Context. The standard dynamo model for the solar and stellar magnetic fields is based on the αΩ mechanism, namely, an interplay between differential rotation (the Ω effect) and a mean electromotive force generated by helical turbulent convection flows (the α effect). There are, however, a number of problems with the α effect and αΩ dynamo models. Two of them are that, in the case of the Sun, the obtained cycle periods are too short and the magnetic activity is not sufficiently concentrated at low latitudes. Aims. We explore the role of turbulent induction effects that may appear in addition to the α effect. The additional effects result from the combined action of rotation and an inhomogeneity of the large-scale magnetic field. The best known of them is the Ω × J effect. We also include anisotropic diffusion and a new dynamo term which is of third order in the rotation vector Ω. Methods. We study axisymmetric mean-field dynamo models containing differential rotation, the α effect and the additional turbulent induction effects. The model calculations are carried out using the rotation profile of the Sun as obtained from helioseismic measurements and radial profiles of other quantities according to a standard model of the solar interior. In addition, we consider a dynamo model for a full sphere which is solely based on the joint induction effects of rotation and an inhomogeneity of the large-scale magnetic field, without differential rotation and the α effect (a δ2 dynamo model). This kind of dynamo model may be relevant for fully convective stars. Results. With respect to the solar dynamo, the inclusion of the additional turbulent induction effects increases the period of the dynamo and brings the large-scale toroidal field closer to the equator, thus improving the agreement of the models with the observations. For the δ2 dynamo working in a full sphere, we find dynamo modes which are steady if the effect of anisotropic diffusion is not included. The inclusion of anisotropic diffusion yields a magnetic field oscillating with a period of the order of the turbulent magnetic diffusion time.