TOPIC 2
OPTIMALITY MODELS IN BEHAVIORAL ECOLOGY

THE USE OF OPTIMALITY MODELS IN BEHAVIOURAL ECOLOGY IS ONE
FORM OF THE EXPERIMENTAL APPROACH (E.G. EXAMPLE OF CROWS
FORAGING ON WHELKS).

A MORE DETAILED LOOK AT THE USE OF OPTIMALITY MODELS -
EXPLOITATION OF FOOD PATCHES (THE MARGINAL VALUE THEOREM)

IT IS ACCEPTED THAT SELECTION FAVOURS INDIVIDUALS THAT LEAVE
THE MOST OFFSPRING. WHEN ANALYZING BEHAVIOURS THAT MAY NOT BE
DIRECTLY ASSOCIATE WITH REPRODUCTIVE OUTPUT IT IS NECESSARY
TO IDENTIFY SOME "CURRENCY", CORRELATED WITH FITNESS, THAT
NATURAL SELECTION CAN OPERATE ON. IN THE CASE OF FORAGING,
THE FITNESS CURRENCY IS THE RATE OF NET ENERGY GAIN. THIS
SHOULD BE CORRELATED WITH FITNESS BECAUSE HIGH RATES OF
ENERGY GAIN WOULD DECREASE THE AMOUNT OF TIME SPENT FORAGING
(DECREASE PREDATION), INCREASE GROWTH RATE OR ENERGY STORES
(INCREASE REPRODUCTIVE OUTPUT, SOCIAL STATUS, ABILITY TO
AVOID PREDATORS), INCREASE TIME AVAILABLE FOR OTHER FITNESS
RELATED ACTIVITIES (NEST BUILDING, TERRITORIAL DEFENSE).

ANIMALS THAT FORAGE BY MOVING THROUGH THEIR ENVIRONMENT
LOOKING FOR FOOD OFTEN ENCOUNTER THEIR FOOD IN MORE OR LESS
DISCRETE PATCHES. I.E. THERE WILL BE A PERIOD OF TRAVEL
BETWEEN PATCHES DURING WHICH NO FEEDING TAKES PLACE. ONCE THE
PATCH HAS BEEN FOUND FEEDING RATE WILL DEPEND ON THE DENSITY
OF FOOD IN THE PATCH.

THE DENSITY OF FOOD IN THE PATCH WILL DECREASE WITH TIME
BECAUSE OF THE REMOVAL OF PREY (PATCH DEPLETION) BY THE
FORAGER.

IF THE FORAGER LEAVES THE PATCH TOO SOON IT WILL HAVE A LOW
RATE OF ENERGY GAIN BECAUSE IT WILL HAVE GIVEN UP A HIGH RATE
OF ENERGY GAIN TO LOOK FOR A PATCH THAT WILL PROBABLY NOT BE
BETTER THAN THE ONE IT JUST LEFT.

IF THE FORAGER STAYS TOO LONG IN A PATCH, IT WILL HAVE A LOW
RATE OF ENERGY GAIN BECAUSE PATCH DEPLETION WILL EVENTUALLY
LEAD TO A LOWER RATE OF ENERGY GAIN THAN IF IT ABANDONED THE
PATCH AND SEARCHED FOR AN UNDEPLETED PATCH.

THERE SHOULD BE SOME RESIDENCE TIME IN THE PATCH (BEHAVIOUR)
THAT MAXIMIZES THE RATE OF ENERGY GAIN.

QUESTION: HOW DO TEST THE HYPOTHESIS THAT AN ANIMAL IS
OPTIMIZING ITS PATCH RESIDENCE TIME.

PREDICTED OPTIMUM RESIDENCE TIME IS DETERMINED GRAPHICALLY:

IF THE PATCH IS REPRESENTATIVE OF OTHER PATCHES THEN THE MEAN
RATE OF ENERGY GAIN IN THE HABITAT IS DETERMINED BY THE SLOPE
OF THE LINE CONNECTING THE ORIGIN TO THE POINT ON THE GAIN
CURVE THAT CORRESPONDS TO THE PATCH RESIDENCE TIME. I.E.,

THE AVERAGE RATE OF ENERGY GAIN FROM THE HABITAT IS:

EXPECTED TOTAL ENERGY GAIN FROM PATCH
-------------------------------------
TRAVEL TIME + RESIDENCE TIME

= SLOPE OF CURVE FROM ORIGIN TO GAIN CURVE, AT GIVEN PATCH
  RESIDENCE TIME.

AVERAGE RATE OF RETURN IS MAXIMIZED BY THE RESIDENCE TIME AT
WHICH THE LINE FROM THE ORIGIN IS TANGENT TO THE GAIN CURVE.
THIS TIME IS THE OPTIMAL PATCH RESIDENCE TIME. (TIME 2 ON THE
FIGURE).


ANOTHER WAY TO STATE THIS RESULT:

ANIMALS SHOULD LEAVE PATCHES WHEN THE INSTANTANEOUS RATE OF
RETURN IN THE PATCH IS EQUAL TO THE EXPECTED RATE OF RETURN
FROM THE ENVIRONMENT.

