Keith Tognetti

The two stage stochastic population model

Abstract The two stage model in which an organism may exist for a constant stage as an egg before it becomes an adult is extended to a stochastic model. In the case where the birth and mortality rates are age independent constants, expressions are developed for the time dependent probabilities, variance and extinction probabilities.

Linking the Calkin-Wilf and Stern-Brocot trees

Links between the Calkin-Wilf tree and the Stern-Brocot tree are discussed answering the questions: What is thejth vertex in thenth level of the Calkin-Wilf tree? and Where is the vertexrslocated in the Calkin-Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin-Wilf tree into the jth entry in the nth level of the Stern-Brocot tree. We also provide a simple method for evaluating terms in the Hyperbinary sequence thus answering a challenge raised in Quantum in September 1997. We also examine successors and predecessors in both trees.

Taylor series expansion of delay differential equations—A warning☆☆☆

Abstract Attempts have been made to approximate the solutions of differential-difference equations using a Taylor series expansion of the lag term and ignoring high order derivatives. It is demonstrated that such a technique may lead to serious errors and that it is, perhaps, easier and certainly more valid to apply numerical methods directly.

Locating terms in the Stern-Brocot tree

In this paper we discover an efficient method for answering two related questions involving the Stern-Brocot tree: What is thejth term in thenth level of the tree? and What is the exact position of the fractionrsin the tree?

A two-stage population model☆

Abstract If an organism may exist for a constant period as an egg before it becomes an adult and if the birth rate per individual and the death rate of an egg and adult are constants, then the number of adults can be represented by the differential delay equation N (t)=γN(t−1)−βN(t) . It is shown that this equation behaves like the Malthusian equation with large t where γ corresponds to the Malthusian birth rate and β corresponds to the Malthusian death rate. With the other parameters held constant the population will inevitably die out if the egg stage is increased beyond a critical value. An expression for the age density function is developed that is shown to asymptotically approach a negative exponential independent of time.

Some extensions of the keyfitz momentum relationship

AbstractA stable population, such that the total birthrateB(t) =Boerot, is abruptly altered by modifying the age-specific birth rate,m(x). The survivor function remains unaltered. The modified population ultimately settles down to a stable behavior, such thatB(t) =B1er1t. It is shown thatB1/B0 = (R0 −R1)/[(r0 −r1)R0Z1], whereR0,R1 are the net reproduction rates before and after the change, and

The two stage integral population model

\bar Z_1

On binary reflected Gray codes and functions

expected age giving birth for the stable population after the change. The age structure and transients resulting from the change are also described. The effect of an abrupt change in the survivor functionl(x) is also investigated for the simple case where the change is caused by alteringl(x) toe−λxl(x). It is shown that the above ratio becomes