A population dynamics model with strong maternal care

A two-sex age-structured nondispersing population dynamics deterministic model is presented taking into account strong maternal and weak paternal care of offspring. The model includes a weighted harmonic-mean type pair formation function and neglects the spatial dispersal and separation of pairs. It is assumed that each sex has pre-reproductive and reproductive age intervals. All adult individuals are divided into single males, single females, permanent pairs, and female-widows taking care of their offsprings after the death of their partners. All pairs are of two types: pairs without offspring under parental care at the given time and pairs taking child care. All individuals of pre-reproductive age are divided into young and juvenile groups. The young offspring are assumed to be under parental or maternal (after the death of their father) care. Juveniles can live without parental or maternal care but they cannot reproduce offsprings. It is assumed that births can only occur from couples. The model consists of nine integro-PDEs subject to the conditions of integral type. A class of separable solutions is studied, and a system for macro-moments evolving in time is derived in the case of age-independent vital ones. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 215–255, April–June, 2006.     

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.     

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.     

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.     

Spectral theory of Hamiltonian systems with almost constant coefficients

We derive the spectral theory for general linear Hamiltonian systems. The coefficients are assumed to be asymptotically constant and satisfy certain smoothness and decay conditions. These latter constraints preclude the appearance of singular continuous spectra. The results are thus far reaching extensions of earlier theorems of the authors. Two-, three- and four-dimensional systems are studied in greater detail. The results also apply to the case of the Dirichlet index and Dirichlet spectrum.     

Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation

In this paper, we consider a dynamical model of population biology which is of the classical Fisher type, but the competition interaction between individuals is nonlocal. The existence, uniqueness, and stability of the steady state solution of the nonlocal problem on a bounded interval with homogeneous Dirichlet boundary conditions are studied.     

Periodic solutions of the Hopf equation

We consider the long-time dynamics of approximate solutions of the boundary-value problem for the Hopf equation on a finite segment. Together with the initial conditions, for instance, we impose the zero Dirichlet conditions on both ends of the segment. In this case, all features of solutions associated with the intersections of characteristics are accumulated on a strip bounded by the vertical characteristics emitted from the boundary points.     

Bifurcation for a reaction–diffusion system with unilateral and Neumann boundary conditions

We consider a reaction–diffusion system of activator–inhibitor or substrate-depletion type which is subject to diffusion-driven instability if supplemented by pure Neumann boundary conditions. We show by a degree-theoretic approach that an obstacle (e.g. a unilateral membrane) modeled in terms of inequalities, introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable. Moreover, this parameter domain is rather different from the known case when also Dirichlet conditions are assumed. In particular, bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know.     

Leslie人口年龄结构模型的修正

Leslie人口年龄结构数学模型建立在没有人口流动的基础上,本文试图建立含人口迁徙因素在内的修正模型,并研究修正模型年龄结构的稳定性.     

LONG-TIME BEHAVIOR OF A CLASS OF REACTION DIFFUSION EQUATIONS WITH TIME DELAYS

The present paper devotes to the long-time behavior of a class of reaction diffusion equations with delays under Dirichlet boundary conditions. The stability and global attractability for the zero solution are provided, and the existence, stability and attractability for the positive stationary solution are also obtained.     

Periodic and Almost Periodic Solutions for a Coupled System of Two Korteweg-de Vries Equations with Boundary Forces

In this paper, we investigate a coupled system of two Korteweg-de Vries equations on a bounded domain. We discuss the long-time behavior of this system with forces on the left Dirichlet boundary conditions. We obtain that if the forces are periodic (almost periodic) with small amplitude, then the solution of the coupled system is periodic (almost periodic).     

Doubly nonlocal reaction–diffusion equations and the emergence of species

The paper is devoted to a reaction–diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of traveling waves is proved in the case of narrow kernels of the integrals. Periodic traveling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin’s definition of species as groups of morphologically similar individuals is related to Mayr’s definition as groups of individuals that can breed only among themselves.     

Dirichlet Problem for a Class of Quasilinear Elliptic Equations

In this paper, the Dirichlet problem for quasilinear elliptic equations is studied. New a priori estimates of the solution and its gradient are obtained. These estimates are derived without any assumptions on the smoothness of the coefficients and the right-hand side of the equation. Moreover, an arbitrary growth of the right-hand side with respect to the gradient of the solution is assumed. On the basis of the resulting estimates, existence theorems are proved.     