THE INSTANTANEOUS RATE OF RETURN IN THE PATCH IS ALSO TERMED
THE "MARGINAL VALUE" OF RETURN. SO THIS MODEL IS CALLED THE
"MARGINAL VALUE THEOREM"

ADDITIONAL PREDICTION FROM THE MODEL:

OPTIMAL PATCH RESIDENCE TIME WILL INCREASE AS THE TRAVEL TIME
BETWEEN PATCHES INCREASES:

EXAMPLES OF USE OF MARGINAL VALUE THEOREM

1. FEEDING OF NESTLINGS IN STARLINGS

STARLINGS SEARCH FOR INSECT PREY ON THE GROUND TO FEED YOUNG.

SEVERAL PREY TRANSPORTED PER TRIP.

RATE OF GAIN OF PREY WHILE EXPLOITING PREY PATCH DECREASES
CONTINUALLY DUE TO DIFFICULTY OF MANIPULATING PREY ALREADY
CAPTURED.

EXPERIMENTAL TEST OF MVT USED ARTIFICIAL FEEDING STATIONS.

PREY DELIVERED ONE AT A TIME FROM A TUBE.

TIME BETWEEN SUCCESSIVE PREY DELIVERIES INCREASED CONTINUALLY
(MIMICS PREY DEPLETION).

IF TRAVEL TIME KNOWN THEN POSSIBLE TO PREDICT THE OPTIMAL
TIME AT WHICH STARLING SHOULD LEAVE FEEDING STATION AND
RETURN TO NEST.

E.G. BEE FORAGING

IGNORE THIS EXAMPLE IN THE TEXT.
IT IS MORE COMPLEX THAN THE ANALYSIS PRESENTED SUGGESTS AND
THE CONCLUSION IS MISLEADING I.E. MAXIMIZING EFFICIENCY AND
NOT RATE OF NET ENERGY GAIN (COULD NOT MEASURE THE ACTUAL NET
RATE OF ENERGY GAIN AS DID NOT KNOW THE ACTUAL ENERGY COSTS
OF LOAD CARRYING).

MORE ON FORAGING

IT IS IMPORTANT TO NOTE THAT WHEN ENERGY IS ASSUMED TO BE THE
FITNESS CURRENCY AND THE OBJECT IS TO TEST AN EVOLUTIONARY
MODEL BASED ON ENERGY MAXIMIZATION, THE PROPER CURRENCY IS
THE RATE OF NET ENERGY GAIN AND NOT THE GROSS RATE OF
ENERGY GAIN (FEEDING RATE).

PREDICTIONS OF OPTIMAL BEHAVIOUR MAY DIFFER DEPENDING
ON THE CURRENCY USED IN THE MODEL (NET VS GROSS) BECAUSE NET
ENERGY INCLUDES THE ENERGY COSTS OF FORAGING.

WHEN THE ENERGY COSTS OF FORAGING ARE CONSTANT OR INDEPENDENT
OF FORAGING ACTIVITY, E.G. WHEN AN ANIMAL IS MOVING AT A MORE
OR LESS CONSTANT RATE THROUGH THE ENVIRONMENT WHILE FORAGING,
THE BEHAVIOUR THAT MAXIMIZES NET ENERGY WILL BE THE SAME AS
THAT WHICH MAXIMIZES THE RATE OF GROSS ENERGY GAIN, AND THE
SIMPLE ANALYSIS USING THE MVT APPLIES.

E.G. LET
C = CONSTANT RATE OF ENERGY OUTPUT DURING FORAGING
t = TIME IN PATCH
T = TRAVEL TIME BETWEEN PATCHES
f(t) = ENERGY GAIN FROM PATCH AFTER TIME t IN PATCH

THE GROSS RATE OF ENERGY GAIN AFTER TIME T+t IS:

f(t)/(T+t)

AND THE BEHAVIOUR THAT MAXIMIZES THE GROSS RATE OF GAIN
(FEEDING RATE) WOULD MAXIMIZE THIS FUNCTION WRT t.

THE NET RATE OF ENERGY GAIN AFTER TIME t IS;

[f(t)-C(T+t)]/T+t

WHICH SIMPLIFIES TO
[f(t)/(T+t)] - C.

AND THE BEHAVIOUR THAT MAXIMIZES THE NET RATE OF ENERGY GAIN
WOULD MAXIMIZE THIS FUNCTION WRT t.

BUT SINCE C IS A CONSTANT (INDEPENDENT OF T OR t) IT WILL
HAVE NO EFFECT ON THE VALUE OF t THAT MAXIMIZES THE FUNCTION
(SAME DERIVATIVE) SO THE SOLUTION WILL BE THE SAME IN BOTH
CASES.

THE SITUATION IS DIFFERENT IF THE ENERGY COSTS OF FORAGING
VARY WITH FORAGING ACTIVITY. IN THIS CASE THE NET RATE OF
ENERGY GAIN WOULD BE

[f(t) - g(t,T,f(t))]/(T+t)

WHICH DOES NOT SIMPLIFY (DIFFERENT DERIVATIVE) AND WOULD
PREDICT A DIFFERENT OPTIMAL BEHAVIOUR.

