Persistence versus extinction under a climate change in mixed environments. (English) Zbl 1330.35216

The author studies the persistence versus extinction of species in the reaction-diffusion equation \[ u_t-\Delta u=f(t,x_1-ct,y,u), \quad t>0, x\in \Omega,\eqno(1) \] which is considered in two frameworks: (1) \(f\) is time-independent and \(x=(x_1,y)\in \Omega=\mathbb{R}\times \omega\), where \(\omega\) is an open, bounded and smooth domain in \(\mathbb{R}^{N-1}\); (2) \(f\) is periodic in \(y\) and \(t\), and \(x=(x_1,y)\in \mathbb{R}^N\). For the first case, the homogeneous Neumann boundary condition is considered, and the traveling front is of the form \(u(t,x)=U(x_1-ct,y)>0\) for \(x\in \Omega\). There exists a threshold value \(c^*\) such that \(U(x_1-ct,y)\) exists and is unique for \(0\leq c<c^*\). Moreover, the species is persistent as \(0\leq c<c^*\), but is extinct as \(c>c^*\). For the second case, the pulsating front is of the form \(u(t,x)=U(t,x_1-ct,y)>0\) for \(x\in \mathbb{R}^N\). Let \(\tilde{\lambda}_1\) be the generalized principal eigenvalue of the linearized operator of the wave profile equation; then \(\tilde{\lambda}_1<0 \) leads to the existence and uniqueness of pulsating front, and also the persistence of the species, while \(\tilde{\lambda}_1\geq 0 \) is corresponding to the extinction of the species. The equation (1) originally comes from a model in population dynamics to study the impact of climate change. Based on this consideration, the author further studies the concentration of the species when the environment outside \(\Omega\) become extremely unfavorable. At the last, a symmetry breaking property of the fronts was presented.


35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
92D25 Population dynamics (general)
35C07 Traveling wave solutions
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