Localised structures in a virus-host model

We investigate both the system of 1D and 2D spatial reaction-diffusion equations with zero Neumann boundary conditions on an spatially extended domain so as to physically model the connectivity between virus and host populations using a ratio-dependent functional response. Thus, using spatial dynamics with numerical support, our main focus is to survey the 1D spatial solution patterns for the coexistent positive steady state. It is found that the Turing bifurcation is subcritical which gives rise to a localised pattern. A two-parameter phase diagram of homogeneous, periodic and localised patterns is achieved analytically and numerically. In particular, pattern formation is explored in the limit where the virus infected population diffuses much more slowly than that of the host population. Our study shows that there exists a localised structure region which is divided into two sub regions by the Belyakov-Devaney transition curve. Depending on the sub region, the generated solution pattern is either an isolated hole or a localised pattern. Under the control of certain system parameters, when an isolated hole pattern occurs an outbreak of a disease may occur. The numerical outcomes for the 2D spatial system confirm the existence of the 1D localised patterns within the corresponding parameter regions. However, in 2D the solution patterns generated are a combination of isolated holes and stripes.