Continuum, partial differential equation models are often used to describe the collective motion of cell populations, with various types of motility represented by the choice of diffusion coefficient, and cell proliferation captured by the source terms. Previously, the choice of diffusion coefficient has been largely arbitrary, with the decision to choose a particular linear or nonlinear form generally based on calibration arguments rather than making any physical connection with the underlying individual-level properties of the cell motility mechanism. In this talk I will discuss a series of individual-level models, which account for important cell properties such as varying cell shape and volume exclusion, and their corresponding population-level partial differential equation formulations. I will demonstrate the ability of these models to predict the population-level response of a cell spreading problem for both proliferative and non-proliferative cases. I will also discuss the potential of the models to predict long time travelling wave invasion rates.
Dispersal and the resulting genetic exchange between populations in spatially heterogeneous environments is typically expected to impede adaptation to local conditions. However, theory suggests some cases where this paradigm breaks down, such as when dispersal provides demographic support and gene flow enhances adaptive capacity to populations experiencing variable population sizes or environmental shifts. A current major driver of environmental change is anthropogenic activities, where humans can both be a source of environmental heterogeneity in space that selects on traits within populations experiencing exchange and a source of environmental shifts in time to which populations must adapt for local persistence. I will present a series of models exploring the potential for a beneficial versus detrimental role of gene flow given anthropogenically-driven global change. First, I will present a model of coral adaptation to climate change, where, given dispersal between populations experiencing different thermal stress, the potential for propagule input to enhance recovery from stressful events outweighs the potential for gene flow to impede adaptation to local thermal conditions. Second, I will present a model of exchange between salmon hatchery and wild populations, where the fitness and demographic consequences of domestication selection in the hatchery critically depend on the relative timing of natural selection, hatchery release, and density dependence in the life cycle. Both of these examples illustrate how a basic science understanding of gene flow can inform conservation management and how models of evolutionary response to global change can inform a basic science understanding of the adaptive role of gene flow.
A general question in the study of the evolution of dispersal is what kind of dispersal strategies can convey competitive advantages and thus will evolve. We consider a two species competition model in which the species are assumed to have the same population dynamics but different dispersal strategies. Both species disperse by random diffusion and advection along certain gradients, with the same random dispersal rates but different advection coefficients. We find a conditional dispersal strategy which results in the ideal free distribution of species, and show that it is a locally evolutionarily stable strategy. We further show that this strategy is also a globally convergent stable strategy under suitable assumptions, and our results illustrate how the evolution of dispersal can lead to an ideal free distribution. The underlying biological reason is that the species with this particular dispersal strategy can perfectly match the environmental resource, which leads to its fitness being equilibrated across the habitat.
Joint work with Chris Cosner and Yuan Lou.
We present a method for obtaining survival and coexistence results for a class of interacting particle systems. This class includes: a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, a model for the evolution of cooperation of Ohtsuki, Hauert, Lieberman and Nowak, and a continuous time version of a non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath and Levin. Each of these, for a range of parameter values, can viewed as a "voter model perturbation," meaning the dynamics are "close" to the dynamics of the voter model, a simple, neutral competition model. The voter model is mathematically tractable because of its dual process, a system of coalescing random walks. We show that when space and time are rescaled appropriately the particle density converges to a solution of a reaction diffusion equation. Analysis of this equation leads in some cases to asymptotically sharp survival and coexistence results, which are qualitatively different from the (pure) voter model case. This work with Rick Durrett and Ed Perkins is closely related to earlier work of Durrett and Neuhaueser on models with rapid stirring.
In the evolving voter model we choose oriented edges (x,y) at random. If the two individuals have the same opinion, nothing happens. If not, x imitates y with probability 1-α, and otherwise severs the connection with y and picks a new neighbor at random (i) from the graph, or (ii) from those with the same opinion as x. Despite the similarity of the rules, the two models have much different phase transitions. This is one example from a large nonrigorous literature on systems where the network structure and the states of the individual in it coevolve.
