Schedule for: 24w5209 - Dynamical Models Inspired by Biology

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Malaria, a deadly disease caused by protozoan Plasmodium parasites, is spread between humans via the bite of infected adult female Anopheles mosquitoes. Over 2.5 billion people live in geographies whose local epidemiology permits transmission of P. falciparum, responsible for most of the life-threatening forms of malaria. The wide-scale and heavy use of insecticide-based mosquito control interventions resulted in a significant reduction in malaria incidence and burden in endemic areas, prompting a renewed quest for malaria eradication. Numerous factors, such as Anopheles resistance to the currently-available insecticides used in mosquito control and anthropogenic climate change, potentially pose important challenges to the eradication efforts. In this talk, I will discuss a genetic-epidemiology mathematical modeling framework for assessing the combined impacts of insecticide resistance and climate change on distribution and burden of malaria mosquitoes and disease. If time permits, I will discuss our modeling effort on assessing the impact of sterile insect technique to control the population abundance of malaria mosquitoes, and explore its utility as an alternative pathway for achieving the malaria eradication objective.

In this study, we investigate a control problem involving a reaction-diffusion partial differential equation (PDE). Specifically, the focus is on optimizing the chemotherapy scheduling for brain tumor treatment to minimize the remaining tumor cells post-chemotherapy. Our findings establish that a bang-bang increasing function is the unique solution, affirming the MTD scheduling as the optimal chemotherapy profile. Several numerical experiments on a real brain image with parameters from clinics are conducted for tumors located in the frontal lobe, temporal lobe, or occipital lobe. They confirm our theoretical results and suggest a correlation between the proliferation rate of the tumor and the effectiveness of the optimal treatment. Joint work with Seyyed Abbas Mohammadi and Mohsen Yousefnezhad. 

A free boundary problem (FBP) consists of a system of PDEs in a domain with unknown boundary, which needs to be solved simultaneously with the unknown boundary of the domain. Such problems are increasing arise in models of bio-medical processes, for example: Cancer growth with treatment aimed at decreasing the growing unknown boundary; a growing plaque in cardiac artery which by blocking the artery will result in heart attack; chronic or diabetic dermal wound which, if not healed in proper time, may require amputation; cartilage shrinkage in rheumatoid arthritis; fungal skin infection which, if not treated, may spread over the whole body. Each if these diseases was modeled as a FBP, and numerical simulations of the model were performed and used to gain understanding, and to make recommendations, for effective treatments in experimental studies or in clinical trials. But what about rigorous analysis, e.g. theorems and proofs? In this talk I will briefly review such models and then proceed to describe mathematical results for simplified version of the models, showing that these results actually capture, in some “generalized” sense, those derived by simulations. I will also mention some open questions.

Periodic phenomena occur naturally due to periodic intake of food. In this talk we shall present our recent work on periodic solutions on two free boundary models in mathematical biology. (1) Atherosclerosis. Plaque formation is a leading cause of death worldwide; it originates from a plaque which builds up in the artery. We considered a simplified model of plaque growth involving LDL and HDL cholesterols, macrophages and foam cells, which satisfy a coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow.  In an earlier work (with Avner Friedman and Wenrui Hao) of an extremely simplified model, we proved that there exist small radially symmetric stationary plaques and established a sharp condition that ensures their stability. In our work with Evelyn Zhao, we look for the existence of non-radially symmetric stationary solutions.  The absence of an explicit radially symmetric stationary solution presents a big challenge to verify the Crandall-Rabinowitz theorem; through asymptotic expansion, we extend the analysis to establish a finite branch of symmetry-breaking stationary solutions which bifurcate from the radially symmetric solutions. This work is further extended (with Xiaohong Zhang, Zhengce Zhang) to include to allow reverse cholesterol transport in the model. Extension in the longitude direction and combined longitude-latitude direction is recently carried out (with Yaodan Huang). A periodic small plaque solution was recently found (with Yaodan Huang). This solution is linearly stable under certain conditions (with Jingyi Liu). (2) Tumor growth. Many models assume tumor cells are immersed in a constant supply of nutrient, for simplicity. We shall present the periodic solution and stability for the radially symmetric case. In particular, we shall establish the existence and uniqueness of the periodic solution in the biologically reasonable case and establish a global attractor in the class of radially symmetric initial data (with Yaodan Huang, Jingyi Liu).  

