Introduction to the course

Introductions

I can start this off. Then we can go around the room, thinking about

what you want to get out of the course
how theory fits into your current research (if you are currently doing research)
any worries/feelings/questions/concerns about the course

Syllabus

Work through syllabus, clarify structure, discuss final project/Case text/homework structure/paper summaries/etc.

syllabus

Course website

Go through the course website and make sure everyone has easy access and is clear on how to get the lecture notes, readings, and homeworks.

Notes

You may want to download R as we will be using it to simulate some of the models we go over. However, we will not be learning how to use R during the course (as this would be a bit much to learn at the same time as the theory), but I definitely encourage folks to check out the resources available for working in R, as scientific research tends to need a bit of programming.

What is theoretical ecology?

Ecology: study of species interactions

Theory: Body of evidence linking a general explanation to a commonly observed phenomenon

Theoretical ecology attempts to explain patterns observed in nature through generalized models

Models

Statistical model: A model exploring the statistical relationships between a response and some set of predictor variables.

can be completely independent of theory
can be focused on prediction or variable importance
requires data

e.g., linear regression

What is the relationship betwen sepal length and sepal length in irises?

data(iris)
mod <- glm(iris$Sepal.Length ~ iris$Sepal.Width)
plot(iris$Sepal.Length ~ iris$Sepal.Width, 
  col=1:3[as.numeric(iris$Species)])
abline(mod)

Phenomenological model: A model describing some phenomenon, often independent of any data.

an idea of how a system behaves given assumptions
incorporates theoretical expectations
incredibly useful when fit or compared to empirical data (where’s the disconnect?)

e.g., exponential growth

Population dynamics with exponential growth

\[ N_{t+1} = N_t \lambda \]

expoGrowth <- function(n, lambda=1.25, times=100){
  nt <- c(n)
  for(i in 1:times){
    nt[i+1] <- nt[i] * lambda
  }
  return(nt)
}

plot(expoGrowth(10), type='l', ylab='Population size', 
  xlab='Time', lwd=2)
lines(expoGrowth(10, lambda=2), col='dodgerblue', lwd=2)

This course will focus pretty much entirely on phenomenological models (as I have defined them)

Also, evolutionary models do not really get emphasized in this book, but this could be good fodder for your paper summaries or final projects if you are interested

Why theoretical ecology?

Theory allows us a lens with which to explain natural systems.

Observation: some set of ecological processes is causing this population to cycle

Conclusion: this population is cycling

Theory: which of the cycle-generating processes could cause cycles?

Generation of novel question: can models create cycles under processes with different ‘strengths’?

Why theoretical ecology?

It provides a useful interface for experimental and observational studies.

Population model: Ricker model to explore population dynamics and the role of demographic/environmental stochasticity

Experiment: fit models to experimental data and find demographic stochasticity important in controlled laboratory experiments

Observation: fit models to empirical time series to explore the relative importance of demographic and environmental stochasticity in natural systems

Why theoretical ecology?

It shows us what ecological interactions could look like in different conditions

Why theoretical ecology?

It provides us a concrete way to explore and explain natural processes

Barriers to a more theory-rich discipline

some ecologists are ‘bad’ at theory. Why? (math/programming/courses)
Is the world too complex to model? No (most of the time)
- Is a complex model useful? No (most of the time)

https://www.cell.com/trends/ecology-evolution/fulltext/S0169-5347(19)30171-5

The promise of theory in ecology

we have lots of patterns, perhaps more patterns than theory?

‘To do science is to search for repeated patterns… The best person to do this [in ecology] is the naturalist who loves to note changes in bird life up a mountainside, or changes in plant life from mainland to island, or changes in butterflies from temperate to tropics’ - Robert MacArthur

note: I 100% disagree with the assertion that there is a ‘best’ type of ecologist, and that the ‘best’ type would be a naturalist

Patterns without theory can be deceiving

distribution of birds in Bismarck Archipelago

https://esajournals.onlinelibrary.wiley.com/doi/10.1890/14-1848.1

Where did Diamond go wrong?

Theory requires tests

There is an inherent feedback in developing some conceptual theory based on observations, and then testing this theory in different locations or in experimental trials.

Breaking the training divide

https://www.cell.com/trends/ecology-evolution/fulltext/S0169-5347(19)30171-5

So, you’re like a ‘modeler’?

The ‘modeler’/‘empiricist’ divide is a misnomer and is silly
The goal of this course is not to pull you away from natural systems, but to give you tools to generalize and explain the dynamics of natural systems

https://royalsocietypublishing.org/doi/10.1098/rsfs.2012.0008

https://www.journals.uchicago.edu/doi/pdfplus/10.1086/717206

Discussion

Codling and Dumbrell. 2012. Mathematical and theoretical ecology:linking models with ecological processes. Interface doi:10.1098/rsfs.2012.0008

Note:

This is an introduction to a special issue organized around ‘Mathematical and theoretical ecology’, so it spends some time trying to tie the papers into a cohesive picture. This is useful for this course, as it gives you an idea of types of theoretical articles.
Additional article some folks might find interesting: Lawton discussing ‘patterns’ back in 1996, muddying some waters while also having some pretty fair points in other places.

The paper was written over a decade ago. In what ways do you think things have changed in the field?

What were your reactions when you read this?

“Without ecological theory, collecting data is a futile and meaningless endeavour. Likewise, producing elegantand beautiful mathematical models of ecological systems without validation against real data is an empty achievement”.

Simplifying assumptions and the mean-field approximation

The authors discuss (bottom left of page 145) the use of mean-field techniques and sometimes when assumptions used in theoretical models may be alright. This is an important point, because we will make oodles of simplifying assumptions. What are some of the tradeoffs in using simplifying assumptions (both from the article and in your own research)?

Generating testable predictions

One role of theory is to generate testable hypotheses. The Rands paper from this issue (discussed on page 147) is a great example. Can we collectively think of other examples of how theoretical models, in the absence of data, have generated testable predictions?

Lines in the sand

Broadly speaking, ecological models can be split into two separate categories; simplistic mathematical models, which offer analytically tractable solutions and the examination of the underlying model properties, and more complex simulation-based models (cannot be solved analytically)

What does it mean to ‘analytically solve’ a model, and how is this useful?
To what extent has this distinction been eroded in more recent years and why/how?

Role of stochastic processes

Ecological systems are very different from physical and chemical systems, notably containing more uncertainty and chaotic dynamics, and are often influenced by stochastic processes.

This gets at two things
1. ecological data are messy (imperfect detection etc.)
2. ecological systems are messy (stochastic birth/death)

To what extent can models incorporate either/or of these processes, allowing theory to easily bridge this data-model divide the authors try to emphasize?

Ecology in the absence of theory

After all, without theory providing testable hypotheses, ecology could become nothing more than data collection for its own sake

The authors are taking a bit of a biased (if not obtuse) view of the field of ecology here (in my opinion). I was curious if this line (towards of the end of article) generated any thoughts from you all.