Motivated by various biological processes, this talk first briefly introduces five typical chemotaxis models: chemotaxis-growth model, chemotaxis-haptotaxis model, chemotaxis-fluid model with signal consumption, chemotaxis-fluid model with chemical production, and chemotaxis model with indirect signal production.
Then this talk focuses a chemotaxis-haptotaxis model for cancer invasion, which describes the mutual interactions between cancer cells, enzymes and extracellular matrix. The system consists of two parabolic PDEs, one of which possesses two cross-diffusion terms reflecting the biased movements of cells due to chemotaxis and haptotaxis, coupled with an ODE. Inspired by some new observations or approaches toward this system, we could discuss the boundedness and asymptotic behavior of the solutions.
This talk also addresses a chemotaxis system with indirect signal production, which models the aggregation behavior of the Mountain Pine Beetle in forest habitat. In the two-dimensional case, it is shown that this system exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, in that it refers to blow-up in infinite time rather than in finite time. In the three-dimensional setting, any arbitrarily small logistic damping is sufficient to prevents blow-up of solutions, which strikingly differs from the classical Keller-Segel system once again.