• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.393-407
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• 2022

In this paper, a diffusive predator-prey system with Beddington DeAngelis functional response and the modified Leslie-Gower type predator dynamics when a prey population is infected is considered. The predator is assumed to predate both the susceptible prey and infected prey following the Beddington-DeAngelis functional response and Holling type II functional response, respectively. The predator follows the modified Leslie-Gower predator dynamics. Both the prey, susceptible and infected, and predator are assumed to be distributed in-homogeneous in space. A reaction-diffusion equation with Neumann boundary conditions is considered to capture the dynamics of the prey and predator population. The global attractor and persistence properties of the system are studied. The priori estimates of the non-constant positive steady state of the system are obtained. The existence of non-constant positive steady state of the system is investigated by the use of Leray-Schauder Theorem. The existence of non-constant positive steady state of the system, with large diffusivity, guarantees for the occurrence of interesting Turing patterns.

• Journal of applied mathematics & informatics
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• v.38 no.5_6
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• pp.375-406
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• 2020

The anti-predator factor due to fear of predator in eco- epidemiological models has a great importance and cannot be evaded. The present paper consists of a modified Lesli-Gower predator-prey model with contagious disease in the predator population only and also consider the fear effect in the prey population. Boundedness and positivity have been studied to ensure the eco-epidemiological model is well-behaved. The existence and stability conditions of all possible equilibria of the model have been studied thoroughly. Considering the fear constant as bifurcating parameter, the conditions for the existence of limit cycle under which the system admits a Hopf bifurcation are investigated. The detailed study for direction of Hopf bifurcation have been derived with the use of both the normal form and the central manifold theory. We observe that the increasing fear constant, not only reduce the prey density, but also stabilize the system from unstable to stable focus by excluding the existence of periodic solutions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.312-326
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• 2021

The prey-predator model is one of the most influential mathematical models in ecology and evolutionary biology. In this study, we considered a modified prey-predator model, which describes the rate of change for each species. The effects of modifications to the classical prey-predator model are investigated here. The conditions required for the existence of the first integral and the stability of the fixed points are studied. In particular, it is shown that the first integral exists only for a subset of the model parameters, and the phase portraits around the fixed points exhibit physically relevant phenomena over a wide range of the parameter space. The results show that adding coupling terms to the classical model widely expands the dynamics with great potential for applicability in real-world phenomena.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.2
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• pp.139-155
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• 2019

In this paper, we study the asymptotic behavior and Hopf bifurcation of the modified Holling-Tanner models for the predator-prey interactions in the absence of diffusion. Further the direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated. Diffusion driven instability of the positive equilibrium solutions and Turing instability region regarding the parameters are established. Finally we illustrate the theoretical results with some numerical examples.

• 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.

• 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 applied mathematics & informatics
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• v.27 no.3_4
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• pp.639-651
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• 2009

In this paper, we study a discrete Leslie-Gower one-predator two-prey model. By using the method of coincidence degree and some techniques, we obtain the existence of at least one positive periodic solution of the system. By linalization of the model at positive periodic solution and construction of Lyapunov function, sufficient conditions are obtained to ensure the global stability of the positive periodic solution. Numerical simulations are carried out to explain the analytical findings.

• The Journal of Fisheries Business Administration
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• v.33 no.2
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• pp.75-98
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• 2002

As a contribution to developing fishery stock assessment, optimum sustainable yield and its international standards such as MSY, MEY, and dynamic MEY for six recommended fisheries are developed using bio-economic models. For selecting the appropriate model, five models - Schaefer, Schnute, Walters and Hilborn, Fox, and CY&P models are tested in effort and catch data of six species. Surprisingly all the models except the CY&P model failed to satisfy statistical standards such as goodness-of-fitness and reliability. Generally, the CY&P model holds good fitness and statistically significant level for all of six fisheries. However, the CY&P model for squid, where the intrinsic growth rate is high, could not explain MSY, MEY, and dynamic MEY appropriately. This study makes a contribution to develop the modified model for the intrinsic growth rate of 1. The reformulated model represents the results reasonably even though the estimated equation has not good fitness. Although most of the CY&P models appear to have good fits and validated results for some cases, these models also seem to be quite sensitive to parameters which means a more stable model should be developed and data should carefully be handled. In particular biological and technical interactions such as multispecies, predator prey relationship, age structure and mortality should be taken into account. In addition, economic factors and fishing efforts such as price, cost, technical change and a reasonable function of fishing input should simultaneously be considered.

• Journal for History of Mathematics
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• v.26 no.2_3
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• pp.197-210
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• 2013

The late 18C Malthus studied population growth for the first time, Verhulst the logistic model in 19C and, after that, the study of the predation competition between two species resulted in the appearance of Lotka-Volterra model and modified model supported by Gause's experiment with bacteria. Instable coexistence equilibrium being found, Solomon and Holling proposed functional and numerical response considering limited abilities of predator on prey, which applied to Lotka Volterra model. Nicholson and Baily, considering the predation between host and parasitoid in discrete time, made a model. In 20C there were developed various models of disease dynamics with the help of mathematics and real data and named SIS, SIR or SEIR on the basis of dynamical phenomena.

• The Journal of Fisheries Business Administration
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• v.25 no.1
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• pp.37-59
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• 1994

Many studies to cope with the present problems of Korean coastal and offshore fisheries has been performed, but these were done partly in necessities and general studies for Korean inshore and offshore fisheries are in early stage. Most of these studies adopted analytical way of approach for each fishery individually and they could not reflect the effect of correlated interaction among fisheries on the several common species/stocks, and thus optimal effort allocation was impossible. To consider general fisheries and optimal effort allocation among competing mixed species, a linear programming (LP) approach is applied in this study and introduced into 16 important inshore and offshore fisheries with 13 constraining species which were chosen by annual yield order. This study is not based on the biological interaction among species (i.e., prey - predator system) but the technological interaction between species and fishing efforts. For the application of LP model in these fisheries, the standardization of fishing efforts through different fishing gears could not be successful and a new way of effort standardization through CPUE for vessel tonnage was originated. Total standardized fishing effort on a particular species i, Ei, is computed as the linear summation of standardized fishing effort generated by each fishery j. That is, (equation omitted) where $f_{j}$ is the total vessel tonnage of fishery j and aij is the coefficients contributing to the standardized fishing effort per ton for species i taken in fishery j. The total fishing effort level on species i due to both directed fishing and by - catch can thus be accounted in the aij's. Optimal effort allocation among the j fisheries may be considered a minimizing problem (minimize $\Sigma$ $f_{j}$), subject to the constraints that standardized fishing effort levels on particular species are maintained at, above, and below certain predefined levels. Fishing effort goals for individual species can be based on various biological and/or economic criteria, i.e., fishing effort level generating maximum sustainable yield and/or maximum economic yield. But in this study the $F_{0.1}$ criteria which was accepted as an approximate level for $F_{mey}$ by Outland and Boerema's (1973) study. The findings of this study are, (1) LP model can be applied to the Korean inshore and offshore fisheries giobally. (2) Through a new way of combining multiple different fisheries' efforts for a particular species together generating standardized fishing effort, Schaefer curve could be applied to the complex system successfully. (3) The results of this study for total reduction scale were mostly the same as those of prior studies, but different much from the individual scales of reduction. This study showed the necessities for exploitation of more concrete parameters to put into consideration of profitability of fisheries and social factors, and this model can be modified according to the actual constraints. Also, considering the age structure of stocks, this model can be developed into better one for better fisheries management.ent.