Ecological Dynamics (I?)

“Mathematical models underpin much ecological theory, […] [y]et most students of ecology and environmental science receive much less formal training in mathematics than their counterparts in other scientific disciplines. Motivating both graduate and undergraduate students to study ecological dynamics thus requires an introduction which is initially accessible with limited mathematical and computational skill, and yet offers glimpses of the state of the art in at least some areas. This volume represents our attempt to reconcile these conflicting demands […] Ecology is the branch of biology that deals with the interaction of living organisms with their environment. […] The primary aim of this book is to develop general theory for describing ecological dynamics. Given this aspiration, it is useful to identify questions that will be relevant to a wide range of organisms and/or habitats. We shall distinguish questions relating to individuals, populations, communities, and ecosystems. A population is all the organisms of a particular species in a given region. A community is all the populations in a given region. An ecosystem is a community related to its physical and chemical environment. […] Just as the physical and chemical properties of materials are the result of interactions involving individual atoms and molecules, so the dynamics of populations and communities can be interpreted as the combined effects of properties of many individuals […] All models are (at best) approximations to the truth so, given data of sufficient quality and diversity, all models will turn out to be false. The key to understanding the role of models in most ecological applications is to recognise that models exist to answer questions. A model may provide a good description of nature in one context but be woefully inadequate in another. […] Ecology is no different from other disciplines in its reliance on simple models to underpin understanding of complex phenomena. […] the real world, with all its complexity, is initially interpreted through comparison with the simplistic situations described by the models. The inevitable deviations from the model predictions [then] become the starting point for the development of more specific theory.”

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I haven’t blogged this book yet even if it’s been a while since I finished it, and I figured I ought to talk a little bit about it now. As pointed out on goodreads, I really liked the book. It’s basically a math textbook for biologists which deals with how to set up models in a specific context, that dealing with questions pertaining to ecological dynamics; having read the above quote you should at this point at least have some idea which kind of stuff this field deals with. Here are a few links to examples of applications mentioned/covered in the book which may give you a better idea of the kinds of things covered.

There are 9 chapters in the book, and only the introductory chapter has fewer than 50 ‘named’ equations – most have around 70-80 equations, and 3 of them have more than 100. I have tried to avoid equations in this post in part because it’s hell to deal with them in wordpress, so I’ll be leaving out a lot of stuff in my coverage. Large chunks of the coverage was to some extent review but there was also some new stuff in there. The book covers material both intended for undergraduates and graduates, and even if the book is presumably intended for biology majors many of the ideas also can be ‘transferred’ to other contexts where the same types of specific modelling frameworks might be applied; for example there are some differences between discrete-time models and continuous-time models, and those differences apply regardless of whether you’re modelling animal behaviour or, say, human behaviour. A local stability analysis looks quite similar in the contexts of an economic model and an ecological model. Etc. I’ve tried to mostly talk about rather ‘general stuff’ in this coverage, i.e. model concepts and key ideas covered in the book which might also be applicable in other fields of research as well. I’ve tried to keep things reasonably simple in this post, and I’ve only talked about stuff from the first three chapters.

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“The simplest ecological models, called deterministic models, make the assumption that if we know the present condition of a system, we can predict its future. Before we can begin to formulate such a model, we must decide what quantities, known as state variables, we shall use to describe the current condition of the system. This choice always involves a subtle balance of biological realism (or at least plausibility) against mathematical complexity. […] The first requirement  in formulating a usable model is […] to decide which characteristics are dynamically important in the context of the questions the model seeks to answer. […] The diversity of individual characteristics and behaviours implies that without considerable effort at simplification, a change of focus towards communities will be accompanied by an explosive increase in model complexity. […] A dynamical model is a mathematical statement of the rules governing change. The majority of models express these rules either as an update rule, specifying the relationship between the current and future state of the system, or as a differential equation, specifying the rate of change of the state variables. […] A system with [the] property [that the update rule does not depend on time] is said to be autonomous. […] [If the update rule depends on time, the models are called non-autonomous].”

“Formulation of a dynamic model always starts by identifying the fundamental processes in the system under investigation and then setting out, in mathematical language, the statement that changes in system state can only result from the operation of these processes. The “bookkeeping” framework which expresses this insight is often called a conservation equation or a balance equation. […] Writing down balance equations is just the first step in formulating an ecological model, since only in the most restrictive circumstances do balance equations on their own contain enough information to allow prediction of future values of state variables. In general, [deterministic] model formulation involves three distinct steps: *choose state variables, *derive balance equations, *make model-specific assumptions.
Selection of state variables involves biological or ecological judgment […] Deriving balance equations involves both ecological choices (what processes to include) and mathematical reasoning. The final step, the selection of assumptions particular to any one model, is left to last in order to facilitate model refinement. For example, if a model makes predictions that are at variance with observation, we may wish to change one of the model assumptions, while still retaining the same state variables and processes in the balance equations. […] a remarkably good approximation to […] stochastic dynamics is often obtained by regarding the dynamics as ‘perturbations’ of a non-autonomous, deterministic system. […] although randomness is ubiquitous, deterministic models are an appropriate starting point for much ecological modelling. […] even where deterministic models are inadequate, an essential prerequisite to the formulation and analysis of many complex, stochastic models is a good understanding of a deterministic representation of the system under investigation.”

