Unformatted text preview: Optimality theory
body of theory used to predict animal behavior under assumption that behavior evolves under natural selection to maximize fitness. Steps to building optimality models 1. state goal of optimization 2. state conditions and constraints of model 3. state terms of model 4. 4 run model & generate predictions d l d 5. test predictions 6. evaluate conditions and terms 7. revise model assumes that it does. Optimality theory does not seek to prove that natural selection shapes behavior, but rather Steps to building model of lift 1. state goal of optimization How to optimize lift in an airplane? e.g. optimize lift 2. state conditions & constraints of model
e.g. no turbulence where air moves over wing 3. state terms of model
e.g. what is considered to affect lift 4. run model & generate predictions
e.g. wing of shape X generates lift of amount Y 5. test predictions
e.g. does shape X generate lift Y?
Lift: component of the total aerodynamic force acting on an airplane that is perpendicular to the relative wind and that for an airplane constitutes the upward force that opposes the downward force of gravity 6. evaluate conditions and terms
e.g. no turbulence over wing 7. revise model Problem of Optimal Patch Time Step 1. State goal of optimization Goal is to optimize patch time time. Patch time is called decision variable variable.
When should a bee leave one patch and move on to the next? 1 Conditions of Patch Time Model Step 2. State conditions of model
x x x x x x x Tt x x x x x xx All foraging models include the following conditions: a. `currency' of fitness b. description of resource environment c. foraging constraints a. currency of fitness: rate of energy gain (E/t) b. resource environment: includes 1 prey type prey occur in patches which: are discrete do not replenish have same prey density travel time of t units (Tt) to move to new patch c. foraging constraints: predator moves at finite speed Energy Gained (E) One more condition in this case:
within patch, amount of energy gained over time follows a fixed, predetermined pattern gain curve Time in patch (Tp) Gain curve specifies how currency of fitness changes over the range of decision variable Tp Energy Gained (E) Restating goal of optimization
For patch of given density and travel time Tt, find patch time, Tp* that maximizes rate of energy gain.
Time in patch (Tp) We will assume gain curve of this form. 2 Step 3. State terms of model marginal value rule A predator should stay in patch until marginal rate of energy gain drops to average rate of energy gain (R*) in habitat We need to know: marginal rate of energy gain average rate of energy gain in the habitat (R*) Step 3. State terms of model marginal value rule A predator should stay in patch until marginal rate of energy gain drops to average rate of energy gain (R*) in habitat We need to know: marginal rate of energy gain average rate of energy gain in the habitat (R*) Marginal rate of energy gain is rate of energy gain in next `microinstant' of time in patch. Marginal rate of energy gain is equal to slope of line tangent to gain curve at a given value of Tp Energy Gained (E) Energy Gained (E) Tp Time in patch (Tp) Tp(1) Tp(2) Tp(3) Time in patch (Tp) Which Tp has highest marginal rate of energy gain? marginal value rule A predator should stay in patch until marginal rate of energy gain drops to average rate of energy gain (R*) in habitat We need to know: marginal rate of energy gain average rate of energy gain in the habitat (R*) marginal value rule A predator should stay in patch until marginal rate of energy gain drops to average rate of energy gain (R*) in habitat We need to know: marginal rate of energy gain average rate of energy gain in the habitat (R*) 3 Average rate of energy gain in the habitat (R*) is equal to: R* = E [Tp*]
Tp* + Tt To solve this kind of problem, we need calculus. But... there is an alternative graphical solution. Oops! There seems to be a problem problem... To determine Tp*, we have to know R*. But to determine R*, we have to know Tp*... Energy Gained (E) Energy Gained (E) Travel time (Tt) Travel time (Tt) Time in Patch (Tp) Time in Patch (Tp) Graphical solution involves extending the xaxis behind the origin, and defining it as the travel time axis. Mark travel time between patches at a point behind the origin. Line extending from that point across the yaxis has slope = E/(Tt + Tp) Energy Gained (E) Energy Gained (E) Yes! No! Travel time (Tt) Time in Patch (Tp) Travel time (Tt) Time in Patch (Tp) Q: What line has slope equal to average rate of energy intake (R*) when patch time is optimal? A: The line just tangent to the curve At that point and only that point, the marginal rate of energy gain equals the average rate of energy gain in the habitat. 4 Energy Gained (E) Energy Gained (E) Travel time (Tt) Tp* Time in Patch (Tp) Travel time (Tt) Tp* Tp*
Time in Patch (Tp) Drop a line down from the tangent to the xaxis and VOILA! we have found the optimal patch time! What happens if it takes longer to get from one patch to another (longer Tt)? Tradeoff!
Patch time model involves a tradeoff between time to find next item in current patch and time needed to travel to next patch. Exact details of tradeoff vary from example to example. Hummingbirds and Flowers
A web exercise on optimal foraging in the form of a game: http://bio150.chass.utoronto.ca/foraging/ Optional Exercise!
For class participation credit, play the Application of Optimal Patch Time Model Copulation duration in dung fly, Scatophaga stercoraria hummingbird game: Design a set of `rules' used to leave a patch. Collect `data' on how the different rules perform. data perform Report results and interpretation in 12 typed pages.
Hints for full credit: Be systematic in how rules vary. Use multiple replicates per rule.
