Stability condition of the steady oscillations in aggregation models with shattering process and self-fragmentation

We consider a system of clusters of various sizes or masses, subject to aggregation and fragmentation by collision with monomers or by self-disintegration. The aggregation rate for the cluster of size or mass $k$ is given by a kernel proportional to $k^{a}$, whereas the collision and disintegration kernels are given by $\lambda k^{b}$ and $\mu k^{a}$, respectively, with $0\le a,b\le 1$ and positive factors $\lambda$ and $\mu$. We study the emergence of oscillations in the phase diagram $(\mu,\lambda)$ for two models: $(a,b)=(1,0)$ and $(1,1)$. It is shown that the monomer population satisfies a class of integral equations possessing oscillatory solutions in a finite domain in the plane $(\mu,\lambda)$. We evaluate analytically this domain and give an estimate of the oscillation frequency. In particular, these oscillations are found to occur generally for small but nonzero values of the parameter $\mu$, far smaller than $\lambda$.