IN THE BEE EXAMPLE, THE LOAD OF HONEY THE BEE HAD TO CARRY
IMPOSED A SIGNIFICANT ENERGY COST OF TRANSPORT ON THE BEE
WHICH INCREASED AS THE LOAD, AND THE TIME SPENT CARRYING THE
LOAD, INCREASED.

THEREFORE WHEN THE BEES DID NOT APPEAR TO BE MAXIMIZING THE
RATE OF GAIN OF NECTAR (GROSS ENERGY) THIS DOES NOT MEAN THAT
THEY WERE NOT MAXIMIZING THE NET RATE OF ENERGY GAIN, THIS
WAS NOT TESTED AS THE COSTS WERE NOT QUANTIFIED.

IN THE STARLING EXAMPLE, THE INSECT FOOD WAS SMALL RELATIVE
TO THE SIZE OF THE ADULT BIRD AND DID NOT IMPOSE A
SIGNIFICANT TRANSPORT COST.

CONCLUDE: MAKE SURE THAT THE CURRENCY INCLUDES THE COSTS OF
THE BEHAVIOUR IF THESE SEEM TO DEPEND ON THE FORAGING
ACTIVITY.

PREY CHOICE

IF ALL PREY ARE THE SAME, THEN THEY SHOULD ALWAYS BE EATEN
(OR AT LEAST PURSUED) WHEN ENCOUNTERED AND THERE IS NO BASIS
IN ENERGETICS FOR CHOOSING AMONG INDIVIDUAL PREY.

BUT IF PREY VARY IN THEIR ENERGY VALUE THEN IT MAY BE
ENERGETICALLY ADVANTAGEOUS TO EXCLUDE SOME PREY FROM THE
DIET.

WHAT CRITERIA DETERMINE IF PREY SHOULD BE INCLUDED IN THE
DIET IF NET ENERGY IS TO BE MAXIMIZED?

EXAMPLE

LET X = THE AVERAGE RATE OF RETURN IN A HABITAT FOR A GIVEN
DIET (COLLECTION OF PREY TYPES)

LET Y = THE AVERAGE TIME IT TAKES TO "HANDLE" A PREY OF TYPE
"A".

LET Z = THE NET ENERGY CONTENT OF PREY OF TYPE "A".

"OPTIMAL DIET RULE"

WHEN A PREY ITEM OF TYPE "A" IS ENCOUNTERED IT SHOULD BE
EATEN IF THAT PROVIDES A HIGHER RATE OF RETURN THAN X.

WHY?

IF PREY IS EATEN, THE RATE OF RETURN WHILE IT IS OCCUPIED
WITH THE PREY WILL BE Z/Y.

IF PREY IS IGNORED, THE FORAGER CAN "EXPECT" TO GAIN ENERGY
AT THE RATE X DURING THE TIME PERIOD Y.

IF BEHAVE SO AS TO MAXIMIZE RATE OF ENERGY GAIN THEN:

EAT IF Z/Y >X; IGNORE IF Z/Y>X.

THE RATIO Z/Y IS TERMED THE "PROFITABILITY" OF PREY "A".

"PREDICTING OPTIMAL DIET"

TO PREDICT THE OPTIMAL DIET BASED ON THE PRINCIPLE OF ENERGY
MAXIMIZATION.

1. RANK ALL PREY BY PROFITABILITY.

2. START WITH A DIET THAT ONLY INCLUDES PREY WITH THE HIGHEST
PROFITABILITY - PREY 1.

3. CALCULATE THE NET RATE OF ENERGY GAIN FOR THIS DIET (NEED
TO ALSO KNOW THE SEARCH TIME FOR EACH PREY)  = X.

4. COMPARE THIS VALUE WITH THE PROFITABILITY OF THE MOST
PROFITABLE PREY NOT IN THE DIET =Pn.

5.  X>Pn?

6. IF NO, THEN ADD PREY TYPE Pn TO THE DIET, n=n+1, AND GO TO
STEP 3.

7. IF YES THEN THE OPTIMAL DIET INCLUDES ALL PREY EXCEPT THE
LAST ONE TESTED (PREY n).

PREDICTION
AS THE RATE OF RETURN IN THE HABITAT DECREASES, LESS
PROFITABLE PREY WILL BE ADDED TO THE DIET.

TEST OF OPTIMAL DIET MODEL

KREBS ET AL. (GREAT TITS)

CAPTIVE BIRD AT FEEDER THAT IS SUPPLIED WITH A CONVEYOR BELT.

SMALL AND LARGE MEAL WORMS MOVE ALONG THE CONVEYOR BELT AND
ARE AVAILABLE FOR 1/2 SEC.

WHEN PREY ARE CAPTURED FROM THE CONVEYOR BELT OTHER PREY
CANNOT BE TAKEN DURING THE HANDLING TIME.

