Solar cycles are studied with the Version 2 monthly smoothed international sunspot number, the variations of which are found to be well represented by a modified logistic differential equation with four parameters: maximum cumulative sunspot number or total sunspot number xm, initial cumulative sunspot number x0, maximum emergence rate r0, and asymmetry . A two-parameter function is obtained by taking and r0 as fixed values. In addition, it is found that xm and x0 can be well determined at the start of a cycle. Therefore, a predictive model of sunspot number is established based on the two-parameter function. The prediction for cycles 4-23 shows that the solar maximum can be predicted with an average relative error of 8.8% and maximum relative error of 22% in cycle 15 at the start of solar cycles if solar minima are already known. The quasi-online method for determining the moment of solar minimum shows that we can obtain the solar minimum 14 months after the start of a cycle. Besides, our model can predict the cycle length with an average relative error of 9.5% and maximum relative error of 22% in cycle 4. Furthermore, we predict the variations in sunspot number of cycle 24 with the relative errors of the solar maximum and ascent time being 1.4% and 12%, respectively, and the predicted cycle length is 11.0 yr (95% confidence interval is 8.3-12.9 yr). A comparison to the observations of cycle 24 shows that our predictive model has good effectiveness.