Part II Population Biology

Nik CunniffeDepartment of Plant Sciences

njc1001@cam.ac.uk

Spatial Heterogeneity IInteractions between individuals

Introduction

• So far have totally ignored spatial effects, butintuition suggests space must be important

• Add spatial heterogeneity to models of: 1. host-parasitoid dynamics (cf. lecture three)2. competition (cf. lecture two)

• Today concentrate on “local” interactions(i.e. between individuals, not populations)

• Key message: Spatial effects can stabilise interactions that would be unstable otherwise

Host-parasitoid interactions• 5% of all Metazoa (Animalia) spp are parasitoids• ~70000 species; 10% of all insects• Parasitoids lay eggs in/on their host (often insects) • Juvenile parasitoids kill host when they emerge• Most parasitoids are found in two orders

• Diptera (two-winged flies)• Hymenoptera (sawflies, bees, wasps, ants)

Why model parasitoids?

• Ecologically important (as so v. widespread)

• Biological pest control

• Simple life cycles• discrete generations• only adult females lay eggs• oviposition immediately follows attack• often only attack larval stage of host

• Easy to study experimentally

Why model parasitoids?

• “Mixture” of predators and parasites• like predators, they kill their victims• like parasites, only require a single host

• Foraging is an ubiquitous interaction: most organisms have to search for something

(food, prey, mates, breeding sites, etc.)

• But these resources are rarely homogeneous

• Qu: How does spatial variability affect dynamics?

Modelling: generic structure

To find the number of hosts next yearNumber of progeny per healthy host

x Number of hosts that escape parasitism

To find the number of parasitoids next yearNumber of new parasitoids per parasitised host

x Number of hosts that are parasitised

Introduce “escape fraction”, f=> Number of hosts that escape parasitism = f N=> Number of hosts that are parasitised = ( 1-f )N

Modelling: generic structure

1

1

( )

(1 ( ))t t

tt

t

t f P

f PHc

H H

P

• t = generation (discrete time model)• Ht = juvenile hosts; Pt = parasitoids• = host rate of increase (density-independent)• c = average surviving progeny per parasitoid• f(Pt) = fraction of hosts that escape parasitism

(written as f(Pt) to emphasise that depends on Pt)Hassell (2000)

Escape fraction: what is f(Pt)?

Do not worry about details of this derivation...only for interest

Let q(t) = Prob(particular host parasitised by time t)

Assume Pt = Total number of parasitoids = Prob(parasitoid lays eggs in a single host per time)Parasitoids attack totally at random

Model

If spend a total of T units of time searching for hosts then

Means the escape fraction f is f = 1 – q =

(1 ) ( ) 1 tPtt

dqq q eP t

dt

( ) 1 1 wheret tPT aPq T e e a T taPe

• Parasitoids encounter hosts randomly (see last slide)

• => escape is Poisson distributed

• What could possibly go wrong?!

Nicholson Bailey model

1

1 (1 )

t

t

a

a

Pt t

tP

t

H H

P cH e

e

( ) taPtf P e

Nicholson & Bailey (1935)

Nicholson Bailey model

• Nicholson Bailey model is unstable

• Due to delayed density dependence (strongly

destabilising)

1

1 (1 )

t

t

aPt t

aPt t

H H e

P cH e

Nicholson & Bailey (1935)

• Stabilising the N-B model was quite the growth industry in the 1970s and 1980s

• Here concentrate on spatial distribution of hostsand consequent aggregation of parasitoids

• Model via proxy of altering the escape fraction, f(Pt)

Stabilising Nicholson Bailey model

1

1 (1 )

t

t

aP

P

t t

ta

t

H H

P cH

e

e

Behavioural aggregation

• This is “behavioural aggregation”

• Two species-dependent mechanisms • attractant (+ve response to chemical stimuli)• arrestant (less dispersal away from “good” areas)

• Some patches have morehosts than

others

• Parasitoids exploit thesepatches preferentially

Behavioural aggregation

• Negative Binomial (May, 1978) modelk = degree of heterogeneity/“clumping parameter”

• Smaller k => more clumping ( tends to Poisson)

Black = dataGrey = -ve Binomial (k=0.28)White = Poisson

Escape fraction ( ) 1k

tt

aPf P

k

k

Spatial heterogeneity stabilises Nicholson Bailey model

Key conclusionIf k < 1, model predicts

stable interaction

solid: stabledotted: unstablea,b,c: parameter sets

Sufficient clumping stabilises “unstable” interactionMay (1978)

