APPLICATION OF LESLIE MATRIX MODELS TO WILD TURKEY POPULATIONS

APPLICATION OF LESLIE MATRIX MODELS TO WILD TURKEY POPULATIONS by Wen-Ching Li Nrth Carlina State University Department f Statistics Bimathematics Graduate Prgram Raleigh 1994

CONTENTS ABSTRACT 1 THE LESLIE MATRIX MODEL 2 Intrductin 2 Mdel structure 2 Parameter estimatin 7 Eigensystem 9 Stable age distributin 11 Applicatin and generalizatin 14 APPLICATION TO EASTERN WILD TURKEY IN lowa Intrductin Objectives Mdel structure Parameter estimatin Result and discussin Future wrk 16 16 17 17 20 23 36

ABSTRACT The Leslie matrix mdel is a ppular and useful methd fr ppulatin prjectin based n age specific survival and fecundity rates. The first part f this prject will describe the therems, prperties, applicatin, and develpment f the Leslie matrix mdel in detail. In the secnd part, specific Leslie mdels are develped fr wild turkeys in Iwa. The Leslie matrix mdel has becme very ppular because f its easy understanding by bilgists and because many theretical prperties can be easily studied using matrix algebra. In the example fr Iwa wild turkey ppulatins, a simple, linear, deterministic mdel based n the Leslie mdel is built. The mdel's simplicity makes it easy t understand and analyze. Hwever, its simplicity als means that it is far frm bilgical reality. There are t few parameters t adequately describe the cmplexity f a real turkey ppulatin. Hwever, it is a gd mdel n which t build mre cmplex and realistic mdels which may allw stchasticity and density dependence f survival and fecundity rates. 1

THE LESLIE MATRIX MODEL Intrductin Births and deathsare age-dependent in mst species except very simple rganisms. It is usually necessary t take accunt f a ppulatin's age structure t get a reliable descriptin f its dynamics. The English bilgist P.R.Leslie (1945) intrduced a mdel which uses the agespecific rates f fertility and mrtality f a ppulatin. He develped the mdel based n the use f matrices. The matrix frm makes the mdel flexible and mathematically easy t study. This basic mdel is an increasingly ppular tl fr describing ppulatin dynamics f plants (Usher 1969) and animals (Yu 1990, Cruse 1987). The Leslie matrix mdel is widely used t prject the present state f a ppulatin int the future, either as an attempt t frecast the age distributin, r as a way t evaluate life histry hyptheses by cnsidering different sets f survival and fecundity parameters. Mdel structure In the simplest Leslie matrix mdel (Leslie 1945), nly females are cnsidered. The female ppulatin can be divided int several categries by age r by size. The classificatin styles and their differences will be discussed later in this sectin. If it's gruped by age, the grup intervals are suppsed t have equal length f time. Fr example, it may be 5 years fr human and 2 years fr large whales (Cullen 1985). We assume that the survival and fecundity rates 2

f each categry are cnstant ver time and therefre nt dependent n ppulatin density. Define nj(t) = the number f females alive in the grup i at time t, Pj = the prbability that a female f grup i at time t will be alive in grup i + 1 at time t+ 1, (t+ 1 means t+ne unit f time), Pi lies between 0 and 1, Fj = the number f daughters brn per female in grup i frm time t t time t+ 1, let i = 1,2,...,x, then nl(t+l) = F1*n](t) + F 2 *n 2 (t) +... + Fx*nx(t) n 2 (t+l) = n 2 (t+ 1) = P1*n)(t) P 2 *n 2 (t) (1.1) These equatins can be represented in matrix frm as n(t+ 1) = A n(t) (1.2) where n(t) is a clumn vectr with cmpnents f the number f individuals in each recgnized categry at time t. A is a nnnegative, square matrix f rder x with all the elements zer except thse in the first rw and in the subdiagnal immediately belw the principal diagnal. 3