HANDLING TIME FOR LARGE AND SMALL PREY IS SIMILAR BUT LARGE
PREY HAVE TWICE THE ENERGY VALUE AND THEREFORE ABOUT TWICE
THE PROFITABILITY.

TESTABLE PREDICTION

WHEN THE MEAN RATE OF RETURN FROM JUST EATING THE LARGE MEAL
WORMS EXCEEDS THE PROFITABILITY OF THE SMALL MEALWORMS, THE
SMALL MEAL WORMS SHOULD BE IGNORED.

WHEN THE DELIVERY RATE OF LARGE MEAL WORMS IS REDUCED TO THE
POINT WHERE THE RATE OF ENERGY GAIN FALLS BELOW THE
PROFITABILITY OF THE SMALL MEAL WORMS, THEN SMALL MEAL WORMS
SHOULD BE EATEN WHEN ENCOUNTERED.

NOTE THAT THE DENSITY OF PREY HAS NO EFFECT ON THEIR
INCLUSION IN THE DIET.

I.E. IF A PREY IS EXCLUDED FROM THE DIET BECAUSE OF LOW
PROFITABILITY IT SHOULD BE EXCLUDED REGARDLESS OF ITS
DENSITY.


ESTIMATING ENVIRONMENTAL VARIABLES

IF FORAGING SO AS TO MAXIMIZE THE NET RATE OF ENERGY GAIN
PROVIDES A SIGNIFICANT ENERGY ADVANTAGE, HOW CAN SELECTION
ACT ON BEHAVIOUR TO PRODUCE THE ABILITY TO MAKE THE DYNAMIC
OPTIMAL "DECISIONS" THAT LEAD TO NET ENERGY MAXIMIZATION.

OTHER THAN SELECTION FOR OMNIPOTENCE, THE ONLY POSSIBILITY IS
BEHAVIOUR THAT ALLOWS ANIMALS TO ESTIMATE ENVIRONMENTAL
VARIABLES THROUGH SAMPLING.

SELECTION WILL FOLLOW THE PATH OF LEAST RESISTANCE, I.E.
PRODUCE THE SIMPLEST EFFECTIVE SAMPLING BEHAVIOUR WITHIN THE
SENSORY & INTEGRATIVE CAPABILITIES OF THE ANIMAL.

THIS SHOULD LEAD TO THE USE OF SIMPLE "RULES OF THUMB" RATHER
THAN COMPLEX SAMPLING BEHAVIOURS AND INFORMATION STORAGE.

E.G. PATCH RESIDENCE TIME

MVT PREDICTS THAT ANIMALS SHOULD LEAVE A PATCH WHEN THEIR
MARGINAL RATE OF RETURN IN A PATCH FALLS TO THE AVERAGE FOR
THE HABITAT.

ANIMAL DON'T DO FORMAL CALCULUS.

ONE WAY TO IDENTIFY POTENTIAL RULES OF THUMB, IS TO IDENTIFY
SIMPLE CORRELATES OF THE VARIABLES THAT HAVE TO BE ESTIMATED.

THE MARGINAL RATE OF RETURN IN A PATCH SHOULD BE PROPORTIONAL
TO THE DENSITY OF PREY IN THE PATCH.

AS DENSITY FALLS, CATCH RATE FALLS.

THEREFORE CATCH RATE CAN BE USED AS A CORRELATE OF DENSITY.

BUT ESTIMATING CATCH RATE ALSO REQUIRES SUBSTANTIAL
INFORMATION.

THE INVERSE OF CATCH RATE (FEEDING RATE) IS THE AVERAGE TIME
BETWEEN CAPTURES.

CATCH RATE = CAPTURES/TIME

1/CATCH RATE = TIME/CAPTURE

TIME/CAPTURE µ 1/DENSITY

THUS BOTH CATCH RATE AND THE DENSITY OF PREY IN A PATCH CAN
BE ESTIMATED FROM THE TIME BETWEEN SUCCESSIVE CAPTURES.

A SIMPLE RULE BASED ON THIS RELATIONSHIP THAT IS CONSISTENT
WITH THE MVT IS TO LEAVE A PATCH AFTER HAVING GONE SOME TIME
T WITHOUT A CAPTURE. ANIMALS FOLLOWING THIS RULE WOULD LEAVE
ALL PATCHES WHEN THEY HAVE BEEN DEPLETED DOWN TO MORE OR LESS
THE SAME LEVEL.

THIS SIMPLE RULE WILL NOT PRODUCE PERFECTLY OPTIMAL BEHAVIOUR
BUT IT PRODUCES A CLOSE APPROXIMATION.

THE USE OF THIS RULE (OR SOMETHING EQUALLY SIMPLE) SHOULD BE
FAVOURED BY SELECTION BECAUSE THE INFORMATION PROCESSING
REQUIREMENTS WOULD BE WITHIN THE CAPABILITIES OF MOST
ANIMALS.

LEAVING A PATCH (OR SWITCHING AMONG BEHAVIOURS) AFTER A
LENGTH OF TIME DURING WHICH NO RESOURCES ARE OBTAINED IS
TERMED USING A "GIVING UP TIME" (GUT) RULE.