Experimental test of these ideas

1

1

1

1 1

k

tt t

k

tt t

aPH H

k

aPP cH

k

Experimental detailsColeoptera spp beetlePteromalid parasitoidTwo environments (50 beans each) - Non-patchy (scattered randomly) - Patchy (each in individual container

with restricted access)

Hassell & May (1988)

Summary (host-parasitoid models)• Behavioural aggregation stabilises unstable model

• Pseudo-interference (wasted effort in high density areas attacking hosts which are already parasitised)

• So some hosts escape each generation

• Allows host population to recover from low levels

• Not the only “solution” to N-B instability, e.g.• density-dependent host reproduction• alternate hosts for parasitoid• other possibilities detailed in Hassell (2000)

Competition

• Second part of lecture concentrates on competition

• Introduce a problem and a theoretical “solution”• paradox of diversity • competition-colonisation trade-off

• Use the models to examine habitat loss and subsequent “Extinction Debt” (cf. conservation)

• But first a reminder of the key results of non-spatialmodels of competition (cf. Lecture Two)

Recall: Lotka-Volterra competition

11 1 2

22 2 2 1

12

1

1

1

dNr N N

dtd

N NN

rdt

Coexistence only if “intra > inter” i.e. 12,21< 1

ij effect of species j on species i

Apologies for small typo in handout

Paradox of diversity

• Strong competition leads to competitive exclusion

• Generic: theory can be taken further (non-spatially)given n species, and k resources, at mostk species can persist (when n>k)

• Paradox of Diversity: many environments havefar fewer distinct resources than species(e.g. a lake with hundreds of species, but only few limiting factors such as nutrients, light, …)

• Spatial heterogeneity is one plausible resolutionfor this paradox (space adds extra niche(s))

Spatial Lotka-Volterra competition

Bolker, Pacala & Neuhauser (2003)

Spatial Lotka-Volterra competition

• Stochastic model with individuals fixed in spaceat the points of a 2D square lattice

• Death rate depends on density of both species inlocal neighbourhood (c.f. non-spatial equations)

• Vacant cells filled by reproduction from a randomindividual within the local neighbourhood

• Behaviour independent of local neighbourhood shape

Bolker, Pacala & Neuhauser (2003)

Spatial Lotka-Volterra competition

Bolker, Pacala & Neuhauser (2003)

Coexistence less likely in spatial model

Where non-spatial models predict coexistence at low density, this spatial model suggests weaker competitor will die out, as individualsare discrete and so willeventually die out locally

Founder effect “probably”goes away entirely…strongercompetitor “wins” eventually

Competition colonisation trade-off

• Strong competition leads to competitive exclusion

• Even stronger version of same message from simple spatial models

• What if competition and colonisation interrelated?

• Begin with “toy” model of two species• Species A is the superior competitor• Species B is the superior coloniser

• Make inter-specific competition v. strong indeed…

Competition colonisation trade-off

• Environment of patches, initially empty (E)

• Model trade-off in competition versus colonisation

• If “superior competitor” A enters a B patch, takes over

• Species B is “superior disperser” (i.e. faster coloniser)

• Canonical example is competition between grasses• A: more energy to roots => better competitor• B: more energy to seeds => better coloniser

Nee and May (1992)

Nee and May model

Nee and May (1992)

A

A A

AB

B

B

A

A Bc EA

c E c

dE

dt

AB

c

e A

e AdA

c EB

dtd

e

B

B

e BE

A

Adt

Bc B

Fraction of Patches:E = emptyA = superior competitorB = superior disperser

Interspecific competitionparticularly strong (A always displaces B)

Nee and May model

Nee and May (1992)

Coexistence,despite v. stronginter-specificcompetition

Nee and May model

Nee and May (1992)

• Both species can persist and coexist

• Requires i.e. B is superior coloniser (or dies less quickly)

• Results generalise to intermediate competition

B A

B A

c c

e e

A

A A

AB

B

B

A

A Bc EA

c E c

dE

dt

AB

c

e A

e AdA

c EB

dtd

e

B

B

e BE

A

Adt

Bc B

Extend to n species: Tilman model

Tilman (1994)

• Paradox of diversity involves many species, not two

• e.g. Cedar Creek Ecosystem Science Reserve

• How do grasslands (single major resource, Nitrogen) support such biodiversity?

