This paper develops a class of continuous-time closed capture-recapture models which incorporate the use of covariates such as environmental variables or an individual's characteristics. The capture intensity is allowed to vary with time, behavioural response and heterogeneity. The heterogeneity effect is modeled as a function of observable covariates but no assumptions regarding the time-varying function are made. The proposed hierarchy of models can be regarded as the continuous version of discrete-time models used in ecological applications. A unified likelihood-based approach is proposed to assess the effect of each possibly time-dependent covariate and to obtain population size estimators. Our model generalizes Yip, Huggins and Lin (1996) to incorporate an animal's behavioural response and to make use of all capture frequency data. The approach also extends Lin and Yip (1999) to a more general semi-parametric approach. Simulation results are presented to show the performance of the proposed estimation procedures. The estimators are applied to a set of capture data for house mouse (Mus musculus) discussed in the literature.