper aims to study the effect of harvested prey species on a Holling type IV prey predator model involving intra-specific competition. Harvesting has a strong impact on the dynamic evolution of a population. This model represents mathematically by nonlinear differential equations. The locally asymptotic stability conditions of all possible equilibrium points were obtained. The stability/instability of nonnegative equilibrium and associated bifurcation were investigated by analysing the characteristic equations. Moreover, bifurcation diagram were obtained for different values of parameters of proposed model. Finally, numerical simulation was used to study the global and rich dynamics of that model.
Discrete Time Prey-Predator Model With Generalized Holling Type Interaction Zac Darcy
This document summarizes a research article that introduces a discrete time prey-predator mathematical model with a generalized Holling type interaction. The model is formulated and its fixed points are determined. The stability of the fixed points is analyzed analytically. Phase diagrams and bifurcation diagrams are drawn numerically to study the behavior of the system for different parameter values. The model is shown to better predict and analyze non-chaotic prey-predator ecological systems compared to models with standard Holling type interactions.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
Discrete time prey predator model with generalized holling type interactionZac Darcy
We have introduced a discrete time prey-predator model with Generalized Holling type interaction. Stability nature of the fixed points of the model are determined analytically. Phase diagrams are drawn after solving the system numerically. Bifurcation analysis is done with respect to various parameters of the system. It is shown that for modeling of non-chaotic prey predator ecological systems with Generalized Holling type interaction may be more useful for better prediction and analysis.
Qualitative Analysis of Prey Predator System With Immigrant PreyIJERDJOURNAL
ABSTRACT: The predator prey system with immigrant prey is introduced and studied through a suitable mathematical model. Existence conditions for interior equilibrium point and their stability is studied under suitable ecological restrictions. Global stability of the system around equilibrium point is also discussed.
This document presents a mathematical model of a prey-predator system involving disease in the prey population. The prey population is divided into susceptible (S) and infected (I) classes. The model accounts for disease transmission between prey, as well as from external sources. It also includes predator switching between feeding on susceptible and infected prey depending on availability. The model is formulated as a system of nonlinear differential equations. Existence, uniqueness, and boundedness of solutions to the system are proven. The local stability of possible equilibrium points - the extinction equilibrium (0), disease-free equilibrium (S*), and endemic equilibrium (S*,I*,Y*) - are analyzed.
The Dynamics of Sokol-Howell Prey-Predator Model Involving Strong Allee Effec...Saad Al-Momen
In this paper, a Sokol-Howell prey-predator model involving strong Allee effect is proposed and analyzed. The existence, uniqueness, and boundedness are studied. All the five possible equilibria have been are obtained and their local stability conditions are established. Using Sotomayor's theorem, the conditions of local saddle-node and transcritical and pitchfork bifurcation are derived and drawn. Numerical simulations are performed to clarify the analytical results.
ECOL203403 – Ecology Populations to Ecosystems Assignment .docxtidwellveronique
Lori B. Ihrig
Copyright 2016 Cengage Learning. All Rights Reserved. May
not be copied, scanned, or duplicated, in whole or in part. Due
to electronic rights, some third party content may be suppressed
from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does
not materially affect the overall learning experience. Cengage
Learning reserves the right to remove additional content at any
time if subsequent rights restrictions require it.
www.MyEbookNiche.eCrater.com
Fundamentals of Management, Eighth Edition
Ricky W. Griffin
Copyright 2016 Cengage Learning. All Rights Reserved.
Fuzzy modeling is well-suited for transforming verbal descriptions of biological systems into mathematical models, making it useful for biomimicry. The document discusses how fuzzy modeling has been used to model animal behaviors like territorial fish and light-orienting planarian worms based on descriptions in scientific literature. Fuzzy modeling represents knowledge through an initial description, fuzzy rulebase, and mathematical model, providing interpretability and a means to verify the original description.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document analyzes a discrete-time host-parasitoid model that incorporates both Allee effects for the host population and aggregation effects for the parasitoid population. The model is described using difference equations. Fixed points and local stability are analyzed mathematically. It is found that adding Allee effects can simplify the population dynamics by making the system less sensitive to initial conditions and compressing the dynamics. The inclusion of Allee effects and aggregation effects provides a more biologically realistic model of host-parasitoid interactions compared to previous models.
This document discusses theoretical ecology, which uses theoretical methods such as mathematical models, computational simulations, and data analysis to study ecological systems. It provides examples of different types of mathematical models used to model population dynamics and species interactions, including exponential growth models, logistic growth models, structured population models using matrices, predator-prey models, host-pathogen models, and competition/mutualism models. It also discusses how theoretical ecology aims to explain a variety of ecological phenomena and how computational modeling has benefited from increased computing power.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Models for a Multi-Agent System Based on Wasp-Like Behaviour for Distributed ...infopapers
D. Simian, F. Stoica, C. Simian, Models for a Multi-Agent System Based on Wasp-like Behaviour for Distributed Patients Repartition, Proceedings of the 9th WSEAS International Conference on Evolutionary Computing, Sofia, Bulgaria, ISBN 978-960-6766-58-9, ISSN 1790-5109, pp. 82-86, May 2008
Metaheuristics Using Agent-Based Models for Swarms and ContagionLingge Li, PhD
This document describes agent-based models used to simulate swarm behavior and emotional contagion. It summarizes a three-zone model of swarm interaction that includes attraction, repulsion, and alignment zones. To improve computational efficiency for large numbers of agents, a linked-cell algorithm is used that only considers interactions between nearby agents. Genetic algorithms are applied to evolve swarm behavior and emotional contagion in response to predator pressure. Particle swarm optimization is also used to model emergency evacuation, aiming to better understand panicked crowd dynamics.
