  Habitat loss, through degradation and destruction of viable habitat, has a
well-documented and undeniable impact on the sustainability of ecosystems [Diaz
et al. (2019), Pimm et al. (2014), Pimm et al. (2000)]. Moreover, most habitat
loss is anthropogenic [Jacobson et al. (2019)]. Understanding the relationships
between varying degrees of habitat degradation, movement strategies, and
population dynamics on species persistence is crucial. We establish a robust
connection between habitat degradation and destruction using a
reaction-diffusion equation framework. Motivated by the recent work [Salmaniw
et al. (2022)], we consider an intrinsic growth rate function that features a
logistic-type growth in the viable habitat but decays at rate $c\geq0$ in the
degraded region(s). In the limit as $c \to \infty$, the solution to the habitat
degradation problem converges uniformly to the solution of a related habitat
destruction problem. When the habitat destruction problem predicts
deterministic extinction, a unique value $c_0$ exists, the extinction
threshold, from which any further habitat degradation leads to deterministic
extinction. This extinction threshold can be bounded below by a constant
depending on the size of the degraded region and the habitat quality in the
undisturbed region. To show these results, we investigate the eigenvalue
problems related to each model formulation and establish a convergence result
between the principal eigenvalues and their associated eigenfunctions,
providing a precise analytical connection between habitat degradation and
destruction in a general setting applicable to any species adopting diffusive
movement strategies.
