Simple population models

This chapter introduces basic concepts in population modeling that will be applied throughout the book. It begins with the oldest example of a population model, the rabbit problem, which was described by Leonardo of Pisa (“Fibonacci”) and whose solution is the Fibonacci series. The chapter then explores what is known about simple models of populations (i.e. those with a single variable such as abundance or biomass). The two major classes are: (1) linear models of exponential (or geometric) growth and (2) models of logistic, density-dependent growth. It covers both discrete time and continuous time versions of each of these. These simple models are then used to illustrate several different population dynamic concepts: dynamic stability, linearizing nonlinear models, calculation of probabilities of extinction, and management of sustainable fisheries. Each of these concepts is discussed further in later chapters, with more complete models.

NEW OPPORTUNITIES LOCAL MEDICAL TREATMENT OF PERIODONTAL DISEASES (MICROBIOLOGICAL RATIONALE)

The article presents the results of a microbiological experiment to study the sensitivity of periodontal pathogens to coniferous polyprenols in the preparation of “Solagift”. The optica density of clinical isolates Streptococcus constellatus; Staphylococcus aureus; Fusobacterium nucleatum; Aggregatibacter actinomycetemcomitans during cultivation with the addition of polyprenol concentrate 1:5 was measured during 3-7 days. Comparison criteria: the change in the phase of adaptation (lag-phase), the phase change of geometric growth, the amplitude of the peak phase of geometric growth, the duration of the stationary phase, the period of the withering away of culture. In comparison with the parameters of periodontal pathogens in the control, the presence of coniferous polyprenols led to a significant decrease in the activity of all microbes according to all criteria, especially Staphylococcus aureus, whose growth was completely suppressed.

Birthday Party Patterns

In this month's problem, students are presented with a geometric growth pattern for table configurations at a birthday party. Students will verbally describe how to extend the pattern, then use their verbal description to generate an expression for the number of tables for different configurations. Each month, this section of the Problem Solvers department showcases students' in-depth thinking and discusses the classroom results of using problems presented in previous issues of Teaching Children Mathematics.

Improved fD(s)∗, fB(s)∗ and fBc from QCD Laplace sum rules

Anticipating future precise measurements of the D- and B-like (semi)leptonic and hadronic decays for alternative determinations of the CKM mixing angles, we pursue our program on the D- and B-like mesons by improving the estimates of [Formula: see text] and [Formula: see text] (analogue to [Formula: see text]) by using the well-established (inverse) Laplace sum rules (LSR) and/or their suitable ratios less affected by the systematics, which are known to N2LO pQCD and where the complete [Formula: see text] nonperturbative condensate contributions are included. The convergence of the PT series is analyzed by an estimate of the N3LO terms based on geometric growth of the coefficients. In addition to the standard LSR variable [Formula: see text] and the QCD continuum threshold [Formula: see text] stability criteria, we extract our optimal results by also requiring stability on the variation of the arbitrary QCD subtraction point [Formula: see text] which reduces the errors in the analysis. We complete the study of the open bottom states by an estimate of [Formula: see text]. Our results summarized in Tables 3 and 4 are compared with some other recent sum rules and lattice estimates.