Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

翻译：移动栖息地模型追踪种群密度，该种群的适宜栖息地因气候变化而移动。尽管该领域以往研究多考虑一维空间，我们通过反应-扩散方程推导并研究了一个空间二维的移动栖息地模型。种群占据整个空间，适宜栖息地为有界区域（种群增长率为正），而其闭包的无界补集为不适宜区域（增长率为负）。两类栖息地之间的界面会移动，表征适宜栖息地向极地方向的迁移。对个体移动行为的精细建模导致密度在界面处出现非标准间断。针对对应的半离散系统，我们证明了恒定迁移速度下的适定性，进而构建了一种隐式-显式混合有限元方法。该方法通过拉格朗日乘子弱施加界面处的跳跃间断。针对静止界面，我们在非共形离散化的共形网格上推导了最优先验误差估计。通过数值收敛测试，我们证明这些结论对移动界面同样成立。最后，我们通过两个生物学实例（一个含曲边区域，另一个含非恒定迁移速度）展示了该混合有限元方法的优势。