Mathematical modelling of Fish Resources Harvesting with Predator at Maximum Sustainable Yield
Kinfe Hailemariam Hntsaa, Zenebe Taka Mengesha (PhD)b*
aDepartment of Mathematics, Aksum University, Axum, Ethiopia, kinfhail@gmail.com
bDepartment of Biology, Aksum University, Axum, Ethiopia, zenebeteka2007@gmail.com
In this study, the population dynamic of fish is considered following Logistic model with the inclusion of harvesting. The prey-predator interaction is also considered with an assumption that the predator population which is completely theoretical and not physically defined has a little effect on the growth of prey population provided that there are no limiting factors other than the predators. This is to say that the prey-predator cycle remains stable as far as other factors are constant in the natural environment. The growth function of the predator population is constructed corresponding to the prey population, and its results showed that the predator population size is either convergent to a finite positive limit, zero or diverges to positive infinity; while the fish population size follows Logistic function and grows to an upper asymptote. Furthermore, the prey-predator interaction is considered with the assumption that the predator population has an effect on the growth of the prey population and the predator population has intra-specific competition for a limited environmental resource. Its result showed that the predator population size did not go to infinity without bound. In both cases the maximum sustainable yield is obtained, numerical simulation and stability analysis of the model are included.
This document proposes a method for estimating k sample survival functions under stochastic ordering constraints. It begins by reviewing existing work on estimating survival functions for two samples and extends this to k samples. The proposed method uses benchmark functions to estimate survival curves in a way that maintains stochastic ordering. It was tested on both uncensored and censored data and was shown to have low mean squared error and bias. The method was also applied to a real-world dataset with results comparable to previous work.
Qualitative Analysis of a Discrete SIR Epidemic Modelijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Stability of Individuals in a Fingerprint System across Force LevelsITIIIndustries
This research studied the question: “Are all
individual’s performance stable in a fingerprint recognition
system?” The fingerprints of 154 individuals, provided at
different force levels, were examined using the biometric
menagerie tool, first coined by Doddington et al. in 1998. The
Biometric Menagerie illustrates how each person in a given
dataset performs in a biometric system, by using their genuine
and impostor scores, and providing them a classification based
upon those scores. This research examined the biometric
menagerie classifications across different force levels in a
fingerprint recognition study to uncover if individuals performed
the same over five force levels. The study concluded that they did
not, and a new metric has been created to quantify this
phenomenon. As a result of this discovery, the new metric,
Stability Score Index is described to showcase the movement of
individuals in the menagerie.
A Novel Method for Prevention of Bandwidth Distributed Denial of Service AttacksIJERD Editor
Distributed Denial of Service (DDoS) Attacks became a massive threat to the Internet. Traditional
Architecture of internet is vulnerable to the attacks like DDoS. Attacker primarily acquire his army of Zombies,
then that army will be instructed by the Attacker that when to start an attack and on whom the attack should be
done. In this paper, different techniques which are used to perform DDoS Attacks, Tools that were used to
perform Attacks and Countermeasures in order to detect the attackers and eliminate the Bandwidth Distributed
Denial of Service attacks (B-DDoS) are reviewed. DDoS Attacks were done by using various Flooding
techniques which are used in DDoS attack.
The main purpose of this paper is to design an architecture which can reduce the Bandwidth
Distributed Denial of service Attack and make the victim site or server available for the normal users by
eliminating the zombie machines. Our Primary focus of this paper is to dispute how normal machines are
turning into zombies (Bots), how attack is been initiated, DDoS attack procedure and how an organization can
save their server from being a DDoS victim. In order to present this we implemented a simulated environment
with Cisco switches, Routers, Firewall, some virtual machines and some Attack tools to display a real DDoS
attack. By using Time scheduling, Resource Limiting, System log, Access Control List and some Modular
policy Framework we stopped the attack and identified the Attacker (Bot) machines
Hearing loss is one of the most common human impairments. It is estimated that by year 2015 more
than 700 million people will suffer mild deafness. Most can be helped by hearing aid devices depending on the
severity of their hearing loss. This paper describes the implementation and characterization details of a dual
channel transmitter front end (TFE) for digital hearing aid (DHA) applications that use novel micro
electromechanical- systems (MEMS) audio transducers and ultra-low power-scalable analog-to-digital
converters (ADCs), which enable a very-low form factor, energy-efficient implementation for next-generation
DHA. The contribution of the design is the implementation of the dual channel MEMS microphones and powerscalable
ADC system.
Typical Measures on Discrete Time Prey-Predator Model with HarvestedPrey MangaiK4
Abstract - Prey-predator model has received much attention during the last few decades due to its wide range of applications. There are many different kinds of prey-predator models in mathematical ecology. The discrete time models governed by difference equations are more appropriate than the continuous time models to describe the prey-predator relations. Thispaper aims to study the effect of harvested prey species on a Holling type IV prey predator model involving intra-specific competition. Harvesting has a strong impact on the dynamic evolution of a population. This model represents mathematically by nonlinear differential equations. The locally asymptotic stability conditions of all possible equilibrium points were obtained. The stability/instability of nonnegative equilibrium and associated bifurcation were investigated by analysing the characteristic equations. Moreover, bifurcation diagram were obtained for different values of parameters of proposed model. Finally, numerical simulation was used to study the global and rich dynamics of that model.