E.G. LIMA'S WOODPECKERS

EXPERIMENTAL TEST OF GUT RULE IN WOODPECKERS SEARCHING FOR
SEEDS HIDDEN IN HOLES DRILLED IN LOGS.

LOGS ALL HAD 24 HOLES AND APPEARED SIMILAR BUT CAME IN 2
TYPES: ONES WITH SEEDS AND ONES WITHOUT SEEDS.

EXPERIMENT ONE, LOGS HAD EITHER 6 OR ZERO SEEDS,

EXPERIMENT TWO,  12 OR 0 SEEDS.

EXP. THREE, 24 OR 0 SEEDS.

IF BIRDS USE A GUT RULE FOR LEAVING PATCHES (LOGS), THEN IT
IS POSSIBLE TO PREDICT HOW MANY HOLES THEY SHOULD SEARCH
BEFORE LEAVING THE PATCH, IN ORDER TO MAXIMIZE THEIR RATE OF
ENERGY GAIN.


FIG.

THEORETICAL PREDICTION FOR EXPERIMENT 1 IS THAT THEY SHOULD
LEAVE A LOG IF THEY HAVE NOT FOUND A SEED IN THE FIRST SIX
HOLES (GUT = 6 HOLES)

EXPERIMENT 2: PREDICTED GUT = 3.

EXPERIMENT 3: PREDICTED GUT = 1.

GUT IS  LOWER FOR EXPERIMENTS 2 & 3 BECAUSE OVERALL RATE OF
RETURN FROM THE ENVIRONMENT IS HIGHER (BETTER PATCHES) &
MARGINAL RETURN WILL EQUAL HABITAT MEAN AFTER A SHORTER PATCH
RESIDENCY TIME.

EXPERIMENTAL RESULTS WERE A CLOSE FIT TO THE PREDICTED VALUES
(6.3; 3.5, 1.7).

THE USE OF RULES OF THUMB BY ANIMALS  WILL DEPEND ON THE
SURVIVAL VALUE OF THE BEHAVIOUR UNDER NATURAL CONDITIONS
(WOODPECKERS EXPLOIT LOGS OF VARIABLE QUALITY, THE SAME
EXPERIMENT MIGHT NOT WORK ON A GOOSE OR A HORSE)

STILL MORE ON FORAGING

ALTERNATIVES TO ENERGY MAXIMIZATION

IT SEEMS CLEAR THAT IF ANIMALS BEHAVE SO AS TO MAXIMIZE THEIR
RATE OF NET ENERGY GAIN, THIS SHOULD LEAD TO A HIGHER
EXPECTED FITNESS THAN WOULD A BEHAVIOUR THAT PRODUCED A LOWER
RATE OF NET ENERGY GAIN, DUE TO CLOSE CONNECTION OF RATE OF
ENERGY GAIN TO GROWTH, SURVIVAL, REPRODUCTION.

HOWEVER, UNDER CERTAIN CONDITIONS MAXIMIZING THE RATE OF NET
ENERGY GAIN (ENET) WILL NOT NECESSARILY  MAXIMIZE EXPECTED
FITNESS.

THIS CAN OCCUR IF THERE ARE ADDITIONAL COSTS OF FORAGING THAT
DIRECTLY INFLUENCE SURVIVAL RATHER THAN ENERGY.

EXAMPLES

1. RISK OF STARVATION WHEN FOOD SUPPLIES ARE VARIABLE.

2. VARIABLE RISK OF PREDATION (DEATH) ASSOCIATED WITH
DIFFERENT FORAGING OPTIONS.

1. RISK OF STARVATION

SUPPOSE THAT THERE ARE TWO HABITATS THAT HAVE SIMILAR
EXPECTED DAILY RATES OF ENERGY GAIN, BUT THAT DIFFER IN THE
VARIABILTY OF THE DAILY RATE OF GAIN.

IF CURRENCY IS THE NET RATE OF ENERGY GAIN, THEN THE HABITATS
SHOULD BE PREFERRED EQUALLY.

BUT IF THERE WERE A SIGNIFICANT PROBABILITY OF ENCOUNTERING A
STRING OF "BAD LUCK" IN THE VARIABLE HABITAT, THE SHORT-TERM
INTAKE RATE COULD FALL BELOW MAINTENANCE REQUIREMENTS.

IN THIS CASE THERE WOULD BE A HIGHER PROBABILITY OF STARVING
TO DEATH OR SUFFERING SOME LONG TERM PHYSIOLOGICAL DAMAGE IN
THE VARIABLE HABITAT.

IN THIS CASE, SELECTION WOULD FAVOUR BEHAVIOURS THAT REDUCED
TIME SPENT IN SUCH HABITATS BECAUSE THEY HAVE A HIGHER DEATH
RATE. THE LESS VARIABLE HABITAT WOULD BE "PREFERRED".

