• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.387-401
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• 2006

This paper deals with the problem of a ratio-dependent prey- predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin's maximal principle.

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.429-444
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• 2012

By using Mawhin continuation theorem and comparison theorem, the existence of periodic solution and persistence for a predator-prey system with diffusion and impulses are investigated in this paper. An example and simulation are given to show the effectiveness of the main results.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1191-1205
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• 2008

We consider a delayed predator prey model. The local stability and Hopf bifurcation results are stated taking the time delay as a control parameter. We apply multiple scale analysis to analyze the effects of additive white noises near the Hopf bifurcation point at the positive interior equilibrium state. The governing equations for the amplitude of oscillations on a slow time scale are derived. We identify the process of amplitude of oscillations and derive its transient properties. We show that oscillations, which would decay in the deterministic system whenever time delay lies below its critical value, persists for long time under the validity of multiple scale analysis.

• 대한수학회지
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• 제46권1호
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• pp.71-82
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• 2009

In this paper, a stage-structured predator-prey system with impulsive perturbations and time delays is presented to investigate the ecological problem of how a pest population and natural enemy population can coexist. Sufficient conditions are obtained using a discrete dynamical system determined by a stroboscopic map, which guarantee that a 'predator-extinction' periodic solution is globally attractive. When the impulsive period is longer than some time threshold or the impulsive harvesting rate is below a control threshold, the system is permanent. Our results provide some reasonable suggestions for pest management.

• 한국시스템다이내믹스연구
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• 제12권2호
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• pp.95-126
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• 2011

This paper aims at developing a System Dynamics model with an augmented predator-prey interaction structure to deal with the population management of roe deer in Jeju, Korea. Although people still regard the creature as one of the important tourist attractions, there has been much debate on the issues of the appropriateness of the population size of roe deers because they have been stigmatized as crop damagers, and roadkill/poaching victims due to their natural habit to move around from the top mountain to the lowland of the island. The model is therefore to incorporate these migrating and grazing behaviors into an augmented Lotka-Volterra model coupling roe deer population in both parts of the island to that of predators and preys of the species. The authors also provide a comprehensive set of dynamic hypotheses and relevant CLD/SFD to understand the population dynamics of roe deer and co-evolving species and perform the steady-state analysis of the proposed equation system to verify the model behavior of the numerical example lastly presented in this paper.

• 대한수학회지
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• 제31권4호
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• pp.545-558
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• 1994

In this paper, we will investigate the existence of positive solutions to the predator-prey interacting system $$ {-\varphi(x, u)\Delta u = uf(x, u, \upsilon) in \Omega {-\psi(x, \upsilon)\Delta\upsilon = \upsilon g(x, u, \upsilon) {\frac{\partial n}{\partial u} + ku = 0 on \partial\Omega {\frac{\partial n}{\partial\upsilon} + \sigma\upsilon = 0. $$ in a bound region $\Omega$ in $R^n$ with smooth boundary, where $\varphi$ and $\psi$ are strictly positive functions, serving as nonlinear diffusion rates, and $k, \sigma > 0$ are constants.

• Journal of applied mathematics & informatics
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• 제41권1호
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• pp.11-31
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• 2023

The paper presents a critical analysis of selected topics related to the modeling of interacting species in which prey has nonlinear reproduction, which is in competition with predator. The mathematical model's stochastic stability is investigated. The method of designing appropriate Lyapunov functions is used to identify permanence conditions among the parameters of the model and conditions for the structure to no longer be extinct. The system's two-dimensional diffusive stability is regarded and studied. The system experiences the process of saddle-node bifurcation by varying the death rate of predator parameter. Further effects of parameters that undergo inherent oscillations are numerically investigated, revealing that as the intensity of predation parameter b is increased, the device encounters non-periodic and damped oscillations.

• ALGAE
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• 제37권2호
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• pp.149-161
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• 2022

Noctiluca scintillans is a heterotrophic dinoflagellate that causes red-colored oceans during the day (red tides) and glowing oceans at night (bioluminescence). This species feeds on diverse prey, including phytoplankton, heterotrophic protists, and eggs of metazoans. Thus, many scientists have conducted studies on the ecophysiology of this species. It is easy to cultivate N. scintillans at a scale of <1 L, but it is difficult to cultivate them at a scale of >100 L because N. scintillans cells usually stay near the surface, while prey cells stay below the surface in large water tanks. To obtain mass-cultured N. scintillans cells, we developed an automatic system for cultivating N. scintillans on a scale of 100 L. The system consisted of four tanks containing fresh nutrients, the chlorophyte Dunaliella salina as prey, N. scintillans for growth, and N. scintillans for storage, respectively. The light intensities supporting the high growth rates of D. salina and N. scintillans were 300 and 20 µmol photons m^-2 s^-1, respectively. Twenty liters of D. salina culture from the prey culture tank were transferred to the predator culture tank, and subsequently 20 L of nutrients from the nutrient tank were transferred to the prey culture tank every 2 d. When the volume of N. scintillans in the predator culture tank reached 90 L 6 d later, 70 L of the culture were transferred to the predator storage tank. To prevent N. scintillans cells from being separated from D. salina cells in the predator culture tank, the culture was mixed using an air pump, a sparger, and a stirrer. The highest abundance of N. scintillans in the predator culture tank was 45 cells mL^-1, which was more than twice the highest abundance when this dinoflagellate was cultivated manually. This automatic system supplies 100 L of N. scintillans pure culture with a high density every 10 d for diverse experiments on N. scintillans.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제17권2호
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• pp.129-138
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• 2013

A spatio-temporal models as systems of ODE which describe two-species Beddington - DeAngelis type predator-prey system living in a habitat of two identical patches linked by migration is investigated. It is assumed in the model that the per capita migration rate of each species is influenced not only by its own but also by the other one's density, i.e. there is cross diffusion present. We show that a standard (self-diffusion) system may be either stable or unstable, a cross-diffusion response can stabilize an unstable standard system and destabilize a stable standard system. For the diffusively stable model, numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross migration response is an important factor that should not be ignored when pattern emerges.

• Kyungpook Mathematical Journal
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• 제48권3호
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• pp.503-514
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• 2008

We investigate a three-species food chain system with Lotka-Volterra functional response and impulsive perturbations. In [23], Zhang and Chen have studied the system. They have given conditions for extinction of lowest-level prey and top predator and considered the local stability of lower-level prey and top predator eradication periodic solution. However, they did not give a condition for permanence, which is one of important facts in population dynamics. In this paper, we establish the condition for permanence of the three-species food chain system with impulsive perturbations. In addition, we give some numerical examples.