• Journal of the Chungcheong Mathematical Society
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• v.33 no.1
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• pp.19-32
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• 2020

We discuss the coexistence of positive solutions to certain strongly-coupled predator-prey elliptic systems under the homogeneous Dirichlet boundary conditions. The sufficient condition for the existence of positive solutions is expressed in terms of the spectral property of differential operators of nonlinear Schrödinger type which reflects the influence of the domain and nonlinearity in the system. Furthermore, applying the obtained results, we investigate the sufficient conditions for the existence of positive solutions of a predator-prey system with degenerate diffusion rates.

• Journal of applied mathematics & informatics
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• v.6 no.3
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• pp.835-850
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• 1999

The present paper deals with the problem of nonselective harvesting in a partly infected prey and predator system in which both the susceptible prey and the predator follow the law of logistic growth and some preys avoid predation by hiding. The dynamical behaviour of the system has been studied in both the local and global sense. The optimal policy of exploitation has been derived by using Pontraygin's maximal principle. Numerical analysis and computer simulation of the results have been performed to investigate the golbal properties of the system.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.695-713
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• 2013

We formulate and study a predator-prey model with non-monotonic functional response type and weak Allee effects on the prey, which extends the system studied by Ruan and Xiao in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001), no. 4, 1445-1472] but containing an extra term describing weak Allee effects on the prey. We obtain the global dynamics of the model by combining the global qualitative and bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and the subcritical Hopf bifurcations, and the homoclinic bifurcation, as the values of parameters vary. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation).

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.955-967
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• 2011

In this paper we have considered a nonautonomous predator-prey model with discrete time delay due to gestation, in which there are two prey habitats linked by isotropic migration. One prey habitat contains a predator and the other (a refuge) does not. Here, we have established some sufficient conditions on the permanence of the system by using in-equality analytical technique. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. We have observed that the per capita migration rate among two prey habitats and the time delay has no effect on the permanence of the system but it has an effect on the global asymptotic stability of this model. The aim of the analysis of this model is to identify the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1827-1840
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• 2013

This paper is concerned with the positive steady states of a diffusive Holling type II predator-prey system, in which two predators and one prey are involved. Under homogeneous Neumann boundary conditions, the local and global asymptotic stability of the spatially homogeneous positive steady state are discussed. Moreover, the large diffusion of predator is considered by proving the nonexistence of non-constant positive steady states, which gives some descriptions of the effect of diffusion on the pattern formation.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.311-326
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• 2005

The paper deals with the problem of selective harvesting in a prey-predator model with predator self limitation. Criteria for local stability and global stability for both the exploited and unexploited system are derived. The effort has been considered as a dynamic variable and taxation as a control instrument to protect the fish populations from over exploitation. Finally, the optimal taxation policy is discussed with the help of control theory.

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.47-55
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• 2016

In this paper, we consider a discrete predator-prey system obtained from a continuous Beddington-DeAngelis type predator-prey system by using the method in [9]. In order to investigate dynamical behaviors of this discrete system, we find out all equilibrium points of the system and study their stability by using eigenvalues of a Jacobian matrix for each equilibrium points. In addition, we illustrate some numerical examples in order to substantiate theoretical results.

• Kyungpook Mathematical Journal
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• v.53 no.4
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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.

• Bulletin of the Korean Mathematical Society
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• v.53 no.1
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• pp.103-117
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• 2016

Abstract. In this paper, we study a modified stochastic predator-prey system with general ratio-dependent functional response. We prove that the system has a unique positive solution for given positive initial value. Then we investigate the persistence and extinction of this stochastic system. At the end, we give some numerical simulations, which support our theoretical conclusions well.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.829-843
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• 2006

We propose a cubic differential system, which can be considered a generalization of the predator-prey models, studied by many authors recently (see [18, 20], for instance) The properties of the equilibrium points, the existences, nonexistence, the uniqueness conditions and the relative positions of the limit cycles are investigated. An example is used to show our theorems are easy to be used in applications.