E.G.
SUPPOSE THAT SELECTION PRODUCES BEHAVIOUR THAT MAXIMIZES
EXPECTED FITNESS AT THE END OF SOME TIME PERIOD, SUCH AS A
DAY.

LET X = EXPECTED FITNESS AT THE BEGINNING OF THE DAY.

AT THE END OF THE DAY, EXPECTED FITNESS WILL BE

X + (ENERGY GAIN DURING DAY)
(PROBABILITY OF SURVIVING FOR ONE DAY)

THIS EXPRESSION IS MAXIMIZED BY MAXIMIZING THE PROBABILITY OF
SURVIVAL (OTHER TERMS ARE CONSTANTS).

IF b UNITS OF ENERGY ARE REQUIRED EACH DAY TO SURVIVE THEN
THE PROBABLILTY OF DAILY SURVIVAL IS (1-PROBABILITY OF
GETTING LESS THAN b UNITS).

FOR VALUES OF b THAT ARE LESS THAN THE EXPECTED RETURN (SAME
FOR BOTH) THE PROBABILITY OF GETTING LESS THAN b UNITS OF
ENERGY (DEATH) WILL BE HIGHER IN THE MORE VARIABLE HABITAT.

WHEN THIS IS THE CASE, EXPECTED FITNESS WILL BE GREATER IN
THE LESS VARIABLE HABITAT.

FOR VALUES OF b THAT ARE GREATER THAN THE EXPECTED RETURN,
THE PROBABILITY OF GETTING LESS THAN b (DEATH) WILL BE HIGHER
IN THE LESS VARIABLE HABITAT.

WHEN THIS IS THE CASE, EXPECTED FITNESS WOULD BE MAXIMIZED BY
CHOOSING THE MORE VARIABLE HABITAT.

THE AVOIDANCE OF VARIABLE HABITATS THAT HAVE A RELATIVELY
HIGH PROBABILITY OF STARVATION IS TERMED BEING "RISK AVERSE".

CHOOSING A MORE VARIABLE HABITAT IS TERMED BEING "RISK
PRONE".

"RISK SENSITIVE" BEHAVIOUR CAN BE OBSERVED IN ANIMALS.

E.G. FORAGING JUNCOS (CARACO ET AL.)

TWO ARTIFICIAL HABITATS
#1 - AVERAGE REWARD SIZE = 3
		0 (50%) OR 6 (50%)
#2 - FIXED REWARD SIZE = 3
		(NO VARIATION)

AT 1 DEG NEITHER "HABITAT" PROVIDED AN EXPECTED RATE OF
RETURN SUFFICIENT TO MEET ENERGY DEMANDS (b>MEAN RETURN)

JUNCOS TENDED TO CHOOSE HABITAT #1 (HIGH VARIABILITY)

AT 19 DEG BOTH HABITS PROVIDED AN EXPECTED RETURN THAT WAS
SUFFICIENT TO MEET ENERGY DEMANDS (b<MEAN RETURN).

JUNCOS TENDED TO CHOOSE HABITAT #2 (LOW VARIABILITY).

BUT ESTIMATING VARIANCE IS MORE DIFFICULT THAN ESTIMATING
MEANS.

A SIMPLE GUT RULE WOULD ALSO CAUSE ANIMALS TO LEAVE VARIABLE
HABITATS SOONER AS THEY WOULD HAVE A RUN OF "BAD LUCK"
SOONER. IN ADDITION, AS OVERALL RATE OF ENERGY GAIN FALLS,
THE GUT WILL INCREASE. THIS WILL REDUCE THE VARIATION IN THE
RATE OF RETURN IN THE VARIABLE PATCH (BY TAKING LARGER
SAMPLES) AND INCREASE THE RELATIVE AMOUNT OF TIME SPENT IN
VARIABLE PATCHES, BUT IT WOULD NOT PRODUCE ABSOLUTELY LONGER
RESIDENCE TIMES IN THE MORE VARIABLE PATCHES.

CURRENCIES OF FITNESS

WHY USE (NET ENERGY GAIN AT END OF TIME PERIOD)(PROBABILITY
OF SURVIVING TIME PERIOD) AS A MEASURE OF FITNESS?

I.E. EXPECTED FITNESS = (E)(P)

SIMPLEST CASE

SUPPOSE THAT FORAGING BEHAVIOUR IS CONTROLLED BY A SINGLE
GENE.

AT TIME 0 WITHIN A HAPLOID POPULATION THERE ARE 2 GROUPS
(EACH OF SIZE N) OF CONSPECIFICS.

GROUP 1 HAVE AN ALLELE THAT PRODUCES BEHAVIOUR PATTERN "1",
GROUP 2 HAS AN ALLELE THAT PRODUCES BEHAVIOUR PATTERN "2".

AT TIME T ALL INDIVIDUALS REPRODUCE BY RELEASING THEIR
GAMETES INTO THE ENVIRONMENT AND THEN THE ADULTS DIE.

TOTAL PRODUCTION OF GAMETES IS A FUNCTION OF THE AMOUNT OF
NET ENERGY THAT IS ACQUIRED DURING TIME T.