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Understanding mechanisms of coexistence is a central topic in ecology. Mathematical analysis of models of competition between two identical species moving at different rates of symmetric diffusion in heterogeneous environments show that the slower mover excludes the faster one. The models have not been tested empirically and lack inclusions of a component of directed movement toward favorable areas. To address these gaps, we extended previous theory by explicitly including exploitable resource dynamics and directed movement. We tested the mathematical results experimentally using laboratory populations of the nematode worm, Caenorhabditis elegans. Our results not only support the previous theory that the species diffusing at a slower rate prevails in heterogeneous environments but also reveal that moderate levels of a directed movement component on top of the diffusive movement allow species to coexist. Additionally, we have expanded our work to test the outcomes of different movement strategies in a various of fragmented and toxincant environments. For instance, we combine mechanistic mathematical modeling and laboratory experiments to disentangle the impacts of habitat fragmentation and locomotion. Our theoretical and empirical results found that species with a relatively low motility rate maintained a moderate growth rate and high population abundance in fragmentation. Alternatively, fragmentation harmed fast-moving populations through a decrease in the populations’ growth rate by creating mismatch between the population distribution and the resource distribution. Our study will advance our knowledge of understanding habitat fragmentation's impacts and potential mitigations, which is a pressing concern in biodiversity conservation.

This talk concerns an equation of Fisher-KPP type with a Keller-Segel chemotaxis term. The goal is to determine the effect of strong repelling chemotaxis on propagation. We provide an almost complete picture of the asymptotic dependence of the traveling wave speed on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence, in certain asymptotic regimes, to traveling waves of the porous medium Fisher-KPP equation and to those of a hyperbolic Fisher-KPP-Keller-Segel equation. The talk is based on joint work with C. Henderson and Q. Griette.

Atherosclerosis, the hardening of arteries due to plaque accumulation, is a leading cause of disability and premature death in the United States and worldwide. In this talk, I will present a highly nonlinear and highly coupled PDE model that describes the growth of arterial plaque in the early stage of atherosclerosis. The model incorporates LDL and HDL cholesterols, macrophage cells, and foam cells, with the interface separating the plaque and blood flow regions being a free boundary. I will discuss our findings on the existence of finite branches of symmetry-breaking bifurcation solutions. Furthermore, we have proved that the first bifurcation point for the system corresponds to the n=1 mode. Since plaque in reality is unlikely to be strictly radially symmetric, our results could be instrumental in explaining the asymmetric shapes of plaque.

Myxococcus xanthus, a social bacterium, exhibits fascinating collective behaviors such as rippling and swarming, where cells self-organize into complex patterns. This talk presents new biological data on these behaviors, featuring high-resolution analyses of cell movements and reversals. Based on these observations, we derive a kinetic model that identifies a key factor that facilitates the emergence of rippling patterns. Additionally, we introduce a 2D agent-based model that links bacterial reversals to congestion through dynamic motility regulation. This model provides a framework that accurately captures the two patterns observed in the data. The model also highlights the role of background anisotropy in pattern formation. This work is a collaboration with Vincent Calvez (Laboratoire de Mathématiques de Bretagne Atlantique), Tâm Mignot and Jean-Baptiste Saulnier (Laboratoire de Chimie Bactérienne - Marseille).

Vector-borne diseases (VBDs) are diseases primarily transmitted to humans and other animals by blood-feeding arthropods, such as mosquitoes, ticks, and bugs. Common VBDs include malaria, dengue fever, Rift Valley fever, Chagas disease, schistosomiasis, and Lyme disease. They constitute a serious threat to global health. For example, malaria alone caused 249 million cases and 608,000 deaths globally in 2022. Human migration and tourism have driven the spread of VBDs wider, faster and longer, as evidenced in recent outbreaks like the 2015-16 Zika virus epidemic. Patch models are widely used to describe the spatial spread of infectious diseases in discrete spaces. Quite a few studies have shown that population dispersal can strengthen or weaken the persistence of VBDs and make disease eradication more challenging. However, it is unclear how human movement affects the host prevalence (proportion of hosts being infected). In this talk, based on a generalized Ross-Macdonald model, we will present some preliminary results on this topic.

