predator-prey interaction in structured models glenn ledder j. david logan university of...
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Predator-Prey Interaction in Structured Models
Glenn LedderJ. David Logan
University of [email protected]
Simple Holling type II dynamics
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prey
pred
ation
per
pre
dato
r
Simple Holling Type II Dynamics
• Time is split between searching and feeding
U – prey density R(U) – overall predation rates – search speed h – handling time
------- = --------- · --------- · ------- foodtotal t
search ttotal t
spacesearch t
foodspace
UshRfUR ),()(
search ttotal t
feed ttotal t--------- = 1 – -------
hRhRf 1),(
shU
sUR
1
Simple Holling Type III Dynamics
• Time is split between searching and feeding• Searching is density-dependent
U – prey density R(U) – overall predation rates – maximum search speed h – handling time
UUd
sUhRfUR
),()( hRhRf 1),(
22
2
22
2
2
2
UD
SU
UAUD
SU
hsUUd
sUR
Simple Holling Type II Dynamics
• Time is split between searching and feeding
U – prey density R(U) – overall predation rates – search speed h – handling time
UshRfUR ),()( hRhRf 1),(
shU
sUR
1
Holling type II—Multiple Prey
• Time is split between searching and feeding
Ui – prey densities Ri(Ui) – overall predation rates
si – search speeds hi – handling times
iiii UshRfUR ),()( jjRhhRf 1),(
jjj
iii Uhs
UsR
1
Full Model – 1 Predator Class
Ui – prey densities Ri(Ui) – overall predation rates
si – search speeds hi – handling times ci -- conversion
G – per capita growth M – per capita mortality
PUhs
UsUUG
dt
dU
jjj
iiiii
i
1
)(
PPMPUhs
Usc
dt
dP
jjj
iii )(1
Multiple Prey and PredatorsUi – prey densities Pk – predator densities
sik – search speeds hik – handling times
cik – conversion efficiencies
kj jjkjk
kiikiii
i
Uhs
PUsUUG
dt
dU
1)(
kkkk
j jjkjk
i iikikk PPMPUhs
Usc
dt
dP)(
1
Continuous Structurex – prey class y – predator classu(x, t) – prey density p(y, t) – predator densitys(x, y) – search speed h(x, y) – handling timeI(y) – set of prey consumed by predators of size yK(x) – set of predators capable of consuming prey of size xdF(x, y, t) – predation of (y, y+dy) on (x, x+dx)
iiii UshRfUR ),()( jjRhhRf 1),(
jjj
iiii Uhs
PUsf
1