I look at the interactions between heterogeneities and delayed negative feedback in systems which admit stationary persistent structures. The former can cause pinning and stabilize neutrally stable dynamics while the latter can induce several types of dynamics instabilities and motion. I show that the time-scale of the negative feedback and the amplitude of the heterogeneities interact to produce qualitatively different sequences of bifurcations. The models are motivated by dynamics of neurons in the rodent hippocampus during navigation. This work is joint with Rodica Curtu, Carina Curto, and Vladimir Itskov.
Organisms reproduce in environments that vary in both time and space. Even if an individual currently resides in a region that is typically quite favorable, it may be optimal for it to "not put all its eggs in the one basket" and disperse some of its off spring to locations that are usually less favorable because the eff ect of unexpectedly poor conditions in one location may be o set by fortuitously good ones in another. I will describe joint work with Peter Ralph and Sebastian Schreiber (both at University of California, Davis) and Arnab Sen (Cambridge) that combines stochastic diff erential equations, random dynamical systems, and even a little elementary group representation theory to explore the eff ects of diff erent dispersal strategies.
Most ecological interactions occur in the context of fine-scale spatial and temporal heterogeneity, a.k.a., "patchiness". In many relevant ecological applications, the primary data sources (e.g. remote sensing), the primary predictive modeling approaches (e.g. biogeochemical or resource management models) and the most important ecological outcomes (e.g., total or "mean-field" populations) do not resolve or explicitly depend on fine-scale patchiness, but nonetheless are strongly affected by unresolved patch dynamics. In this talk, I will consider large-scale characteristics of interacting populations in which spatial and temporal heterogeneity is either imposed by external environmental forcing or arises autonomously from social interactions such as schooling and swarming. In many such populations, secondary population characteristics emerge that operate over larger spatio-temporal scales than primary patch dynamics and that strongly affect ecological outcomes. I will discuss some examples in which analysis of these secondary characteristics may improve interpretation and prediction of unresolved patch dynamics in data and models.
In joint work with Brett Melbourne we have studied highly replicated spatial population dynamics of flour beetles in a lab setting. I will describe the results of experiments on single species and spatial spread, and corresponding models. The models have to incorporate stochasticity of different forms to provide a good match to the data. In particular, demographic heterogeneity, fixed differences among individuals, are critical for understanding the dynamics.
For the past decade, Internet worms (a type of malicious software similar to a virus) spreading through networks have been using biological strategies, such as hierarchical dispersal and adaptive strategies, to spread more efficiently among susceptible computers. There is a direct analogy between susceptible computers on the Internet and susceptible hosts in community-structured populations.
Our measurements show that the Internet is an incredibly clustered heterogeneous environment when measured according to the dispersal strategy used by worms. We have used these measurements to build an epidemiological simulation model of the entire Internet (4.29 billion hosts, with roughly 2 million susceptible) efficient enough to run on an ordinary desktop computer. A worm which would have a basic reproduction ratio far less than one and therefore be quite unsuccessful at spreading using simple random dispersal strategies can be very successful by exploiting the large variance or clustering of vulnerable computers among subnetworks in the Internet. With the new Internet addressing scheme (IPv6) currently being rolled out, these issues will be amplified by many orders of magnitude.
This will be something of an introductory talk that considers two types of spatial models used in population biology, and connections between them. Interacting particle systems can be thought of as "microscopic" level descriptions of populations, including interactions between discrete individuals and stochasticity. Reaction-diffusion equations provide deterministic models that can be thought of as "macroscopic" versions of particle systems through scaling limits. We will discuss the basic ideas behind this connection, treat a few examples, and try to understand the extent to which the two types of models predict the same behavior.
In the seventies, biologists Maynard Smith and Price used concepts from game theory to describe animal conflicts. Their work is at the origin of the popular framework of evolutionary game theory. Space is another component that has been identified as a key factor in how communities are shaped. Spatial game models are therefore of primary interest for biologists and sociologists. There is however a lack of analytical results in this field. The objective of this talk is to explore the framework analytically through a simple spatial game model based on interacting particle systems (agent-based models). Our results indicate that the behavior of this process strongly differs from the one of its non-spatial mean-field approximation, which reveals the importance of space in game theoretic interactions.
