Ectotherms in Changing Environments. A State Variable Approach

In August 2015, I started working on this book, supported by an OPUS (Opportunities for Understanding through Synthesis) grant from the Division of Environmental Biology of the National Science Foundation. The purpose of such grants is to allow time and resources for senior people to synthesize a body of research done over many years. My objective is to develop a kind of "lessons learned" from my work on salmonids, southern ocean krill, and insect parasitoids and tephritid flies for understanding how we can predict the responses of ectotherms to changing environments (temperature, precipitation, competitors, predators, etc). My goal is to show how State Dependent Behavioral and Life History Theory, as implemented by Stochastic Dynamic Programming, can be used with other tools used to study the response of organisms to changing environments. The book will ultimately be published by the University of Chicago Press. On this page, I will provide updates on progress. (Most recent update: June 2016.)

Completed Chapters

Chapter 1 (First Draft): Evolutionary Ecology and Environmental Change. I begin with a discussion of what is means to do science (motivated in part by my experiences in the Whaling Case in the International Court of Justice) and my understanding of synthesis, emphaszing the importance of a plurality of approaches. I then turn to the role of State Dependent Life History Theory (SDLHT), as implemented by Stochastic Dynamic Programming (SDP), where I emphasize the importance of the intercollation of models and empirical studies, evolutionary endpoints (which is what SDLHT provides), and multiple time scales. I discuss the roles of theory and basic, derived, and applied science and explain what it means to work in Pasteur's Quadrant (see D. Stokes Pasteur's Quadrant. Basic Science and Technological Innovation , Brookings Institution Press 1997). Since I believe that we make the most progress in mathematical biology by considering case studies -- actual biological systems -- from which general principles and techniques emerge, I discuss the importance of case studies, using salmonids, krill, and parasitoids as examples (along with a variety of other species, which appear in this chapter and throughout the rest of the book). Since all ectotherms are linked by the Thermal Performance Curve (TPC), I spend a bit of time looking at such curves for a wide variety of species (a figure with 26 panels).

I explain why I use the simplest models possible in this book. First, when connecting SDP models to other modeling tools, the situation becomes complicated quite quickly, so a simple SDP model has advantages. Second, there are now (as opposed to 1988, when Mangel and Clark [ Dynamic Modeling in Behavioral Ecology , Princeton University Press] was published, or even 2000, when Clark and Mangel [ Dynamic State Variable Models in Ecology. Methods and Applications , Cambridge University Press] was published) many papers with detailed and specific SDP mo dels are available. I explain why I do not spend much time on connecting SDP models and data, since once again there is a now a rich ecological literature of that subject (Hilborn and Mangel. 1997. The Ecological Detective , Princeton University Press, is still a nice introduction to such ideas). Finally, I explain why I do not consider dynamic games in this book (once again because of the complexity of dynamic games, but point readers to the literature.)

Chapter 2 (First Draft): The Canonical Equations of State Dependent Life History Theory. In a 2015 paper in the Bulletin of Mathematical Biology, on the general subject of 'what has mathematics done for biology?', I introduced two canonical equations for SDLHT. This chapter begins with a motivation for the canonical equations via the Euler-Lotka equation (perhaps the canonical equation of population biollogy). I derive equations for expected individual lifetime reproduction and the growth rate of a population following a given schedule of survival and fecundity. This raises the question of how we unpack those schedules by linking envrionment, physiology, and natural selection. I derive two canonical equations of activity choice (one of which Colin Clark and I called the patch selection model in our 1988 book), the canonical equation of resource allocation (which basically corresponds to oviposition by a solitary parasitoid), and the canonical equation for complex life histories (in which organisms substantially change when moving from one life stage to another, as in animals that metamorphose).

I then explain how we use forward simulation of optimally derived behaviors of life history decisions to compare model predictions with empirical observations, and this requires numerical solution (which has changed greatly in the 30 years since Colin and I published, nearly simultaneous with the paper of Alasdair Houston and John McNamara, our first paper on this topic). Although I offer actual Rscript in the book, I also offer pseudocode (e.g. for readers who do not work in R) and discuss the promise and perils of pseudocode.

There will be three kinds of models in this book. First, I will develop SDP models completely, showing pseudocode and numerical results. Second, I will describe the models and results of other colleagues who have worked on relevant problems. Third, I will describe potential case studies -- leading the reader to the edge of modeling, where the reader than then take off.

I then describe the von Bertalanffy (vB) growth model for a semelparous organism and Ray Beverton's theory of Growth, Maturity, and Longevity (GML), from which the notion of life history invariants arose. I show how the canonical equations are modified when the vB growth model is used for the dynamics of state and then turn to the Kleiber-Brody model of growth for an iteroparous organism and show how the canonical equations are modified. I discuss mechanism in more detail and the the role of sensitivity analysis.