Osteoporosis is a disease characterized by a loss of bone mass, which leads to increased fragility and a higher likelihood of fractures. Cellular senescence is the permanent arrest of the normal cell cycle while maintaining cell viability. The number of senescent cells increases with age. Since osteoporosis is an age-related condition, it is natural to consider the extent to which senescent cells contribute to bone density loss and osteoporosis. In this talk, we use a mathematical model to address this question. We also evaluate senolytic drugs, such as fisetin and quercetin, which selectively eliminate senescent cells, and assess their efficacy in reducing bone loss.

Berestycki, Roquejoffre, and Rossi introduced a reaction-diffusion system for populations that have a distinguished ‘road’ on which they move quickly but do not reproduce. The goal is to understand invasion behavior (fronts). This model has attracted enormous interest in the decade since it was introduced, with a nearly complete picture in the case of a straight road. In this talk, I will discuss a joint work with Adrian Lam in which we provide an optimal control perspective on this problem. This gives a natural interpretation of the front in terms of balancing speed on the road and growth in the field, and it lets us easily deduce that ‘bent’ line case, which was previously not well-understood, is a simple consequence of the straight line case and some elementary geometry.

In this talk, we focus on the pattern formation of a chemotactic cell aggregation model with a mechanism that density suppresses motility. The model exhibits four types of cell aggregation patterns: single-point peaks, hot spots, cold spots, and stripes, depending on the parameters and mean density. The analysis is performed in two ways. First, traditional instability analysis reveals the existence of two critical densities. This local analysis shows patterns emerge if the initial mean density lies between the two values. Second, a phase separation method using van der Waals’ double well potential reveals that pattern formation is possible in a bigger parameter regime that includes the one identified by the local analysis. This non-local analysis shows that pattern formation occurs beyond the parameter regimes of the classical local instability analysis.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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 prove existence theorems for a class of nonlocal multispecies advection-diffusion models with an arbitrary number of coexistent 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. This is joint work with Valeria Giunta (Swansea), Thomas Hillen (Alberta) and Jonathan Potts (Sheffield)

An intriguing phenomenon, called dispersal-induced growth (DIG) in literatures, occurs when populations, that would become extinct when either isolated or well mixed, are able to persist by dispersing in the habitats. This somewhat counter-intuitive effect of dispersal has attracted attentions in both theoretical and empirical studies. In this talk we will discuss some potential underlying mechanisms of DIG by investigating the qualitative properties of the principal eigenvalues for second order elliptic or parabolic operators, mainly focusing on the dependence of principal eigenvalues on diffusion rates.

We explore the use of group inverses, a type of generalized inverses, to quantify heterogeneity in metapopulation dynamics. Specifically, we introduce a novel approach to understanding how spatial and temporal variability influence population persistence and growth rates. The analysis focuses on ecological models where habitat heterogeneity plays a significant role, offering insights into how environmental variations, resource distribution, and migration patterns affect population stability and resilience.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The evolution of mutualism between hosts and symbiont communities plays and essential role in maintaining ecosystem function and thus should have a profound effect during range expansion. In particular, the presence of mutualistic symbionts at the leading edge should enhance the propagation of the host and the overall symbiont community. Here we develop a theoretical framework that captures the eco-evolutionary dynamics of resource exchange between host symbionts and their dispersal in space. We provide quantitative insights into how the evolution of resource exchange may shape community strucure during range expasion. Parasitic symbionts receive the same amount of resources from the host as mutualistic symbionts, but at lower cost. This selective advantage is strengthened with resource availability (i.e., with host density), promoting mutualism at the range edges, where host density is low, and parasitism in the core of the range, where host desnity is higher. Host growth depends on the overall benefit provided by the symbiotic community, and is maximal at the expansion edges, where symbionts are more mutualistic. The expansion of host-symbiont communities is pulled by the hosts, but pushed by the symbionts. The spatial selection also influences the speed of spread. In particular, hosts with low dependence on their symbionts, or host-symbiont communities with high symbiont density at their core (e.g., resulting from more mutualistic hosts) or at their leading edge (e.g., resulting from symbiont inoculation) enhance the speed of spread into new territories.

