Behavioral Ecology and Individual-based Modeling Lecture Notes for Beijer Institute Advanced Course on Ecological Modeling Snata Fe Institute, December 1998 Louis J. Gross The Institute for Environmental Modeling University of Tennessee, Knoxville gross@tiem.utk.edu Copyright 1998. Louis J. Gross Behavioral Ecology deals in a broad sense with how the actions of individuals are affected by environmental influences, biotic and abiotic. It is closely related to ethology, the attempt to explain observed behaviors through a combination of immediate causation, development, function and evolution. Key question: How are behaviors related to the procurement of resources (food, mates, space) and allocation of these (e.g. as affected by social structure, sex ratio, life history pattern, etc.)? Specific questions: How do organisms choose what foods to eat, where to search for them, and how to carry out this search? In a patch view of the world, the issue is how long to spend in a patch and which patches should be ignored. How do groups form and what determines the size and structure of groups? What is the relationship between group size, risk of predation, and foraging efficiency? What behaviors are learned and under what circumstances do organisms learn? Here learning may be defined as a change in behavior due to experience. What factors cause different mating systems to have evolved, what affects sex ratios, why is sexual reproduction so prevalent, and what affects the type of parental care observed? Under what circumstances does cooperation evolve? When are supposedly altruistic behaviors adaptive? What are the requisite conditions for social structure to exist? What limits aggressive behavior within populations? What causes different life histories to have evolved? When is iteroparity versus semelparity expected? How are movements affected by trade-offs between foraging requirements, mating requirements, and predation avoidance? When is territoriality expected? Basic approaches: Comparative: developing and evaluating theories for differences in behavior based upon observed comparisons between different species under similar or divergent ecological conditions. Example: an analysis of home range size based upon metabolic needs and thus body weight of the individuals of a species. Many allometric relations are observed in such contexts. Mechanistic: Relate behavioral characteristics of individuals to the effects of environmental factors through biophysical analysis. Example: an analysis of spatial movement based upon the physiological response of receptors to various stimuli. Optimization: Assume there are underlying fitness criteria which affect what characteristics evolve. Ignore genetic constraints and use a phenotypic approach to determine which characteristics are "optimum" under particular choices for fitness measures. Examples: Optimal foraging theory in which what organisms to include in a diet or how long to spend in a patch is determined by maximizing the rate of energy gain. Some Optimization Approaches: Generally need to describe a set of possible decis ions, a set of constraints on these decisions, and a currency (pay-off) to measu re the relative benefit of different decisions. Average rate maximization: Diet choice Here the objective is to determine which prey to include in a diet based upon differences in the prey and the assumption that the objective is to maximize the net energy gain rate per unit time. h(i) = expected handling time for prey of type i e(i) = expect net energy gain from a prey of type i r(i) = rate at which prey of type i are encountered p(i) = probability that prey type i will be attacked upon encounter Then Rate of net energy gain is R = SUM (i=1,n) ( p(i) r(i) e(i) } over (1 + Sum {i=1,n) {p(i) e(i)h(i) } where s = cost of the search per unit time. Then to maximize R with a little bit of work can show that all p(i)'s are either 0 or 1 and we determine whether a particular prey is included by ranking the prey types so that e(1)/h(1) > e(2)/h(2) > ... and include all prey types in the diet until SUM (i=1,j+1) ( p(i) r(i) e(i) } over (1 + Sum {i=1,j+1) {p(i) e(i) h(i) } is greater than e(j+1)/h(j+1) This says that prey are included in the diet until the net rate of energy gain from adding the next ranked prey item is less than energy gain rate expected from the items already in the diet. Other general optimization approaches include Evolutionary Stable Strategies (a game theoretic approach) which offers great insight in situations of frequency dependence (e.g the behaviors displayed by an individual depend upon the frequency of encounters with other behaviors) and Dynamic decision analysis (based upon statistical decision theory) which applies dynamic programming methods to time dependent behaviors (e.g. the behavior displayed depends upon the history of previous behaviors). Main historical trends of Population Regulation- after den Boer and Reddingus (1996) Regulation and Stabilization Paradigms in Population Ecology): Mechanistic view (Lotka, Volterra) - populations as differential equations - treating them as made up of many, many individuals, each with small effect on the aggregate - a mechanistic view that grew out of physical models. In this view, one can still break down populations (as Lotka did extensively for age structure) but maintain the same assumptions about large sizes within each age group. Regulation or engineering view (Nicholson) - population density is in balance, e.g. adjusted to prevailing conditions, positing a "controlling