Discrete Time Prey-Predator Model With Generalized Holling Type Interaction Zac Darcy
This document summarizes a research article that introduces a discrete time prey-predator mathematical model with a generalized Holling type interaction. The model is formulated and its fixed points are determined. The stability of the fixed points is analyzed analytically. Phase diagrams and bifurcation diagrams are drawn numerically to study the behavior of the system for different parameter values. The model is shown to better predict and analyze non-chaotic prey-predator ecological systems compared to models with standard Holling type interactions.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
Discrete time prey predator model with generalized holling type interactionZac Darcy
We have introduced a discrete time prey-predator model with Generalized Holling type interaction. Stability nature of the fixed points of the model are determined analytically. Phase diagrams are drawn after solving the system numerically. Bifurcation analysis is done with respect to various parameters of the system. It is shown that for modeling of non-chaotic prey predator ecological systems with Generalized Holling type interaction may be more useful for better prediction and analysis.
Qualitative Analysis of Prey Predator System With Immigrant PreyIJERDJOURNAL
ABSTRACT: The predator prey system with immigrant prey is introduced and studied through a suitable mathematical model. Existence conditions for interior equilibrium point and their stability is studied under suitable ecological restrictions. Global stability of the system around equilibrium point is also discussed.
This document presents a mathematical model of a prey-predator system involving disease in the prey population. The prey population is divided into susceptible (S) and infected (I) classes. The model accounts for disease transmission between prey, as well as from external sources. It also includes predator switching between feeding on susceptible and infected prey depending on availability. The model is formulated as a system of nonlinear differential equations. Existence, uniqueness, and boundedness of solutions to the system are proven. The local stability of possible equilibrium points - the extinction equilibrium (0), disease-free equilibrium (S*), and endemic equilibrium (S*,I*,Y*) - are analyzed.
The Dynamics of Sokol-Howell Prey-Predator Model Involving Strong Allee Effec...Saad Al-Momen
In this paper, a Sokol-Howell prey-predator model involving strong Allee effect is proposed and analyzed. The existence, uniqueness, and boundedness are studied. All the five possible equilibria have been are obtained and their local stability conditions are established. Using Sotomayor's theorem, the conditions of local saddle-node and transcritical and pitchfork bifurcation are derived and drawn. Numerical simulations are performed to clarify the analytical results.
ECOL203403 – Ecology Populations to Ecosystems Assignment .docxtidwellveronique
Lori B. Ihrig
Copyright 2016 Cengage Learning. All Rights Reserved. May
not be copied, scanned, or duplicated, in whole or in part. Due
to electronic rights, some third party content may be suppressed
from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does
not materially affect the overall learning experience. Cengage
Learning reserves the right to remove additional content at any
time if subsequent rights restrictions require it.
www.MyEbookNiche.eCrater.com
Fundamentals of Management, Eighth Edition
Ricky W. Griffin
Copyright 2016 Cengage Learning. All Rights Reserved.
Fuzzy modeling is well-suited for transforming verbal descriptions of biological systems into mathematical models, making it useful for biomimicry. The document discusses how fuzzy modeling has been used to model animal behaviors like territorial fish and light-orienting planarian worms based on descriptions in scientific literature. Fuzzy modeling represents knowledge through an initial description, fuzzy rulebase, and mathematical model, providing interpretability and a means to verify the original description.
International Journal of Engineering Research and DevelopmentIJERD Editor
This document analyzes a discrete-time host-parasitoid model that incorporates both Allee effects for the host population and aggregation effects for the parasitoid population. The model is described using difference equations. Fixed points and local stability are analyzed mathematically. It is found that adding Allee effects can simplify the population dynamics by making the system less sensitive to initial conditions and compressing the dynamics. The inclusion of Allee effects and aggregation effects provides a more biologically realistic model of host-parasitoid interactions compared to previous models.
This document discusses theoretical ecology, which uses theoretical methods such as mathematical models, computational simulations, and data analysis to study ecological systems. It provides examples of different types of mathematical models used to model population dynamics and species interactions, including exponential growth models, logistic growth models, structured population models using matrices, predator-prey models, host-pathogen models, and competition/mutualism models. It also discusses how theoretical ecology aims to explain a variety of ecological phenomena and how computational modeling has benefited from increased computing power.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Models for a Multi-Agent System Based on Wasp-Like Behaviour for Distributed ...infopapers
D. Simian, F. Stoica, C. Simian, Models for a Multi-Agent System Based on Wasp-like Behaviour for Distributed Patients Repartition, Proceedings of the 9th WSEAS International Conference on Evolutionary Computing, Sofia, Bulgaria, ISBN 978-960-6766-58-9, ISSN 1790-5109, pp. 82-86, May 2008
Metaheuristics Using Agent-Based Models for Swarms and ContagionLingge Li, PhD
This document describes agent-based models used to simulate swarm behavior and emotional contagion. It summarizes a three-zone model of swarm interaction that includes attraction, repulsion, and alignment zones. To improve computational efficiency for large numbers of agents, a linked-cell algorithm is used that only considers interactions between nearby agents. Genetic algorithms are applied to evolve swarm behavior and emotional contagion in response to predator pressure. Particle swarm optimization is also used to model emergency evacuation, aiming to better understand panicked crowd dynamics.