F i F 2 F 3 F X - 1 F x P 1 a 0 a a A = a P 2 0 a a a 0 P 3 a a (1.3) a 0 0 P X - 1 0 A is knwn as the "Leslie matrix" based n age classificatins r the "ppulatin prjectin matrix" mdeled n stage classificatins (Lefkvitch 1965, Grenendae et al. 1988, Cruse et al. 1987). The prjectin matrix was intrduced by Lefkvitch (1965). The stages in the prjectin matrix mdel are nt necessarily related t age. Its main assumptin is that all individuals in a given stage are subject t identical mrtality, grwth, and fecundity schedules. These matrices are mre cmplicated in appearance than the Leslie matrices, but have almst the same analytical methd. Fig. 1.1 shws the life cycle graph depending n different decmpsitin f the life cycle. The arrws in the graph indicate the transitins which are pssible fr an individual frm ne time t the next. The Leslie matrix crrespnding with the life cycle graph (A) in Fig 1.1 is just the same as (1.3) and much simpler than the prjectin matrix fr the stage-classified ppulatin, (8) f Fig 1.1, btained as fllws: 4

A F u F 12 F 13 F 14. F 1X P 21 P 22 P 23 0 0 0 P 32 P 33 P 34 0 0 0 0 P X - 1 x-2 P X - 1 x-l P X - 1 x 0 0 P x x-l P xx (1. 4) A.,/, J, F, 1F 2 Fx ~Pl Age class Age class P 2 Px- 1 Age class,- -, ~ 1 2... x B.,.l J, F ll l -.J P 21 F l2 Fix Stage Stage P 32.. Pxx- 1 Stage ;, I... ~ J 1 2 P 23..... r- Px- 1 x IJ' ~ r x Fig 1.1 Life cycle graphs fr tw chices f categries. (A) Age-classified ppulatin. (B) One example f stage-classified ppulatin. 5

In the stage-classified mdel, the pssibility f an individual f grup i mving backwards t the previus grup, mving frwards t the next grup, r staying the same grup at next time perid is cnsidered. The chice f categries is a very imprtant step in establishing a ppulatin matrix mdel (Grenendael et al. 1988). Since smetimes it is hard t determine age-specific parameters f the mdel fr sme rganisms with variable develpment, stage categries may be mre apprpriate than age categries (e.g. fr turtles, Cruse 1987). Fr ther rganisms like mammals and birds, their develpment is mre dependent n age than size and therefre age categries are used frequently. If age and size are bth imprtant and significant interactins exist between them, a twdimensinal mdel using bth stages and age classes simultaneusly can be develped(yu 1990). In Yu' s study, develpmental stages are as meaningful as chrnlgical age classes in the crn earwrm (CEW) management. He divided the CEW ppulatin int five stages and split the first stage int fur age grups, the remaining stages int six age grups, except taking the last stage as a single age grup. Thus, his prjectin matrix is a 23*23 matrix taking bth stage and age int accunt. In this prject, I will mainly discuss the prperties f an age-classified matrix mdel based n frm (1.3). The mdel has linear, time invariant functins which prject the current state f the ppulatin int the next state. The female ppulatin alne will be cnsidered. 6

Parameter estimatin Survival prbability Pi The values f Pi can be btained frm a life table r experimental data. It is usually assumed that (Leslie 1945, Ple 1974) L i + 1 p.=- ~ L ~ (1.5) where Li = the number alive in the age grup i t i+1 in the statinary r life table age distributin. Pi is assumed t be cnstant ver a unit f time and nt dependent n the ttal number in the ppulatin. Actually, seasnal differences and density dependence f the survival rate d exist, but we assume that they are nt significant enugh t cunt. Fecundity rates F j F; is determined nt nly by the average number f female births per mther per time perid, m j in the life table, but als by infant survival rates (Leslie 1945, Grenendael et al. 1988). Ppulatins can be divided int tw cases f breeding systems, 'birth-flw' ppulatin with cntinuus reprductin and 'birth-pulse' ppulatins with discrete reprductin (Caughley 1977). Ppulatins, e.g. human, f the first case prduce ffspring at a rate almst cnstant thrughut the year and thse f the secnd case prduce ffspring ver a restricted seasn, e.g. 7

blue whale. In birth-flw ppulatins, the estimatin f F i invlves integratin f cntinuus fertility and infant mrtality functins ver the time interval (Leslie 1945). Fr birth pulse ppulatins reprductin is during a shrt perid f time per year. It is imprtant t knw the exact breeding perid with respect t the time step in birth-pulse ppulatins. Tw different ways f cunting the ppulatin, census just befre r after reprductin, will influence the estimatin f F i (Cullen 1985). Sme studies cunt the ppulatin just befre reprductin and we illustrate that here. Define M j = average number f female ffspring per female in categry i brn between t and t+i, S = the ffspring survival rate between birth and the time when they are cunted as part f the ppulatin, then F i = S * M i 1 = 1,2,...,x (1,6) F i, the age specific fecundity rate, is assumed t be cnstant and nt dependent n density just as the age specific survival rate, Pi' is. 8

