Time and space Time and space ----

Brett MelbourneEli GoldwynDerin WyshamOthers mentioned later

Stochastic dynamics of invasive Stochastic dynamics of invasive spreadspread (Brett)(Brett)

AcknowledgmentsAcknowledgments

Assistants:Claire KoenigDavid SmithRoselia VillalobosMotoki Wu.

NSF Biological Invasions IGERTDGE 0114432

NSF DEB 0516150

Classic spreadClassic spread

Missing ingredientMissing ingredient

With some exceptions, stochasticity

Stochastic spreadStochastic spread

Stochasticity (→ variance in speed)Population growth & dispersalDemographic, environmental, genetic

Repeat an invasion: differentNature: one realization

Real invasions can't be repeatedMany times, identical conditionsLaboratory microcosms

Now on to persistence (work with Now on to persistence (work with LooLoo Botsford)Botsford)

TwoTwo approachesapproaches

Deterministic network modelsMetapopulation models with patches that behave more or less like sources

There are many issues that may There are many issues that may affect affect extinction and persistenceextinction and persistence

Habitat requirements

Habitat heterogeneity

Species interactions Economic issues

Dispersal

Fishing pressures

Ocean currents

Human considerations

There are different ways to There are different ways to incorporate these issuesincorporate these issues

Habitat requirements

Habitat heterogeneity

Species interactions Economic issues

Dispersal

Fishing pressures

Ocean currents

Human considerations

Simulation based models

Analytic models for

general principles

The BahamasThe Bahamas

What is structure of models with population dynamics What is structure of models with population dynamics and discrete spaceand discrete space

Focus on ability of target species to Focus on ability of target species to increase when rare, which is one way increase when rare, which is one way to look at persistence/extinctionto look at persistence/extinctionYes, this leaves out a lotYes, this leaves out a lotBut is a necessary first stepBut is a necessary first step

Growth when rareGrowth when rare

Think of role of RThink of role of R00 in age structured in age structured modelsmodels

Growth rate given by largest Growth rate given by largest eigenvalueeigenvalueBut RBut R00 in age structured models has in age structured models has biological interpretation and says when biological interpretation and says when growth is positivegrowth is positive

Is there a similar idea for spatial Is there a similar idea for spatial models?models?

Model structureModel structure--density independent density independent (for persistence)(for persistence)-- Hastings and Botsford PNAS (2006)Hastings and Botsford PNAS (2006)

p12

Production ai,Survival of settling larvae bi

p21

p23p13

p32p31

1 2

3

connectance

Single species, multiSingle species, multi--patchpatch math math stuffstuff

the persistence problem:the analysis of the stability properties of the matrix C, with entries cij=bjpijai , where ai is the approximate per capita larval production in habitat patch i, bj is the fraction of larvae arriving in habitat patch jthat successfully settle (until censusedthe following year) when the species is rare, and pij is the probability that a larva produced in habitat j ends up in habitat i .

Network only matters if no single Network only matters if no single patch persists, so assume thatpatch persists, so assume that

Q=C-Iqij=cij=bjpijai (‘exchange’ terms)Assume no single patch persists

Network persistence criteria on next slide

01<−= iiiiii apbq

13322

3223

3311

3113

2211

2112

332211

213213

332211

312312 >++++qqqq

qqqq

qqqq

qqqqqq

qqqqqq

Network persistence for 2 and 3 Network persistence for 2 and 3 patch systems patch systems –– have results for have results for arbitrary number of patchesarbitrary number of patches

12211

2112 >qqqq

Mean lifetimes

Rate of settlement ofother patches

Return to original patch after 3 years 2 patch subnetworks

01<−= iiiiii apbq

qij=cij=bjpijai

4 patch condition analogous, but has extra terms and new features

Structure with four patchesStructure with four patches

Can have two independent two patch subnetworks

Persistence possible if either one, or both, subnetworks persistPersistence possible if three or four patch systems persist

So need to check 2 patch subsystems, and also the four patch conditions

Model structureModel structure--density independent density independent (for persistence)(for persistence)--

p12

Production ai,Survival of settling larvae bi

p21

p23p13

p32p31

1 2

3

connectance

Connectivity Patches that are sinks (even lifePatches that are sinks (even life--cycle sinks) may be essential for the cycle sinks) may be essential for the persistence of a network.persistence of a network.

If there is a two patch network If there is a two patch network (patches labeled 1 and 2) that is not (patches labeled 1 and 2) that is not persistent, then adding a third patch persistent, then adding a third patch may ensure persistence even if the may ensure persistence even if the third patch is a life cycle sink, with third patch is a life cycle sink, with qq3333 < 1, if the third patch acts as a < 1, if the third patch acts as a vital link by contributing enough vital link by contributing enough connectivity.connectivity.

