predator-prey interaction in structured models glenn ledder j. david logan university of...

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Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln [email protected]

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Page 1: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

Predator-Prey Interaction in Structured Models

Glenn LedderJ. David Logan

University of [email protected]

Page 2: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

Simple Holling type II dynamics

0

5

10

15

20

25

30

35

0 50 100 150

prey

pred

ation

per

pre

dato

r

Page 3: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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

Page 4: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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

Page 5: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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

Page 6: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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

Page 7: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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

Page 8: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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

Page 9: Predator-Prey Interaction in Structured Models Glenn Ledder J. David Logan University of Nebraska-Lincoln gledder@math.unl.edu

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