There is a long history of research on the mathematical modeling of animal populations, largely built on diffusion models. The classical literature, however, is inadequate to explain observed spatial patterning, or foraging and anti-predator behavior, because animals actively aggregate. This lecture will discuss models of animal aggregation, and the role of leadership in collective motion. It will also explore models of the evolution of collective behavior, and implications for the optimal design of robotic networks of interacting sensors, with particular application to marine systems.
In this talk I will outline first passage time analysis for animals undertaking complex movement patterns while searching for prey. I will extend the analysis to complex heterogeneous environments to assess the effects of man-made linear landscape features on functional responses in wolves searching for elk. We developed a mechanistic first passage time model, based on an anisotropic elliptic partial differential equation, and used this to explore how wolf movement responses to seismic lines influence the encounter rate of the wolves with their prey. (This work is joint with Hannah McKenzie, Evelyn Merrill and Ray Spiteri)
Spatial patterns of genetic variation are clearly indicative of past dispersal and migration processes, but performing formal inference with spatial models in population genetics has been challenging and fairly limited. In this lecture I will overview several areas of recent progress, some using model-based approaches and others using informal exploratory approaches. Particular attention will be given to the insights that can be gained from the spatial distribution of rare variants, as well as spatial assignment approaches. The examples will include data from humans and migratory birds.
The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. For this we analyze a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems has been done by simulation or non-rigorous methods such as pair approximation. I will describe a model based on the general voter model perturbations recently studied by Cox, Durrett, and Perkins (2011) which allows us to rigorously and explicitly compute evolutionarily stable strategies. A main difficulty in this case is to extend the earlier work in three or more dimensions to the more complicated two-dimensional case, which is the natural setting for this problem. This is joint work with Rick Durrett.
This talk will review three recent results about persistence of spatially structured populations and the spatial spread of populations in the presence of stochasticity. For the first part of this talk, I discuss the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable. An application to the coupled-Ricker model will be given. For the second part of the talk, I present results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise coupled with dispersal can rescue locally extinction prone populations. For the final part of the talk, I present invasion speed formulas for models combining state-structured local demography (e.g., an integral or matrix projection model) with general dispersal kernels, and stationary temporal variation in both local demography and dispersal kernels. Using these results, I will show that random temporal variability in dispersal can accelerate population spread. More surprisingly, demographic variability can further accelerate spread if it is positively correlated with dispersal variability.
Migration is a widely used strategy for dealing with seasonal environments, yet little work has been done to understand what ultimate factors drive migration. Here I will present joint work with Iain Couzin, where we have developed a spatially explicit, individual-based model in which we can evolve behavior rules via simulations under a wide range of ecological conditions to answer two questions. First, under what types of ecological conditions can an individual maximize its fitness by migrating (versus being a resident)? Second, what types of information do individuals use to guide their movement? We find that different types of migration can evolve, depending on the ecological conditions and availability of information.
Postharvest diseases, especially those caused by fungi, can cause considerable damage to harvested apples in controlled atmosphere storage. Fungicides are used to control the disease, but resistance to fungicides is increasing and there is pressure by consumers and ecologists to reduce reliance on chemical controls. There is some evidence that physical conditions related to orchard management are predictive of postharvest disease incidence, and so the first line of defense against postharvest disease should involve best practices in orchards. In this work, we develop and analyse mathematical models to understand the dispersal of spores in the orchard, the initial infection level of fruit entering storage, and the epidemiology of the disease once the apples are in storage. We focus on conditions in the Okanagan Valley, where summers are dry and fungal spore presence is generally low. This leads to a mathematical problem where we are attempting to quantitatively and deterministically evaluate conditions surrounding rare events, that is, infection of fruit, and the fundamental stochasticity of the problem is crucial.
Work done in collaboration with L. Nelson, K.A. Williams, and M. Tutot.
