PowerPoint Presentation

Feral Cats & Bird Extinctions on Islands

Mathematical Biology Modeling GroupSchool of Mathematical Sciences &School of Biological SciencesSerge Bloch, New York TimesSeptember 28, 2009

J. Fielder

*Introduce ourselves

One of the key messages of this talk is that collaborative groups that work across disciplines in the sciences can often accomplish more than individuals from one discipline.I am a poor mathematicianI remain mathematically semiliterate

~ E. O. Wilson ~Naturalist, 1994, page 122M. Hayes

Jennie SummerallNational Portrait Gallery, Washington, D. C.Oil on canvas...I have succeeded to some extant in theoretical model building by collaborating with mathematical theoreticiansMy role was to suggest problems to be addressed, to combine my intuition with theirs, and to lay out empirical evidence unknown to them. They were my intellectual prosthesis and I theirs.

Along with Robert MacArthur who was a very skilled mathematician, Wilson was the founder of Island Biogeography theory.

Todays lab doesnt deal with island biogeography per se, but does deal with ecological issues that occur on islands. Key Goals of Todays LabShow how math can help explain ecological processes.

Demonstrate how biologists and mathematicians can collaborate to address problems that neither group could address on their own.Overview of Todays LabProblem with feral cats on islands and a possible solutionWhat role can math play?A gentle intro to mathematical modeling in ecology and its use in evaluating management strategies

~ Take a break ~

Go to computer lab for a simulation of population dynamics on islandsWrap upThe Problem with Feral Cats on IslandsKerguelen Islands (Chapuis 1995)5 domestic founder catsHow many cats will there be in 25 years?Before you answer, lets take a closer look at the island

en.wikipedia.org glennmci.brinkster.net

http://www.chthonic.f9.co.uk/b3ta/underage.jpg

Introduced predators can decimate naive prey populations. Several reasonsbut one of the most important issues is that prey animals have often not adapted to the presence of predators, so have not learned how to cope with predators.

Kerguelen Island is in the south Indian Ocean is an example of the ecological problems feral cats (or other introduced predators) can cause on islands. This island has a well established feral cat population from 5 original cats that were ships cats. This population of cats lives on sea birds and also feral rabbits, which were also introduced to the island.

7,200 sq km equivalent to an island that is 50 miles long and 50 miles wide. For comparison, this roughly 70 percent of the area of the Island of Hawaii (the Big Island in the Hawaiin island chain).

Heres an example:

Slide is animated. 5 domestic founder cats shows when the slide comes up.Ask: how many cats do you think there will be in 25 years?Give them 15 seconds and then have them pair up and talk with a partner about their guessAsk: how many think between 0 and 50? 50-100? 100-500? 500-1000? 1000-5000? 5000-10, 000? 10, 100-20,000? 20000-30000? 30000-50000? More than 50000?Give them the answer (bring it in with the animation) and then quickly compare to their estimates

Ask: How many birds a year do you think 30,000 cats can kill?Give them 15 seconds and then have them pair up and talk with a partner about their guessGet a sense of their answers and then give them the answer (bring it in with the animation) and then quickly compare to their estimates

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Port-aux-FranaisCapital settlement of Kerguelen Island, French Territory en.wikipedia.org/wiki/Port-aux-Franais 60 winter inhabitants up to 120 summer inhabitants 5 founder cats?... 25 years later30,000 cats Birds killed per year? > 3 million per year! This is a view of the capital settlement of Kerguelen Island. This is in the subarctic and at first glance I might not guess that this would be an environment conducive to the rapid growth of a feral cat population.

60 winter inhabitants (humans )120 summer inhabitants Control of Feral CatsCulling & trapping are expensiveTime-consumingLabor intensiveControversialDisease

NYTimes

www.wildliferesearchmanagement.com.au Share thoughts from readings:Pair upDecide who will go firstWill have 2 minutes to share your response to the prompt and the articleThen switch and other person will have 2 minutesAsk you to share something the other person said7DiseasesFeline Immunodeficiency Virus=FIV

Feline Leukemia Virus=FeLV

www.morgansplace.comToday we are going to focus on introducing disease as a way of reducing the cat population. There are multiple diseases that affect felines and are a possibility for controlling or eradicating feral cats. Two possibilities that were discussed in the paper that initially sparked this project are: FIV and FeLV.

