A Mathematical Approach to Territorial Pattern Formation

Territorial behavior is widespread in the animal kingdom, with creatures seeking to gain parts of space for their exclusive use. It arises through a complicated interplay of many different behavioral features. Extracting and quantifying the processes that give rise to territorial patterns requires both mathematical models of movement and interaction mechanisms, together with statistical techniques for rigorously extracting parameters from data. Here, we give a brisk, pedagogical overview of the techniques so far developed to tackle the problem of territory formation. We give some examples of what has already been achieved using these techniques, together with pointers to where we believe the future lies in this area of study. This progress is a single example of a major aim for 21st century science: to construct quantitatively predictive theory for ecological systems

How memory of direct animal interactions can lead to territorial pattern formation

Mechanistic home range analysis (MHRA) is a highly effective tool for understanding spacing patterns of animal populations. It has hitherto focused on populations where animals defend their territories by communicating indirectly, e.g. via scent marks. However, many animal populations defend their territories using direct interactions such as ritualised aggression. To enable application of MHRA to such populations, we construct a model of direct territorial interactions, using linear stability analysis and energy methods to understand when territorial patterns may form. We show that spatial memory of past interactions is vital for pattern formation, as is memory of ‘safe’ places, where the animal has visited but not suffered recent territorial encounters. Additionally, the spatial range over which animals make decisions to move is key to understanding the size and shape of their resulting territories. Analysis using energy methods, on a simplified version of our system, shows that stability in the non-linear system corresponds well to predictions of linear analysis. We also uncover a hysteresis in the process of territory formation, so that formation may depend crucially on initial space-use. Our analysis, in 1D and 2D, provides mathematical groundwork required for extending MHRA to situations where territories are defended by direct encounters

Doctor of Philosophy

dissertationWe investigate how worker rules and battle conditions affect the overall behavior of territorial battles, battle initiation, and community structure in ants of the genus Tetramorium. Battles are modeled with partial differential equations describing how workers move through the battle and return to the nest to recruit. The probability of battle initiation was investigated for different worker rules and initial conditions using a two-dimensional individual-based simulation. We extend the model for battles to find a distribution of territory sizes for a one-dimensional community and include a model for colony founding, growth, decline, and death. We find that the class of workers in the battle using information to make decisions affects battle dynamics with a Hopf bifurcation, where battles change from having steady state solutions to oscillations, occurring when 34% or more grappling workers are using information. Larger colonies are able to control more territory than the break even point proportional to colony size. Initial conditions followed by recruitment response have the strongest effect on the probability of battle initiation, suggesting that some outside stimulation may be required to initiate a battle. Total ants in the community are sensitive to the lifespan of the colony and how many workers survive from one year to the next. Diversity and total number of colonies were sensitive to recruitment rate and which ants used information in the battle with lower diversity and total colonies when recruitment is high and information is used by grappling workers in the battle

Movement ecology, survival, and territorial dynamics in Canada lynx (Lynx canadensis) over a cyclic population decline

Dissertation (Ph.D.) University of Alaska Fairbanks, 2023As the quintessential predator-prey cycle, research into Canada lynx (Lynx canadensis) and snowshoe hares (Lepus americanus) have led to many discoveries in population biology; however, much remains unknown about the nuances of their populations. In this dissertation, I examined the ways in which population cycles influence survival, reproduction, territoriality, and connectivity. First, I examined ways in which climate change induced shifts in fire regimes have affected lynx persistence in their current refugia. Lynx mainly sought out areas of higher hare density and lower cover, such as intermediately aged coniferous and deciduous forests. This type of forest was predicted to persist in the near future, so long as fire intervals remained higher than current levels. Secondly, I investigated how landscape connectivity varied as a function of dispersal status and survival. We found that although the landscape was physically well connected based on resident lynx, it was even more so given dispersing lynx tolerance of poor habitat. This was dampened by survival declines in dispersing lynx over the course of a population crash to a near complete loss of connectivity. Thirdly, I assessed the degree to which dispersal, reproduction, and survival patterns were consistent with those displayed by populations exhibiting a traveling wave. My results supported the hypothesized westward moving population wave, but one mediated by differential survival and spatially varying reproduction rather than directionally-biased dispersal. Additionally, these characteristics were consistent with lynx as driving a similar population wave in snowshoe hare. Finally, I applied a novel mathematical approach to parameterizing advection-diffusion equations to examine how territorial formation occurs at population highs. I found evidence for hierarchical formation of territories in available space, with boundaries defined by preferred habitat. This methodology was a considerable improvement over previous descriptive methods typically used to define territories, as evidenced by the model's ability to predict territorial annexation following sudden vacancy following harvest. These results underscore the importance of maintaining population refugia and existing physical connectivity for the duration of a population downturn, likely on wildlife refuges and national parks across the state, even as the impacts of climate change remain small in the near future.US Fish and Wildlife Service Inventory and Monitoring and Science Applications, Tetlin National Wildlife Refuge, David Burnett Dunn Memorial Endowment and Erich Follmann Memorial Student Research SupportChapter 1: Introduction -- Chapter 2: Complex seasonal patterns of habitat use by a keystone mesopredator in boreal forest landscapes shaped by fire -- Chapter 3: Functional connectivity of Canada lynx habitats in the North American boreal region -- Chapter 4: Evidence for a survival driven traveling wave of a boreal predator -- Chapter 5: Linking conspecific interactions and habitat selection through a mechanistic home range model to examine drivers of territoriality in an Arctic mesocarnivore -- Chapter 6: General conclusion