Does the diffusion DM-DE interaction model solve cosmological puzzles?
  We study dynamics of cosmological models with diffusion effects modeling dark
matter and dark energy interactions. We show the simple model with diffusion
between the cosmological constant sector and dark matter, where the canonical
scaling law of dark matter $(\rho_{dm,0}a^{-3}(t))$ is modified by an additive
$\epsilon(t)=\gamma t a^{-3}(t)$ to the form
$\rho_{dm}=\rho_{dm,0}a^{-3}(t)+\epsilon(t)$. We reduced this model to the
autonomous dynamical system and investigate it using dynamical system methods.
This system possesses a two-dimensional invariant submanifold on which the
DM-DE interaction can be analyzed on the phase plane. The state variables are
density parameter for matter (dark and visible) and parameter $\delta$
characterizing the rate of growth of energy transfer between the dark sectors.
A corresponding dynamical system belongs to a general class of jungle type of
cosmologies represented by coupled cosmological models in a Lotka-Volterra
framework. We demonstrate that the de Sitter solution is a global attractor for
all trajectories in the phase space and there are two repellers: the
Einstein-de Sitter universe and the de Sitter universe state dominating by the
diffusion effects. We distinguish in the phase space trajectories, which become
in good agreement with the data. They should intersect a rectangle with sides
of $\Omega_{m,0}\in [0.2724, 0.3624]$, $\delta \in [0.0000, 0.0364]$ at the
95\% CL. Our model could solve some of the puzzles of the $\Lambda$CDM model,
such as the coincidence and fine-tuning problems. In the context of the
coincidence problem, our model can explain the present ratio of $\rho_{m}$ to
$\rho_{de}$, which is equal $0.4576^{+0.1109}_{-0.0831}$ at a 2$\sigma$
confidence level.
