As another semester at the University of Wisconsin – La Crosse reaches its halfway point, it’s time to start preparing for my spring class – MTH 265: Mathematical Models in Biology. This is a course that only has a first-semester calculus prerequisite, meaning that it is unlike many of the mathematical biology courses around the world (which often require differential equations and/or linear algebra as a prerequisite).
My thought process when teaching this course is that the students likely do not have the mathematical background to fully appreciate the breadth and depth that mathematical biology has to offer. Whether it’s the global and/or asymptotic stability of equilibria to difference equations, principal component analysis applied to multivariate data, or Markov Processes applied to Allele frequencies, research-level mathematical biology requires mathematical flexibility and maturity. However, most of the students in my MTH 265 class are not mathematics majors. Many will be researchers or practitioners of the life sciences, though, meaning that they will have to interact in a meaningful way with mathematicians, statisticians and computer scientists at some point during their careers. Thus, my goal for the course eventually became to give a survey of many different topics pertaining to mathematical biology during the 15-week course. This way, they will know that a solution (possibly) exists to their quantitative problems (even if they may not be able to come up with it themselves). Simply knowing such a solution exists allows one to approach the right people for collaborations, and keeps the math-biology interface a fruitful one.
Survey courses are fairly common in graduate work, but students in their second semester of mathematics are pretty new to reading mathematics. Thus, to cover the material in 15 weeks, I created a collection of videos as a part of an inverted, or “flipped” classroom. Videos appear to be a medium that reaches current students better than (or in conjunction with) traditional textbooks. Students were asked to view these videos prior to class, while during class they were assigned groups in which they worked on “case studies” that took the duration of the hour. I provided assistance with the case studies, as well as any homework questions the students had.
The term “flipped” comes from the way the course is structured relative to a traditional course, where lectures occur during the regular class period (where the professor is present but the student engagement is low) and homework/case studies occur outside of the classroom space (where demands on the student are high, but direct help from the professor is not immediately available).
This course has been a great success. Some of the things we’ve learned from flipping the course can be found in this paper, and were used in a section of Grand Valley State professor Robert Talbert’s new book on flipped learning in the college classroom. I owe a great deal of my ideas to the Mathematical Association of America, especially their Project NExT program. The progress we’ve made as educators even in the short time (six years) I’ve been a mathematics professor has me excited for what is to come for the future.
Recently, I’ve been exposed to situations where I am trying to model discrete, binary events (i.e., 0 or 1 like heads-or-tails). My knee-jerk response has been: use a logistic regression or another model with a binomial outcome. The jack-of-all trades generalize linear model usually servers me well in these situations. However, my recent events have had continuous-time predictors. Although Cox proportional hazards model can be used if the event is something like survival, this did not seem appropriate for my situation because I had multiple events occurring per individual. Enter in continuous-time, discrete events.
A Poisson regression is similar to a binomial if the probability of an even occurring is small enough. Enter in a Poisson regression as a method for modeling animal behavior. I first saw this in a mathematical statistics paper describing models animal movements, but found another paper by some of the co-authors that was more accessible. From this, I learned I needed to use the following version of the Poisson regression:
y ~ Poisson(μ)
μ = τ exp( β x’).
I was able to program this in Stan, by adopting code I found online. This model can also be modified to treat individuals a random effect (and prevent pseudo-replication) if the data allows or requires it.
In my second post for Quantitative Dynamics I’m going to discuss a topic that I have studied since my graduate work at the University of Nebraska. In 2010 Brigitte Tenhumberg, Richard Rebarber, Diana Pilson and I embarked on a journey studying the long-term, stochastic dynamics of wild sunflower, a disturbance specialist plant population that uses a seed bank to buffer against the randomness of disturbances.
Because the seeds of disturbance specialist plants cannot germinate without a soil disturbance, there are many periods of time for which these populations will have zero or few above-ground plants, and hence no new members of the population from one season to the next. As such, much like a freelance worker with uncertain pay, a seed bank (account) is necessary for long-term viability.
In our work (which you can find here, here and here) we created an integral projection model with stochasticity modeling 1) the presence of a disturbance and 2) the depth of a disturbance. We found through mathematical analyses and simulations that the presence of disturbances increased population viability (as you would expect), but the intensity, depth and autocorrelation of disturbances had a different effect on populations depending on their viability. For populations that were viable, increasingly intense and positively-autocorrelated disturbances enhanced long-term population sizes, whereas when populations were near extinction levels both dynamics were actually harmful to population viability. These results were novel and surprising. You can find my blog post on the topic in The American Naturalist as well.
In subsequent work we would like to study transient dynamics of such systems. Transient dynamics, to this point, have not garnered the attention of long-term dynamics in stochastic systems. However, my friend Iain Stott and colleagues have gotten the ball rolling in that direction, and it’s only a matter of time.
Often, people collect data for time with replication. For example, the LTRM program collects fish, aquatic vegetation, and water quality data through time. Multiple samples are collected for each year. However, these observations are not independent and failure to consider this would be pseudoreplication. Aggregating (or taking the mean) of data within a year can be one method to prevent pseudoreplication. Aggregating comes with a trade-off of losing information about the raw data. State-space models may be a method to recover this information.
State-space models describe a true, but unknown and un-measurable “state” (e.g., the “true” population of catfish in the Upper Mississippi River) and the observation error associated with collecting the data. Kalman Fileters can be used to fit these model such as the MARSS package in R can be used to fit these models.
We were interesting in comparing state-space models from the MARSS package to other methods such as simple linear regression and auto-regressive models (publication here). Using simulated data and observed data from the LTRM, we found that the simpler models performed better than the state-space models likely because the LTRM data was not long enough for the state-space models.
Matrix population models describe populations as discrete life-, size-, or age-stages. Scientists apply these models to understand population ecology and guide conservation. However, some species have continuous life histories. For example, thistles grow continuously as presented within this paper.
Fish also grow continuously. We sought to understand how different management approaches could be used to control grass carp. This species impacts native ecosystems by out-competing native fish. Mangers were interesting in evaluating the use of YY-males to control populations. YY-males work because they spawn and only produce male offspring. Thus, it is possible in theory to cause a population to crash by biasing the sex-ratio.
We constructed an integral projection model for grass carp and compared different yy-male release methods. We found the life history of grass carp does not work well with the YY-male strategy because the species lives long and females produce many offspring.
Howdy! I’m Eric Eager, and I’m an associate professor of mathematical biology at the University of Wisconsin – La Crosse. I’m also a data scientist for Pro Football Focus and Orca Pacific. In my first post for Quantitative Dynamics, I’m going to discuss a topic near and dear to my heart: Branching processes (thanks Sebastian Schreiber for teaching me these five years ago).
Branching processes are a great bridge between the continuous-space population models that permeate the ecological literature (e.g. Caswell 2001, Ellner, Childs and Rees 2016) and the individual-based realities that drive ecological systems (Railsback and Grimm 2011). All branching process models specify an absorbing state (usually extinction in ecology) and model the probability of reaching the absorbing state by creating an iterative map from one generation to the next. This allows you to work with a model whose space is in a set of discrete values (individual-based), but with a resulting model that’s a difference equation (traditional ecological models).
The most famous example of a branching process is the Galton-Watson process. Francis Galton was concerned about the eventual fate of surnames (a quaint artifact of the past), especially among the aristocracy. Below are a couple of videos I made, one deriving the Galton-Watson process and one solving it. Enjoy!