Modeling in Behavioral Ecology
Dr. Mike Heithaus
MS361
Office Hours: Give me a ring!
(305) 919-5234
Course Structure
• Initial meetings
• Start developing your own model
• Suggested books
– Haefner (2005) Modeling biological systems: principles and applications
– Grimm and Railsback (2005) Individual-based modeling and ecology
– Mangel and Clark (2000) Dynamic state variable models in ecology
– Dugatkin and Reeve (1998) Game theory and animal behavior
– Hilborn and Mangel (1997) The ecological detective: confronting models with data
What is a model?
• A description of a system
– System – any collection of interrelated objects
– Object – some unit upon which observations can be made
– “. . . systems are anything humans wish to discuss and models are one tool that facilitates that discussion.” (Haefner 2005)
– Metaphorical descriptions of nature (Hilborn and Mangel 1997)
Why model?
• Three uses of models in science
– Understanding a system
• Use knowledge of inputs and outputs to infer system characteristics
– Prediction about some future state
• Use knowledge of the parts of a system and their stimuli to account for responses
– Control or manipulation of a system (Decisions)
• Design a system to obtain a desired output
Why model?: Island biogeography
• Understanding:
– Develop quantitative hypotheses about how the number of species on an island should change with rates of immigration and extinction
• Prediction
– Predict how long an island to recover after disturbance or how many species will be supported by an island
• Decision/Control
– Aid in the design of island-like reserves to maximize the number of species
Why model: an ecologists perspective
• The utility of models often comes from the predictions that it generates that can be tested
Types of models
• Conceptual or Verbal (Qualitative)
– Descriptions in language
• Diagrammatic (Qualitative)
– Graphical representations of objects and relationships
• Physical
– a real mock-up
• Formal (Quantitative)
– Mathematical model
Types of mathematical models I
• Scientific
– Describes how nature might work and then proceeds to a set of predictions relating dependent and independent variables
• Statistical
– Describes relationships among variables without any explanation for why an interaction occurs
Types of mathematical models II
• Mechanistic models
– Explicit representation of mechanistic processes; also called a process-oriented model
• Descriptive models
– No representation of mechanistic processes
Types of mathematical models III
• Static
– Responses to input variables do not change over time
• Dynamic
– Responses may change over time
Types of mathematical models IV
• Continuous
– Time may take any value
• Discrete
– Time is only expressed as an integer
Types of mathematical models V
• Spatially homogeneous
– No spatial structure to model
• Spatially heterogeneous
– Objects have a position in space
– May be:
• Discrete if space is represented in blocks or cells that are homogeneous
• Continuous if every point in space is different
Types of mathematical models VI
• Stochastic
– Allows random fluctuations
• Deterministic
– All parameters are constant
Constraints on models
• Trade-offs are inherent in modeling: you can’t simultaneously maximize all three of the major properties of models
– Realism
– Accuracy of outputs
– Generality
• How you solve the tradeoffs depends on the goal of your model
Model tradeoffs
• Prediction: don’t need much generality, good accuracy and realism
– Does this depend on the type of prediction?
• Understanding: need generality and some reality, but accuracy less important
• Control: need reality, but virtually no generality
Misuse of models
• Models are sometimes given greater weight than is appropriate
– Not all models are equal
• Models don’t “prove” anything
– A particular problem with the interpretation of ecosystem models like ECOSIM these days
• No model is “correct” – there can be a “best” model or several models may be equally likely
– The presence of competing models is critical, otherwise a poor one may continue to be used
What is behavioral ecology?
• The study of the adaptive nature of behavior in relation to ecological conditions
• Why do animals do what they do?
• Concerned with ultimate questions – why and how? Not what?
Key Concepts in Behavioral Ecology
• Optimality
– Organisms are optimized by natural selection
1. fitness is a function of design (e.g. behavior)
2. selection tends to maximize fitness
3. like begets like (heritable genetic variation)
– Therefore, given enough time and raw material, selection will lead to optimal design (behavior)
Optimality
– Optimization model
• Hypothesis about how things work
• Provides quantitative predictions that can be tested by observations or experiments
– Optimization modeling steps
1. Specify constraints in a system
a. Possible phenotypes (options) – allowable outcomes: STRATEGY SET
b. State equations: morphology and physiology – constraints on behavior
2. Hypothesize an optimization criteria (what is being maximized or minimized)
fitness (the ultimate optimization)
surrogate of fitness (e.g. net energy intake, lifetime energy intake)
measure “indifference” – how much will be given up for something else – UTILITY FUNCTION (mean vs variance; food vs distance from cover)
3. Assumption about heredity
a. Behavior passed on from one generation to the next (genetics of behavior are largely unstudied)
Optimality
– Optimality models
• Optimality modeling does not test whether nature optimizes, just the constraints and assumptions of the model
• Optimality is a modeling technique, not what animals try to do.