Eigensystem One advantage f using Leslie matrices is the flexibility f their mathematical frmulatin. By applying linear algebra techniques t the mdel, several bilgical phenmena can be interpreted. Cnsider the hmgeneus discrete-time system (1.2), n(t+ 1) = A net), where A IS a square matrix (1.3). If A has the frmula, (1. 7) then AI,...,Axare called the eigenvalues f A with I AI I > I A2 I ~ I A3 I ~... ~ I Ax I and el'0..,exare the crrespnding linearly independent eigenvectrs. The eigenvalue Al f greatest magnitude is the dminant eigenvalue. Then the slutins t n(t+ 1) = A net) have the frm (Ducet and Slep 1992), nl(t) = all All + a l2 A2' + n2(t) = a21 All + a 22 A2 1 + (1.8) njt) = axl All + ax2 A 1 2 +... + a xx A/ Fr large t, althugh the influence f A2.3...,x des nt necessarily disappear except that their abslute values are smaller than 1, the first term with the dminant eigenvalue will grw faster I than the thers. Therefre, nj(t) will becme similar t the single term aila I. Each age class eventually will grw expnentially at the rate f the dminant eigenvalue, AI, per time perid. The system (1.2) is asympttically stable if and nly if the eigenvalues f A all have 9

magnitude less than ne (Luenberger 1979), that js, I A; I < I fr every i. The state vectr n(t) will tend t an equilibrium pint (the rigin) fr any initial cnditin if the system is asympttically stable. Once the system state vectr is equal t an equilibrium pint, it will remain equal t that fr all future time. The system is called marginally stable jf n eigenvalues have magnitude greater than ne but ne r mre is exactly equal t 1. eigenvectr is an equilibrium pint. If unity is an eigenvalue f A, then the crrespnding If unity is nt an eigenvalue f A, the rigin is the nly equilibrium pint f the system (Luenberger 1979). Therefre, if unity is the dminant eigenvalue, the state vectr eventually will be a specific multiple f the assciated eigenvectr and wn't change in future. It means that ppulatin size will remain the same fr each age class after it accmplishes that pint. When the dminant eigenvalue AI = 1, we say the ppulatin is statinary. Each eigenvalue defines nt nly a characteristic grwth rate but als a characteristic frequency f scillatin (Luenberger 1979). Fr a discrete-time system, n scillatins are derived frm a psitive eigenvalue, in ther wrds, scillatins are due t a negative r cmplex eigenvalue with perid 2 r with a larger perid respectively (Ducet et al. 1992). Sme eigenvalue-eigenvectr therems fr Leslie matrices are useful in determining the develpment f a ppulatin (Cullen 1985). (1) A Leslie matrix has at least ne psitive real eigenvalue. (2) If there are at least tw cnsecutive age classes that are fertile, a psitive real dminant eigenvalue always exists. 10

Stable age distributin Accrding t the utstanding prperties f the dminant eigenvalue, a ppulatin grwing based n the Leslie matrix will have an age cmpsitin which is entirely determined by its Leslie matrix and des nt depend n the initial cmpsitin after a perid f time. Then frm sme year n, next year's ppulatin will be a multiple f this year's. This age cmpsitin is called a stable age distributin. (1. 9) The multiple Al is the dminant eigenvalue f the Leslie matrix A (1.3) and if e l IS the assciated eigenvectr, the stable age distributin can be btained frm el' Let (1.10) and v = VI + v 2 + + vx' then the vectr S is called the stable age distributin f the ppulatin (Cullen 1985). s= (1.11) 11

