Page 42 - Contributed Paper Session (CPS) - Volume 6

CPS1484 Neela A Gulanikar et al.
2. Methodology
2.1 Pair Formation
For modeling the number of marriages we plan to use the pair formation
models or marriage functions. The homogeneous functions used to model the
rates of pair-formation are commonly referred to as marriage functions. Most
pair-formation models have been developed to study the dynamics of
heterosexual populations that only include one single group of males and
females. Heterogeneity, in a heterosexually-mixing population, is usually
introduced by dividing the population of interest into subgroups (within each
sex) based on attributes of interest to the modellers or the scientists (e.g., age,
education, etc.).
In population theory Birth, death within certain ranges can be described by
linear processes (Hadeler et al. 1988). The marriage or pair formation is an
essentially nonlinear phenomenon. Pair-formation function is nonlinear,
homogeneous of degree one and has certain monotonicity properties. An
appropriate modeling the pair formation process has been called the two-sex
problem".
Kendall (1949), Keyfitz (1972), Fredrickson (1971), MacFarland (1972), and
Pollard (1973,Ch.7) have discussed ordinary differential equations and
integrals describing the age structure. Hadeler et al. (1988) have developed an
approach to homogeneous evolution equations, and their theory provides the
appropriate framework for pair formation models. The process of pair
formation is essentially nonlinear. They have suggested various functional
forms for the rate of pair-formation or marriage function using harmonic
mean, geometric mean and minimum function. Marriage function is a function
Ψ of the population sizes of single males M and single females F. The basic
properties satisfied by the marriage function are,
• Ψ(M,F) ≥ 0
• Ψ(M + u,F + v) ≥ Ψ(M,F), for u,v ≥ 0
• Ψ(λM,λF) = λ Ψ(M,F), for λ ≥ 0
• Ψ(M,0) = Ψ(0,F) = 0
Hadeler et al. (1988) given the following functions:
Harmonic mean function (HMF): (, ) = 2 ( )
+
Geometric mean function (GMF): (, ) = √
Minimum function (MF): (, ) = {, }
M,F describe the population sizes of single males and single females
respectively, ρ > 0
Schmitz (2000) has also discussed pair formation using typical marriage
functions, the Minimum Function (MF) and the Harmonic Mean Function
(HMF).

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