Jose Antonio Carrillo
Title: TBC
Valeria Giunta
Title: Bifurcations, pattern formation and multi-stability in non-local models of interacting species
Abstract: In nature, every individual, be it a cell or an animal, inspects its territory before moving. The process of acquiring information from the environment is typically non-local, i.e. individuals have the ability to inspect a portion of their territory. In recent years, a growing body of empirical research has shown that non-locality is a key aspect of movement processes, while mathematical models incorporating non-local interactions have received increasing attention for their ability to accurately describe how interactions between individuals and their environment can affect their movement, reproduction rate and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movements generated by non-local species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide variety of spatio-temporal patterns that are able to reproduce segregation, aggregation and chase-and-run behaviours commonly observed in real systems. I will also show the existence of parameter regions where multiple stable solutions coexist and hysteresis phenomena.
Kevin Painter
Title: Phenotype-structured models for biological movement and invasion.
Abstract: Phenotypic diversity refers to the range of traits or characteristics across a population, where for cells this could be the levels of signalling, movement and growth activity, etc. Clearly, the phenotypic distribution – and how this changes over time and space - could be a major determinant of population-level dynamics. For instance, across a cancerous population, variations in movement, growth, and ability to evade death may determine its growth trajectory and response to therapy. In this talk I’ll briefly discuss the history of structured population models, and the extension of classic (cell) movement models to include phenotype heterogeneity. The resulting nonlocal models – which we call phenotype-structured partial integro-differential equations (PS-PIDEs) – can display rich dynamics, including concentration phenomena and travelling waves. Motivated by experimental observations of phenotypic diversity in chemotactic bacteria waves, I will focus on a phenotype-structured model of chemotaxis invasion that incorporates a trade-off: less chemotactic cells are more proliferative and vice versa. Under this scenario, travelling waves arise whereby the locally dominant phenotypic trait varies across the wavefront. But the exact form of structuring strongly depends on the extent to which phenotypic change is driven by the local surroundings, reinforcing the importance of environment on the structuring of invading populations.
Jonathan Potts
Title: Nonlocal advection-diffusion for modelling organism space use and movement
Abstract: How do mobile organisms situate themselves in space? This is a fundamental question in both ecology and cell biology but, since space use is an emergent feature of movement processes operating on small spatio-temporal scales, it requires a mathematical approach to answer. In recent years, increasing empirical research has shown that non-locality is a key aspect of movement processes, whilst mathematical models have demonstrated its importance for understanding emergent space use patterns. In this talk, I will describe a broad class of models for modelling the space use of interacting populations, whereby directed movement is in the form of non-local advection. I will detail various methods for ascertaining pattern formation properties of these models, fundamental for answering the question of how organisms situate themselves in space, and describe some of the rich variety of patterns that emerge. I will also explain how to connect these models to data on animal and cellular movement.