MS03 - ECOP-04

Nonlinearity and Nonlocality: Complex Dynamics in Models of Animal Movement

Tuesday, July 15 at 10:20am

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Alex Safsten (University of Maryland), Abba Gumel

Description:

Mathematical models, of various types, have been developed and used to understand animal movement and assess animal dynamics under various constraints of resources, predation, and/or changes in environmental and climate conditions. Specifically, these models have, more recently, focused on studying pertinent phenomena such as the impacts of animal memory, learning, fear, disease spread, invasion, climate change, and heterogeneous resource distribution on animal movement and dynamics. This minisymposium brings together researchers to discuss recent advances and challenges on the design and rigorous analysis of mathematical models for animal movement, particularly highlighting the role and effect of nonlinear and nonlocal terms in these models, and demonstrating analytical tools and methods which can be used to study them.

Thomas Hillen

University of Alberta

"Go-or-Grow Models in Biology: a Monster on a Leash"

Go-or-grow approaches represent a specific class of mathematical models used to describe populations where individuals either migrate or reproduce, but not both simultaneously. These models have a wide range of applications in biology and medicine, chiefly among those the modeling of brain cancer spread. The analysis of go-or-grow models has inspired new mathematics, and it is the purpose of this talk to highlight interesting and challenging mathematical properties. I present new general results related to the critical domain size and traveling wave problems, and I demonstrate the high level of instability inherent in go-or-grow models. We argue that there is currently no accurate numerical solver for these models, and emphasize that special care must be taken when dealing with the 'monster on a leash'' (joint work with R. Thiessen, M. Conte, T. Stepien).

Mark Lewis

University of Victoria

"Nonlocal Multispecies Advection-Diffusion Models"

Nonlocal advection is a key process in a range of biological systems, from cells within individuals to the movement of whole organisms. Consequently, in recent years, there has been increasing attention on modeling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modeling the nonlocality. One common formalism is the aggregation-diffusion equation, a class of advection-diffusion models with nonlocal advection. This was originally used to model a single population but has recently been extended to the multispecies case to model the way organisms may alter their movement in the presence of coexistent species. Here we analyze behaviour in a class of nonlocal multispecies advection-diffusion models with an arbitrary number of coexisting species. We give methods for determining the qualitative structure of local minimum energy states and analyze the pattern formation potential using weakly nonlinear analysis and numerical methods. Joint work with Valeria Giunta (Swansea), Thomas Hillen (Alberta) and Jonathan Potts (Sheffield)

Rebecca Tyson

University of British Columbia Okanagan Campus

"The Importance of Exploration: Modelling Site-Constant Foraging"

Foraging site constancy, or repeated return to the same foraging location, is a foraging strategy used by many species to decrease uncertainty and risks. It is often unclear, however, exactly how organisms identify the foraging site. Here we are interested in the context where the actual harvesting of food is first preceded by a separate exploration period. In this context, foraging consists of three distinct steps: (1) exploration of the landscape (site-generation), (2) selection of a foraging site (site- selection), and (3) exploitation (harvesting) through repeated trips between the foraging site and ”home base”. This type of foraging has received scant attention in the modelling literature, leading to two main knowledge gaps. First, there is very little known about how organisms implement steps (1) and (2). Second, it is not known how the choice of implementation method affects the outcomes of step (3). Typical outcomes include the foragers’ rate of energy return, and the distribution of foragers on the landscape. We investigate these two gaps, using an agent-based model with bumble bees as our model organism foraging in a patchy resource landscape of crop, wildflower, and empty cells. We tested two different site-generation methods (random and circular foray exploration behaviour) and four different site-selection methods (random and optimizing based on distance from the nest, local wildflower density, or net rate of energy return) on a variety of outcomes from the site-constant harvesting step. We find that site-selection method has a high impact on crop pollination services as well as the percent of crop resources collected, while site-generation method has a high impact on the percent of time spent harvesting and the total trip time. We also find that some of the patterns we identify can be used to infer how a given real organism is identifying a foraging site. Our results underscore the importance of explicitly considering exploratory behaviour to better understand the ecological consequences of foraging dynamics. Joint work with Sarah A. MacQueen, Clara F. Hardy, and W. John Braun.

Chris Cosner

University of Miami

"Mean Field Games and the Ideal Free Distribution"

The ideal free distribution in ecology was introduced by Fretwell and Lucas to model the habitat selection of animal populations. It is based on the idea that individuals can assess their fitness at any location, making allowances for crowding, and will move to optimize it. In the context of the evolution of dispersal, movement strategies that can produce an ideal free distribution have been shown to be evolutionarily stable from the viewpoint of adaptive dynamics in many modeling contexts. In this paper, we revisit the ideal free distribution from the viewpoint of a habitat selection game in ecology. We specifically use the approach of mean field games, as introduced by Lasry and Lions. In that approach, an individual agent using a given strategy competes with the “mean field” of the strategies used by other agents. We find that the population density of agents converges to the ideal free distribution for the underlying habitat selection game, as cost of control tends to zero. Our analysis provides a derivation of ideal free distribution in a dynamical context.

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Annual Meeting for the Society for Mathematical Biology, 2025.