The “wave-pinning” model is a two-component reaction-diffusion system in which mass-conservation and bistability lead to the formation of a moving interface which eventually stops (or is “pinned”). The model was originally proposed in 2008 by Y. Mori, A. Jilkine, and L. Edelstein-Keshet to describe the polarization of Rho GTPases. Using formal asymptotic methods one can derive an ODE for the interface motion from which its speed and final configuration can be determined. In this talk, I will provide an overview of the wave-pinning model and recent work by Adrian Lam, Yoichiro Mori, and myself rigorously justifying its formal asymptotic analysis.

Seed dispersal underlies important plant ecological and evolutionary processes, including gene flow, population dynamics, and diversity. While seed dispersal is a complex and context-dependent process, I emphasize that we can further our understanding of seed dispersal’s contribution to plant population dynamics and ecosystem functions using multidisciplinary and interdisciplinary perspectives. I will present two vignettes from on-going work to understand the movement ecology of seeds and demographic consequences of seed dispersal under global change. With a mechanistic understanding of seed dispersal, we will better understand the evolutionary ecology of seed dispersal, the influence of global change on seed dispersal processes, and potential cascading effects to ecosystems.

In this talk, we will review several free boundary problems related to species invasion/competition.  We will also introduce an approach for deducing Stefan-type problems to better understand the meaning of various parameters from a modeling perspective. Additionally, we will revisit and discuss some free boundary problems from the literature.

Mathematical models and simulations have emerged as valuable tools to analyze the spread and control of infectious diseases. In differential equation epidemic models, transmission mechanisms that describe the interaction of susceptible (S) and infected (I) people play an essential role. FIn this talk, we examine the role of nonlinear transmission mechanism of the for $S^pI^q$, $00$ in the disease spread of a diffusive-epidemic models. $q$ being greater than zero indicates that the transmission rate increases with $I$, which is typical as more infected individuals are present. Our results reveal the intricate role played by $p>0$ on the dynamics of solutions to the epidemics models.

Natural selection often operates simultaneously at multiple levels of organization, producing conflicts between traits or behaviors favored for selection within and among competing groups. It is possible to formulate models for this tension between levels of selection using evolutionary game theory, exploring the evolutionary tradeoff between the individual-level incentive to cheat and a collective incentive to establish cooperative groups of individuals. In this talk, we will discuss a class of hyperbolic PDE models to describe these scenarios of multilevel selection, capturing the tug-of-war between levels of selection using an advection term describing individual-level competition based on personal payoff and a nonlocal term describing competition between groups based on the average payoff of group members. We will discuss the characterization of the long-time behavior for these models, showing that cooperation can persist in the population provided that between-group competition is sufficiently strong relative to competition within groups. Interestingly, we find that this model displays a long shadow cast by lower-level selection, as the long-time population may achieve a level of cooperation much lower than the socially optimal of cooperation even in the limit of infinitely strong group-level competition. This talk is based on joint work with Yoichiro Mori.

A primary driver of species extinctions and declining biodiversity is loss and fragmentation of habitats owing to human activities. Many studies spanning a wide diversity of taxa have described the relationship between population density and habitat patch area, i.e., the density-area relationship (DAR), as positive, neutral, negative or some combination of the three. However, the mechanisms responsible for these relationships remain elusive. In this talk, we will discuss implementation of a reaction-diffusion framework with absorbing boundary conditions to model a habitat specialist dwelling in islands of habitat surrounded by a hostile matrix. We consider patches with both a convex and non-convex geometry. Our results show overall DAR structure can be either 1) positive, 2) positive for small areas and neutral for large, or 3) hump-shaped, i.e., positive for area below a threshold and negative for area above. We will also discuss comparison of our theoretical results with two empirical studies. Close qualitative agreement between theoretical and observed DAR indicates that our model gives a reasonable explanation of the mechanisms underpinning DAR found in those studies.