Tw different Leslie matrices may have the same AI and the same stable age cmpsitin, that is, given a value f AI and a stable age vectr, mre than ne Leslie matrix can be determined (Pielu 1977). Fr example, 0. 3 1, 6 6~. 0] x= 00.5 0 [ 0.5 and.7 1.91~. 5] y= 0.5 0 0.5 X and Y bth have the same dminate eigenvalue, AI = 1.5, and the same stable vectr S =[0.69 0.23 0.08]T. Since they are actually unlike, they differ in ther eigenvalues. The apprach t S depends n hw much larger I AI I is than I A 2 I,..., I Ax I. The larger the difference between them, the faster the ppulatin mves tward stability. Therefre, X and Y have varius appraching rates t the stable age cmpsitin. The cnvergence twards the stable age cmpsitin ccurs nly if AI is "strictly" larger than the ther eigenvalues. Beginning with an unstable r stable age distributin, a ppulatin in an unlimited envirnment will reach a stable age distributin in time, whether increasing, decreasing, r staying at the same density (Ple 1974). Unfrtunately, there is an exceptin t this rule (Ple 1974, Ducet and Slep 1992). When there exists ne f the negative r cmplex eigenvalues has the same abslute value as AI, the cntributin f All will never 12

dminate the ther cntributins and the ppulatin will scillate frever. This happens when the entries f the first rw f Leslie matrix are all zer except the rightmst ne. Fr example (Ple 1974), 0 0 6 1 - A= 2 0 0 0 1-3 0 The eigenvalues f A are and The dminant eigenvalue is 1, s the ppulatin will neither increase r decrease as a trend. Nte that A 2 = 0 3 0 0 0 2 1 a a - 6 A 3 =I and s frth. The ppulatin will scillate with perid 3 time units and never apprach stability unless it begins with a stable age distributin. 13

Applicatin and Generalizatin In the riginal Leslie matrix mdel, the last age class is assumed t be remved frm the ppulatin after a time unit. S the entry with rw x and clumn x is 0 in the matrix (1.3). Actually, individuals in the last age class may nt die after ne time unit and their survival prbability and fecundity rate will just stay the same until they die. It indicates that we allw members f the last grup t live and reprduce fr sme years. Besides, lder individuals are assumed t be essentially the same as thse f the last age grup. Let P x = the prbability that an individual f grup x at time t will alive and stay in the grup x at time t+ 1. Then the Leslie matrix can be written as F 1 F 2 F 3 F X - 1 F x P 1 a a a a A= a P 2 0 0 0 a a P 3 a a (1.12 ) a a a P X - 1 P x and the last equatin f (1.1), n x (t+l) = Px-l nx_l(t), is replaced by which is an aggregatin f all remaining age classes. 14

The basic Leslie matrix mdel is a deterministic mdel. The parameters, survival and reprductive rates, are cnstant ver time. In real ppulatin, thse parameters may change with many factrs. In the cases f human ppulatin, the survival rates, especially amng the elderly, will change with new medical advances r better feeding habits, and the fecundity rates will be affected by changing scial attitudes tward marriage and family. The effect f harvest and seasnal differences n survival prbabilities shuld be cnsidered fr animal r plant ppulatins. If the parameter values fr years are randmly generated frm a specific prbability distributin, the mdel will be stchastic. Only females are cnsidered in Leslie's mdel (1945). In bisexual ppulatins, the basic mdel can be mdified t keep track f bth sexes (Pielu 1977, Cullen 1985). The Leslie matrix in its classical frm makes n allwance fr density dependent ppulatin grwth and it is fundamentally linear. Cnsidering the effect f density dependence n fertility and survival rates will turn the mdel int a nnlinear ne (Pielu 1977). The size f the current ppulatin will play an imprtant rle in determining thse parameters and the initial size has a lasting effect n the ppulatin's future chances f reprducing and surviving. 15

APPLICATION TO EASTERN WILD TURKEY IN lowa Intrductin The eastern wild turkey (Meleagris gallpav silvestris), ne f five subspecies, inhabits rughly the eastern half f the United States. Turkey hunting has a substantial ecnmic effect in many rural cmmunities. It nt nly brings in revenue because f the actual turkey hunting, but it als enhances the develpment f the related industries f turkey-hunting clthes and equipment. Imprvement f the knwledge f turkey ppulatin dynamics is imprtant t frmulate hunting regulatins and ther turkey management practices. Much research abut the rates f reprductin, mrtality, and survival, and the mvement f wild turkeys has been dne (Dicksn 1992). Hwever, there were few mdels established fr evaluating turkey ppulatins. Fr a specific cmbinatin f survival and reprductive rates, a mdel may tell us whether a wild turkey ppulatin will grw, decline, r remain stable. The survival rates will bviusly als depend n the level f harvest. By changing the harvest level we can see the effect n the ppulatin size and cnsider changes t the hunting regulatins and enhance the management f wild turkeys. A Leslie matrix mdel is develped fr the ppulatin dynamics f eastern wild turkeys in Iwa. The parameters used in the mdel are based n Iwa wild turkey studies (Suchy et al. 16

