One theory, many models of habitat suitability
Kenneth B. Vernon Scientific Computing and Imaging Institute, University of Utah
The Box model
All models are wrong but some are useful.
- George Box
Is the Box model of models right or wrong?
All Cretans are liars.
- Epimenides of Crete
Is Epimenides telling the truth or lying?
Diagnosis
A more charitable interpretation would be that all statistical models are wrong, meaning they come with some degree of uncertainty.
But there are other kinds of models…
Models
Both of these are models:
note: Medians have probably wobbled a lot over the last 250 years, but there is a sense in which the rules are eternal…
Rulebooks
The Laws of Cricket are…
- True. There may be disagreement about their precise content, but they cannot be systematically wrong.
- General. They apply to every game, even to backyard cricket where some rules may be ignored.
- Necessary. If something is to count as an instance of the game, then the rules must apply to it (again, even if some rules are broken).
The Charnov model
Optimality models are analogous to rulebooks for games.
- Eric Charnov
tbf: Charnov never suggested the view of rulebooks that I just offered…
Actual Quote
From Charnov and Orians (1973), unpublished monograph:
Building optimality models requires at least two things — first, the choice of a goal (what is being optimized?), then the choice of the game that the organism will take part in. The game gives the constraints or rules within which the organism must operate.
Inference
We observe Simon hit a ball with a strange flat board…
INTERPRETATION:
“Simon must be playing cricket.”
We classify his action as a move in cricket.
INFERENCE:
“What are the Laws of Cricket again?”
Our interpretation sets off a cascade of entailments:
- Law 1: The players. A cricket team consists of eleven players, including a captain…
- Law 16: The result. The side which scores the most runs wins the match…
- Law 18: Scoring runs. Runs are scored when the two batters run to each other’s end of the pitch…
Explanation
Why did Simon hit the ball?
THE GOAL:
“Because he wants his team to score more points.”
What Aristotle called a final cause.
THE RULE:
“Because the ball was bowled to him.”
Do not ask me to explain cricket batting strategies.
THE GAME:
“Because he is playing cricket.”
Sometimes called naive action explanation.
note: In behavioral ecology, we are typically concerned with rule-explanations involving descriptions of the environment.
Principle of Charity
SYNONYMS: Adaptationism, Optimization Analysis, Rational Accommodation.
When seeking to explain intentional behavior, we revise these descriptions until we arrive at a set that makes what the agent does rational in its circumstances.
Science
But enough with philosophical stage setting, let’s get on with the science!
Habitat selection
An organism must choose one habitat from a finite set of alternatives.
Over time, these decisions pile up. The result is a species distribution.
Two Models
of habitat selection…
Optimality model
The Ideal Free Distribution (IFD) defines general rules of habitat selection that structure individual decisions.
Reason forward from utility to choice.
Statistical models
Species Distribution Models (SDMs) describe the aggregate distributional effect of actual or observed habitat selection by individuals.
Reason backward from choice to utility.
Ideal Free Distribution
Let \(E_{i}\) be net energetic output of habitat \(i \in 1, ..., K\) at density \(n_{i}\).
SUITABILITY:
\(E_{i}/n_{i}\)
The payoff to each individual is the average productivity.
SELECTION RULE:
\(\text{choose } i^{*} = \arg\max_{i} \; E_{i}/n_{i}\)
An individual always prefers the habitat with the highest payoff.
EQUILIBRIUM:
\(E_{i}/n_{i} = E_{j}/n_{j} \;\; \forall_{i,j} \mid n_{i}, n_{j} > 0\)
Individuals are indifferent when payoffs are equal across occupied habitats.
IFD:
\((n_{1}, \dots, n_{K})\)
The equilibrium implies a habitat distribution.
Habitat suitability
The baseline suitability \(B_{i}\) with density-dependent effects \(f_{i}(n_{i})\).
\[
E_{i}/n_{i} = B_{i} - f_{i}(n_{i})
\]
ENERGY:
\(E_{i} = n_{i}(B_{i} - f_{i}(n_{i}))\)
Net energy output, with \(n_{i}B_{i}\) and \(n_{i}f_{i}(n_{i})\) the aggregate output and cost.
EXTERNALITY:
\(f_{i}(n_{i}) = B_{i} - E_{i}/n_{i}\)
The difference between the baseline and realized suitability, the average cost.
Density dependence
Let \(AP = E/n\) and \(MP = dE/dn\), then…
\[
AP - MP = n f'(n)
\]
with \(n f'(n)\) being the total effect on the population of one more individual.
Negative dependence
Decreasing returns to scale: \(MP < AP \iff f'(n) > 0\)
Positive dependence
Increasing returns to scale: \(MP > AP \iff f'(n) < 0\)
Species Distribution Models
GOAL: Model how a population is distributed across a set of habitats:
\[
E\left[\frac{n_{i}}{N}\right] = p_{i}
\]
where \(p_{i}\) is the expected share of the population in habitat \(i\) and is a function of environmental characteristics:
\[
p_i = g(X_i), \quad \sum_{i} p_{i} = 1
\]
note: (i) These shares correspond to relative productivity, \(E_{i}/Q\). (ii) You can use any kind of statistical model you want to estimate \(g\).
Discrete habitats
For \(K\) discrete habitats, each individual independently selects one habitat with probability \(p_{1}, \dots, p_{K}\). Then
\[
(n_{1}, \dots, n_{K}) \sim \mathrm{Multinomial}(N; p_{1}, \dots, p_{K})
\]
and
\[
E\left[\frac{n_{i}}{N}\right] = p_{i}
\]
\[
p = \textrm{softmax}(X\beta)
\]
note: This is just one example of a parametric model for discrete habitats.
Presence-only data
CHALLENGE: The species abundance, \(N\), is rarely if ever known.
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Less than ~6.5% of the Grand Staircase-Escalante National Monument has been subject to archaeological survey (as of 2020).
Ambiguous preferences
CHALLENGE: Absolute measurements of habitat conditions are ambiguous.
It matters what the available alternatives are.
Different preferences
CHALLENGE: Species operate under different constraints.
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Example
Archaeological populations of foragers and farmers in the GSENM.
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Hand waving…
A popular way to handle these challenges is with a Poisson process model like MaxEnt.
You type up some R code, run it, and…
Foragers and farmers
Variation in habitat preferences…
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But…
do species always choose their habitats?
It is tempting to distinguish between Species Distribution and Habitat Suitability models, letting the former be merely correlative and leaving the latter to have the force of necessity implied by IFD rules.
I am fine with this view so as long as we accept that the moment we interpret the behavior as habitat selection, the cascade of IFD entailments must inevitably follow.
If it was all mere correlation, intentional behavior would be incoherent, sort of like…
Acknowledgments
- Simon “Waldorf” Brewer
- Brian “Statler” Codding
