• 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).

• Honam Mathematical Journal
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• v.30 no.3
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• pp.521-533
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• 2008

We investigate a periodically forced Lotka-Volterra type predator-prey system with impulsive perturbations - seasonal effects on the prey, periodic releasing of natural enemies(predator) and spraying pesticide at the same fixed times. We show that the solutions of the system are bounded using the comparison theorems and find conditions for the stability of a stable prey-free solution and for the permanence of the system.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.353-373
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• 2013

In this paper, a class of delayed predator-prey models of prey migration and predator switching is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.575-587
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• 2006

In this paper, we provide 3 detailed and explicit procedure of obtaining some regions of attraction for the positive steady state (assumed to exist) of a well known Lotka-Volterra type predator-prey system. Also we obtain the sufficient conditions to ensure that the positive equilibrium point of a well known Lotka-Volterra type predator-prey system with a single discrete delay is globally asymptotically stable.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.781-802
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• 2011

Of concern in the paper is a Holling-Tanner predator-prey model with modified version of the Leslie-Gower functional response. Dynamical behaviours such as stability, permanence and Hopf bifurcation have been carried out deterministically. Using the normal form theory and center manifold theorem, the explicit formulae determining the stability and direction of Hopf bifurcation have been derived. The deterministic model is extended to a stochastic one by perturbing the growth equation of prey and predator by white and colored noises and finally the mean square stability of the stochastic model systems is investigated analytically. An extensive quantitative analysis has been performed based on numerical computation so as to validate the applicability of the proposed mathematical model.

• 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.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.193-207
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• 2010

In this paper, a predator-prey model with stage structure and distributed delay is investigated. Mathematical analyses of the model equation with regard to boundedness of solutions, nature of equilibria, permanence, extinction and stability are performed. By the comparison theorem, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Taking the product of the per-capita rate of predation and the rate of conversing prey into predator as the bifurcating parameter, we prove that there exists a threshold value beyond which the positive equilibrium bifurcates towards a periodic solution.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.251-261
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• 2010

This paper deals with a predator-prey model with Holling type III response function and cross-diffusion subject to the homogeneous Neumann boundary condition. We first give a priori estimates (positive upper and lower bounds) of positive steady states. Then the non-existence and existence results of non-constant positive steady states are given as the cross-diffusion coefficient is varied, which means that stationary patterns arise from cross-diffusion.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.75-85
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• 2011

In this paper, using the energy estimates and the bootstrap arguments, the global existence of classical solutions for a prey-predator model with non-monotonic functional response and cross-diffusion where the prey and predator both have linear density restriction is proved when the space dimension n < 10.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.16 no.4
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• pp.249-256
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• 2012

In this work, we analyze the spatial patterns of a predator-prey system with cross diffusion. First we get the critical lines of Hopf and Turing bifurcations in a spatial domain by using mathematical theory. More specifically, the exact Turing region is given in a two parameter space. Our results reveal that cross diffusion can induce stationary patterns which may be useful in understanding the dynamics of the real ecosystems better.