We consider a population playing two-strategy symmetric games on a grid. A discrete Gaussian kernel rules out the interactions and information collection between individuals. We study how frequency and spatial structure of the population change over time for Prisoner's Dilemma game under four different update rules. Simulation results for these rules show frequencies of the games for kernels with different deviations. It is conjectured that, if the deviation of the kernel is large enough, bifurcation diagram of mean-field dynamics and spatial simulations agree.
Demographers recognize that a population's probability of extinction is affected by three main categories of stochasticity: environmental stochasticity, demographic stochasticity, and, more recently, demographic heterogeneity (i.e. among-individual variation in vital rates). Demographic heterogeneity increases the risk of extinction in density dependent populations more than environmental stochasticity does, but if it's left out of the statistical model, variation due to demographic heterogeneity is erroneously lumped in with environmental variation (Melbourne & Hastings 2008).
Here, we quantify environmental stochasticity and demographic heterogeneity to improve predictions of population viability in a long term observational study of marked individuals of Heliconia acuminata in the Amazon. The number of shoots and number of inflorescences per plant were monitored yearly for 12 years in 10 plots, 4 of which were forest fragments.
Fragmentation alters forest hydrology and wind patterns in ways that we hypothesize will create more spatial and temporal environmental heterogeneity. To test this hypothesis, we used mixed models to quantify variability in vital rates on multiple scales and habitats: yearly, individual, and landscape scales, in fragmented and continuous forest. We quantified uncertainty in the variability using MCMC sampling and likelihood profiling. All models were fit in AD Model Builder.
In growth models, years were the most variable unit, with more among year variability in continuous forest than fragments (92% MCMC support). This could be due to El Nino events. Plots were the second most variable unit, with more among plot variability in fragments than in continuous forest (93% MCMC support). Estimates of individual variability were near zero; this matched bias observed in our simulations. Variation on the individual scale may be due to processes such as canopy openings that are too temporary to detect in this data.
Survival varied equally among years and plots with more variability in continuous forest than in fragments.
Reproduction varied greatly among individuals, approximately equivalently in continuous and fragmented forest. Among year variation in reproduction was greater in continuous forest than in fragments. Random deviates in the reproduction model did not appear to be symmetric around zero. There were many small negative deviates and a long tail of large positive deviates representing individuals that reproduce more than average. This variability among individuals will cause the population level reproduction rate to be more variable than expected from the variance in our zero inflated Poisson distribution (Fox & Kendall 2003).
Work done in collaboration with Emilio Bruna and Benjamin Bolker.
Spatiotemporal stochastic simulations on the mesoscopic scale offer a great computational challenge as typical intra-event times in biochemical systems are on the order of nanoseconds. The requirements to handle realistic geometries, including pre- and post-processing in three space dimensions, make it challenging to develop new simulation methods targeting realistic applications. URDME simplifies this by implementing an interface to a mature, external geometry and mesh handling software that is used to define the model and analyze the results. The core simulation routines are logically separated from the interface, making URDME easy to extend such that more efficient solvers may be evaluated and validated.
Biological species (viruses, bacteria, parasites, insects, plants, or animals) replicate, mutate, compete, adapt, and evolve. In evolutionary game theory, such a process is modeled as a so-called evolutionary game. We describe the Nash equilibrium problem for an evolutionary game and discuss its computational complexity. We discuss the necessary and sufficient conditions for the equilibrium states, and derive the methods for the computation of the optimal strategies, including a specialized Snow-Shapley algorithm, a specialized Lemke-Howson algorithm, and an algorithm based on the solution of a complementarity problem on a simplex. Computational results are presented. Theoretical difficulties and computational challenges are highlighted.
This is joint work with Wen Zhou and Zhijun Wu.