TYPE 1 INDIVIDUALS THAT SURVIVE TO THE END OF TIME T ACQUIRE
X UNITS OF ENERGY. THERE IS A RISK ASSOCIATED WITH TYPE 1
BEHAVIOUR SUCH THAT ONLY A PROPORTION P1 OF GROUP 1
INDIVIDUALS SURVIVE TO TIME T.

SIMILARLY GROUP 2 INDIVIDUALS THAT  SURVIVE ACQUIRE Y UNITS
OF ENERGY AND A PROPORTION P2 OF THESE SURVIVE TO THE END OF
TIME T.

TOTAL PRODUCTION OF TYPE 1 GAMETES (ALLELE 1) AT TIME T IS:

(N)(P1)(X)

TOTAL PRODUCTION OF TYPE 2 GAMETES (ALLELE 2) AT TIME T IS:

(N)(P2)(Y)

IN THIS CASE THE "EXPECTED FITNESS" OF INDIVIDUALS CARRYING
ALLELE 1 AT TIME 0 IS MEASURED BY THE EXPECTED PRODUCTION OF
GAMETES PER INDIVIDUAL, OR

(N)(P1)(X)/N = (P1)(X)

SIMILARLY FOR INDIVIDUALS CARRYING ALLELE 2 EXPECTED FITNESS
IS

(N)(P2)(Y)/N = (P2)(Y)

IN OTHER WORDS "EXPECTED FITNESS" IS THE PRODUCT OF THE
EXPECTED ENERGY GAIN AND THE PROBABILITY OF SURVIVING TO THE
END OF THE TIME PERIOD.


WHY DO ANIMALS BEHAVE SO AS TO MAXIMIZE THEIR EXPECTED
FITNESS (A STATISTICAL PARAMETER) INSTEAD OF THEIR OWN
PERSONAL SURVIVAL.

I.E. WHAT IS THE ADVANTAGE OF A BIG PAYOFF IF THERE IS A GOOD
CHANCE OF DYING IN THE PROCESS:

WHAT WOULD BE YOUR CHOICE IF FORCED TO CHOOSE BETWEEN TWO
ALTERNATIVES

1. HAVE 10 OFFSPRING IF YOU SURVIVE; PROBABILITY OF SURVIVAL
IS 0.3 (EXPECTED FITNESS= 3)

2. HAVE 2 OFFSPRING IF YOU SURVIVE; PROBABILITY OF SURVIVAL =
1 (EXPECTED FITNESS= 2)

PRESUMABLY YOU WOULD CHOOSE ALTERNATIVE 2 WITH THE LOWER
EXPECTED FITNESS BUT HIGHER PROBABILITY OF PERSONAL SURVIVAL.

BUT NATURAL SELECTION WOULD FAVOUR ANIMALS THAT BEHAVE
ACCORDING TO ALTERNATIVE 1.

SINCE BEHAVIOUR IS GENERALLY CONTROLLED BY GENES, THE EFFECT
OF SELECTION ON BEHAVIOUR CAN BE DETERMINED BY LOOKING AT THE
EFFECT OF SELECTION ON THE GENES THAT CONTROL BEHAVIOUR.

IN THE EXAMPLE ABOVE THERE WERE N COPIES OF EACH ALLELE IN
THE POPULATION AT TIME 0.

AT THE END OF TIME T THERE WERE

(N)(P1)(X) COPIES OF ALLELE 1

AND

(N)(P2)(Y) COPIES OF ALLELE 2

INDIVIDUALS WITH BEHAVIOUR 1 WILL INCREASE IN FREQUENCY
(FAVOURED BY SELECTION) IF THE FREQUENCY OF ALLELE 1
INCREASES.

THIS WOULD OCCUR IF THE RATIO OF THE NUMBER OF TYPE 1 ALLELES
TO TYPE 2 ALLELES IN THE GAMETES PRODUCED AT TIME T IS
GREATER THAN 1, OR

(N)(P1)(X)/(N)(P2)(Y) > 1

OR (P1)(X) > (P2)(Y)

THAT IS, IF THE EXPECTED FITNESS OF TYPE 1 INDIVIDUALS
IS GREATER THAN THE EXPECTED FITNESS OF TYPE 2
INDIVIDUALS.

THIS WILL OCCUR EVEN IF THE PROBABILITY OF SURVIVAL FOR TYPE
1 BEHAVIOUR IS LOWER THAN THAT FOR TYPE 2 BEHAVIOUR, IF THE
EXPECTED RETURN P1 IS SUFFICIENTLY LARGER THAN P2.

IN OTHER WORDS NATURAL SELECTION CANNOT PRODUCE BEHAVIOUR
THAT MAXIMIZES THE SURVIVAL OF AN INDIVIDUAL INDEPENDENT OF
THE INDIVIDUAL'S REPRODUCTIVE OUTPUT.

CONSEQUENTLY, ANIMALS CONTINUOUSLY PERFORM RISKY BEHAVIOURS
THAT REDUCE THEIR PROBABILITY OF INDIVIDUAL SURVIVAL
BECAUSE THESE BEHAVIOURS INCREASE THEIR EXPECTED
FITNESS (A STATISTICAL MEASURE).