Mathematical modelling of Fish Resources Harvesting with Predator at Maximum Sustainable Yield
Kinfe Hailemariam Hntsaa, Zenebe Taka Mengesha (PhD)b*
aDepartment of Mathematics, Aksum University, Axum, Ethiopia, kinfhail@gmail.com
bDepartment of Biology, Aksum University, Axum, Ethiopia, zenebeteka2007@gmail.com
In this study, the population dynamic of fish is considered following Logistic model with the inclusion of harvesting. The prey-predator interaction is also considered with an assumption that the predator population which is completely theoretical and not physically defined has a little effect on the growth of prey population provided that there are no limiting factors other than the predators. This is to say that the prey-predator cycle remains stable as far as other factors are constant in the natural environment. The growth function of the predator population is constructed corresponding to the prey population, and its results showed that the predator population size is either convergent to a finite positive limit, zero or diverges to positive infinity; while the fish population size follows Logistic function and grows to an upper asymptote. Furthermore, the prey-predator interaction is considered with the assumption that the predator population has an effect on the growth of the prey population and the predator population has intra-specific competition for a limited environmental resource. Its result showed that the predator population size did not go to infinity without bound. In both cases the maximum sustainable yield is obtained, numerical simulation and stability analysis of the model are included.
This document proposes a method for estimating k sample survival functions under stochastic ordering constraints. It begins by reviewing existing work on estimating survival functions for two samples and extends this to k samples. The proposed method uses benchmark functions to estimate survival curves in a way that maintains stochastic ordering. It was tested on both uncensored and censored data and was shown to have low mean squared error and bias. The method was also applied to a real-world dataset with results comparable to previous work.
Qualitative Analysis of a Discrete SIR Epidemic Modelijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Stability of Individuals in a Fingerprint System across Force LevelsITIIIndustries
This research studied the question: “Are all
individual’s performance stable in a fingerprint recognition
system?” The fingerprints of 154 individuals, provided at
different force levels, were examined using the biometric
menagerie tool, first coined by Doddington et al. in 1998. The
Biometric Menagerie illustrates how each person in a given
dataset performs in a biometric system, by using their genuine
and impostor scores, and providing them a classification based
upon those scores. This research examined the biometric
menagerie classifications across different force levels in a
fingerprint recognition study to uncover if individuals performed
the same over five force levels. The study concluded that they did
not, and a new metric has been created to quantify this
phenomenon. As a result of this discovery, the new metric,
Stability Score Index is described to showcase the movement of
individuals in the menagerie.
A Novel Method for Prevention of Bandwidth Distributed Denial of Service AttacksIJERD Editor
Distributed Denial of Service (DDoS) Attacks became a massive threat to the Internet. Traditional
Architecture of internet is vulnerable to the attacks like DDoS. Attacker primarily acquire his army of Zombies,
then that army will be instructed by the Attacker that when to start an attack and on whom the attack should be
done. In this paper, different techniques which are used to perform DDoS Attacks, Tools that were used to
perform Attacks and Countermeasures in order to detect the attackers and eliminate the Bandwidth Distributed
Denial of Service attacks (B-DDoS) are reviewed. DDoS Attacks were done by using various Flooding
techniques which are used in DDoS attack.
The main purpose of this paper is to design an architecture which can reduce the Bandwidth
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eliminating the zombie machines. Our Primary focus of this paper is to dispute how normal machines are
turning into zombies (Bots), how attack is been initiated, DDoS attack procedure and how an organization can
save their server from being a DDoS victim. In order to present this we implemented a simulated environment
with Cisco switches, Routers, Firewall, some virtual machines and some Attack tools to display a real DDoS
attack. By using Time scheduling, Resource Limiting, System log, Access Control List and some Modular
policy Framework we stopped the attack and identified the Attacker (Bot) machines
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electromechanical- systems (MEMS) audio transducers and ultra-low power-scalable analog-to-digital
converters (ADCs), which enable a very-low form factor, energy-efficient implementation for next-generation
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shear studs located in the crack-concentration zones due to large hogging moments sustain significantly smaller
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connection appears also to be closely related to the change in the tensile strain of rebar according to the increase
of the load. Further experimental and analytical studies shall be conducted considering variables such as the
reinforcement ratio and the arrangement of shear connectors to achieve efficient design of the shear connection
in composite beams subjected to hogging moment.
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previous century the Geological Research Authority of Sudan (GRAS) in cooperation with BRGM utilized
satellite data of Landsat TM using spectral ratio technique to map possible mineralized zones in the Red Sea
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applied to Arbaat area and a signature of alteration zone was detected. The alteration zones are commonly
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confirmed the existence of stock work of gold bearing quartz in the alteration zone. Another type of gold
mineralization that was discovered using remote sensing is the gold associated with metachert in the Atmur
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Study on the Fused Deposition Modelling In Additive ManufacturingIJERD Editor
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plastic, PLA and nylon. Recently the use of gold, bronze and wood has also been implemented. The complexity
factor of this process is 0% as in any object of any shape and size can be manufactured.