Organisms routinely locate targets in complex environments. They can do this by following gradients in the strength of sensory signals, provided such gradients are available and reliably lead toward targets. But in many natural settings this is not the case. We propose a model of search-decision making when sensory signals are infrequent, exhibit large fluctuations, and contain little or no directional information. Our approach simultaneously models an organism's intrinsic movement behavior (e.g. L ´evy walk) while allowing this behavior to be adjusted based on sensory data. We find that including even a simple model for signal response can dominate other features of random search and greatly decrease the time needed to find targets. In particular, we show that a lack of signal is not a lack of information. Searchers that receive no signal can quickly abandon target-poor regions and concentrate search efforts near targets. These phenomena naturally give rise to the area-restricted search behavior exhibited by many searching organisms.
The compartmental, Susceptible-Infected-Recovered (SIR) model is an important paradigm for understanding and preventing the spread of infectious diseases through populations. We study the dynamics of stochastic SIR-type reactions on complex contact networks with adaptive topologies and interconnected, multi-network coupling, including meta-population models with multi-scale structure and contact networks coupled to information networks. We explore strategies for mitigating the spread of disease in disordered systems with the intent of developing robust techniques for precluding large epidemics while preserving network function. The models are explored with stochastic, Gillespie simulation and analytic techniques, including directed bond percolation and dynamic mean-field theory. Some preliminary findings are offered in simple models with focus on future work and current, unfinished investigations.
A discrete agent-based model on a periodic lattice of arbitrary dimension is considered. Agents move to nearest-neighbor sites by a motility mechanism accounting for general interactions, which may include volume exclusion. An approximate partial differential equation describing the average occupancy of the agent population is derived systematically. A diffusion equation arises for all types of interactions and is nonlinear except for the simplest interactions. In addition, multiple species of interacting subpopulations give rise to an advection-diffusion equation for each subpopulation. This work extends and generalizes previous specific results, providing a construction method for determining the transport coefficients in terms of a single conditional transition probability, which depends on the occupancy of sites in an influence region. These coefficients characterize the diffusion of agents in a crowded environment in biological and physical processes. (A detailed account of the work has appeared in Physical Review E 84 (2011) 041120.)
Work done in collaboration C.J. Penington and K.A. Landman.
Much of the earth's microbial biomass resides in sessile, spatially structured communities such as biofilms and microbial mats, systems consisting of large numbers of single celled organisms living within self-secreted matrices made of polymers and other molecules. As a result of their spatial structure, these communities differ in important ways from well-mixed (and well-studied) microbial systems such as those present in chemostats. Here we consider a widely used class of 1D biofilm models in the context of a description of their basic ecology.
It is argued via an exclusion principle resulting from competition for space that these models lead to restrictions on ecological structure. As a result of the exclusion principle, it is argued that some form of downward mobility, against the favorable substrate gradient direction, is needed at least in models and possibly in actuality.
Bacteria in colonies coordinate gene regulation through the exchange of diffusible signal molecules known as autoinducers. This "quorum signaling" often occurs in physically heterogeneous and spatially extended environments such as biofilms. Under these conditions the space and time scales for diffusion of the signal limit the range and timing of effective gene regulation. We expect that spatial and temporal patterns of gene expression will reflect physical environmental constraints, the solubility and mobility of the autoinducer signal, as well as nonlinear transcriptional activation and feedback within the gene regulatory system. We have combined experiments and modeling to investigate how these spatiotemporal patterns develop. We embed engineered plasmid/GFP quorum sensor strains or wild type strains in a long narrow agar lane, and then introduce autoinducer signal at one terminus of the lane. Diffusion of the autoinducer initiates reporter expression along the length of the lane, extending to macroscopic distances of mm-cm. Underlying nonlinearities in the quorum regulatory system leads to spatial and temporal patterns that are qualitatively different from the simpler spreading that can observed for a diffusing dye . These patterns are captured quantitatively by a mathematical model that incorporates logistic growth of the population, diffusion of autoinducer, and nonlinear transcriptional activation. Our results show that a diffusing quorum signal can coordinate gene expression over distances of order 1 cm on time scales of order 10 hours.
