• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권3호
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• pp.329-342
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• 2008

There have been many experimental and observational evidences which indicate the predator response to prey density needs not always monotone increasing as in the classical predator-prey models in population dynamics. Holling type functional response depicts situations in which sufficiently large number of the prey species increases their ability to defend or disguise themselves from the predator. In this paper we investigated the stability and instability property for a Holling type predator-prey system of a generalized form. Hopf type bifurcation properties of the non-diffusive system and the diffusion effects on instability and bifurcation values are studied.

• Kyungpook Mathematical Journal
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• 제48권3호
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• pp.515-527
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• 2008

We investigate a three species food chain system with a Holling type IV functional response and impulsive perturbations. We find conditions for local and global stabilities of prey(or predator) free periodic solutions by applying the Floquet theory and the comparison theorems.

• 대한수학회논문집
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• 제24권2호
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• pp.215-226
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• 2009

For a class of Holling type II food chain systems with biological and chemical controls, we give conditions of the local stability of prey-free periodic solutions and of the permanence of the system. Further, we show the system is uniformly bounded.

• Kyungpook Mathematical Journal
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• 제49권4호
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• pp.763-770
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• 2009

We investigate the dynamical properties of a Holling type I predator-prey model, which harvests both prey and predator and stock predator impulsively. By using the Floquet theory and small amplitude perturbation method we prove that there exists a stable prey-extermination solution when the impulsive period is less than some critical value, which implies that the model could be extinct under some conditions. Moreover, we give a sufficient condition for the permanence of the model.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제16권3호
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• pp.151-167
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• 2012

In this paper, we investigate the dynamic behaviors of a one-prey and two-predator system with Holling-type II functional response and defensive ability by introducing a proportion that is periodic impulsive harvesting for all species and a constant periodic releasing, or immigrating, for predators at different fixed time. We establish conditions for the local stability and global asymptotic stability of prey-free periodic solutions by using Floquet theory for the impulsive equation, small amplitude perturbation skills. Also, we prove that the system is uniformly bounded and is permanent under some conditions via comparison techniques. By displaying bifurcation diagrams, we show that the system has complex dynamical aspects.

• Kyungpook Mathematical Journal
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• 제53권4호
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• pp.647-660
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• 2013

In the paper, a two-prey one-predator system with defensive ability and Holling type-II functional responses is investigated. First, the stability of equilibrium points of the system is discussed and then conditions for the persistence of the system are established according to the existence of limit cycles. Numerical examples are illustrated to attest to our mathematical results. Finally, via bifurcation diagrams, various dynamic behaviors including chaotic phenomena are demonstrated.

• Kyungpook Mathematical Journal
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• 제60권2호
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• pp.319-334
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• 2020

We propose a mathematical model with Holling type II functional response, to study the dynamics of vaccination. In order to make our model more realistic, we have incorporated the recruitment of infected individuals as a continuous process. We have assumed that vaccination cannot be perfect and there is always a possibility of re-infection. We have obtained the existence of a disease free and endemic equilibrium point, when the recruitment of infective is not considered and also obtained the existence of at least one endemic equilibrium point when recruitment of infective is considered. We have proved that if R_v < 1, disease free equilibrium is locally asymptotically stable, which leads to the elimination of the disease from the population. The persistence of the model has also been established. Numerical simulations have been done to establish the results obtained.

• Journal of the Korean Data and Information Science Society
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• 제20권1호
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• pp.211-217
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• 2009

In this paper, we are studying the property for permanence of a three species food chain system with impulsive perturbations and Holling type II functional response, species which is important concept or property in ecological systems. Specially, we give the conditions for the permanence of this system. To do it, we consider the comparison method which is typical skill happened in impulsive differential inequalities. In addition, we reaffirm our results by using a numerical example.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권1호
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• pp.39-63
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• 2019

In this paper, we study the effect of Cytotoxic T Lymphocyte (CTL) and antibody immune responses on the virus dynamics with both virus-to-cell and cell-to-cell transmissions. The infection rate is given by Holling type-II. We first show that the model is biologically acceptable by showing that the solutions of the model are nonnegative and bounded. We find the equilibria of the model and investigate their global stability analysis. We derive five threshold parameters which fully determine the existence and stability of the five equilibria of the model. The global stability of all equilibria of the model is proven using Lyapunov method and applying LaSalle's invariance principle. To support our theoretical results we have performed some numerical simulations for the model. The results show the CTL and antibody immune response can control the disease progression.

• Journal of applied mathematics & informatics
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• 제41권4호
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• pp.723-739
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• 2023

In this paper, we developed three theoretical models based on prey and predator that exhibit holling-type response functions. In both a fuzzy and a crisp environment, we have provided a mathematical formulation for the prey predator concept. We used the signed distance method to defuzzify the triangular fuzzy numbers using the alpha-cut function. We can identify equilibrium points for all three theoretical models using the defuzzification technique. Utilizing a variational matrix, stability is also performed with the two species model through three theoretical models. Results are presented, followed by discussion. MATLAB software is used to provide numerical simulations.