## 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.

### MSC:

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

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