The effect of boundary condition on critical niche for stage-structured population

Critical niche and survival bifurcation are key ecological issues for understanding how habitat extent determines population persistence. The associated threshold can thus be spatially quantified as the critical domain size, which refers to the minimum habitat size required for long-term population persistence. Given that most species possess highly structured life cycles, a discrete-time stage-structured population model, which is governed by an impulsive reaction-diffusion equation in a bounded domain, is thus investigated to explore population survival under different boundary conditions and derive the critical domain size for geometric domains including n-dimensional hypercubes and spheres. The theoretical analysis of population persistence across varying geometric configurations is established by the corresponding eigenvalue problems. In particular, for symmetric domains such as spheres, the critical domain size problem is still formulated within the Lie group framework, and its analytical characterization is thus obtained in terms of Bessel functions. Furthermore, by employing principal eigenvalue theory and the method of upper and lower solutions, the critical domain size is confirmed to trigger a survival bifurcation, which distinguishes population extinction in subcritical habitats from population persistence in supercritical ones. Finally, the classic application of our results to protected area design is also demonstrated and validated through numerical simulations.