Wave propagation for a reaction-diffusion model with a quiescent stage on a 2D spatial lattice
The following 2D lattice spatially discrete reaction-diffusion system with a quiescent stage is considered \[ \begin{cases} {\mathrm d u_{i,j}(t) \over \mathrm d t} = (\Delta_2 u)_{i,j} + f(u_{i,j}(t)) - \gamma_1 u_{i,j}(t) + \gamma_2 v_{i,j}(t), \\ {\mathrm d v_{i,j}(t) \over \mathrm d t} = \gamma_1 u_{i,j}(t) - \gamma_2 v_{i,j}(t), \end{cases} \] where \(u_{i,j}(t), v_{i,j}(t) \in \mathbb{R}, i,j \in \mathbb{Z}, t>0\) and \((\Delta_2 u)_{i,j}:= d[u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i, j-1} - 4u_{i,j}]\), and finally \(f\) is a mono-stable reaction term. The aim is to investigate the phenomenon of the biological invasion of a single-species population with two separate mobile and stationary parts. To this aim, for every direction \(\theta \in \mathbb R\), the existence of a minimal wave speed \(c^*(\theta)\) is proved, such that traveling wave solutions exist if and only if the wave speed is greater or equal than \(c^*(\theta)\). Moreover, the asymptotic behavior, monotonicity and uniqueness of traveling wave solutions are also studied. Finally, the effects of the direction of propagation \(\theta\) and the transfer rates \(\gamma_1, \gamma_2\) on the minimal wave speed are discussed.
spatially discrete reaction-diffusion systems
scientific article; zbMATH DE number 5838196