Formulating mathematical models for population dynamics and spread of invasive insects, in heterogeneous landscapes, has garnered much attention in recent years. In particular, these models have been developed on fragmented landscapes, since fragmentation may lead to extinction and poor conditions between habitats may slow the spread of a population. Inherent to this problem is the assumption of how a population disperses across an interface. It was widely accepted that the density and flux of a population density must be continuous across an interface. Ovaskinanen and Cornell recently showed that, under certain conditions, the density may not be continuous.
To study these phenomena, we formulate an integrodifference equation (IDE) on a periodic landscape, consisting of good and bad patches. To investigate the effect of different movement assumptions on persistence and spread, we first derive several dispersal kernels to implement into the IDE. We then derive explicit formulae for the critical size of a good patch necessary for persistence, and compare how the different assumptions about behaviour at an interface affect persistence. The effect of these assumptions on the spread of the population is briefly discussed.
Several systems in human physiology are classified as "excitable systems", including cardiac cells, within which specialized structures called "calcium release units" (CRUs) exhibit both threshold excitation and a refractory period. The CRUs are coupled both by a global periodic forcing signal and by spatial nearest-neighbor interactions. A significant amount of noise resulting from the stochastic nature of the subcellular elements involved serves to decouple the CRUs. We introduce a probabilistic cellular automata model that replicates these features of the cardiac cell. We first analyze the model using a mean-field approach and derive conditions under which the expected excitation rate can undergo a bifurcation to period-2 behavior (mimicking the pathophysiological condition known as "alternans"). We then apply a version of local structure theory to account for the spatial correlation that results from neighbor-to-neighbor coupling, and predict the emergence of synchronized clusters of excitation. The clusters represent portions of the cardiac cell which are "in phase", so that during period - 2 behavior they excite nearly simultaneously.
Cell division is a complex process requiring the cell to have many internal checks so that division may proceed and be completed correctly. Failure to divide correctly can have serious consequences, including progression to cancer. During mitosis, chromosomal segregation is one such process that is crucial for successful progression. Accurate segregation of chromosomes during mitosis requires regulation of the interactions between chromosomes and spindle microtubules. If left uncorrected, chromosome attachment errors can cause chromosome segregation defects which have serious e ffects on cell fates. In early prometaphase, where kinetochores are exposed to multiple microtubules originating from the two spindle poles, there are frequent errors in kinetochore-microtubule attachment. Erroneous attachments are classi ed into two categories, syntelic and merotelic. In this paper we consider a stochastic model for a possible function of syntelic and merotelic kinetochores, and we provide theoretical evidence that the erroneous kinetochore-microtubule attachments can contribute to lessening the stochastic noise in the time for completion of the mitotic process in eukaryotic cells.
Spatial epidemics described by the Quadratic Contact Process on a square, cubic, or hypercubic lattice (d=2, 3 or more dimensions) involves: (i) spontaneous recovery of sick individual at lattice sites with rate p; and (ii) infection of healthy individual at a rate proportional to the number of diagonal sick neighbor pairs.
This model displays a nonequilibrium discontinuous phase transition from an infected state to an all-healthy state as p increases above p_e(d). However, it also exhibits a non-trivial generic two-phase coexistence of these states for a finite range, p_f(d) < p < p_e(d), spanned by the orientation-dependent stationary points for planar interfaces. Our interface dynamic analysis from kinetic Monte Carlo simulation and from discrete reaction-diffusion equations (dRDEs) obtained from truncations of the exact master equation, reveals that p_e(f) ∼ 0.2113765 + c_e(f)/d as d →∞. The dRDEs display artificial propagation failure absent due to fluctuations in the stochastic model, and the propagation failure regimes are amplified for increasing d.
Non-mean-field behavior of catalytic conversion reactions in narrow pores is controlled by interplay between fluctuations in adsorption-desorption at pore openings, restricted diffusion, and reaction. Behavior is captured by generalized hydrodynamic formulation of the reaction-diffusion equations (RDE). These incorporate an appropriate description of chemical diffusion in mixed-component quasi-single-file systems, which is based on a refined picture of tracer diffusion. The RDE elucidate the non-exponential decay of the steady-state reactant concentration into the pore and anomalous scaling of the reactant penetration depth.