Spyware triggering system by particular string valueIJERD Editor
This computer programme can be used for good and bad purpose in hacking or in any general
purpose. We can say it is next step for hacking techniques such as keylogger and spyware. Once in this system if
user or hacker store particular string as a input after that software continually compare typing activity of user
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A Blind Steganalysis on JPEG Gray Level Image Based on Statistical Features a...IJERD Editor
This paper presents a blind steganalysis technique to effectively attack the JPEG steganographic
schemes i.e. Jsteg, F5, Outguess and DWT Based. The proposed method exploits the correlations between
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Secure Image Transmission for Cloud Storage System Using Hybrid SchemeIJERD Editor
- Data over the cloud is transferred or transmitted between servers and users. Privacy of that
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Application of Buckley-Leverett Equation in Modeling the Radius of Invasion i...IJERD Editor
A thorough review of existing literature indicates that the Buckley-Leverett equation only analyzes
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equation to estimate the radius invasion of the displacing fluid during waterflooding. The model is also adopted
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Gesture Gaming on the World Wide Web Using an Ordinary Web CameraIJERD Editor
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Hardware Analysis of Resonant Frequency Converter Using Isolated Circuits And...IJERD Editor
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exclusive dead band in micro seconds so as to avoid the simultaneous firing of two pairs of IGBT’s where one
pair switches off and the other on for a slightest period of time. Hence, the isolator ckt here is associated with
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Simulated Analysis of Resonant Frequency Converter Using Different Tank Circu...IJERD Editor
LLC resonant frequency converter is basically a combo of series as well as parallel resonant ckt. For
LCC resonant converter it is associated with a disadvantage that, though it has two resonant frequencies, the
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working at this resonant frequency. LLC resonant converter existed for a very long time but because of
unknown characteristic of this converter it was used as a series resonant converter with basically a passive
(resistive) load. . Here, it was designed to operate in switching frequency higher than resonant frequency of the
series resonant tank of Lr and Cr converter acts very similar to Series Resonant Converter. The benefit of LLC
resonant converter is narrow switching frequency range with light load[6] . Basically, the control ckt plays a
very imp. role and hence 555 Timer used here provides a perfect square wave as the control ckt provides no
slew rate which makes the square wave really strong and impenetrable. The dead band circuit provides the
exclusive dead band in micro seconds so as to avoid the simultaneous firing of two pairs of IGBT’s where one
pair switches off and the other on for a slightest period of time. Hence, the isolator ckt here is associated with
each and every ckt used because it acts as a driver and an isolation to each of the IGBT is provided with one
exclusive transformer supply[3]. The IGBT’s are fired using the appropriate signal using the previous boards
and hence at last a high frequency rectifier ckt with a filtering capacitor is used to get an exact dc
waveform .The basic goal of this particular analysis is to observe the wave forms and characteristics of
converters with differently positioned passive elements in the form of tank circuits. The supported simulation
is done through PSIM 6.0 software tool
Amateurs Radio operator, also known as HAM communicates with other HAMs through Radio
waves. Wireless communication in which Moon is used as natural satellite is called Moon-bounce or EME
(Earth -Moon-Earth) technique. Long distance communication (DXing) using Very High Frequency (VHF)
operated amateur HAM radio was difficult. Even with the modest setup having good transceiver, power
amplifier and high gain antenna with high directivity, VHF DXing is possible. Generally 2X11 YAGI antenna
along with rotor to set horizontal and vertical angle is used. Moon tracking software gives exact location,
visibility of Moon at both the stations and other vital data to acquire real time position of moon.
“MS-Extractor: An Innovative Approach to Extract Microsatellites on „Y‟ Chrom...IJERD Editor
Simple Sequence Repeats (SSR), also known as Microsatellites, have been extensively used as
molecular markers due to their abundance and high degree of polymorphism. The nucleotide sequences of
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crucial. However, Microsatellites repeat count is compared, if they differ largely, he has some disorder. The Y
chromosome likely contains 50 to 60 genes that provide instructions for making proteins. Because only males
have the Y chromosome, the genes on this chromosome tend to be involved in male sex determination and
development. Several Microsatellite Extractors exist and they fail to extract microsatellites on large data sets of
giga bytes and tera bytes in size. The proposed tool “MS-Extractor: An Innovative Approach to extract
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locate Microsatellites and extracts them.
Importance of Measurements in Smart GridIJERD Editor
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measurement challenges, development of the Smart Meters and the critical parameters of electric energy to be
monitored for improving the reliability of power systems has been discussed.
Study of Macro level Properties of SCC using GGBS and Lime stone powderIJERD Editor
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Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
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solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
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train and test our model. The results of our experiments show that our CNN-LSTM method is much better
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Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
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Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
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ML Based Model for NIDS MSc Updated Presentation.v2.pptx
H1064251
1. International Journal of Engineering Research and Development
e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com
Volume 10, Issue 6 (June 2014), PP.42-51
Bifurcation Analysis of Prey-Predator Model with
Harvested Predator
Vijaya Lakshmi. G.M1, Gunasekaran. M2,*, Vijaya. S3
1Department of Mathematics, Indira Institute of Engineering and Technology,Thiruvallur-631 203.
2PG Department of Mathematics, Sri Subramaniya Swamy Government Arts College, Tiruttani-631 209.
3PG& Research Department of Mathematics, Annamalai university, Chidambaram-608 002.
Abstract:- This paper aims to study the effect of harvested predator species on a Holling type IV Prey-Predator
model involving intra-specific competition. Prey-predator model has received much attention during the last few
decades due to its wide range of applications. There are many kind of prey-predator models in mathematical
ecology. The Prey-predator models governed by differential equations are more appropriate than the difference
equations to describe the prey-predator relations. Harvesting has a strong impact on the dynamic evolution of a
population. This model represents mathematically by non-linear differential equations. The locally asymptotic
stability conditions of all possible equilibrium points were obtained. The stability/instability of non-negative
equilibrium and associated bifurcation were investigated by analysing the characteristic equations. Moreover,
bifurcation diagrams were obtained for different values of parameters of proposed model.
Keywords:- Prey-Predator model, Holling type functional response, Harvesting, Bifurcation.
I. INTRODUCTION
The prey-predator model with differential equations give rise to more efficient computational models
for numerical simulations and it exhibits more plentiful dynamical behaviours than a prey-predator model with
difference equations of the same type. There has been growing interest in the study of Prey-Predator models
described by differential equations. In ecology, predator-prey or plant herbivore models can be formulated as
differential equations. It is well known that one of the dominant themes in both ecology and mathematical
ecology is the dynamic relationship between predators and their prey. One of the important factors which affect
the dynamical properties of biological and mathematical models is the functional response. The formulation of
a predator-prey model critically depends on the form of the functional response that describes the amount of
prey consumed per predator per unit of time, as well as the growth function of prey [1,15]. That is a functional
response of the predator to the prey density in population dynamics refers to the change in the density of prey
attached per unit time per predator as the prey density changes.