Increasing evidence shows that evolution may take place during species range expansion. In such processes, dispersal ability tends to be selected for at the leading edge, ultimately increasing species' spreading speeds. However, in many organisms across different taxa, higher dispersal comes at the cost of fitness, producing different evolutionary outcomes at the leading edge. Using reaction-diffusion equations and adaptive dynamics, we provide new insights on how such evolutionary processes take place. We show how evolution may drive species' phenotypes at the leading edge to be such that asymptotic spreading speed is maximized, and that phenotypic plasticity in dispersal may be selected for or against in different dispersal-reproduction trade-off scenarios. We provide some possible future directions of research, as well as present other systems where the framework may be applied.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

One of the longest-running research questions in ecology is why so many species can coexist while competing for so few resources. As such, there are many theories with supporting evidence that describe mechanisms that drive tradeoffs among species strategies. And when species exist that specialize in different areas of these tradeoffs they can often coexist stably, in theory. As we work to build more and more mechanistic models of ecological systems (for use in predicting the pace of climate change among other things), being able to predict the specializations of different species, the differences among species that allow them to coexist, would allow us to predict the diversity and resulting ecosystem function from the environment and physiological constraints of species alone, without knowledge of detailed species parameters. In this talk, I will show how we have struggled to find biologically realistic models that indeed lead to the emergence of species diversification that allows for their coexistence. I will show an area of promising results using successional tradeoffs in trees and end with a question of whether that diversification might depend on the limits to optimization due to the realities of ecological systems that allow many imperfect (but difficult to predict) species to coexist.

Populations consist of individuals living in different states and experiencing temporally varying environmental conditions. Individuals may differ in their geographic location, stage of development (e.g. juvenile versus adult), or physiological state (infected or susceptible). Environmental conditions may vary due to abiotic (e.g. temperature) or biotic (e.g. resource availability) factors. As survival, growth, and reproduction of individuals depend on their state and the environmental conditions, environmental fluctuations often impact population growth. Here, we examine to what extent the tempo and mode of these fluctuations matter for population growth. We model population growth for a population with $d$  individual states and experiencing $N$ different environmental states. The models are  switching, linear ordinary differential equations $x'(t)=A(\sigma(\omega t))x(t)$ where $x(t)=(x_1(t),\dots,x_d(t))$ corresponds to the population densities in the $d$ individual states, $\sigma(t)$ is a piece-wise constant function representing the fluctuations in the environmental states $1,\dots,N$, $\omega$ is the frequency of the environmental fluctuations, and $A(1),\dots,A(n)$ are Metzler matrices representing the population dynamics in the environmental states $1,\dots,N$. $\sigma(t)$ can either be a periodic function or correspond to a continuous-time Markov chain. Under suitable conditions, there exists a Lyapunov exponent $\Lambda(\omega)$ such that $\lim_{t\to\infty} \frac{1}{t}\log\sum_i x_i(t)=\Lambda(\omega)$ for all non-negative, non-zero initial conditions $x(0)$ (with probability one in the random case). For both  random and periodic switching, we derive analytical first-order and second-order approximations of $\Lambda(\omega)$ in the limits of slow  ($\omega\to 0$) and fast ($\omega\to\infty$) environmental fluctuations. When the order of switching and the average switching times are equal, we show that the first-order approximations of $\Lambda(\omega)$ are equivalent in the slow-switching limit, but not in the fast-switching limit. Hence, the mode (random versus periodic) of switching matters for population growth. We illustrate our results with applications to a simple stage-structured model and a general spatially structured model. When dispersal rates are symmetric, the first order approximations suggest that population growth rates increase with the frequency of switching -- consistent with earlier work on periodic switching. In the absence of dispersal symmetry, we demonstrate that $\Lambda(\omega)$ can be non-monotonic in $\omega$. In conclusion, our results show that population growth rates often depend both on the tempo ($\omega$) and mode (random versus deterministic) of the environmental fluctuations. This work is in collaboration with Pierre Monmarch\'{e} (Institut universitaire de France) and \'{E}douard Strickler (Universit\'{e} de Lorraine).

