Stochastic differential equations are introduced and some of their properties are described. Equivalence of SDE systems is explained. Commonly used numerical proce- dures are discussed for computationally solving systems of stochastic differential equations. A procedure is described for deriving stochastic differential equation (SDEs) from associated discrete stochastic models for randomly-varying problems in biology. The SDEs are derived from basic principles, i.e., from the changes in the system which occur in a small time interval. In the derivation procedure, a discrete stochastic model is rst constructed. As the time interval decreases, the discrete stochastic model leads to a system of It^o stochastic differential equations. Several examples illustrate the procedure. In particular, stochastic differential equations are derived for predator- prey, competition, and epidemic problems. In addition, for certain problems such as a size-structured population, it is shown how stochastic partial differential equations can be derived through replacing Wiener processes in the SDE system with appropriate Brownian sheets.
A brief introduction is presented to basic stochastic epidemic models. Several useful epidemiological concepts such as the basic reproduction number, herd immunity, and the final size of an epidemic are dened in term of deterministic epidemic models. Then three well-known stochastic modeling formulations are introduced, discrete-time Markov chains (DTMC), continuous-time Markov chains (CTMC), and stochastic differential equations (SDE). Some of the important differences between deterministic and stochastic models and between the stochastic modeling formulations are discussed in relation to SIS and SIR epidemic models. Computation of the quasistationary distribution, probability of an outbreak, and the final size distribution are illustrated in stochastic epidemic models. In addition, methods for derivation, analysis, and numerical simulation are discussed for stochastic models. As a final example, an SDE epidemic model with vaccination is formulated, which has applications to pertussis (whooping cough).
Mathematical models predict that in environments that are heterogeneous in space but constant in time, there will be selection for slower rates of unconditional dispersal, including specifically random dispersal by diffusion. However, some types of unconditional dispersal may be unavoidable for some organisms, and some organisms may disperse in ways that depend on environmental conditions. In some cases, models predict that certain types of conditional dispersal strategies may be evolutionarily stable within a given class of strategies. For environments that vary in space but not in time those strategies are often the ones that lead to an ideal free distribution of the population using them, provided that such strategies are available within the class of feasible strategies.
Problems in the evolution of dispersal have been addressed from two complementary mathematical viewpoints, namely game theory and mathematical population dynamics. This talk will describe some results and open problems from the viewpoint of spatially explicit models in population dynamics, specifically reaction-diffusion-advection models. Some of the results and problems are related to the evolutionary stability of dispersal strategies leading to an ideal free distribution and the mechanisms that might allow organisms to realize such strategies.
Ecological systems can be studied as ensembles of locally interacting individuals within heterogeneous environments. Because many ecological interactions are nonlinear, they can promote spatial and temporal heterogeneity in the distribution of abundance over scales ranging from individuals to whole continents. Understanding patterns of heterogeneity is key to understanding the response of key ecosystem services (e.g. productivity of commercial species) to stressors (e.g. climate change, harvesting). The lecture will provide a survey of the main quantitative formalisms used in ecology to predict the onset of spatiotemporal patterns of abundance, with an emphasis on their ecological relevance. I will cover concepts of self-organization, criticality and synchrony, and link individual characteristics to emergent spatial structures over large spatial scales. I will then apply these concepts to marine ecosystems where spatiotemporal dynamics can be captured through the analysis of spatial synchrony. I will detail some recent work showing how asynchrony can be maintained over continental scales in an ensemble of locally coupled ecological oscillators. Such regional asynchrony can be tested in natural systems and provide important insights for the design of marine reserve networks.
Project: Functional responses and phase dynamics of predator-prey systems
Some ecological interactions such as predator-prey relationship and disturbance dynamics can lead to spontaneous fluctuations of abundance. As we scale-up these local fluctuations, we need to understand conditions that can lead to heterogeneous distribution of abundance among local communities. This is a complex problem that has been addressed using coupled-oscillator dynamics to predict conditions leading to synchrony among fluctuating populations and communities. There are few studies in ecology that have analysed coupled ecological communities using the formalism of coupled oscillators and of the resulting phase dynamics. Goldwyn and Hastings (2008) have provided an analysis of a specific predator-prey model showing how separation of temporal scales between the predator and the prey can lock the phases of local predator-prey oscillators and drive spatiotemporal heterogeneity. One remaining question is the generality of these results in relation to alternate functional responses characterizing the relationship between prey abundance and per capita predation rate. Explore the existence of phase-locked solution to a 2 patch Rosenzweig-MacArthur predator prey model when the predator has a type III (prey refuge) functional response.
Natural selection favors traits that maximize fitness. However, the fitness effects of a particular behavioral strategy often depend on the behavior of other individuals. Evolutionary game theoretical models are used to model the evolution of strategic social behavior. A strategy is said to be a Nash equilibrium if it is a best response to itself, and an evolutionarily stable strategy (ESS) if it cannot be invaded by a rare, alternative tactic. I will present several game theoretical models used in evolutionary biology, including the Prisoner's Dilemma and Hawk-Dove game and games against the field. I will discuss a variety of methods to find Nash equilibriums and ESS and discuss the application of game theoretical models to biological problems of cooperation and conflict and the spatial distribution of organisms.