I close this chapter with a little bit of the philosophy of science (i.e. what exactly is this thing called science, and why and how are we doing it?).

Chapter 3 (First Draft): Incorporating Basic Thermal Physiology into the Canonical Equations . I begin with the simple metabolic theory of ecology, in which temperature dependent rates have the Arrhenius form k(T)=k_0 exp(-E_a/k_B T) where k(T) is the rate at temperature T, k_0 is a pre-exponential factor, E_a activation energy, and k_B is Boltzmann's constant. This function clearly cannot account for peaked TPCs, but is a good starting point. I show how to incorporate it, as temperature dependent metabolic cost, into the canonical equation for activity choice and thus allow us to predict changes in activity in response to temperature.

I then use the canonical equation for allocation for model a parasitoid in a doubling changing world -- one in which temperature is rising and natural host densities are replaced by agricultural hosts densities. This work is motivated by models of Damien Denis et al for changing pupal allocations with temperature and empirical studies of Michal Segoli and Jay Rosenheimn on leaf hopper parasitoids in vineyards or natural settings. Finally, we return to vB growth, and see that if the vB rate parameter has the MTE form of temperature dependence then the widely observed phenomenon of organisms being larger at colder temperatures immediately emerges.

This chapter also contains bits of history, such as the crisis of the common currency (which readers with long knowledge of behavioral ecology may recall, and which SDLHT resolved)and the Lack clutch size and its extensions. An appendix includes a description of linear interpolation for the dynamic programming equation and forward iteration.

Chapter 4 (First Draft): Within and Between Season Time Scales: Linking SDP to Population Models . Because they focus on the vital rates that are directly relevant to population dynamics, SDP models can naturally be synthesized with those of population dynamics. One reason for doing this is that environmentally induced changes in survival and/or reproduction do not necessarily lead to a straightforward change in population size.

To begin, I review both stage structured and age structured models. For example, a simple stage structured model has three stages (birth, juvenile, reproductive/mature) and a simple age-structured model has constant mortality across all ages except the birth age and density dependence in the birth age. After providing an introduction to these models suitable for somebody who has never worked with them, I explore some of the classic properties such as i) the eigenvalues and stable state distribution of the stage-structured model and ii) the steady states of the age-structured model, the form of density dependence in reproduction, age-dependent specific fecundity, and the contributions of large females to the population.

I then show how the canonical equations for activity choice and allocation can be used to compute the entries (survival within a stage, survival and transition between stages, and fecundity) for a stage-structured matrix model, so that we are able to predict how those entries will change with temperature (assuming that the MTE applies), and thus predict how the dominant eigenvalue of the matrix will change with temperature. I show that these predictions differ than what we would obtain by simply measuring the entries at one temperature and then applying the MTE to scale them; this is due to the facultative nature of SDLHT.

I then show how the canonical equation for allocation was used by my student EJ Dick in his PhD thesis [Modeling The Reproductive Potential of Rockfishes (Sebastes spp.) ] to show how age or size dependent mass specific fecundity emerges from SDLHT.

Chapter 5 (First Draft): Connecting to Quantitative Genetics -- The Norm of Reaction and the Breeder's Equation The methods of Quantitative Genetics (QG) also provide a framework for studying the response of ecotherms to changing environments. Two key tools of QG are the norm of reaction, which shows the phenotypic expression of genotypes in differing environments, and the breeder's equation, which shows how the mean value of a phenotypic trait changes in response to selection.

In this chapter, I first review these concepts (including showing a wide variety of norms of reactions) and then show how SDLHT can be used to predict the norm of reaction for a TPC with respect to temperature, the norm of reaction for von Bertalanffy growth, and the norm of reaction with respect to food and temperature using for the canonical equation of activity choice. I then show how it is possible to use SDLHT to compute fitness for use in the breeder's equation, using data on metabolic rate for crucian carp as a motivation.

I conclude with a discussion of the evolution of the norm of reaction, which will become important in the chapter on transgenerational effects.

Chapter 6 (First Draft): Cues, Conflicting Cues, and Information as a State Variable

Since one cannot think about information and cues without using probability models, I give a brief reminder of the rule about probability, and develop Bayes theorem. We then flex Bayesian muscles by considering a nesting male wrasse who needs to determine if a potential satellite male will be a good helper or not, based on the cues that the potential helper gives (this question is motivated by the work of Suzanne Alonzo).