1983, Dicksn 1992), and the ppulatin characteristics will be interpreted by the mdel analysis. Objectives 1. T build a mdel in rder t predict the ppulatin size and age structure f Iwa wild turkeys in the future. 2. T see the effect f harvest level (percent f ttal ppulatin) n the ppulatin age structure and grwth. 3. T find the mst apprpriate wild turkey hunting seasns. Mdel structure An age-classified mdel is built here. There culd be several stages in the age structure f the wild turkey ppulatin. A three-stage mdel is chsen in rder t simplify the mdeling prcedure. The first categry is "pults", aged frm 0 t 1, the secnd categry is "yearlings", aged frm 1 t 2, and the last categry is " adults", aged 2 and lder. Reprductin ccurs frm yearlings nwards. The time unit is ne year. Only females are cnsidered in this mdel. Let Pi = the prbability that a female f grup i at time t will be alive in grup i+ 1 fr i= 1,2 r will stay in grup i fr i =3 at time t+ 1. F j = the number f daughters brn per female in grup i frm time t t time t+ 1. i = 1, 2, r 3. 17

The life cycle graph is as fllws, l I r Pults 1 PI Yearlings 2 P 2 Adults 3,. i \ 10- The state variables are the numbers f pults, yearlings, and adults, dented as n l, n 2, and n3 respectively. Then, n 1 (t+l) = F z n z (t)+f 3 n 3 (t) n z ( t +1) = PIn I ( t) (2. 1 ) n 3 (t+l) = P z n Z (t)+p 3 n 3 (t) A matrix frm can be derived frm thse equatins. n(t+l)==an(t) (2.2) I.e. n 1 (t+l) 0 F z F 3 n 1 (t) n z (t+l) PI 0 a n 3 (t+l) 0 Pz P 3 n 3 ( t) * n z ( t) (2.3) 18

The Leslie matrix A, a square matrix f rder 3, is defined in this three-age-class mdel as A= (2.4) Estimating parameters annually we verlk sme cmplicatins, fr example, different mrtality rates in different seasns. A number f studies n estimating the survival and reprductive prcesses were well dne fr wild turkeys. It is nt t hard t estimate the parameters, Pi and F j, t use in the mdel. This is the reasn why an age-classified mdel is chsen rather than a stage (size)-classified mdel in this prject. Assumptins The basic assumptins are as fllws: 1. The survival and reprductive rates are cnstant ver time within each age-class and are therefre nt dependent n ppulatin density. 2. The individuals in each grup always g t next grup frm ne time t the next except thse in the last age-class which may stay fr sme years befre they die. 3. The cnditins f each individual in the same stage are similar, that is, all individuals in that stage have the same survival and reprductive rates. In practice survival and fecundity d vary frm year t year due t envirnmental factrs. If the Leslie mdel is still applied using averages there is a degree f apprximatin invlved. In practice survival and fecundity may als be density dependent. 19

Parameter estimatin Survival prbability Pi 1983). The survival rates are derived frm a study n Iwa wild turkey ppulatins (Suchy's et al. Pl=0.445 P 2 0. 616 P 3 0.610 They are the average estimates f the data frm 1977 t 1981. Fecundity rates F j Frm Dicksn (1992), sme infrmatin abut Iwa turkey ppulatins is imprtant fr estimating the reprductive rates. Estimates f nesting and renesting rates (Table 2.1) indicates that a higher prprtin f adult hens attempt t nest. Clutch size (Table 2.2) is als larger fr adult hens than fr yearlings. Since reprductin starts frm yearling-stage, the value f F I is zer. RecalI that F j shuld be determined nt nly by the number f daughters brn per female per time perid but als by infant survival prbabilities (hatching success). Hatching success = number eggs h~tching = 0.85 clutch s~ze fr successful nests. 20