Proposed Project –Evolutionary Dynamics of Games With Positive Frequency and Density Dependence: Evolutionary game theory has been used to understand strategic foraging behavior, such as the decision to forage in a particular patch (the ideal free distribution and related models) or to steal food from others (the producer-scrounger model and related models). This often involves 'games against the field', in which there is no specific opponent or partner, but in which the payoff to a particular strategy depends on what everyone else is doing. Usually, payoffs are assumed to be negatively frequency-dependent; as the frequency of a particular strategy increases, the payoff to individuals using that strategy decreases. However, biologists have observed that many strategies have positively frequency dependent payoffs, with individual payoff increasing with the frequency of the strategy, at least over some range of frequencies. Positive density-dependence is also observed, a phenomenon known as the 'Allee effect'. However, few evolutionary game theory models have investigated the influence of positive frequency dependence and Allee effects on the dynamics of games against the field. The goal of this project is to model the evolutionary dynamics of games such as the ideal free distribution or producer-scrounger model when payoffs are positively frequency dependent over some range of frequencies. This may include use of replicator dynamic models, population ecological models and adaptive dynamics.
I will start with a discussion of some real world examples, and then return to thinking about (relatively) simple population models that can exhibit dramatic shifts in dynamics either without external changes, or with small external changes. The mathematical tools will essentially be drawn from ordinary differential and difference equations (dynamical systems).
I will discuss the extensions of these ideas to more complex systems.
Afternoon activity – I assume students will be able to solve the model equations I discuss in matlab (or some equivalent) and they will be asked to do so – the matcont package provides a nice front end. I will get details.
Small Group Project:
(based on Hastings, A., and Wysham, D.B. (2010) above)
In that paper we present solutions of a variety of ecological models relative to the detection of regime shifts (and rapid changes in dynamics). There is much to explore further in this topic. In particular, there are a variety of other models that should (and could) be analyzed to see if they exhibit similar kinds of dynamics. I would suggest starting with, but not limiting the analysis to some of the following:
Optimal control of ordinary differential equations will be introduced. Formulation of a system with control, choosing an objective functional, control analysis and numerical solutions will be included. Some epidemic, immunology and other population models will be illustrated. Practice with a simple control problem to be calculated by hand will given as a class exercise. Numerical solutions and solution dependence on parameters will be illustrated in the afternoon lab work.
Project: Consider the paper by Choisy and Rohani on Harvesting can increase severity of wildlife disease epidemics.
Formulate an optimal control problem with harvesting and corresponding costs and investigate the effects of harvesting. Compare your results with the results in Choisy and Rohani. One can use modifications of the model in their paper or choose another model. For your model, complete necessary conditions for an optimal control and run some numerical simulations for illustrations.
Macroevolution is the study of evolution or biodiversity on a long time scale, in other words, the evolutionary history of speciations and extinctions. In order to incorporate the stochastic effects of evolutionary events, one often uses branching models that generate the new speciation events and the extinction of species. Mathematical models used to describe these events are branching processes, in particular birth and death processes. The ultimate goal of using mathematical models for macroevolution is to identify which features of the observed data may have arisen by chance and which require a biological explanation.
In this lecture we will cover the basics of stochastic processes needed for such models, and describe the distribution of various quantities that will help us make a correspondence between these models and data. In addition to the one dimensional population size process, we will also discuss the more complicated aspects of phylogenetic tree structures, such as tree shape, generated by such models. We will begin with a review on birth and death processes and branching processes in general. We will discuss the random trees generated by these models and discuss various relevant statistics. Finally, if time permits, we will conclude with a discussion of hierarchical models on random trees that describe the grouping of species trees into trees of genera, families, etc.
Transfer functions are a useful tool for analyzing the effects of moderate spatial and/or temporal variation in model parameters. By the end of the day, you should have a grasp of basic Fourier analysis and be able to "read" transfer functions to tell how changes in parameters amplify or damp population response to variation and increase or decrease any response lags or spatial shifts. If there is time, we will consider stochastic environments, defined only by their spatial and/or temporal autocorrelation functions.
Project: Choose a simple analytical model that you are working on or are interested in and add spatial or temporal variation.
Afternoon Lab Session: PDF
A severe cholera outbreak began in Haiti in late October 2010 and continues to this day. We will use this situation to motivate and discuss some basic aspects of mathematical epidemiology. In particular, we will discuss the basic reproduction number, spatial "patch" models, and the effects of different patch properties and connectivity patterns on disease dynamics.
Mini-project Description: This project examines cholera dynamics in Haiti on the Department (Province) level. Each Department is a patch, modeled in isolation as a simple extension of an SIR model. One way to connect the patches is to use "gravity" models, where patches influence one another phenomenologically through a force of infection which depends upon the population sizes and inter-patch distance. An alternative approach is to link patches mechanistically, for example through the movement of people. This project will compare the disease dynamics resulting from these two types of models. Specifically, we will focus on how the basic reproduction number depends upon connectivity patterns which are reasonable for Haiti.