Luc Viatour Males defend dung patties as territories Females lay eggs and larvae develop in patties Female is `patch' Male's `decision variable' is copulation duration `Currency of fitness' is % eggs fertilized 5 Copulation duration in dung flies
% Eggs fertilized Increasing ratio of males to females should increase search time for next female... ... which should increase duration of copulation with current female. predicted Search & guard time ( 156. 5 min) Time in copula (min) observed
C. Hedgcock We tested prediction in our lab using walnut flies. Tradeoff between sperm transferred to one female and sperm for next female AlonsoPimentel and Papaj (1996) Birds and mealworms (Cowie 1977) Effect of Sex Ratio on Mating Duration
Copulation Duration (sec) D
n = 16 n=7 System: Great Tits feeding on mealworms hidden in sawdustfilled cups on artificial tree. Cups could be covered or not; covers had to be removed by birds before food was obtained. 1000 800 600 400 200 0
n=2 n = 31 n = 25 <1 1 >1 >2 >3 Local Sex Ratio (male:female) What does cover do to conditions of model?
Energy Gained (E) Fig. 10.10 Travel time (Tt) Time in Patch (Tp) Adding a cap can be considered to extend travel time. 6 Results Placement of cover increased time needed to obtain mealworms... ... and residence increased at a given patch... ... meeting predictions! Crows and whelks: BE SURE TO READ! READ! Birds adjust height of drop so as to minimize the total height from which they drop the whelk.
Pages 210211 in TEXT Let's Review!
Energy (E) marginal value rule states: A predator should stay in patch until its marginal rate of energy gain drops to the average rate of energy gain in the entire habitat Time in Patch (Tp) Gain curve shows how net energy gained changes as a forager's time in a patch increases. Energy Gained (E) Energy (E) Travel time (Tt) Time in patch (Tp) The marginal rate of energy gained for any Tp is defined by slope of a line tangent to gain curve at Tp It has units E/Tp Time in Patch (Tp) The average rate of energy gained in the habitat (R) for any Tp is defined by the slope of a line extending from Tt to the gain curve at Tp. It has units E/(Tp+Tt) 7 Energy (E) Do you get it?
What happens if you add competitors to an entire habitat of patches? p
Travel time (Tt) T p* Time in Patch (Tp) Can you illustrate using graphical model? The optimal patch time occurs where slope of line defining marginal rate of energy gain = slope of line defining average rate of energy gained This occurs at point where line drawn from travel time is tangent to gain curve. Next model... The Basic Diet Choice Model Problem: When should a forager specialize?
x y x yx x x x y y Conditions of the model: Two (2) prey: Energy content (cal): Handling time (sec): Rate of energy intake: Time spent searching: Encounter rate: 1 (beans) and 2 (rice) E1 > E2 h1 < h2 E1/h1 > E2/h2 Ts 1, 2 Decision Variable: Resource Envmt: Envmt: Constraints: # prey types taken density of prey types animal eats one at a time Currency of Fitness: rate of energy gain (so Ts 1 = number of Type 1 prey found) Specialist rate of energy intake: Specialize when E/T [specialist] > E/T [generalist] 1E 1 E Ts 1E1 = = T Ts + Ts 1h1 1 + 1h1
Generalist rate of energy intake: 1 E1 1E1 + 2 E 2 > 1 + 1h1 1 + 1h1 + 2 h 2
Rearranging... 1E1 + 2 E 2 E Ts 1E1 + Ts 2 E 2 = = T Ts + Ts 1h1 + Ts 2h 2 1 + 1h1 + 2 h 2 1 1 >( E1 h 2)  h1 E2 8 1 1 >( E1 h 2)  h1 E2 Fig. 10.6 An Unintuitive Prediction Emerges!
Specialization does not depend on the rate of encounter with prey of type 2. LOBSTERS AND HOT DOGS!! Challenging Assumptions
Assumption: "Rate of energy intake is maximized." Counterexample: Moose feed on both aquatic and terrestrial plants, even though terrestrial plants have more energy per unit weight than aquatic ones, and even though terrestrial plants are very abundant. Why? Because aquatic plants satisfy a requirement for sodium sodium. Fig. 10.12 Are moose maximizing rate of energy intake or rate of sodium intake? Belovsky, 1978
Inta of aquatic plants (g) ake Intake of aquatic plants (g) energy minimum constraint line energy minimum constraint line ?
sodium minimum constraint line sodium minimum constraint line rumen maximum constraint line Intake of terrestrial plants (g) ? rumen maximum constraint line Intake of terrestrial plants (g) The mixture of aquatic and terrestrial plants is a function of energy, sodium and room in their rumen. The moose must eat a mixture of aquatic and terrestrial plants that lies within the grey area. But where within that area? 9 Intak of aquatic plants (g) ke energy minimum constraint line Myths about optimal foraging theory Assumes that animals know algebra or calculus. NO!
sodium minimum constraint line rumen maximum constraint line ** Attempts to show that animals are perfectly adapted. NO!! Proves that behavior is shaped by natural selection. FOR PETE'S SAKE... NO! Intake of terrestrial plants (g) In fact, the moose attempts to maximize energy within the constraints of sodium needs and rumen volume. 10 ...
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 Spring '09
 Papaj
 Optimal foraging theory, energy gain, Prey switching, Energy Gained

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