In recent years, one of the important Predator – Prey models with the functional response is the Holling
type – IV, originally due to Holling which has been extensively studies in many articles [4-6, 11]. Two species
models like Holling type II, III and IV of predator to its prey have been extensively discussed in the literature
[2-6,9,16]. Leslie-Gower predator- prey model with variable delays, bifurcation analysis with time delay, global
stability in a delayed diffusive system has been studied [8,12,14]. Three tropic level food chain system with
Holling type IV functional responses , the discrete Nicholson Bailey model with Holling type II functional
response and global dynamical behavior of prey-predator system has been revisited [7,10,11,13]. The purpose
of this paper is to study the effect of harvested predator species on a Holling type IV prey predator model
involving intra-specific competition. We prove that the model has bifurcation that is associated with intrinsic
growth rate. The stability analysis that we carried out analytically has also been proved.
The period-doubling bifurcations and period-halving bifurcations exhibited by the differential
equations can be attributed to the fact that ecological communities show several unstable dynamical states,
which can change with very small perturbation. This paper is organized as follows: In section 2 we introduced
the model. In section 3, the equilibrium points and the local stability conditions of the trivial and axial
equilibrium points were investigated by using the theorem when the prey population in system (3) is subject to
an Holling type IV functional response. In section 4 we analysed the local and dynamical behaviour of the
interior equilibrium point, when the prey population in system (3) is subject to an Holling type IV functional
response. In section 5, some numerical simulations, dynamical behaviour of the system and bifurcation
diagrams supporting the theoretical stability results were shown in which the plots are generated by using
MATLAB software. Finally, the last section 6, is devoted to the conclusion and remarks.
42
2. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
In this paper we consider the following Lotka-Volterra Prey- Predator system:
p x
43
( ) ( )
(1)
( )
dx
xq x yp x
dt
dy
yp x y
dt
here x(0), y(0) 0,
Where x and y represent the prey and predator density, respectively. p(x) and q(x) are so-called
predator and prey functional response respectively. , 0 are the conversion and predator`s death rates,
respectively. If ( )
mx
p x
a x
refers to as Michaelis-Menten function or a Holling type – II function, where
m 0 denotes the maximal growth rate of the species and a 0 is half-saturation constant. Another class of
response functions are Holling type-III and Holling type-IV function, in which Holling type – III function is
2
mx
p x
2 ( )
a x
and Holling type-IV function is
mx
2 ( )
a x
. The Holling type – IV function otherwise
known as Monod-Haldane function which is used in our model. The simplified Monod-Haldane or Holling type
– IV function is a modification of the Holling type-III function. In this paper, we focus on effect of harvested
predator species on a Holling type IV prey-predator model involving intra-specific competition and establish
results for boundedness, existence of a positively invariant and the locally asymptotical stability of coexisting
interior equilibrium.
II. THE MODEL
The prey-predator systems have been discussed widely in the many decades. In the literature many
studies considered the prey-predator with functional responses. However, considerable evidence that some prey
or predator species have functional response, because of the environmental factors. It is more appropriate to add
the functional responses to these models in such circumstances. For example a system is suggested in (1),
where x(t) and y(t) represent densities or biomasses of the prey-species and predator species, respectively;
p(x) and q(x) are the intrinsic growth rates of the predator and prey respectively; and are the death
rates of prey and predator respectively.
If
2 ( )
1
mx
p x
x
and q(x) ax1 x , in p(x) assuming a 1 in general function, where a is the
half-saturation constant in the Holling type IV functional response, then Eq.(1) becomes
1
1
2
2
(2)
1
my
x x a x
x
mx
y y
x
Here a,, and m are all positive parameters.
Now introducing harvesting factor on predator with intra-specific competitions, the Eq. (2) becomes
2
1
2 0
(3)
1
my
x x a bx
x
e mx
y y y q E
x
With x(0), y(0) 0 and , , ,m,a,b,e, 0 q and E are all positive constants.
Where a is the intrinsic growth rate of the prey population; is the intrinsic death rate of the predator
population; b is strength of intra-specific competition among prey species; is strength of intra-specific
competition among predator species; m is direct measure of predator immunity from the prey; is maximum
3. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
attack rate of prey by the predator , e represents the conversion rate, E is harvesting effort and finally 0 q is
the catchability coefficient. The catch-rate function 0 q E is based on the catch-per-unit-effort (CPUE).
III. EXISTENCE AND LOCAL STABILITY ANALYSIS WITH PERSISTENCE
In this section, we first determine the existence of the fixed points of the differential equations (3), and
then we investigate their stability by calculating the Eigen values for the variation matrix of (3) at each fixed
point. To determine the fixed points, the equilibrium is the solution of the pair of equations below:
x A x A x A x A x A x A
y B x B x B x B
b a b
A A A
, , ,
a q E b a m
mq E
A A A
, ,
b a b a
B B B B
44
2
2 0
0
1
(4)
0
1
my
x a bx
x
e mx
y y q E
x
By simple computation of the above algebraic system, it was found that there are three nonnegative fixed points:
(i) 0 E 0,0 is the trivial equilibrium point always exists.
a
E
(ii) 1 ,0
b
is the axial fixed point always exists, as the prey population grows to the carrying capacity in
the absence of predation.
(iii) * *
2 E x , y is the positive equilibrium point exists in the interior of the first quadrant if and only if
there is a positive solution to the following algebraic nonlinear equations
We have the following polynomial with fifth and third degree.
