Mathematical Models in Genetics and Population Inheritance

Mathematical genetics provides an important quantitative framework for understanding heredity, inheritance, genetic variation, and population evolution. In the present study, several mathematical models associated with genetics are investigated using probability theory, matrix methods, combinatorial techniques, and nonlinear difference equations. The paper mainly focuses on inheritance models, Hardy–Weinberg equilibrium, phenotype and genotype distributions, mutation-selection balance, blood group inheritance, sex-linked characteristics, and inbreeding models. Genetic matrices corresponding to dominant, hybrid, and recessive genotypes are formulated to describe the transmission of hereditary traits across successive generations. The Hardy Weinberg law is derived as an equilibrium condition under random mating, demonstrating the constancy of gene frequencies in stable populations. Mathematical expressions governing phenotype ratios and multiple allele systems are also developed, together with applications to human blood groups. The study further investigates models of genetic improvement through crossbreeding and elimination of recessive traits. Selection and mutation processes are analysed using nonlinear difference equations, and the corresponding equilibrium and stability conditions are discussed in detail. Different inbreeding models, including selfing, sib-mating, implanting, and brother sister mating, are examined to understand the reduction of heterozygosity and long-term genetic variability. The analysis shows that mathematical models play a significant role in explaining hereditary mechanisms, evolutionary dynamics, and population stability. These models possess important applications in agriculture, biotechnology, medicine, animal breeding, and genetic engineering, particularly in the development of disease-resistant and high-yielding species. The present work demonstrates the effectiveness of mathematical techniques in studying complex biological inheritance systems and genetic evolution.