Bacterial populations in natural conditions are expected to experience stochastic environmental fluctuations, and in addition, environments are affected by bacterial activities since they consume substrates and excrete various chemicals. We here study possible outcomes of population dynamics and evolution under the repeated cycle of substrate-rich conditions and starvation, called the "feast-famine cycle," by a simple stochastic model with the trade-off relationship between the growth rate and the growth yield or the death rate. In the model, the feast (substrate-rich) period is led by a stochastic substrate addition event, while the famine (starvation) period is evoked because bacteria use the supplied substrate. Under the repeated feast-famine cycle, the bacterial population tends to increase the growth rate, even though that tends to decrease the total population size due to the trade-off. Analysis of the model shows that the ratio between the growth rate and the death rate becomes the effective fitness of the population. Hence, the functional form of the trade-off between the growth and death rate determines if the bacterial population eventually goes extinct as an evolutionary consequence. We then show that the increase of the added substrate in the feast period can drive the extinction faster. Overall, the model sheds light on nontrivial possible outcomes under repeated feast-famine cycles.