* 5 4 3 2
5 4 3 2 1 0
* 3 2
3 2 1 0
(5)
Where
2
5 2 2 4 2 2 3 2 2
e m e m e
m
0 0
2
2 2 2 1 2 2 0 2 2
e m e m e m e
m
and
3 2 1 0 , , ,
m m m m
Remark 1: There is no equilibrium point on y axis as the predator population dies in the absence of its prey.
Lemma: For values of all parameters, Eqn.(3) has fixed points, the boundary fixed point and the positive fixed
point x * , y * x * , y * , where
satisfy
2
1
2 0
(6)
1
my
a bx
x
e mx
y q E
x
4. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
Now we study the stability of these fixed points. Note that the local stability of a fixed point x, y is
determined by the modules of Eigen values of the characteristic equation at the fixed point.
The Jacobian matrix J of the map (3) evaluated at any point x, y is given by
a a
( , ) (7)
a
(i.e.,) 0 E is sink point, otherwise unstable if
, 0 E is non-hyperbolic point if
45
11 12
21 12
J x y
a a
Where
2
my x
11 2
2
1
2
1
a a bx
x
mx
; 12 1
x
2
2
e my 1
x
21 2
2
1
a
x
e mx
a y q E
; 22 2 0 2
1
x
and the characteristic equation of the Jacobian matrix J x, y can be written as
2 p x, y q x, y 0 ,
Where
11 22 p x, y a a , 11 22 12 21 q x, y a a a a .
In order to discuss the stability of the fixed points, we also need the following lemma, which can be easily
proved by the relations between roots and coefficients of a quadratic equation.
Theorem: Let
2 F() P Q. Suppose that F 1 0 , 1 2 , are two roots of F() 0. Then (i)
1
| |1 and | |1 if and only if F 1 0 andQ 1;
1 2 (ii) | |1 and | |1 (or | |1 and | |1) if and only if F 1 0 ;
1 2 1 2 (iii) | |1 and | |1 if and only if F 1 0 and Q 1;
1 2 (iv) 1 and | |1 if and only if F 1 0 and P 0,2 ;
1 2 (v) and are complex and | |1 and | |1 if and only if
2 1 2 2 P 4Q 0 andQ 1.
Let 1
and 2 be two roots of (7), which are called Eigen values of the fixed point x, y . We recall some
definitions of topological types for a fixed point x, y . A fixed point x, y is called a sink if 1 | |1
and 2 | |1, so the sink is locally asymptotic stable. x, y is called a source if 1 | |1 and 2 | |1, so the
source is locally un stable. x, y
is called a saddle if 1 | |1 and 2 | |1 (or 1 | |1 and 2 | |1). And
x, y is called non-hyperbolic if either 1 | |1 and 2 | |1.
Proposition 1: The Eigen values of the trivial fixed point 0 E 0,0 is locally asymptotically stable if
0
1
a 1, E
q
and also 0 E is
0
1
a 1, E
q
saddle point if
0
1
a 1, E
q
.
0
1
a 1, E
q
Proof: In order to prove this result, we estimate the Eigen values of Jacobian matrix J at 0 E 0,0 . On
substituting x, y values in (7) we get the Jacobian matrix for 0 E
5. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
, 0 E is non-hyperbolic point
46
0
0
0
(0,0)
a
0
J
q E
Hence the Eigen values of the matrix are 1 2 0 =a, = q E
Thus it is clear that by Theorem, 0 E is sink point if 1,2 | |1
,that is 0 E is locally
0
1
a 1, E
q
asymptotically stable. 0 E is unstable (i.e.,) source if 1,2 | |1
.
0
1
a 1, E
q
And also 0 E is saddle point if 1 2 | |1, | |1
0
1
a 1, E
q
.
if 1 2 | |1 or | |1
0
1
a 1, E
q
a
E
Proposition 2: The fixed point 1 ,0
b
is locally asymptotically stable, that is sink
if
2 2
2 2
0
1
1 and
eab m a b
a E
q a b
; 1 E is locally unstable, that is source
if
2 2
2 2
0
1
1 and
eab m a b
a E
q a b
; 1 E is a saddle point if
2 2
2 2
0
1
1 and
eab m a b
a E
q a b
and 1 E is non-hyperbolic point if
either
2 2
2 2
0
1
1 or
eab m a b
a E
q a b
.
Proof: One can easily see that the Jacobian matrix at 1 E is
2
2
1
2 2 0
,0
0
ab m
a
a a b
J
b eab m
Eq
a b
Hence the Eigen values of the matrix are
eab
m
a Eq
| |= , | |=
1 2 2 2 0 a
b
By using Theorem, it is easy to see that, 1 E is a sink if
2 2
2 2
0
1
1 and
eab m a b
a E
q a b
;
6. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
my x
mx
my x e mx
1
* *
Tr J a bx y Eq
( ) 2 2 = +
2 2 * 0 1 2
2 2
* * 2 2 * * *
my x e mx e m x y x
1
1
* *
Det J a bx y q E
( ) 2 2
= .
2 2 * 0 3 1 2 3
x x x
1 1 1
my x e mx
1
* *
a 2 bx , 2
y Eq
1 2 2 * 0
47
1 E is a source if
2 2
2 2
0
1
1 and
eab m a b
a E
q a b
; 1 E is a saddle
if
2 2
2 2
0
1
1 and
eab m a b
a E
q a b
; and 1 E is a non-hyperbolic if either
2 2
2 2
0
1
1 or
eab m a b
a E
q a b
.
Remark 2: If 2
2 2 Tr(J ) Det(J ) 0, then the necessary and sufficient condition for linear stability
are 2 2 Tr(J ) 0 and Det(J ) 0.
