The authors study a system generalizing the classical Keller-Segel model with the logistic term to the case of diffusions defined by fractional powers of Laplacians \[ \begin{aligned} u_t & =-\mu\Lambda^\alpha u+(u\Lambda^{\beta-1}H v)_x+ru(1-u),\\ \tau v_t & =-\nu\Lambda^\beta v-\lambda v+u,\end{aligned} \] with the periodic boundary conditions, i.e. on the circle. Here, \(H\) denotes the Hilbert transform and \(\Lambda=(-\Delta)^{1/2}\). There are many interesting questions concerned with this mathematical model motivated by biological models of chemotaxis with growth. For instance, even the local-in-time existence, regularity and continuation of solutions questions are quite intriguing for certain values of nonnegative parameters \(\mu\), \(\nu\), \(r\), \(\lambda\). Then, the authors thoroughly discuss large data regularity of solutions for \(\alpha>1\), global-in-time existence of solutions when \(r>0\), and the long time asymptotics of solutions described by maximal attractors. There is also shown a bound on the number of peaks for \(u\) and \(v\) which might be an evidence of spatio-temporal chaotic behaviour of solutions. Numerical experiments are given which strongly suggest either merging of peaks, or transition to chaos, or blowup behaviour of solutions.