YOU MAY OR MAY NOT FIND IT A SOBERING THOUGHT THAT YOUR OWN
BRAIN TREATS YOU AS A STATISTIC AND DOES NOT REALLY HAVE YOUR
BEST INTERESTS AT HEART.

2. RISK OF PREDATION AND FORAGING

IN ADDITION TO THE RISK OF STARVATION ASSOCIATED WITH EITHER
HIGHLY VARIABLE OR LOW QUALITY ENVIRONMENTS, THERE IS A RISK
OF PREDATION ASSOCIATED WITH FORAGING FOR MANY ANIMALS. THE
RISK OF PREDATION IS OFTEN HIGHER WHILE FORAGING THAN IT IS
WHEN ANIMALS ARE INACTIVE.

THIS IS BECAUSE FORAGING MAY INCREASE EXPOSURE (E.G.
VISIBILITY) TO PREDATORS, INCREASE THE DISTANCE FROM A SAFE
HAVEN, OR DECREASE THE LIKELIHOOD OF DETECTING A PREDATOR
BECAUSE OF THE PREOCCUPATION WITH SEARCHING FOR FOOD.

IF ANIMALS FORAGE SO AS TO MAXIMIZE EXPECTED FITNESS
(FUNCTION OF BOTH ENERGY GAIN AND PROBABILITY OF SURVIVAL)
THEN ANIMALS SHOULD AVOID FORAGING AREAS WITH A HIGH RISK OF
PREDATION IF ALL FORAGING AREAS OFFER SIMILAR RATES OF
ENERGY GAIN OR IF AREAS WITH HIGHER RISK OF PREDATION HAVE
LOWER RATES OF ENERGY GAIN.

WHEN THE AREAS OFFERING THE HIGHER RATES OF ENERGY GAIN ALSO
HAVE THE HIGHER PREDATION, THE USE OF THESE AREAS WILL DEPEND
OF THE FORM OF THE RELATIONSHIP BETWEEN ENERGY GAIN AND RISK.

FIG.

E.G.
SQUIRRELS FEEDING ON COOKIES ON PARK BENCHES CAPTURE SINGLE
COOKIE AND RETURN TO TREE TO EAT. WHEN FINISHED RETURN TO
BENCH FOR ANOTHER COOKIE. RATE OF ENERGY GAIN WOULD BE HIGHER
IF STAYED AND ATE COOKIES CONTINUALLY, BUT PREDATION RATE
WOULD BE HIGHER AS WELL.

FIG.

E.G.
JUVENILE BLUEGILL SUNFISH GROW FASTEST WHEN FORAGING IN THE
OPEN ON BENTHOS BUT IN THE PRESENCE OF PREDATORS THEY FORAGE
IN COVER AND EXPERIENCE LOWER GROWTH RATES (BUT HIGHER
SURVIVAL).

FIG.


E.G.
BIRDS AT FEEDER ARE LESS VIGILANT (SPEND MORE TIME FORAGING
AND LESS TIME SCANNING FOR PREDATORS) ON COLD DAYS.

NUTRIENT CONSTRAINTS

IN SOME CASES OF FORAGING IT IS NOT SUFFICIENT TO JUST
MEASURE THE RATE OF ENERGY GAIN AS THERE MAY BE CERTAIN
MINIMAL REQUIREMENTS FOR CERTAIN NUTRIENTS THAT MUST BE MET.

SUCH REQUIREMENTS ARE TERMED "CONSTRAINTS" BECAUSE THEY LIMIT
THE FORAGING OPTIONS AVAILABLE. HOWEVER, FORAGING BEHAVIOUR
MAY STILL LEAD TO ENERGY MAXIMIZATION WITHIN THIS LIMITED
SUBSET OF OPTIONS.

E.G. MOOSE FORAGING

FIG.

OPTIMALITY MODELS IN GENERAL

USEFUL BECAUSE THEY FORCE US TO THINK EXPLICITLY ABOUT HOW
NATURAL SELECTION MIGHT HAVE GENERATED A PARTICULAR BEHAVIOUR
PATTERN.

BUILDING OF THE MODEL REQUIRES THAT WE DEFINE A CURRENCY OF
FITNESS BEFORE PREDICTIONS CAN BE MADE.

IF THE OUTCOME OF A TEST OF AN OPTIMALITY MODEL IS SUCCESSFUL
THAN WE HAVE AN UNDERSTANDING OF HOW SELECTION OPERATES IN A
PARTICULAR CASE, AND THIS MAY BE MORE GENERALLY APPLICABLE TO
OTHER SPECIES AND OTHER BEHAVIOURS.

IF THE PREDICTIONS ARE NOT SUPPORTED BY THE TEST WE CAN USE
THE INFORMATION TO RE-EXAMINE  THE MODEL, MODIFY ITS
ASSUMPTIONS ABOUT THE CURRENCY OF FITNESS AND GENERATE NEW
PREDICTIONS.