Global dynamics of diffusive animal movement models under habitat degradation, destruction, and fragmentation: eigenvalue problems and geometry at the landscape scale

• Author / Creator

  Salmaniw, Yurij

• Habitat loss is a significant problem and the leading cause of species extinctions and decreases in biodiversity worldwide. This issue is primarily driven by human activities, making it crucial to understand the consequences of our actions as a global community. The empirical study of habitat loss is complex, expensive, and time-consuming. Moreover, habitat is being lost at an accelerating rate, and so there is a substantial need for a timely assessment of the impacts of habitat loss. Fortunately, mathematical modelling allows us to assess the impacts of different aspects of habitat loss in a general setting, applicable to numerous species in various environmental scenarios.
  In this dissertation, we delve into three critical aspects of habitat loss: habitat degradation, habitat destruction, and habitat fragmentation. By employing the framework of reaction-diffusion equations, we investigate the global dynamics of single and multi-species models under different forms of habitat loss. This analysis includes a detailed exploration of the global dynamics of time-dependent single and multi-species models using tools from the theory of partial differential equations and the theory of monotone flows. We develop robust modelling frameworks specific to habitat loss processes derived from a careful consideration of the ecological definitions of these forms of habitat loss. Of significance is a rigorous, analytical connection between the degradation and destruction formulations through an asymptotic limit. This connection between habitat degradation and destruction appears to be the first of its kind, establishing a connection consistent with the observation that the level of degradation of different habitats lies on a spectrum, ranging from intact to destroyed.
  A central object of study in this analysis is the so-called principal eigenvalue, which, in our context, provides a theoretical net growth rate of the population, at least for small population sizes. By examining these principal eigenvalues, we can assess habitat fragmentation’s impact independently of or in conjunction with habitat degradation and destruction. This results in a fitness index intimately dependent on the arrangement and geometry of the degraded/destroyed regions. Compared to existing measures of habitat fragmentation, our approach offers a mechanistic and species-oriented tool, providing unique insights into the effect of fragmentation on diffusive species while also providing a robust framework for translation to other environmental factors or
  non-diffusive movement mechanisms.
  Importantly, this framework is versatile enough to be applied to both the landscape
  and patch scales. It is essential to note that fragmentation is often studied at the patch scale, which can lead to misleading conclusions regarding the overall impact of habitat fragmentation. Since habitat fragmentation is sometimes measured empirically in ways incompatible with our fitness index metric, we also examine the impacts of habitat fragmentation on the theoretical total abundance of a population, considered for both single- and multi-species models.
  In all assessment methods, we consider fragmentation as an arrangement and a process. By doing so, we take into account the spatial distribution of habitats and the changes that may occur over time, relating more closely to habitat fragmentation as it occurs in the natural world. In turn, this allows us to analyze habitat loss’s consequences more comprehensively and generate more accurate conclusions about the effects on species populations.
  This dissertation presents a detailed and thorough investigation of habitat loss and its impacts on species, focusing on the connections between habitat degradation, destruction, and fragmentation. By employing mathematical modelling and the frame- work of reaction-diffusion equations, we are able to study the global dynamics of single and multi-species models under different scenarios of habitat loss. This work provides numerous assessment tools and viable hypotheses which can be tested and verified, either experimentally or in the field, giving precise mechanisms and, ultimately, answers to some of the most pressing questions surrounding the complex effects of habitat loss.

• Subjects / Keywords

  • partial differential equations
  • habitat loss
  • reaction-diffusion equations
  • eigenvalue problems
  • habitat degradation
  • habitat destruction
  • habitat fragmentation

• Graduation date

  Fall 2023

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-6w3t-0n76

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.