IV. LOCAL STABILITY AND DYNAMIC BEHAVIOUR AROUND
INTERIOR FIXED POINT 2 E
Now we investigate the local stability and bifurcations of interior fixed point 2 E . The Jacobian matrix at 2 E is
of the form
2
2
2
2
2
2
*
*
*
2 *
*
* *
2
* *
*
*
2 * 0
*
1
2
1 1
( , ) (8)
1
2
1 1
a bx
x x
J x y
e my x e mx
y q E
x x
Its characteristic equation is 2
2 2 F() Tr(J ) Det(J ) 0 where Tr is the trace and Det is the
determinant of the Jacobian matrix 2 J (E ) defines in Eq.(8), (by Lemma) where
2
2
* *
2
x x
1 *
1
and
2
2 2
* *
2
2
* *
2
x x
1 *
1
e 2 m 2 x * y *
x
*
and 3 3
2
2
*
1
1
x
By Remark 2, 2 E is stable if 1 2 0 and 1 2 3 . 0 that is on solving we get,
7. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
* 2 2 * * * *
e mx 2 y e
m x y 1
x
q x q q x a bx x my x
* 2 * * * * *
48
2 E is stable if
bx * a
y
*
* 3 * 2
* *
E > 2
0 0
ex yx ex y
q q
(9)
and
2
2
2 2 2
0 0
E <
1 1 2 1
1
(10)
If both equations (9) and (10) are satisfied, then the interior equilibrium point will be stable.
V. NUMERICAL SIMULATION
The global dynamical behaviour of the non-linear model system (3) in
the positive octant is investigated numerically. Under the bifurcation analysis of the model (3), very rich and
complex behaviours are observed, presenting various sequences of period-doubling bifurcation leading to
chaotic dynamics or sequences of period-halving bifurcation leading to limit cycles.
The prey-predator system (3) with the effect of harvested predator species on a Holling type IV
functional response, intra-specific competition exhibits a variety of dynamical behaviour in respect of the
population size. We first plotted the diagrams for the prey system with various intrinsic growth rates. The
Fig.(1) shows that stabilized prey density first bifurcates 2 cycles, 4 cycles, forms a little chaos and then forms
chaotic band with intrinsic prey growth rate 0 to 4 in the absence of the predator. That is, period-doubling
bifurcation leading to chaotic dynamics. Next we introduce predator, then for various predator values.
Fig. (1)
a=0 to 4, b=0.2, =0.5, m=0.75 in the absence of predator
Fig.(2)-(4) shows the prey growth rates bifurcates 2 cycles, 4 cycles and then settles down to a stable fixed
point.
8. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
49
Fig. (2)
a=0 to 4, b=0.2, =0.5, m=0.75 and
y=7
Fig. (3)
a=0 to 4, b=0.2, =0.5, m=0.75 and
y=9
Fig. (4)
a=0 to 4, b=0.2, =0.5, m=0.75 and
y=10
Fig.(5) shows prey growth rate which leads to period-halving bifurcation leading to limit cycles.
Fig. (5)
a=0 to 4, b=0.2, =0.5, m=0.75and y = 13
Next we generated the bifurcation diagrams for predator growth rate with immunity. For various immunity
parameter values, m = 0 to 5 and various values of prey x , harvesting effort E and catchability coefficient 0 q ,
Fig.(6)-(8) shows that the predator growth rate bifurcates 2 cycles and settles down to a stable fixed point.
9. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
50
Fig. (6)
m=0 to 5, e=0.4, =0.5, x =0.5,
y =5, 0 q =0.5 and E =1.5
Fig. (7)
m=0 to 5, e=0.4, =0.5, x =10,
y =5, 0 q =0.5 and E =1.5
Fig. (8)
m=0 to 5, e=0.4, =0.5, x =1.5,
y =5, 0 q =0.5 and E =2
Fig.(9) and Fig.(10) shows clearly the evidence of the route to chaos through the cascade of period-halving
bifurcation respectively for the prey values x =1.5 and x =6.
Fig. (9)
m=0 to 5, e=0.4, =0.5, x =1.5,
y =5, 0 q =0.5 and E =3
Fig. (10)
m=0 to 5, e=0.4, =0.5, x =6,
y =5, 0 q =0.5 and E =3
The above plots have been generated by using MATLAB 7 software.
VI. CONCLUSION
In this paper, we have investigated the complex behaviours of two species prey- predator system as a
set of differential equations with the effect of harvested predator species on a Holling type IV functional
response and intra-specific competition in the closed first quadrant, and showed that the unique positive fixed
point of system (3) can undergo bifurcation and chaos. Bifurcation diagrams have shown that there exists much
more interesting dynamical and complex behaviour for system (3) including periodic doubling cascade, periodic
windows and chaos. All these results showed that for richer dynamical behaviour of the prey-predator
differential equation model (3) under periodical perturbations compared to the difference equation model. The
system is examined via the techniques of local stability analysis of the equilibrium points from which we obtain
the bifurcation criterion.
The numerical simulation of the population size shows a succession of period-doubling bifurcations
leading up to chaos. That is from Fig. (1)-(10), we observed the period-doubling route to chaos for the intrinsic
growth of prey population parameter and period-halving route to limit cycle for the strength of immunity growth
10. Bifurcation Analysis of Prey-Predator Model with Harvested Predator
rate among the predator species with harvesting effort and catchability coefficient. Thus it is observed that even
a small variation in parameters a
with zero predators and m with e
51
and 0 q may cause a shift form limit
cycles to chaos and vice-versa respectively. This study gives support to the view that two species prey-predator
model with a system of differential equations are able to generate unpredictable and complex behaviour with
small perturbations in parameters.
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