Need connectivity and life history Need connectivity and life history parameters to get persistenceparameters to get persistence

Simple model shows that merely Simple model shows that merely preserving habitat types where a species preserving habitat types where a species is now found not enough to ensure is now found not enough to ensure persistencepersistence

SelfSelf--connectivity would depend on size of connectivity would depend on size of habitats preservedhabitats preservedModel shows that species can be found Model shows that species can be found currently in locations that do not contribute to currently in locations that do not contribute to persistencepersistenceModel shows how to calculate persistence in Model shows how to calculate persistence in general general

For Bahamas, connectivity estimatesFor Bahamas, connectivity estimatesPhysical oceanography (Olsen)Physical oceanography (Olsen)Genetics (Palumbi)Genetics (Palumbi)

Extensions and commentsExtensions and comments

Model presented was for annual species, Model presented was for annual species, extension to longer lived species extension to longer lived species straightforwardstraightforwardSize of patches enters into self seeding Size of patches enters into self seeding term, term, ppiiii. Think of original patch size . Think of original patch size results (e.g., results (e.g., SkellamSkellam))Have ways of going from continuous Have ways of going from continuous space to discrete space.space to discrete space.Breaking up one patch into two but Breaking up one patch into two but keeping overall self seeding and keeping overall self seeding and communication with other patches the communication with other patches the same maintains the same persistence same maintains the same persistence criteria.criteria.

MetapopulationMetapopulation modelsmodels

Age distribution of patches with density q(a) where a is the time since disturbance

Persistence for general Persistence for general metapopulationmetapopulation modelmodel

Rate of change of the fraction of occupied patches, p, is given by the rate of colonization, C, minus the rate ofextinction, E. Rate of colonization is m times the fraction of empty patches times the fraction of occupied patches, so C =mp(1 – p). The general persistence condition is stillm >1/AThe per-patch colonization rate must be greater than the inverse of the mean patch

Distribution affects persistenceDistribution affects persistence Distribution affects persistenceDistribution affects persistence

Way out are sources

ConclusionsConclusions

Have developed approaches for spatial Have developed approaches for spatial persistence persistence –– details!!details!!Realize that Realize that stochasticitystochasticity, etc will be important, etc will be importantBut general principles do emerge from simple But general principles do emerge from simple modelsmodelsProvides basis for more complete understandingProvides basis for more complete understandingBuilds on source sink ideasBuilds on source sink ideasImportance of dispersal in conjunction with Importance of dispersal in conjunction with sourcesource--sink ideassink ideas

Role of dispersal clearly identifiedRole of dispersal clearly identifiedDifficulties with measuring dispersalDifficulties with measuring dispersal

Temporal ScalesTemporal Scales vary widely in vary widely in ecological systemsecological systems

Climatic shiftsIce age time scales – but only 100 generations of redwoodsDecadal oscillations in the North East Pacific

Anthropogenic effects –Global climate changeYearly scales

SeasonalityAgricultural systems

Management scales of years to decadesFisheriesInvasive speciesEcosystem services

Combining space and time further Combining space and time further complicates the resultscomplicates the results and illustrates and illustrates problems with focus on asymptotic problems with focus on asymptotic behaviorbehavior

Coupled logistic equations2 species predator-prey model in 2 patchesOne parasitoid, two host models

Hastings and Higgins, 1994

How common is the transient How common is the transient behavior just described?behavior just described?

That is what we will now look atRequires attention to both mathematics and biologyBiological issues are important

Notion of ‘regime shifts’ (Carpenter)Response to climate change?

Movement plus dynamics (Hastings, Movement plus dynamics (Hastings, 19931993, Ecology, Ecology))

Step 1 (local dynamics):

Step 2 (exchange a fraction of the population):

REPEAT (D is fraction exchanged, r as before)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

f(X)45 degree linecobweb

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

f(X)45 degree linecobweb

Long term dynamics

A is in phase solutions, B is out of phase solutions

Long term outcomes

Perfectly out of phase

Almost in phase

Black ends up as B, white ends up as A

Now expand the lower left corner by a factor of 1000

But what do the dynamics look like But what do the dynamics look like on ecologically realistic time scales?on ecologically realistic time scales?