Table 2.1 Nesting rates and nest success fr Iwa wild turkey hens. (Dicksn 1992) First nest Renest nesting nest renesting nest rate success rate success yearling 0.42 0.56 0.12 0.00 adults 0.97 0.38 0.32 0.52 Table 2.2 Clutch size fr Iwa wild turkey hens. (Dicksn 1992) clutch SIze First nest Renest yearlings 8.8 7.0 adults 9.4 9.1 Using the values in Table 2.1, Table 2.2, and the hatching success, F 2 and F 3 can be estimated as fllws, 21

F 2 estimatin F 2 = number f female pul ts number f females = number f female eggs * number f female pul ts number f females number f female eggs = female eggs f first nesting + female eggs f renesting number f females * number f femal e pul ts number f female eggs (0.42) (0.56) (1) (~) +0 =0.880 1 2 *0.85.E 3 estimatin Use the similar way as abve. (0.97) (0.38) (1) (2...:..!)+(0.97) (0.62) (0.32) (0.52) (1) (1...:1.) 2 2 1 * 0.85 =1.860 Therefre, the riginal Leslie matrix f Iwa wild turkey ppulatin in this prject is btained. A= 0.445 0.8801.860 0 0.616 0.610 (2.5) 22

Result and discussin 1. Stability I used a FORTRAN prgram t find eigenvalues and eigenvectrs f the Leslie matrix A. The initial values f state variables are n1(o) = the initial number f female pults in the ppulatin = 580.00 n2(o) = the initial number f female yearlings in the ppulatin = 122.70 n,(o) = the initial number f female adults in the ppulatin = 156.10 which are the 1977-78 data derived frm Suchy et al. (1983). Fr the riginal matrix (2.5), eigenvalues are A)=1.15, A2,3=-0.27±0.40i and I Al I =1.15>1 which shw that the ppulatin is unstable. Actually, sme parameters will fluctuate at randm in the real situatin. Fr example, nest success fluctuates widely frm year t year in weather cnditins r rainfall (Dicksn 1992). T study the sensitivity f the mdel t individual parameter values and t manage the turkey ppulatins, I will change the parameter values ne by ne until I get the stable situatin. I emphasize that this is dne t investigate mdel prperties but smetimes the results may nt be bilgically reasnable. Let Al be the dminant eigenvalue. Case1 The adult reprductive rate, F3, is decreased frm 1.86 t 0.80. The Leslie matrix changed t be A= 0.445 0.880 0.800 a 0 0.616 0.610 23

Then eigenvalues are Al =0.999, A2=0.05, A3=-0.42, I Ai I < 1. It is asympttically stable. In this case, a big change in F3 is required t btain the stable situatin. Case2 The pult survival rate, PI' is decreased frm 0.445 t 0.260. Other parameters remain the same values as in the riginal matrix. Thatis A= 0.260 0.880 0.616 1. ~60] 0.610 The eigenvalues f this mdel are Al =0.998, A2. 3 =-0.19±0.35i and I Ai I < 1. It is a stable system. Case3 The yearling survival rate, P2' is decreased frm 0.616 t 0.28. Other parameters remain the same values as in the riginal matrix. Then Al =0.996, A2=0.02, A 3 =-0.40. Case4 The adult survival rate, P 3, is decreased frm 0.61 t 0.15. Other parameters remain the same values as in the riginal matrix. Then A( =0.997, Av =-0.42±0.52i. Case5 It's hard t get a stable state if nly F2 is decreased. I A( I is still bigger than I when F 2 had been decreased t 0.01 which is t small fr the yearlings fertility. Case6 I tried t change bth F 2 and F3 simultaneusly. Then when F 2 is reduced frm 0.88 t 0.40 and F3 is changed frm 1.86 t 1.13, AJ =0.999, A2,3 =0.19±0.4li. This is mre bilgically reasnable than case 1. There are several ways t change the values f F 2 and F3 in this case. In casel,2,3,4,and 6, the abslute values f Ai are less than ne. Thse systems are asympttically stable and because unity is nt an eigenvalue f A, the rigin is the nly equilibrium pint f system. The ppulatin is ging t be extinct after a lng run. 24

