Evolutionary dynamics models in biometrical genetics supports QTL $$\times $$ × environment interactions

The process of development of quantitative trait locus (QTL) involves interactions between many factors, both environmental and genetic, in which many genes interact often in no additive pathways together and with environment. Integration of the mathematical, statistical and biological aspects of these subjects has made important and interesting results. In this review, mathematical methods offered to study the QTL \(\times \) environment interactions. The topic is circumscribed, going from basic selection equations to models of evolution of QTLs. Discrete and continuous time mathematical models and subsequently, QTL modelling were introduced with and without environmental interactions. The mathematical models derived here showed that the gradients of mean fitness which have revealed in studies by many researchers had a basic role in mathematical genetics, evolutionary aspects of biometrical genetics and QTL analysis. QTL \(\times \) environment interactions were studied mathematically including fitness components too. It was revealed that QTL \(\times \) environment interactions in fitness could generate a balancing selection. Also, QTL analysis could be used to calculate the geometry of the phenotype landscape. In this paper, models applied in biometrical genetics corresponds to QTL analysis and matched with results from other researchers. The originality of this synthesis is the evolutionary modelling of QTL \(\times \) environment interactions which can be used to investigate the extinction or stability of a population. Also to emphasize that although some scientific subjects like Brownian motion, quantum mechanics, general relativity, differential geometry, and evolutionary biometrical genetics were apparently different subjects, but the mathematical models were the backbone of these branches of science. This implies that such matters in nature have probably common and elegant basis. The perspective of the subject of this paper in future will be a new and interesting branch of interdisciplinary science.