• Korean Journal of Mathematics
• /
• v.13 no.2
• /
• pp.161-176
• /
• 2005

We consider a model of population dynamics whose mortality function is unbounded. We note that the regularity of the solution depends on the growth rate of the mortality near the maximum age. We propose Gauss-Legendre methods along the characteristics to approximate the solution when the solution is smooth enough. It is proven that the scheme is convergent at fourth-order rate in the maximum norm. We also propose discontinuous Galerkin finite element methods to approximate the solution which is not smooth enough. The stability of the method is discussed. Several numerical examples are presented.

• Communications of the Korean Mathematical Society
• /
• v.18 no.4
• /
• pp.767-779
• /
• 2003

We consider a model of population dynamics whose mortality function is unbounded. We approximate the solution of the model using a discontinuous Galerkin finite element for the age variable and a backward Euler for the time variable. We present several numerical examples. It is experimentally shown that the scheme converges at the rate of $h^{3/2}$ in the case of piecewise linear polynomial space.

• Communications of the Korean Mathematical Society
• /
• v.18 no.3
• /
• pp.569-580
• /
• 2003

The Lotka-McKendrick equation which describes the evolution of a single population under the phenomenological conditions is developed from the well-known Malthus’model. In this paper, we introduce the Lotka-McKendrick equation for the description of the dynamics of a population. We apply a discontinuous Galerkin finite element method in age-time domain to approximate the solution of the system. We provide some numerical results. It is experimentally shown that, when the mortality function is bounded, the scheme converges at the rate of $h^2$ in the case of piecewise linear polynomial space. It is also shown that the scheme converges at the rate of $h^{3/2}$ when the mortality function is unbounded.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.14 no.3
• /
• pp.189-200
• /
• 2010

A numerical method is proposed and analyzed to approximate a mathematical model of age-dependent population dynamics with spatial diffusion. The model takes a form of nonlinear and nonlocal system of integro-differential equations. A finite difference method along the characteristic age-time direction is considered and primal mixed finite elements are used in the spatial variable. A priori error estimates are derived for the relevant variables.

• Journal of applied mathematics & informatics
• /
• v.25 no.1_2
• /
• pp.35-50
• /
• 2007

In this work, optimal harvesting policy for the predator-prey system of three species with age-dependent and diffusion is discussed. Existence and uniqueness of non-negative solution to the system are investigated by using the fixed point theorem. The existence of optimal control strategy is discussed and optimality conditions are obtained. Our results extend some known criteria.

• Communications of the Korean Mathematical Society
• /
• v.22 no.4
• /
• pp.623-640
• /
• 2007

The Lotka-McKendrick model which describes the evolution of a single population is developed from the well known Malthus model. In this paper, we introduce the Lotka-McKendrick model. We approximate the solution to the model using hp-discontinuous Galerkin finite element method. The numerical results show that the presented hp-discontinuous Galerkin method is very efficient in case that the solution has a sharp decay.

• Korean journal of applied entomology
• /
• v.58 no.4
• /
• pp.347-354
• /
• 2019

Temperature-dependent traits of Laodelphax striatellus, rice stripe virus vector, were investigated at 10 constant temperatures (12.5, 15.0, 17.5, 20.0, 22.5, 25.0, 27.5, 30.0, 32.5, and 35.0 ± 1℃) under a fixed photoperiod (14/10-hr light/dark cycle). Unit functions for the oviposition model were estimated and implemented into a population dynamics model using DYMEX. The longevity of L. striatellus adults decreased with increasing temperature (56.0 days at 15.0℃ and 17.7 days at 35.0℃). The highest total fecundity (515.9 eggs/female) was observed at 22.5℃, while the lowest (18.6 eggs/female) was observed at 35.0℃. Adult developmental rates, temperature-dependent fecundity, age-specific mortality rates, and age-specific cumulative oviposition rates were estimated. All unit equations described adult performances of L. striatellus accurately (r² =0.94~0.97). After inoculating adults, the constructed model was tested under pot and field conditions using the rice-plant hopper system. The model output and observed data were similar up to 30 days after inoculation; however, there were large discrepancies between observed and estimated population density after 30 days, especially for 1^st and 2^nd instar nymph densities. Model estimates were one or two nymphal stages faster than was observed. Further refinement of the model created in this study could provide realistic forecasting of this important rice pest.