Explosive growth in the availability of animal movement tracking data is providing unprecedented opportunities for investigating the linkages between behavior and ecology over large spatial scales. Cognitive movement ecology brings together aspects of animal cognition (perception, learning, and memory) to understand how animals’ context and experience influence movement and space use, affording insights into encounters, territoriality, migration, and biogeography, among many other topics. Such datasets provide a rich source of inspiration for mathematical modeling. Here I will discuss several recent and ongoing models concerning the ways in which different kinds of learning and memory shape spatial dynamics with specific attention to movement paths, migration, and consumer-resource matching.

Social insects represent core ecological components of ecosystems, with high biomasses, particularly in the tropics. Dispersal is a fundamental process that influences colony establishment, genetic diversity, population dynamics, ecological interactions, and sociality. Consequently, dispersal drives ecosystem processes such as pollination (bees, wasps), predation (ants, wasps), soil turnover (termites, ants), and seed dispersal (ants). By understanding their dispersal and sociality, we gain insights into the evolutionary success and ecological roles of social insects. In this talk, I will provide a few examples of our work on developing mathematical models to explore the impacts of dispersal on social insect colonies at different colony stages across varied time scales. I hope that we can use the platform of our workshop to discuss how future mathematical models and theory should continue to explore the complex interplay among dispersal strategies, sociality, and environmental factors. This will help us better predict and manage the dynamics of social insect populations and their impacts on our ecosystems.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Roughly speaking, a population is said to exhibit an ideal free distribution (IFD) on a spatial region if all of its members can and do distribute themselves in a way that optimizes fitness, allowing for the effects of crowding. In models in which habitats are spatially heterogeneous but temporally constant, a remarkable confluence of adaptive dynamics, mathematics, and ecological theory shows that across a range of modeling formats, populations that can achieve an IFD as an equilibrium have an evolutionary advantage that can be understood as one alternative of the reduction principle. In general, there is an expectation that habitats will vary in both space and time, and recent work has progressed on defining an IFD in contexts wherein there is explicit or implicit seasonal variation such that a population exhibiting an IFD retains an evolutionary advantage. In the temporally constant case, an IFD can frequently be attained on the basis of local information. In the temporally varying context, it seems that nonlocal information in space and time is needed. This leads to very interesting theoretical and applied questions about perception, memory and learning. In this talk we will give an overview of these results and give an example of the evolution of dispersal by memory and learning in integrodifference models.

Classical reaction-diffusion-advection models assume that populations are not structured by movement pattern, so that all individuals in a species or subspecies move in the same way, and members of a species or subspecies only produce offspring of the same subspecies. There are at least three ways that populations can violate those assumptions. Individuals can change behavior, for example switching between flying and walking or search and resource exploitation. Populations can be stage structured and individuals at different stages can have different movement patterns. Individuals of one subspecies can produce offspring of another subspecies by genetic mutation or recombination. These phenomena lead to models for populations whose members can switch between different movement modes. Such models can have properties that are different from those for populations where all individuals move in the same way. Stage structured models may not satisfy the reduction principle, which says that slower dispersal is advantageous, and holds for many types of models with a single movement mode. Models with switching typically lead to systems that are cooperative at low densities but competitive at high densities, which creates some mathematical issues. This talk will describe some models of this type and some of their possible properties.

Tulips have captivated human interest for centuries, with their vibrant colors and unique shapes. Particularly striped tulips have been highly popular, leading to the “tulipomania” in the Dutch Golden Age. But how do the tulips get their stripes? Maybe Turing can help?

