Remarks on the Stability of Some Size-Structured Population Models VI: The Case When the Death Rate Depends on Juveniles Only and the Growth Rate Depends on Size Only and the Case When Both Rates Depend on Size Only

We continue our study of size-structured population dynamics models when the population is divided into adults and juveniles, started in El-Doma (to appear 1) and continued in El-Doma (to appear 2). We concentrate our efforts in two special cases, the first is when the death rate depends on juveniles only and the growth rate depends on size only, and, the second is when both the death rate and the growth rate depend on size only. In both special cases we assume that the maximum size for an individual in the population is infinite. We identify three demographic parameters and show that they determine sufficient conditions for the (in)stability of a nontrivial steady state. We also give examples that illustrate the stability results. The results in this paper generalize previous results, for example, see Calsina, et al. (2003), El-Doma (2006), and El-Doma (2008)

The Principle of Linearized Stability for Size-Structured Population Models

The principle of linearized stability for size-structured population dynamics models is proved giving validity to previous stability results reported in, for example, El-Doma (2008-1). In particular, we show that if all the roots of the characteristic equation lie to the left of the imaginary axis then the steady state is locally exponentially stable, and on the other hand, if there is at least one root that lies to the right of the imaginary axis then the steady state is unstable. We also point out cases when there is resonanc

A size-structured population dynamics model of Daphnia

AbstractThe stability of a size-structured population dynamics model of Daphnia coupled with the dynamics of an unstructured algal food source is investigated for the case where there is also an inflow of newborns from an external source. We determine the steady states and study the stability of the nontrivial steady states. We also identify a demographic-algae parameter that determines a condition for the stability

Stability Analysis for an SEIR Age-Structured Epidemic Model Under Vaccination

An SEIR age-structured epidemic model is investigated when susceptible and immune individuals are vaccinated indiscriminately and the force of infection of proportionate mixing type. We determine the steady states and obtain an explicitly computable threshold condition, and then study the stability of the steady states

Remarks on the Stability of Some Size-Structured Population Models V: The Case When the Death Rate Depends on Adults Only and the Growth Rate Depends on Size Only

We continue our study of size-structured population dynamics models when the population is divided into adults and juveniles, started in El-Doma (To appear). We concentrate our efforts in the special case when the death rate depends on adults only, the growth rate depends on size only and the maximum size for an individual in the population is infinite. Three demographic parameters are identified and are shown to determine conditions for the (in)stability of a nontrivial steady state. We also give examples that illustrate the stability results. The results in this paper generalize previous results, for example, see Calsina, et al. (2003), El-Doma (2006), and El-Doma (2008)

Global Stability Results of an SIS Age-Structured Epidemic Model with Vertical Transmission

An SIS age-structured epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility, mortality and cure rates depend on age and the force of infection of proportionate mixing assumption type. We determine the steady states and prove the global stability for the endemic equilibriums

Global Stability Results and Well Posedness of an SI Age-Structured Epidemic Model with Vertical Transmission

An SI age-structured epidemic model for a vertically as well as horizontally transmitted disease is investigated when the fertility and mortality rates depend on age and the force of infection of proportionate mixing assumption type. We prove the well posedness of the model as well as the global stability for endemic equilibriums

Remarks on the Stability of Some Size-Structured Population Models IV: The General Case of Juveniles and Adults

The stability of some size-structured population dynamics models is investigated when the population is divided into adults and juveniles. We determine the steady states and study their stability. We also give examples that illustrate the stability results. The results in this paper generalize previous results, for example, see Calsina, et al. (2003), El-Doma (2006), Farkas, et al. (2008), and El-Doma (2008 a)

Visualization of Some Energy Levels of Even-Even Nuclei

The properties of the lowest excited states of atomic nuclei offer a very sensitive test for nuclear structure theories. To identify and distinguish the shape phase transitions, one needs the variation behaviors of the energy ratios with respect to the neutron or proton number of the nuclei. The possibility of generating images of the variation behaviors of energy levels and energy ratios by exploiting the large body of data on stable and near-stable nuclei, whose accumulation over the past decades now permits, prompted us to produce this atlas which illustrates in a simple and effective way these behaviors. The atlas represents the experimental data of the energy levels and energy ratios of the 2_1^+, 4_1^+, 2_2^+ and 4_2^+ states of 645 even-even nuclei. The interpolation surfaces of the values of energy levels and energy ratios, E(2,1+), E(2,2+), E(4,1+), E(4,2+), E(4,1+)/E(2,1+), E(4,2+)/E(2,2+), E(2,2+)/E(2,1+), E(4,2+)/E(4,1+), E(4,2+)/E(2,1+) and E(4,1+)/E(2,2+), through all points of the even-even nuclear landscape are presented. The variation of these values with respect to each chain of isotopes and isotones are displayed using two-dimensional figures. Visualization of the data through visual imagery has been considered an effective way for connecting abstract and concrete ideas as well as allowing the emergence of the significant patterns. Moreover, we will see that it is possible, in many if not most cases, to understand the detailed results of complex calculations of the nuclear structure theories with an absolute minimum of formalism and often by inspection. The introduction of this atlas contains a brief account of some topics of present-day interest in theoretical nuclear structure. These topics can be used to explain the existence of a few typical patterns of nuclear spectra as well as some of the systematic changes in these patterns over sequences of nuclei. Researchers working on nuclear theory will find this atlas handy to understand the nuclear structure theories