Both behave in some similar ways, but also have some differences. For the purposes of this talk, were only going to consider FeLV.8

Feline Leukemia Virus (FeLV)Retrovirus

Found worldwide

Spread by coming in contact with saliva, urine, feces, and nasal secretionsAmerican Association of Feline Practitioners and Cornell Feline Health Center, 2006We are going to focus only on FeLV for todays in an effort to introduce biological modeling and avoid overwhelming you.Feline Leukemia is a retrovirus, which means that the virus produces an enzyme, reverse transcriptase, which permits them to insert copies of their own genetic material into that of the cells they have infected Virus is found worldwide, but the prevalence of infection varies greatly depending on their age, health, environment, and lifestyle. (in U.S., about 2-3% of all cats are infected), rates rise when the cats immune system is compromised (ill, young, etc.))FeLV is spread by coming in contact with saliva, urine, feces, and nasal secretions. Typically this is via social behavior, such a grooming9Impossible to accurately predict the life expectancy of a cat with FeLVSigns of the diseaseLoss of appetite, weight lossPoor coat conditionEnlarged lymph nodes, may lead to cancerPersistent diarrhea and feverInflammation of the gumsSeizures, behavior changes, and neurological disordersPrimary stageConcentrated in bloodstreamSome individuals become immuneSecondary stageLater stage, infection in bone marrow and tissuesPoint of no returnLeads to state of immune deficiency and cancerAmerican Association of Feline Practitioners and Cornell Feline Health Center, 2006

FeLV CharacteristicsLife expectancy typically depends on the age and health of the cat, but since it ultimately breaks down the immune system, it is impossible to accurately predict the life expectancy of a cat infected with FeLV It is possible for the cat to show little sign of viral infection in the early stages of the disease, but as weeks, months, or even years pass, the infected cat may develop the following symptoms: Loss of appetite, weight lossPoor coat conditionEnlarged lymph nodes, may lead to cancerPersistent diarrhea and feverInflammation of the gumsSeizures, behavior changes, and neurological disordersThere are 2 viremic stages:Primary stageConcentrated in bloodstreamSome individuals become immuneSecondary stageLater stage, infection in bone marrow and tissuesPoint of no return, cats will be infected for the remainder of their lives

10QuestionsWhat would you like to know before you introduce feline leukemia into the cat population on an island?

http://www.petcareloop.com/images/twitter_question_mark.png?1255219975I gave you a few background facts on FeLV, but what other things would you like to know before you introduce this virus into a cat population on an island?

Give them 1 minute to think on their own and 2 minutes to talk to a partner.Each persons job is to present a question their partner asked.Solicit questions from the classMay want to record on a chalkboard/whiteboard

This should generate all the relevant stuff, but some of the questions that the model can answer are listed below. Want to make sure they get brought upDoes introducing FeLV actually decrease the cat population and increase the bird population?Does introducing the disease actually do what you want it to do?How long will it take for the cat population to fall below a certain thresholdHow long will it take for the bird population to grow to a certain thresholdHow deadly is the virus? How fast does it spread?

This is where the math comes in.we can create a mathematical model of this situation and use it to try to answer these biological questions.11Some of Our QuestionsHow deadly is the virus? How fast does it spread?Does introducing the disease actually do what you want it to do?How long will it take for the bird population to grow to recover?

Possible questionsonly use if not generating questions themselves.12Modeling in BiologyA tool for understanding Population growth or declineEcological systems and processWhat variables in a system are importantTransmission of diseaseMigration of a populationHabitat utilizationHow modeling can be used in Biology

How does modeling tie into their projectshelp from Brandon, Emily and Mark on this piece.

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Lives in old-growth forests1985-2003 19+ years of data11,432 birds marked32,054 capture events

This is complex data! Cost > $30,000,000

Analyzed data using a collaborative workshop of biologists, statisticians, & mathematicians. Example: Modeling Northern Spotted Owl Population Data

Types of Questions in Modeling Cat Bird PopulationsWhat information do we need to build a model? (whats important? whats not?)What questions can the model answer? (what can the model predict?)What questions cant be answered by the model?

Ask the students to categorize the questions into these 3 categories.

What questions do we need to answer to build the model?How fast does the disease spreadWhat questions can the model answer?How long does it take for the bird population to recover (once a population size is assigned to the what recover means)What questions cant be answered by the model?How cost effective is this approach?

This is where the math comes in.we can create a mathematical model of this situation and use it to try to answer these questions.

15Modeling Populations2 ApproachesMeasuring Total populationCounts of animals, bacteria, trees, cellsAided by samplingCan be challenging/impossible to get accurate estimatesChanges in populationBirth rates, death rates, infection rates, migrationDescribe interaction between speciesOur focus has been on modeling populationsthat can mean populations of animals or bacteria or treespopulation has a wide definition here.

When looking at populations, we have two basic choices.

Do we look at total population or changes in population?

Can do both, but it turns out the looking at changes in population and using that to estimate total population turns out to be easier in part because many biological factors such as birth rates, death rates, rates of disease spread are really talking about change.

Populations are dynamic so trying to count everyone is challenging..since have to keep track of births, deaths, immigration and emigration.

Advantage to measuring change is that there is a whole branch of mathematics called differential equations that studies change.