Based on empirical data, a cellular automata (CA) simulation model was developed for competition between floating (FAV) and submersed (SAV) aquatic vegetation in which an insect biocontrol agent consumes FAV biomass. In the absence of biocontrol, at low nutrient concentrations, the SAV excludes the FAV, while the reverse happens at high concentrations. At intermediate concentrations, alternative stable states occur. When the biocontrol agent, a weevil, is added, a dynamic regular striped pattern of rock-scissors-paper (weevil-FAV-SAV) formed and persisted for over 10,000 days despite stochastic disturbances in the form of added adult weevils. At some point in the simulation, which varies depending on the random number initiator, an apparently insignificant spatial deviation in small set of pixels triggers an instability that grows rapidly until the striped pattern has been replaced by a chaotic-appearingpattern. The CA model provides a unique opportunity to study exactly how the spatial instability develops and spreads as a butterfly effect. This research is important in revealing thedetailed mechanisms by which a long transient striped pattern transitions to an irregular pattern, offering valuable insights intohow spatial pattern form and change.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Alzheimer’s disease (AD) affects over 5 million people in the United States, presenting an urgent public health challenge. Personalized treatment strategies offer a promising pathway for improving patient outcomes, yet they require innovative methods to analyze the growing body of electronic brain data. In this talk, I present a mathematical modeling approach to capture the progression of AD clinical biomarkers, integrating patient-specific data to enable personalized predictions and guide optimal treatment strategies. Our model will be validated using a multi-institutional dataset of AD biomarkers, offering tailored predictions for individual patients.

The theory of "Additional food" states that a predator's efficacy in keeping a pest population in check, can be improved by providing the predator with additional food. In this talk we survey recent results that show how such biocontrol works if mechanisms such as predator competition, drift and dispersal pressures are taken into consideration. We aim to apply our results to the control of the pestiferous Soybean Aphid, and are motivated by climate change, as well as the current STRIPS project, a conservation practice at Iowa State University.

Specialists species thriving under specific environmental conditions in narrow geographic ranges are widely recognised as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse toward a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. Here, I study the eco-evolutionary dynamics of a sexually reproducing specialist species in a two-patch quantitative genetic model with moving optima. Thanks to a separation of ecological and evolutionary time scales and the phase-line study of the selection gradient, I derive the critical environmental speed for persistence, which reflects how the existence of a refugium and the cost of dispersal impact extinction patterns. Contrary to analogous studies featuring a homogeneous environment, the critical speed for persistence in a fragmented environment can be a non-increasing and discontinuous function of the strength of selection. Moreover, the analysis provides key insights about the dynamics that arise on the path towards this refugium. I show that after an initial increase of population size, there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch. Besides, when selection for local adaptation is strong, this habitat switch passes through an evolutionary “death valley” that can promote extinction for lower environmental speeds than the critical one and can lead to dynamics of evolutionary rescue

The ideal free distribution was originally introduced by Fretwell and Lucas as a Nash equilbrium of a habitat selection game. Since then, it has been proved to be linked with evolutionarily stable strategies (or evolutionary endpoints) in a number of situation. It has also been shown to arise if the agents use some local information in their dispersal strategy. In this talk, we will talk about a generalization of the ideal free distribution if the environment is varies periodically with time, by discussing a suitable notion of fitness for individuals residing in such environment. Next, we also discuss a recent result showing that in the context of mean field games. This is a novel framework of a multi-agent problem introduced by Lasry, Lions and collaborators, wherein the individual agents are modeled by a controlled diffusion process where they seek to optimize a payoff functional that is also dependent on a mean field term (describing the average behavior of other agents). We will give a brief review of IFD and MFG, and show that in the stationary situation, the population of agents approaches the ideal free distribution as the cost of control tends to zero, thus providing an alternative derivation of how ideal free distribution can arise in the dynamical situation of a mean field game. This is joint work with R.S. Cantrell, C. Cosner and I. Mazari.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will briefly report our recent progress beyond Feng, Lewis, Wang and Wang. 2022. A Fisher–KPP model with a nonlocal weighted free boundary: analysis of how habitat boundaries expand, balance or shrink. Bulletin of Mathematical Biology, 84(3), p.34.

Since the famous result that a "slower disperser wins" competition with a faster disperser (with otherwise identical demography), many theoretical studies have tried to find mechanisms by which faster or intermediate dispersal is beneficial for a population. One such mechanism is downstream drift in a single river reach. In such "advective" environments, intermediate or high dispersal can evolve, depending on boundary conditions. In this talk, I will review some of these results and present novel results on the evolution of dispersal in a small network of three connected river reaches. I will show that the outcome of the evolution of dispersal depends on the geometry of the network, such as the lengths and cross-sectional areas of the three reaches. This is joint work with Olga Vasilyeva.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.