The model Central place foraging

1 model

1.1 basic math: single stage of processing
1.2 multiple components , multiple stages of processing
1.3 assumptions
1.4 predictions

the model
basic math: single stage of processing

the effect of flattening utility curve,



u
(
t
)


{\displaystyle u(t)}

whilekeeping constant procurement , processing times



(

x

0


,

x

1


)


{\displaystyle (x_{0},x_{1})}

. when difference between field processing , transporting whole items



(

y

1


,

y

0


)


{\displaystyle (y_{1},y_{0})}

reduced, should expect increase in transport time atwhich processing occur. forager should process items when transport time central place exceeds threshold. (adapted metcalfe , barlow 1992.)

the goal of field processing model forager maximize return rate per roundtrip home base patch. model typically solves amount of travel time makes worthwhile process resource stage. determine this, need relate benefit of processing , time spent processing travel time. let

z
=


{\displaystyle z=}

point on transport-time axis field processing become profitable

x

0


=


{\displaystyle x_{0}=}

time procure unprocessed resources

x

1


=


{\displaystyle x_{1}=}

time procure , process load of resources

y

0


=


{\displaystyle y_{0}=}

utility of load without field processing

y

1


=


{\displaystyle y_{1}=}

utility of load field processing

the relationship specified by:

z
=




y

0



x

1


−

y

1



x

0





y

1


−

y

0







{\displaystyle z={\frac {y_{0}x_{1}-y_{1}x_{0}}{y_{1}-y_{0}}}}

with values utility , time of processed



(

y

1


,

x

1


)


{\displaystyle (y_{1},x_{1})}

, unprocessed loads



(

y

0


,

x

0


)


{\displaystyle (y_{0},x_{0})}

, can solve



z


{\displaystyle z}

. right hand side of equation proportion of relative utility*time utility. 2 conditions must satisfied. first, processed load must have higher utility unprocessed load. second, return rate of unprocessed load must @ least return rate processed load. formally,

if




x

1


>

x

0




{\displaystyle x_{1}>x_{0}}






y

1


>

y

0




{\displaystyle y_{1}>y_{0}}

.

if




y

0


<

y

1




{\displaystyle y_{0}<y_{1}}

,






y

0



x

0




≥



y

1



x

1






{\displaystyle {\frac {y_{0}}{x_{0}}}\geq {\frac {y_{1}}{x_{1}}}}

.

multiple components , multiple stages of processing

many resources have multiple components can removed during processing increase utility. multistage field processing models provide way calculate travel thresholds each stage when resource has more 1 component. 1 increases utility per load, time needed procure complete load increases.

the benefit of each stage of processing is:

y

j


=




∑

i
∈

s

j





a

i



b

i





∑

i
∈

s

j





b

i







{\displaystyle y_{j}={\frac {\sum _{i\in s_{j}}a_{i}b_{i}}{\sum _{i\in s_{j}}b_{i}}}}

where

a

j


=


{\displaystyle a_{j}=}

utility of resource component j

b

j


=


{\displaystyle b_{j}=}

proportion of package composed of resource component j prior processing

y

j


=


{\displaystyle y_{j}=}

utility of load @ field-processing stage j

the cost in terms of time each stage of processing is:

x

j


=

(


l

p

∑

i
∈

s

j





b

i





)


(
m
+

∑

i
∉

s

j





d

i


)



{\displaystyle x_{j}=\left({\frac {l}{p\sum _{i\in s_{j}}b_{i}}}\right)\left(m+\sum _{i\notin s_{j}}d_{i}\right)}

where

d

j


=


{\displaystyle d_{j}=}

time required remove resource component j

l
=


{\displaystyle l=}

weight of optimal load size transport

p
=


{\displaystyle p=}

weight of unmodified resource package

m
=


{\displaystyle m=}

time required handle each resource package

x

j


=


{\displaystyle x_{j}=}

total handling , processing time required reach each stage j of processing

now these values can used calculate




z

j




{\displaystyle z_{j}}

, travel threshold processing stage j. in addition resource multiple components, same model generalizes resource multiple stages, each of composed of multiple resources, each of can removed independently of each other (i.e., no additional cost). model can further generalized case multiple components additional costs can removed in multiple stages of processing through recursion.

assumptions

transport decay curves demonstrate reduction in return rates (cal/hour) experienced central place forager function of round trip travel time. travel threshold field processing models,




z

j




{\displaystyle z_{j}}

represents travel time @ processing next stage provide higher return rates, indicated intersection of decay curves 2 sequential stages of processing. shaded areas represent optimal extent of processing travel time increases.

this model rests on number of assumptions. important listed here.

individuals attempt maximize rate of delivery per round trip * packages have @ least 2 components different utilities
the optimal load size less or equal resources available
time spent away camp comes opportunity cost, time spent in camp not. there no cost processing in camp.

predictions

there 3 key predictions field processing model.

transport decay curves demonstrate reduction in return rates (cal/hour) experienced central place forager function of round trip travel time.