16Describing Rates MathematicallySeem complicatedCan be analyzed numerically and graphicallyEasier to align with biological factors

Advantage to measuring change is that there is a whole brand of mathematics called differential equations that studies change.

17Exponential Growth Model

Graph of Total PopulationEquation of Total PopulationEquation of Population ChangeThis is the classic exponential growth model.

We can describe this mathematically with the equation where we put in a time and the equation spits out the population at that time.

Or, we can describe this in its corresponding form as a differential equation where can put in a population at a given time and differential equation tells us how much we expect the population to change over the next time step.

Graphic will be interactive, allowing control of B0 and r

Can usually get hands on DE but not on the solution

DEs generally easier: solution is exponential, corresponding DE is linear18Logistic Growth Model

Graph of Total PopulationEquation of Total PopulationEquation of Population ChangeMore realistic situation is logistic growth, in which there is a bound (carrying capacity) on the size of the population.

Note that the equation describing the population of birds is considerably more complicated than in the exponential model, but that the differential equation description just has an added factor that limits the growth rate.

Graphic will be interactive, allowing control of B0, r, and K. What do you expect to happen if we change r now? if we change B0? If we change K?

Can often get an equation for change when cant get the time dependent equation19

Cat-Bird DynamicsRemember, we are in the situation where we have both cat and bird populations which interact. The number of cats impacts the birds and vice versa. So we need models for each population and we need to link them somehow to model the impact of one population on the other.20Tweety-Batman to the rescue!(along with differential equations)

If we could just call in Tweety-Batman to come in and conquer/wipe out/keep under control the bad cats we would! Our equivalent is to introduce a cat disease.21Predator-Prey with Logistic Growth

BirdsCatsNote that without the circled term, the birds eqn is just the previous logistic growth equation. The circled term accounts for the birds being eaten by the cats.The cats equation accounts for the bird because the circled part is the carrying capacity which is determined by the number of birds available for consumption.

The cats eqn also represents logisitic growth, but the carrying capacity of the cats depends on the size of the bird population.

The fact that the bird and cat populations are interdependent makes this a what is sometimes known as a coupled system of DEs.

Next slide will be interactive graphics of B vs t, C vs t, and C vs B22Try an Example

Starting Values

Break off into group and have a mock problem to use the logistic growth equation

Birds/cats graph with starting point and goes point to point by plugging in info(cats=12, birds=2mil; cats=15, birds=1.8 mil)23Graph of (Cat, Bird) Trajectory

CatsBirds24What happens if a cat disease is introduced to the system?

BirdsCatsWe need to decide exactly which disease model well use! Once we have decided, we can add a slide that shows the more complicated system (well keep the slide hidden unless someone asks about it), and it will also determine whats behind the graphics on the next slide.25The S-I-R Disease Equations

Describes how the number of susceptible, infected, and recovered individuals changes with time26The State Diagram for FeLVA modified S-I-R modelX+Y+Z = total populationSusceptibleXDeath1-

ImmuneZDevelop SymptomsYbmmViremic(very brief)

bmMarks Diagram attempt:

Here are the parameters of the FeLV model:X = Number of susceptible individualsThe viremic stage is assumed to be very brief and we assume that individuals who become infected rapidly transition into the Y or Z category. Y = Number of infected individuals that develop symptoms. This stage is also called persistently viremic. Individuals who develop symptoms are considered too sick to reproduce and give birth, so b = 0 and is not shown in the diagram. The average time to death for this category is 2 years.Z = Immune individuals. These individuals are not infectious, have a normal life span, are immune for life, and give birth to susceptible individuals at the normal rate for uninfected individuals. = Transmission rate (or coefficient). This rate assumed that = Proportion of individuals who become infected and die b = Birth ratem = Intrinsic death rate (which is assumed to be unaffected by the FeLV virus) = mortality rate (once infected) due to the FeLV infection

We could also add two more parameters as follows:

= harvest rate due to licensed hunting of the cats, if this was used as a control measurev = Vaccination rate of susceptible cats, if some cats were vaccinated

Where would you add these parameters in the model we have here?

27Differential equations for FeLV

Maybe say something along the lines this is where the collaboration with mathematicians comes in handy 28To the Computer Lab!

29Conclusion What is modeling in biology, and how can it be used as a tool for understanding ecological systems?

Biological influences on a system that go into building a modelbirth rate, death rate, disease, im/em-igration, food supply, etc.

For the conclusion, maybe answer talk about how we saw the first question answered for the feral cat problem and talk in general about how biological models are built and how they can be used.

Also talk about the usefulness of interdisciplinary groups (mathematicians, statisticians and biologist) working together to understand biological systems30Wrapping it upYahoo article and worksheet

www.flickr.com/photos/bostich/3048320868/

So to wrap it up, we have a Yahoo article for you to read and a worksheet for you to fill out31