In many oviparous species, the incubation temperature of the egg determines the sex of the offspring. This is known as temperature-dependent sex determination (TSD). The probability of the hatched offspring being male or female varies across the incubation temperature range. This leads to the appearance of different TSD patterns in species such as FM pattern where females are predominately born at lower temperature and males at higher temperature, FMF pattern where the probability of female being born is higher at extreme temperatures and of the male being born is high at intermediate temperatures. We analyze an enzymatic reaction system proposed in the literature involving sex hormones with positive feedback effect to understand the emergence of different TSD patterns. The nonlinearity in the model is accounted through temperature sensitivity of the reaction rates affecting the catalytic mechanism in the reaction system. We employ a dynamical systems approach of singularity theory and bifurcation analysis to divide the parameter plane of temperature sensitivities into different regions where different TSD patterns are observed. Bifurcation analysis in association with the delineation of the parameter space for different TSD pattern has led to the identification of a subspace where all the TSD patterns observed in nature can be realized. We also show how modulation of the sex hormone in the species can be used to change the probability of occurrence of a specific sex, thereby preventing the extinction of endangered species. © 2020, Society for Mathematical Biology.