A Population Dynamics Model with Parental Care

A model for an agestructured unlimited population dynamics with parental care of offspring is presented (migration of individuals is not taken into account). The model consists of six partial integrodifferential equations for single males, single females, pairs with offspring under parental care, pairs without offspring under parental care, and offspring of the male and female sex. A class of separable solutions is constructed.     

Large-Time Behavior in Population Dynamics with Offspring Production at Fixed Ages,Child Care,and Spatial Dispersal

We present a density-dependent population dynamics model with age-dependence, child care, and spatial dispersal. The population consists of the young (under maternal care), juvenile, and adult (producing offsprings at fixed ages or of post-reproductive age) classes. Death moduli of the juvenile and adult individuals are decomposed into two-term sums. The first sum represents the death rate by natural causes and by those that do not depend on the population spatial density, while the other one describes the environmental influence depending on the spatial density of the juvenile and adult individuals. The steady-state and a class of separable solutions are considered, and the large-time behavior of separable solutions is analyzed for the stationary vital rates. The asymptotic behavior of nondispersing semelparous species is also examined.     

An Age-Structured Population Dynamics Model with Females' Pregnancy and Child Care

We present a deterministic model for an age-structured population dynamics taking into account females' pregnancy, maternal care of offsprings, and environmental pressure with or without spatial migration. The model is based on the age-density notion for a group formed by a mother and her offsprings under maternal care. A harmonic-mean-type mating function of sexes without formation of permanent pairs is used. It is assumed that each sex has the pre-reproductive, reproductive, and post-reproductive age intervals. All adult individuals are divided into males, single females, fertilized females, and females taking child care. Individuals of post-reproductive age belong to the group of singles. All individuals of pre-reproductive age are divided into the young and juvenile groups. Only young offsprings are assumed to be under maternal care. Juvenile individuals can live without maternal care. The model consists of integro-PDEs subject to the conditions of integral type. The existence and uniqueness theorem is proved in the case of unlimited population. Separable solutions and their long-time behavior are studied for the limited nondispersing population. In the case of random migration two types of separable solutions and their long-time behavior for the homogeneous Dirichlet and Neumann boundary conditions are studied. In the case of directed migration in one-dimensional domain with special initial and Dirichlet boundary conditions, the unlimited invasive population dynamics is studied. In particular, an explicit formula for the migration front is given.     

Product desorption rate influence on catalytic reactivity of spatially inhomogeneous surfaces

We investigate numerically a phenomenological mathematical model of unimolecular reactions proceeding on inhomogeneous planar surfaces in the two-dimensional space case taking into account: the bulk diffusion of the reactant from the bounded vessel toward the adsorbent and the product bulk one from the adsorbent into the same vessel, the adsorption and desorption of reactant particles, long-range surface diffusion of the adsorbate, and a slow product desorption from the adsorbent. Simulations were performed using the finite difference technique. The influence of the long-range surface diffusion and product desorption rate on the kinetics of processes catalysed by inhomogeneous surfaces with different arrangements of reactive and nonreactive adsorption sites are studied.     

Numerical study of long-range surface diffusion influence on catalytic reactivity of spatially inhomogeneous planar surfaces

A phenomenological (mean-field) mathematical model of unimolecular reactions proceeding onto inhomogeneous planar surfaces is presented and investigated numerically in two-dimensional in space case taking into account the adsorption and desorption of reactant particles, long-range surface diffusion of the adsorbed particles, and an instantaneous product desorption from an adsorbent. The model also involves the bulk diffusion of the reactant from the bounded vessel towards the adsorbent and the product bulk one from the adsorbent into the same vessel. Simulations were performed using the finite difference technique. The influence of the long-range surface diffusion of adsorbed particles on the kinetics for processes catalyzed by inhomogeneous surfaces with a different arrangement of reactive and non-reactive adsorption sites is studied.     

A pair-formation model with a discrete set of offsprings and child care

We present a discrete newborns set-based deterministic model for a two-sex population structured by age and marital status. The model includes the spatial migration, a weighted harmonic mean-type pair formation function, and strong parental care and neglects the separation of pairs. Each sex has pre-reproductive and reproductive age intervals. All adult (of reproductive age) individuals are divided into single males, single females, and permanent pairs. All pairs are of two types: pairs without offsprings under parental care at the given time and pairs taking care of their young offsprings. All individuals of pre-reproductive age are divided into young (under parental care) and juvenile (offsprings who can live without parental care but cannot produce offsprings) groups. It is assumed that births can only occur from couples and after the death of any of the pair partner all young offsprings of this pair die. The model consists of integro-partial differential equations subject to the conditions of integral type. The number of these equations depends on the biologically possible maximal newborns number of the same generation produced by a pair. A class of separable solutions is studied for this model. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 93–129, January–March, 2007.     

Positive Solution of a Nonlinear Parabolic System Arising in Grain Drying

Coupled system of nonlinear parabolic equations for grain drying is proposed and the existence and uniqueness theorem of classical solutions is proved by using the upper and lower solutions technique. The long-time behaviour of the solution is also investigated.