Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Lotka-Volterra Model: Predator-Prey Dynamics and Stability, Study notes of Forestry

The lotka-volterra model, a mathematical representation of predator-prey relationships. The model examines the interactions between prey (h) and predators (p), their population growth rates, and the stability of the system. Tanner (1975) discussed the model's stability, focusing on the critical point where predator and prey isoclines cross, which can result in stable or unstable equilibria. The document also discusses the impact of predator resource limitations and prey self-limitation on the system.

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

koofers-user-aq3
koofers-user-aq3 šŸ‡ŗšŸ‡ø

5

(1)

10 documents

1 / 49

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Lotka
‐
Volterra Model
Lotka
Volterra
Model
H = number of
p
re
y
dH/dt
=
rH
‐
bHP
py
r = prey population growth rate
b = attack rate
dH/dt
r
H
bHP
P = number of predators
dP/dt =
cHP
c = predator population growth
rate due to predation
k = rate of
p
redator decline in
dP/dt
=
cHP
‐
p
absence of prey
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20
pf21
pf22
pf23
pf24
pf25
pf26
pf27
pf28
pf29
pf2a
pf2b
pf2c
pf2d
pf2e
pf2f
pf30
pf31

Partial preview of the text

Download Lotka-Volterra Model: Predator-Prey Dynamics and Stability and more Study notes Forestry in PDF only on Docsity!

Lotka

‐Volterra Model

Lotka

Volterra

Model

H = number of prey

dH/dt =

r H

bHP

p^

y

r = prey population growth rateb = attack rate

dH/dt

r^

H

bHP

P = number of predators

dP/dt =

cHP

k P

c = predator population growthrate due to predationk = rate of predator decline in

dP/dt =

cHP

k^

P

p

absence of prey

Lotka

‐Volterra Model

Lotka

Volterra

Model

dH / dt < 0dP / dt < 0

dH / dt < 0dP / dt > 0

P

r/b P

k/c

dH / dt > 0dP / dt > 0

dH / dt > 0dP / dt < 0

H

StabilityStability

•^

Tanner (1975 Ecology 56:855)

•^

Explored features of this model to find generalproperties, particularly model stability

•^

Does the ā€œcritical pointā€ where predator andprey isoclines cross produce a:p

y^

p

-^

stable equilibrium (ā€œfocus pointā€)

-^

limit cyclelimit cycle

-^

unstable

•^

predator growth / prey growth rates (s/r)

•^

predator growth / prey growth rates (s/r)(note

c

s

Tanner (1975)Tanner

Stable focus whenthe critical pointfalls to the right offalls to the right ofthe prey zeroisocline peak for all values of s/r

Tanner (1975)Tanner

When the criticalpoint falls to theleft of the prey zeroleft of the prey zeroisocline peak, 2)

limit cycle

if s/r

small

Tanner (1975)Tanner

When the criticalpoint falls to theleft of the prey zeroleft of the prey zeroisocline peak, 3) unstable focus

if

s/r small and K isvery large –

y^

g

extinction; nocoexistence

Tanner (1975)Tanner

Once again, sincethe critical pointfalls to the right offalls to the right ofthe prey zeroisocline peak, a stable focus resultsfor all values of s/r

Tanner (1975)Tanner

Again, since thecritical point falls tothe right of thethe right of theprey zero isoclinepeak, a stable results for all valuesof s/r

Tanner (1975)Tanner

The preypopulation can get

Unstable focus

ā€œstuckā€ at very lowdensity unlesspredation rates

Stable focus

predation rates drop substantially ,called a predatorpitpit

Stable focus ā€œP

d t

Pitā€

ā€œP

redator Pitā€

Tanner 1975Tanner

•^

Complex model behavior nearly any outcome!Complex

model behavior, nearly any outcome!

•^

So what? Is this useful?

Tanner 1975Tanner

•^

Hypothesized that stable prey species wereHypothesized

that stable prey species were

either strongly self

‐limited (e.g., by

territoriality) or the prey population growthterritoriality), or the prey population growthrate was less than that of the predator^ –

Prey growth rate appeared higher (s/r < 1) for:– Prey growth rate appeared higher (s/r < 1) for:

•^

sparrow hawk / house sparrow and

-^

Mink / muskratMink

/ muskrat

-^

And

both prey species thought to be self

‐limited

(sparrows: food or breeding sites; muskrats:( p

g^

territories)

Tanner 1975Tanner

•^

Hypothesized that stable prey species wereHypothesized

that stable prey species were

either strongly self

‐limited (e.g., by

territoriality) or the prey population growthterritoriality), or the prey population growthrate was less than that of the predator^ –

Prey growth rate appeared similar(s/r = 1) for– Prey growth rate appeared similar(s/r = 1) for

•^

Lynx / snowshoe hare

-^

Hare and lynx show cyclesHare and lynx show cycles

Model assumptionsModel

assumptions

•^

No time lags

•^

No prey refuges

•^

Predator searching constant, not affected byPredator

searching constant, not affected by

external factors

•^

No differences in prey susceptibility

•^

No differences in prey susceptibility

Optimal Foraging TheoryOptimal

Foraging Theory

-^

How does a predator choose which prey to hunt forHow

does a predator choose which prey to hunt for

and for how long?

-^

Theory developed to identify the optimal choices

y^

p^

y^

p

based on profitability of prey items or foragingpatches where

profitability = energy / handling time

-^

The optimal diet or foraging patches are thosemaximizing profitability

-^

Perfect match unlikely because animals must explorechoices to learn profitabilities and profitabilitieschange through time