factor" or "density governing factor" (now thought of as a density dependent factor) which produced a balance. There is an inherent equilibrium, with a stabilizing feedback driving population densities back to this equilibrium following perturbations - a cybernetic view, in which population regulation is taken for granted. Systems view (von Bertalanffy) - there are general principles, laws and models that apply to systems with many components, irrespective of the details of these components, which are applicable to the hierarchical levels in ecology. These laws are not necessarily derivable from a reductionist view. Natural History view (Andrewartha and Birch) - population dynamics is the result of a complex interplay between the properties of the organisms themselves and the variables in the environment. A view of populations made up of many small interaction groups, with which most individuals interact across their lifespan. So distribution and abundance determined by variations in localized environmental factors which determine the organisms growth and survival in these localized groups. An Alternative (not in den Boer and Reddingius): Individual-Based View - here we take the reductionist view that the properties of populations can be derived from the complex of interactions between individuals, environment, and other species. Thus it is not just localized interaction groups which determine population dynamics, but the entire complex of individual characteristics which vary through the population. It is a view in which it is possible for rare individuals to have significant impacts on population-level phenomena. Classical Approaches: Differential equations - assume a well-mixed, homogeneous population of identical individuals. Link together to form community models based upon standard interactions of competition, predator-prey, commensalism, etc. Age (now size/stage as well) structured matrix models: Disaggregate into groups of assumed identical individuals, and track dynamics of population as affected by structure Discrete generation models: based upon difference equations with either a population totally aggregated, or with some limited set of sub-classes (e.g. size/stage). Partial differential equation models for structured populations (McKendrick von-Foerster equation) in which there is a continuous age (or size, physiological status, etc.) variable, inflow from births, and outflow to deaths. Spatial models: patch occupancy (statistical mechanics of lots of identical patches), explicit patch (metapopulation) models, reaction-diffusion-advection models. Limitations of Classical Aggregated Approaches: These were generally designed for theory development, not necessarily to be realistic or precise. As such, the focus ignored individual differences except for characteristics that were presumed to have great influence on vital rates (fertilities and survivorship). So even structured models are highly aggregated relative to individual differences within a population. So limitations include: Ignoring individual differences in most characteristics, including physiology Difficulty taking account of small population sizes in which individual differences become more significant Can deal with spatial variation only in a fairly general manner, not easily able to take account of the kinds of spatially explicit information available from remote sensing data Do not readily deal with external, abiotic driving factors, since the historical focus was on biotic effects rather than abiotic ones and dealing with non-autonomous dynamical systems is much more difficult (and therefore less interesting to mathematicians) than dealing with autonomous ones about which you have some possibility of proving general theorems. Individual-based models: designed to ameliorate some of the above limitations - two general classes: Physiologically structured models (i-state distribution models as defined by Caswell and John) - extensions of the McKendrick von-Foerster approach to take into account detailed bioenergetic formulations associated with physiological structuring of a population in addition to age/size etc. Follows "individuals" as cohorts or characteristics of a partial differential equation Spatially-explicit models (i-state configuration models as defined by Caswell and John) - follow the states of many individuals which differ in a variety of characteristics and these variations affect the rules by which the individuals behave. Can be followed explicitly in space (so location is a state variable). Allows for analysis of the effects of rare individuals which can have a large impact, and for linkage to abiotic forcing and spatial detail such as that available through a GIS Spatially-Explicit Individual-Based Models: An Introduction Goal: Deal with spatial variation in underlying environmental and habitat factors and evaluate how these affect populations and communities through time. Objectives: 1. Consider individual variations in factors such as sex, size, age, health, social status, etc. 2. Include spatially-explicit information on habitat, roads, topography, etc. and the effects these have on individual behavior. 3. Provide mechanism for interactions between individuals. 4. Allow for dynamic coupling of habitat components to organisms through direct feedback of organism behavior on appropriate habitat conditions, such as reducing available forage due to effects of individuals. 5. Provide mechanism to take into account detailed behavioral and physiological information when available. 