– An animal doesn’t consciously think they are trying to maximize energy intake, they respond to proximate cues (e.g. gut fullness) but the end result is optimization
• If test of model does not fit predictions of the model, refine the hypothesis and repeat the process – note: this is not circular, it is attempting to zero in on how systems work
Optimality
– Fitness function
• Relationship between the magnitude of expression of a behavior or trait and the fitness associated with it
• A way to generate predictions about what behaviors should be used
• Determining an optimum doesn’t tell us how a population will evolve towards the optimum or the costs of suboptimal behavior – need to know the shape of the fitness function
Optimality
– Dynamic optimization models (state-variable models)
• Tell us the optimal policy at a given state and time
– For example: what habitat should a bird use based on its energy reserves if habitats vary in risk and food
• There must be an end time T
• Based on a terminal fitness function – the expected fitness of an animal in a given state at time T and a dynamic programming equation (DPE) that incorporates state-dependent survival probabilities and transitions in state variables
• Uses backwards iteration (start with terminal function then calculate possible ways to get there from T-1 and so on)
• Assumes that optimal decisions are made in each cell
• Can look at cost of inappropriate decisions
• Once optimal decisions have been determined, can use forward simulations to determine likely outcomes
Game theory
– Optimal policy depends on behavior of others
– Optimization in a frequency dependent cast
• e.g. interactions between predators and prey or between prey
– Analyzes the outcomes of behavioral decisions when these outcomes depend on the behavior of other players
– Predicts the individual’s behavior based the best estimates of
• the other contestant’s response
– The rewards or costs of various outcomes
ESS and game theory
– Evolutionarily Stable Strategy (ESS)
• Nash equilibrium in which no “invading” strategy or mutation can invade the populations and do better than other individuals in the population
• Nash Equilibrium = no individual can do better by unilaterally switching tactics
• May involve mixed strategies within individuals
– Evolutionarily stable state (ESSt)
• Fixed frequencies of pure strategies in the population (e.g. 60% of individuals sneak, 40 % of individuals guard)
ESS and game theory
– There may be more than one ESS for a population and the one attained may depend on history.
• There may even be a better ESS but the population can’t get there from here (complex adaptive landscape) so an ESS can be a local optimum but not a global optimum
– When solving for an ESS need to set up starting conditions, alternate strategies
– Compare fitness of specified strategies : note, you only find the ESS against specified strategies!
Key concepts in behavioral ecology
• Principle of Allocation
– Probably the most important concept in behavioral ecology
– An organism only has so much of a currency (e.g. time, energy)
• How are these allocated among behaviors?
– Underlies the idea of tradeoffs
– All problems in behavioral ecology come down to allocation and tradeoffs
The place of models in behavioral ecology
• Generate testable hypotheses
• Makes assumptions obvious
• Forces identification of important components of a system
• Serve as starting point for iterative work
Some major types of models in Behavioral Ecology
• Optimality models
• Game theoretical models
• Dynamic state variable models
• Genetic algorithms
– Technique to solve optimization problems
– Mimic natural selection
– Do not represent actual genetics!
– Used at the population level
– Include density and frequency dependence
Building a model
• Identify a problem
• Identify important factors and interactions
– A graphical model can be a good first step
• Determine mathematical relationships between factors
– Build the model
• Solve or implement model
• Confront model with data
• Modify model based on empirical evidence and complete iterations
– Depending on the question, this may serve to decrease generality and increase realism
Building a dynamic state variable model
• Identify a problem
• Choose basic time period (t and T)
• Specify Decision
• Identify Constraints (e.g. mortality rate)
– be sure not to make decisions a constraint
• Add dynamics (e.g. probability of finding food)
• Choose Optimization Criteria
• Specify Terminal Fitness Function
• Develop Dynamic Programming Equation
• Code (sometimes Excel works!)
• Run Forward Iterations