Choose r=3.8, D=0.15, and three sets of initial conditions Follow population sizes through time for different choices of initial conditionsRed dot is current population levels, line comes from the previous population levels

AllAll three three icic’’ss togethertogether

Analytic treatment of transients in coupled Analytic treatment of transients in coupled patches (Wysham & Hastings, Bull Math patches (Wysham & Hastings, Bull Math BiolBiol, (2008); submitted), (2008); submitted)

Depends on understanding of crisesCan study cases when an attractor ‘collides’ with another solution as a parameter is changedTypically produces transientsCan look at how transient length scales with parameter values

Start with 2 patches and Ricker local dynamics

Dynamics of other multispecies Dynamics of other multispecies systems in time and space also systems in time and space also introduce important questionsintroduce important questionsSpatial synchrony is an important question in ecology as a way to understand dynamics and for its direct importance

Hare-lynx in CanadaMasting (simultaneous flowering of plants)Voles (and other small mammals)

Persistence across space and time

Time series of total weekly measles notifications for 60 towns and cities in England and Wales, for the period 1944 to 1994; the vertical blue line represents the onset of mass vaccination around 1968. (Levin, Grenfell, Hastings, Perelson, Science 1997)

Spatial synchronySpatial synchrony

Moran effectInternal dynamicsBut what about time scale?Look at a system of one predator and one prey in two discrete patches

Predator-prey dynamics

Predator-prey dynamics

Interchange –Random movement

Analytic treatment of coupled predatorAnalytic treatment of coupled predator--prey (Goldwyn & Hastings (2008) prey (Goldwyn & Hastings (2008) TheorTheor Pop Pop BiolBiol, (2008) submitted, (2008) submitted

Use ideas from coupled oscillators derived in the context of neuroscience (Ermentrout, Izhekivich, Hoppensteadt)Use different time scales in predator prey systemsLook at dynamics of oscillations

More species lead to more questionsMore species lead to more questions

Coexistence of multiple species is a fundamental question Does explicit space matter, or is it just ‘space’ that permits coexistence?Two hosts-one parasitoid

AnaphesAnaphes flavipesflavipes(Hymenoptera: (Hymenoptera: MymaridaeMymaridae)) is a is a typical parasitoidtypical parasitoid

A. flavipes female on host egg.PHOTO: PHOTO: USDA, APHIS, PPQ, Niles Plant Protection Center

A. flavipes early pupal stage within host. Red compound eyes are the first visible feature.PHOTO: USDA, APHIS, PPQ, Niles Plant Protection Center

A. flavipes late pupal stage within host. Note the darkened body.PHOTO: USDA, APHIS, PPQ, Niles Plant Protection Center

Mathematical descriptionMathematical description starts from starts from a classic modela classic model

Large number of discrete patchesDiscrete time description of dynamics with one parasitoid and two hostsSince Nicholson and Bailey (1930’s)

Host that supports the higher parasitoid level eliminates the other host

Mathematical descriptionMathematical description needs to needs to match naturematch nature

Large number of discrete patchesDiscrete time description of dynamics with one parasitoid and two hostsSince Nicholson and Bailey

Host that supports the higher parasitoid level eliminates the other hostYet, in nature very complex host-parasitoid webs exit

Lewis, Owen T., Memmott, Jane, Lasalle, John, Lyal, Chris H.C., Whitefoord, Caroline & Godfray, H. Charles J.Structure of a diverse tropical forest insect–parasitoid community.Journal of Animal Ecology 71 (5), 855-873.doi: 10.1046/j.1365-2656.2002.00651.x

Mathematical descriptionMathematical description

Large number of discrete patchesDiscrete time description of dynamics with one parasitoid and two hostsSince Nicholson and Bailey

Host that supports the higher parasitoid level eliminates the other hostYet, in nature very complex host-parasitoid webs exit

Answer may be in transient dynamics

For many combinations of parameters there is long transient coexistence, although asymptotic solution is never coexistence (King and Hastings, 2003)

Transient dynamics are shown for a laboratory population of Tribolium, as reproduced with permission from Cushing et al.For one replicate (a), the population numbers (of larvae, pupae and adults) go through a period of time of approximate constancy, and then the dynamics change so that a two-point cycle is observed.For the other replicate (b), no transient dynamics are observed. This demonstrates that, even in a simple laboratory system, transient dynamics can be observed and that different dynamics are observed on a different timescale.

(Hastings, 2004)

Where do transients show up?Where do transients show up?

EverywhereInteresting mathematical and biological questionsRaises important issues about time scales

ConclusionsConclusions

Importance of biologically interpretable resultsFuture work is focused on

other growth rates (easy, done), relating to data from the Bahamas (Claire Paris)stochasticity (really hard, unless stochastic effects are small, and in some cases the effects are definitely large)Multiple habitat requirements or uses