“Faced with an update rule or a balance equation describing an ecological system, what do we do? The most obvious line of attack is to attempt to find an analytical solution […] However, except for the simplest models, analytical solutions tend to be impossible to derive or to involve formulae so complex as to be completely unhelpful. In other situations, an explicit solution can be calculated numerically. A numerical solution of a difference equation is a table of values of the state variable (or variables) at successive time steps, obtained by repeated application of the update rule […] Numerical solutions of differential equations are more tricky [but sophisticated methods for finding them do exist] […] for simple systems it is possible to obtain considerable insight by ‘numerical experiments’ involving solutions for a number of parameter values and/or initial conditions. For more complex models, numerical analysis is typically the only approach available. But the unpleasant reality is that in the vast majority of investigations it proves impossible to obtain complete or near-complete information about a dynamical system, either by deriving analytical solutions or by numerical experimentation. It is therefore reassuring that over the past century or so, mathematicians have developed methods of determining the qualitative properties of the solutions of dynamic equations, and thus answering many questions […] without explicitly solving the equations concerned.”

“[If] the long-term behaviour of the state variable is independent of the initial condition […] the ‘end state’ […] is known as an attractor. […] Equilibrium states need not be attractors; they can be repellers [as well] […] if a dynamical system has an equilibrium state, any initial condition other than the exact equilibrium value may lead to the state variable converging towards the equilibrium or diverging away from it. We characterize such equilibria as stable and unstable respectively. In some models all initial conditions result in the state variable eventually converging towards a single equilibrium value. We characterize such equilibria as globally stable. An equilibrium that is approached only from a subset of all possible initial conditions (often those close to the equilibrium itself) is said to be locally stable. […] The combination of non-periodic solutions and sensitive dependence on initial conditions is the signature of the pattern of behaviour known to mathematicians as chaos.

“Most variables and parameters in models have units. […] However, the behaviour of a natural system cannot be affected by the units in which we chose to measure the quantities we use to describe it. This implies that it should be possible to write down the defining equations of a model in a form independent of the units we use. For any dynamical equation to be valid, the quantities being equated must be measured in the same units. How then do we restate such an equation in a form which is unaffected by our choice of units? The answer lies in identifying a natural scale or base unit for each quantity in the equations and then using the ratio of each variable to its natural scale in our dynamic description. Since such ratios are pure numbers, we say that they are dimensionless. If a dynamic equation couched in terms of dimensionless variables is to be valid, then both sides of any equality must likewise be dimensionless. […] the process of non-dimensionalisation, which we call dimensional analysis, can […] yield information on system dynamics. […] Since there is no unique dimensionless form for any set of dynamical equations, it is tempting to cut short the scaling process by ‘setting some parameter(s) equal to one’. Even experienced modellers make embarrasing blunders doing this, and we strongly recommend a systematic […] approach […] The key element in the scaling process is the selection of appropriate base units – the optimal choice being dependent on the questions motivating our study.”

“The starting point for selecting the appropriate formalism [in the context of the time dimension] must […] be recognition that real ecological processes operate in continuous time. Discrete-time models make some approximation to the outcome of these processes over a finite time interval, and should thus be interpreted with care. This caution is particularly important as difference equations are intuitively appealing and computationally simple. […] incautious empirical modelling with difference equations can have surprising (adverse) consequences. […] where the time increment of a discrete-time model is an arbitrary modelling choice, model predictions should be shown to be robust against changes in the value chosen.”

“Of the almost limitless range of relations between population flux and local density, we shall discuss only two extreme possibilities. Advection occurs when an external physical flow (such as an ocean current) transports all the members of the population past the point, x [in a spatially one-dimensional model], with essentially the same velocity, v. […] Diffusion occurs when the members of the population move at random. […] This leads to a net flow rate which is proportional to the spatial gradient of population density, with a constant of proportionality D, which we call the diffusion constant. […] the net flow [in this case] takes individuals from regions of high density to regions of low density” […] […some remarks about reaction-diffusion models, which I’d initially thought I’d cover here but which turned out to be too much work to deal with (the coverage is highly context-dependent)].