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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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?