  In this work, we study dynamic properties of classical solutions to a
homogenous Neumann initial-boundary value problem (IBVP) for a two-species and
two-stimuli chemotaxis model with/without chemical signalling loop in a 2D
bounded and smooth domain. We successfully detect the product of two species
masses as a feature to determine boundedness, gradient estimates, blow-up and
$W^{j,\infty}(1\leq j\leq 3)$-exponential convergence of classical solutions
for the corresponding IBVP. More specifically, we first show generally a
smallness on the product of both species masses, thus allowing one species mass
to be suitably large, is sufficient to guarantee global boundedness, higher
order gradient estimates and $W^{j,\infty}$-convergence with rates of
convergence to constant equilibria; and then, in a special case, we detect a
straight line of masses on which blow-up occurs for large product of masses.
Our findings provide new understandings about the underlying model, and thus,
improve and extend greatly the existing knowledge relevant to this model.

Analysis of PDEs