Population model with diffusion and supplementary forest resource in a two-patch habitat

A mathematical model is proposed to study the role of supplementary self-renewable resource on species population in a two-patch habitat. It is assumed that the density of forest resource biomass is governed by the logistic equation in both the regions but with the different intrinsic growth rate but the same carrying capacity in the entire habitat. It is further assumed that the densities of species population is also governed by the generalized logistic equations in both the regions but with different growth rates and carrying capacities. It is shown that the steady state solutions are positive, monotonic and continuous under both reservoir and no-flux boundary conditions. The linear and non-linear asymptotic stability conditions of non-uniform steady state are compared with the case of the model with and without diffusion in a homogeneous habitat.     

Dirichlet–Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.     

WILDLIFE RESERVES,POPULATIONS, AND HUNTING OUTCOME WITH SMART WILDLIFE

We consider a hunting area and a wildlife reserve and answer the question: How does clever migration decision affect the social optimal and the private optimal hunting levels and population stocks? We analyze this in a model allowing for two‐way migration between hunting and reserve areas, where the populations’ migration decisions depend on both hunting pressure and relative population densities. In the social optimum a pure stress effect on the behavior of smart wildlife exists. This implies that the population level in the wildlife reserve tends to increase and the population level in the hunting area and hunting levels tend to decrease. On the other hand, the effect on stock tends to reduce the population in the wildlife reserve and increase the population in the hunting area and thereby also increase hunting. In the case of the private optimum, open‐access is assumed and we find that the same qualitative results arise when comparing a situation with and without stress effects, but of course at a higher level of hunting. We also show that when net social benefits of hunting dominate the net social benefits of populations, wildlife reserves are optimally placed in areas of low carrying capacity and vice versa.     

Generalized impedance boundary conditions for the maxwell equations as singular perturbations problems

In this paper the Maxwell equations in an exterior domain with generalaized impedance boundary conditions of Engquist-Nédélec are considered. The particular form of the assumed boundary conditions can be considered to be a singular perturbation of the Dirichlet boundary conditions. The convergence of the solution of the Maxwell equations with these generalized impedance boundary conditions to that of the corresponding Dirichlet problem is proven. The proof uses a new integral equations method combined with results dealing with singular perturbation problems of a class of pseudo-differential operators.     

USING RESERVES TO PROTECT FISH AND WILDLIFE SIMPLIFIED MODELING APPROACHES

ABSTRACT. This paper investigates theoretically to what extent a nature reserve may protect a uniformly distributed population of fish or wildlife against negative effects of harvesting. Two objectives of this protection are considered: avoidance of population extinction and maintenance of population, at or above a given precautionary population level. The pre‐reserve population is assumed to follow the logistic growth law and two models for post‐reserve population dynamics are formulated and discussed. For Model A by assumption the logistic growth law with a common carrying capacity is valid also for the post‐reserve population growth. In Model B, it is assumed that each sub‐population has its own carrying capacity proportionate to its distribution area. For both models, migration from the high‐density area to the low‐density area is proportional to the density difference. For both models there are two possible outcomes, either a unique globally stable equilibrium, or extinction. The latter may occur when the exploitation effort is above a threshold that is derived explicitly for both models. However, when the migration rate is less than the growth rate both models imply that the reserve can be chosen so that extinction cannot occur. For the opposite case, when migration is large compared to natural growth, a reserve as the only management tool cannot assure survival of the population, but the specific way it increases critical effort is discussed.     

Determination of the non-linear reaction term in a coupled reaction-diffusion system

In a reaction-diffusion model, the problem of determination of the reaction term is of immense importance. Sometimes the form of the reaction is assumed with certain unknown parameters which are later determined from experimental data. The aim of the present study is to propose an iterative method of determination of the reaction term in a general form without involving parameters and analyze the basic problem of convergence of the iterative process. The equations considered are coupled one dimensional react ion-diffusion system with nonlinear interactive terms along with initial conditions and boundarydata. The overspecified Dirichlet data are used for the determination of the reaction term.     

Acceleration of the Process of Entering Stationary Mode for Solutions of a Linearized System of Viscous Gas Dynamics. I

The problem of construction of control Dirichlet boundary conditions accelerating the convergence of the corresponding solution to its steady state for given initial conditions is studied for the linearized system of differential equations approximately describing the dynamics of viscous gas. The algorithm is described and estimates of convergence rate are presented for the differential case.