Front propagation for a jump process model arising in spacial ecology. (English) Zbl 1085.35017
The model studied by the authors is an integro-differential equation taking the form \(u_t=-u+\lambda(1-u)K*u\) in the space \({\mathbb R}^d,\) where \(\lambda>1\) is a constant, \(K*u\) stands for the convolution, and \(K\geq 0\) is a function such that \(\int_{{\mathbb R}^d} K(z)\, dz=1.\) It is shown that, for certain bounded solutions \(u\) of the afore mentioned equation, the region \(\Omega(t):=\{x\in{\mathbb R}^d:u(x,t)=(\lambda-1)/\lambda\}\) of colonized habitat expands and propagates at large scales with a special speed. The authors call this the phenomenon of front propagation.
Reviewer: Sen-Zhong Huang (Hamburg)
MSC:
35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |
45G10 | Other nonlinear integral equations |
45M05 | Asymptotics of solutions to integral equations |
92D40 | Ecology |