I then describe the various ways of thinking about information, emphasizing that the simpler the tool the more likely it is to deliver the goods. So, I begin with a sliding memory model for information, motivated by work that Bernie Roitberg and I did on rose hip flies Rhagoletis basiola . I then discuss classical Bayesian approaches (the conjugate prior) fully developing the Poisson-Gamma and Beta-Binomial models; after that I briefly discuss the normal-normal model and how Tachiki and Koizumi (Am Nat 2016) used it to model the norm of reaction in threshold traits in Masu salmon. I conclude by showing how we can account for changing worlds by weighting posterior parameters by a fogetting parameter.

Since another way to think about information -- that is relevant for population processes particular -- is entropy, I discuss the ideas of entropy and classical information theory, illustrating the ideas with the entropy of an age structured population as discussed in Chapter 4 and show that entropy allows us to characterize with a single number the age structured population. This is particularly useful when there is anthropogenic mortality, since entropy allows us to characterize an effective rate of mortality, which is what the natural mortality would have to be for an undisturbed population to have the same entropy as the one experiencing anthropogenic mortality. This actually gives us ways of designing the anthropogenic intervention. I close this section with a discussion of John Thompson's wonderful work on a greya moth that lays its eggs in seeds and the egg parasitoid of the moth. John showed that the distribution of eggs maximizes the entropy -- the uncertainty -- that the parasitoid faces.

I then show how to put information into state dependent life history theory. First, I revisit the canonical equation for allocation with an uncertain end time, updating the end time as the season progresses. Second, I show how Bernie Roitberg and I used the sliding memory window to study the acceptance of previously parasitized hosts by the rose hips fly. Third, I discuss information, development, and the long0-term effects of early life experiences, show-casing the excellent work of Frankenhuis and Panchanathan (2011 onwards) and English et al (Am Nat 2016) respectively. Fourth, I return to the canonical equation for allocation and show how we can compute the value of information for a parasitoid searching for hosts and learning about the world.

I conclude the chapter with a brief discussion of what is coming up in Chapter 8: using modern Bayesian methods to learn about thermal performance curves

Chapter 7 (First Draft): Plasticity and Trans-Generational Plasticity (TGP)

In this chapter, I address questions such as when will plasticity or TGP not be seen even it if there there, if the role ofTGP is to give a head-start on the development of traits, what are the best ones, what is the attenuation of TGP across generations, and what are the consequences of plasticity and TGP for persistence in a changing environment? I also assess the tension between general conclusions from specific models and specific conclusions from general models.

I begin with a study of plasticity in offspring size and number, exploring extensions of the canonical equation for allocation when individuals have the ability to adjust both offspring number and size, only one of them, neither of them. I use the offspring size model of Smith and Fretwell for fixed offspring size and compare first period fitness of the different strategies, with a goal of determining when we would see within generation plasticity.

I then turn to within generation thermal plasticity, motivated by experiments on Atlantic salmon Salmo salar , using a Ratkowsky Thermal Performance Curve (TPC), which allows us to consider a tradeoff between performance at high temperature and at low temperature.

This naturally leads into thermal TGP and I describe work done by my collaborators and me on sheepshead minnows Cyprinodon variegatus , which leads to a discussion of thermal norms of reaction generated by the Ratkowsky TPC -- and how easily one can be mislead when determining the norm of reaction.

Developing a state dependent life history model for TGP using the Ratkowsky TPC requires both a growth model and an informational model for updating information about temperature. The latter leads to the normal-normal Bayesian model (which I delayed from the previous chapter), which I develop in careful detail since there are many small tricks to the analysis. I develop state dependent life history models in which temperature is known, and in which it is unknown but estimated by the organism and in which the organism allocates between growth and moving the TPC to match the temperature. Forwrd iteration allows us to get to the heart of the TGP questions.

I then describe models for constitutive and morphological defenses developed by Colin Clark and Drew Harvell. In these models, organisms allocate resources to growth, reproduction, and defense. The output of the model is the optimal allocation to these as a function of the environment --including the rate at which predators arrive -- and physiological parameters.

I close with a potential case study of mortality drive plasticity in the smolt size in salmonids. In general, the probability that a smolt survives downstream migration, ocean entry, and sbusequent time in the ocean to be able to return to reproduce is a sigmoidal function of smolt size. I give examples of some of these functions, both theoretical and empirical, and then argue that the community of predators determines the location parameter of the sigmoid (the value of smolt size at which the probability of survival to return is half of its maximum value); this allows one to set up a model in which females returning to spawn may provide information to their offspring about predator risks that they latter will experience.

Likely Chapters

My plan is to have the following additional chapters (of course, plans change):

Chapter 8: The Origin of Peaked Thermal Performance Curves

Chapter 9: Communicating to the Non-Expert

Chapter 10: Having a Career in Pasteur's Quadrant

As I complete these, I will add summaries to this page.