Extend to n species: Tilman model

Tilman (1994)

R* is equilibrium Nitrogen concentration in monoculture (lower for better competitors)

Evidence for competitioncolonisationtrade-off

Extend to n species: Tilman model

Tilman (1994)

• Similar underlying idea to Nee & May model

• Rank species in competitive hierarchy

• Species higher up the hierarchy always displace weaker competitors

• But superior competitors colonise more slowly

• Model pi, proportion of sites occupied by species i, (convention is that species 1 is the best competitor)

Extend to n species: Tilman model

Tilman (1994)

1

11

1i i

j j ij

i i jj

ii

dp

dtmpp pp c pc

• Colonise space empty/filled by inferior competitors

• Die (for simplicity assume all species at same rate, m)

• Outcompeted by all superior competitors

• Take cn > cn-1 > … > c2 > c1 (plus other conditions)

Extend to n species: Tilman model

Tilman (1994)

12n-2n-1n

Empty Space

Sup

erio

r co

loni

sers

Sup

erio

r co

mpe

titor

s

1

11

1i i

j j ij

i i jj

ii

dp

dtmpp pp c pc

………..

Extend to n species: Tilman model

Tilman (1994)

Two species Four species

Forty species(different initialconditions)

Forty species

Extend to n species: Tilman model

Tilman (1994)

• Competitive hierarchy and competition-colonisationtrade-off are sufficient to explain paradox of diversity

• Coexistence of an arbitrary number of species

• Important consequences for conservation

• In particular can investigate the effect of habitat fragmentation/loss by reducing number ofhabitable patches

• Does coexistence still occur, and if not, which speciesbecome extinct first?

•Much conservation interest in habitat loss/fragmentation

•Back to Nee & May two species model

•Assume only fraction h of the habitat is habitable

Conservation and habitat loss

A

A A

AB

B

B

A

A Bc EA

c E c

dE

dt

AB

c

e A

e AdA

c EB

dtd

e

B

B

e BE

A

Adt

Bc B

Nee and May (1992)

Conservation and habitat loss

Nee and May (1992)

A (solid) superior competitor, but poor coloniserB (dotted) superior coloniser, but poor competitorh fraction of habitat removed

“Weedy” species more likely to survive habitat loss

Extinction debt

Tilman, May, Lebman and Nowak (1994)

•Results generalise to the n species model of Tilman

•Extinctions increase as more habitat is removed•Superior competitors go first•Often after a long time…this is the Extinction Debt

(i.e. real effects of habitat loss felt in long-term)

How well does this work in practice?Mammalian Carnivores

A number of poor competitors, but good colonisers (African wild dogs, cheetahs,…) threatened with extinction

However superior competitors (lions, hyenas, leopards, …) persist

The exact reverse of theoretical predictions

Aggressive superior competitors force weaker speciesto edges of habitats/reserves, where they face increased mortality (from humans, mainly)

Summary (competition)

• Simple spatial models replicate (in fact enhance)paradox of diversity

• However competition colonisation trade-off promotes coexistence in pseudo-spatial models

• And can be generalised to allow coexistence ofarbitrary number of species, resolving paradox

• Theory suggests an Extinction Debt…current actionshave long term consequences (and that superiorcompetitors go first)

Summary (overall)

• Spatial heterogeneity in foraging effort tends tostabilise natural enemy interactions

• Spatial effects important determinant of long-termoutcome of competition relationships

• Coexistence greatly enhanced by the competition colonisation trade-off

• The common thread…spatial structure is important in an individual’s search for “enemy-free space”

References

1. Nicholson & Bailey (1935) Proceedings of Zoological Society of London 1:559-5582. May (1978) Journal of Animal Ecology 47:833-8433. Hassell (2000) Journal of Animal Ecology 69:543-5664. Hassell & May (1988) Annales Zoologici Fennici 25:55-685. Mills & Getz (1996) Ecological Modelling 92:121-1436. Bolker et al. (2003) American Naturalist 162:135-1487. Neuhauser (1998) Notices of the AMS 48:1304-13148. Crawley & May (1987) Journal of Theoretical Biology 125:475-4899. Nee & May (1992) Journal of Animal Ecology 61: 37-4010. Tilman (1994) Ecology 75:2-1611. Tilman et al. (1994) Nature 371:65-6612. May & Hasell (1981) American Naturalist 117:234-261

Underlined bold references are strongly recommended Other references are optional, but interesting as background