6. Estimate larger-scale phenomena (e.g. population/community scale) from actions of individuals Alternatives: 1. Compartment models with underlying spatial grid. These are box diagrams within each spatial cell connected by fluxes between boxes both within a cell and between cells. Ignores individual-variation and formulas for fluxes between cells are typically chosen ad-hoc as it is very difficult to estimate population-scale fluxes except for small organisma for which aggregated models are reasonable. 2. Index models. These apply some simple or complex function to each cell in a spatial grid, using as variables the local habitat variables. Ignores all aspects of individuals and population structure. Approach of Habitat Suitability Indices (HSI), Habitat Evaluation Procedures (HEP), and GAP models. Disadvantages of Individual-Based approaches: 1. Requires detailed knowledge of behavior and physiology, thus is generally appropriate for large, charismatic species, but of limited use in other cases. 2. May require considerable coding expertise to develop as well as considerable computer time to run. 3. Typically requires many simulations to evaluate any particular situation as it is based upon an underlying stochastic model. 4. As with any model, typically requires assumptions about what aspects of behavior are important and what can be ignored. Building Individual-Based Models: 1. Set up a spatial grid, in which each location (or pixel in a computer map) corresponds to a spatial location with a certain spatial extent for each grid cell. Each grid cell then is characterized by a state, with each state corresponding to (a) presence or absence of a given species, (b) number of a given species present in the cell, (c) numbers of each of several species present in the cell, (d) underlying habitat conditions in the cell, etc. 2. Two basic approaches: (i) model how each grid cell changes based upon some set of rules and the states of surrounding grid cells, or (ii) model how each individual organism being considers moves among the grid cells and how the organisms state changes through time. Here (i) is the cellular automata approach, in which each cell is in one of a relatively small number of states and the rules for changes in state depend on the current state and that of nearest neighbors. Here (ii) is a truly individual-based approach, in which individual organisms can move anywhere across the spatial set of grid cells, change their respective states (e.g. size, fat content, number of offspring, etc.) and have their location attached to them as just another state variable. In this approach, each grid cell just combines the individuals located in it at any particular time. Key questions in developing an individual-based model: 1. Setting an appropriate spatial and temporal extents and resolution. 2. Deciding what individual-level state variables should be included and how to model them. 3. How to model movement, growth, mortality and reproduction of individuals. These are not independent issues, since setting the resolution and extent affects all aspects of the model. Appropriate spatial extent and resolution depends upon (a) the activity pattern of an individual over the shortest time period of interest; (b) the availability of spatially-explicit information on habitat preferences, food resources, shelter and other spatial components affecting individual behavior (c) the availability of information to accurately model individual behavior. If activities at the shortest time period of interest all occur within the smallest spatial resolution at which you have habitat information, then there is little advantage to modeling behaviors which occur within that time step. If behaviors over the smallest time period of interest range over several spatial locations for which you have habitat information, then including state variables which measure such behaviors is appropriate. Driving all this is the set of questions you wish the model to address, which will no doubt lead to an appropriate temporal resolution. Activities which occur on a time periods which significantly affect the aspects of individual behavior directly related to the questions of interest should be included, while those with small effect may be ignored or averaged. Two methods for time sequences of behavior: 1. Have a fixed underlying time step at which one censuses all individuals, considering changes in behavior or state variables for all individuals. If this is done sequentially, then care must be taken to randomize the ordering of the individuals chosen in different time periods in order to avoid bias. 2. Event-based models take each individual, determines a time at which the next behavior or state variable change occurs, and ignores that individual until such a time is reached. This may be computationally efficient - every individual need not be tracked each time period. Parallelization: Though their investigation is still rather limited, parallelization methods for individual-based models have been developed and indicate that basic model assumptions made for sequential versions may well need to be modified in order for efficient parallel algorithms to be utilized. The assumptions that may be modified concern the ordering of interactions between individuals, and parallel implementations may well handle these in a more biologically realistic manner than sequential implementations. Therefore there is no inherent reason why one should expect exactly the same model output from sequential and parallel implementations.