• Korean journal of applied entomology
• /
• v.50 no.3
• /
• pp.235-245
• /
• 2011

Temperature-related parameters of Panonychus citri (McGregor) (Acarina: Tetranychidae) development were estimated and a stage-structured matrix model was developed. The lower threshold temperatures were estimated as $8.4^{\circ}C$ for eggs, $9.9^{\circ}C$ for larvae, $9.2^{\circ}C$ for protonymphs, and $10.9^{\circ}C$ for deutonymphs. Thermal constants were 113.6, 29.1, 29.8, and 33.4 degree days for eggs, larvae, protonymphs, and deutonymphs, respectively. Non-linear development models were established for each stage of P. citri. In addition, temperature-dependent total fecundity, age-specific oviposition rate, and age-specific survival rate models were developed for the construction of an oviposition model. P. citri age was categorized into five stages to construct a matrix model: eggs, larvae, protonymphs, deutonymphs and adults. For the elements in the projection matrix, transition probabilities from an age class to the next age class or the probabilities of remaining in an age class were obtained from development rate function of each stage (age classes). Also, the fecundity coefficients of adult population were expressed as the products of adult longevity completion rate (1/longevity) by temperature-dependent total fecundity. To evaluate the predictability of the matrix model, model outputs were compared with actual field data in a cool early season and hot mid to late season in 2004. The model outputs closely matched the actual field patterns within 30 d after the model was run in both the early and mid to late seasons. Therefore, the developed matrix model can be used to estimate the population density of P. citri for a period of 30 d in citrus orchards.

• Korean journal of applied entomology
• /
• v.61 no.4
• /
• pp.577-590
• /
• 2022

Ostrinia scapulalis is one of important pests in leguminous crops, especially red bean. In order to understand the biological characteristics of the insect, we investigated the effects of temperature on development of each life stage, adult longevity and fecundity of O. scapulalis at eleven constant temperatures of 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, and 36℃. Eggs and larvae successfully developed next life stage at most temperature subjected except 7, 10 and 13℃. The developmental period of egg, larva and pupa decreased as temperature increased. Lower and higher threshold temperature (TL and TH) were calculated by the Lobry-Rosso-Flandrois (LRF) and Sharpe-Schoolfield-Ikemoto (SSI) models. The lower developmental threshold (LDT) and thermal constant (K) from egg hatching to adult emergence of O. scapulalis were estimated by linear regression as 13.5℃ and 384.5DD, respectively. TL and TH from egg hatching to adult emergence using SSI model were 19.4℃ and 39.8℃. Thermal windows, i.e., the range in temperature between the minimum and maximum rate of development, of O. scapulalis was 20.4℃. Adults produced viable eggs at the temperature range between 16℃ and 34℃, and showed a maximum number, ca. 416 offsprings, at 25℃. Adult models including aging rate, age-specific survival rate, age-specific cumulative oviposition, and temperature-dependent fecundity were constructed, using the temperature-dependent adult traits. Temperature-dependent development models and adult oviposition models will be useful components to understand the population dynamics of O. scapulalis and will be expected using a basic data for establishing the strategy of integrated pest management in leguminous crops.

• Korean journal of applied entomology
• /
• v.62 no.4
• /
• pp.315-323
• /
• 2023

The brassica leaf beetle, Phaedon brassicae Baly (Coleoptera: Chrysomelidae), is one of the important pests infesting cruciferous vegetables. In order to understand the biological characteristics of the insect, we investigated the effects of temperature on development of each life stage, adult longevity and fecundity of P. brassicae at four constant temperatures of 15, 20, 25 and 27.5℃ for immature life stage and five constant different temperatures of 10, 15, 20, 25 and 27.5℃ for adult stage. Eggs and larvae successfully developed next life stage at temperature tested. The development period of egg, larva, and pupa decreased as temperature increased. Lower developmental threshold (LDT) and thermal constant (K) were calculated using linear regression as 8.7℃ and 344.73DD, respectively. Lower and higher threshold temperature (TL and TH) from egg to adult emergence were estimated by Briere function as 5.3℃ and 40.4℃, respectively. Adults produced eggs at the temperature range between 10℃ and 27.5℃, and showed an estimated maximum number, ca. 627.5 eggs at 21.7℃. Adult oviposition models including aging rate, age-specific survival rate, age-specific cumulative oviposition, and temperature-dependent fecundity were constructed. Temperature-dependent development models and adult oviposition models would be useful components to understand the population dynamics of P. brassicae and to establish the strategy of integrated pest management in cruciferous crops.