Basic math: single stage of processing Central place foraging

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}}}}

.