If unity is the dminant eigenvalue f A, instead f ging t be extinct, the ppulatin will eventually achieve a stable age distributin, which is prprtinal t the crrespnding eigenvectr f unity. This situatin is called the marginal stability. In rder t get the marginal stability, I change the parameter values ne by ne again frm the riginal matrix (Table 2.3). Table 2.3 The eigenvalues f marginally stable ppulatins btained by adjusting parameters separately riginal mdified Al A 2 A 3 value value PI 0.445 0.261928 1-0.195 + -0.195-0.3485i 0.3485i P 2 0.616 0.286669 1-0.394 0.004 P 3 0.610 0.16196 1-0.419 + -0.419-0.52i 0.52i F 3 1.860 0.86559 1-0.394 0.004 2. Sensitivity Analysis I used the change f the ttal ppulatin size based n the change f the parameter as an indicatin f parameter sensitivity. If Q(p) means the ttal ppulatin size (Q) is a functin f the parameter (p), then fr example, 25

5% sensitivity f Q t p = Q(p+5%p) -Q(p-5%p) 2*5%*Q(p) Table 2.4 shws that PI' the pult survival rate, is the mst sensitive parameter. In Suchy et al. (1983), they reasned that the mst sensitive parameters wuld be thse that prduced a statinary ppulatin with the smallest change in parameter value. Recall the data in Table 2.3. Table 2.4 The 5 %, 10%, and 15 % sensitivities f the ttal ppulatin size t each parameter Sensitivities Parameters 5 % 10% 15 % PI 1.4359 1.4361 1.4364 P 2 0.9627 0.9627 0.9627 P 3 1.0219 1.0240 1.0275 F 2 0.4873 0.4873 0.4875 F 3 1.0930 1.0930 1.0930 A decrease f 41 % in pult survival rate (PI) is the smallest reductin f all parameters t prduce statinary ppulatins. S PI shuld be the mst sensitive parameter which is shwn in Table 2.4. Changes in survival rates f female pults have a great effect n the ppulatin's net rate f change. The fecundity rate f adults is the secnd mst sensitive parameter. 26

3. Stable age distributin Recall that any ppulatin beginning with an unstable r stable age distributin will reach a stable age cmpsitin in time, the exceptin being a matrix with the first rw which nly has ne nnzer entry at the rightmst psitin. Frtunately, the turkey mdel is nt that kind f matrix and it will cme t a stable age distributin sner r later depending n the differences between the dminant eigenvalue and the ther nes. Fr the unstable, riginal matrix, the dminant eigenvalue is 1.1533 and the crrespnding eigenvectr is [1.26 0.486 0.551]T. Then the stable age distributin is S=[0.55 0.21 0.24]T. Fig 2.1 shws the simulatin f 15 years. Althugh the ppulatin keeps grwing, the age cmpsitin reaches a cnstant rati after nine years and wn't change in the future. One asympttically stable system is chsen (case 2) t d the simulatin f frecasting the ppulatin size. A= 0.250 0.880 1. 860] 0 0.616 0.610 Fig 2.2 shws the result f fifteen-year simulatin. After a shrt perid (within five years) f scillatin, the age distributin reaches stability althugh the ppulatin is decreasing. We are interested in the statinary situatin. When the dminant eigenvalue becmes ne, nt nly the age cmpsitin appraches a stable state but als the ppulatin size wn't change eventually. Then the ppulatin will never vergrw r die ut. Pick the statinary cases I have run befre and d the simulatin fr each ne. Fig 2.3, 2.4, 2.5, 2.6 shw the result f thse marginally stable systems gtten by adjusting PI' P 2, P 3, F 3 respectively. There are scillatins in all f them but it des nt take a lng time. 27

a aa ('t) pults yearlings adults Q) N '0 :: a C\I 1a ::::J Q. Q. a aa ~ /... /.. ' /..../.. ',...,~...,,...,.. ' ----:-:.:::... -.-:-::.-::..':.:':;/.. ~.-:-:-:'... :- /... /.. ' / / /...' ' /... 0--4..--.-----------r--------r-------~ 5 year 10 15 / Fig 2.1 Simulatin f riginal ppulatin mdel 28

LO Q) V N "00 c: "ca ::J 0 a. 0 8. C") pults yearlings adults C\I /', / '----- /"" ----- /"" -------,................. 5 10 15 year Fig 2.2 Simulatin f the asympttically stable ppulatin mdel btained by adjusting P1 29

LO 0 Q) 'I::t N en c: 0 pults 1a... ::l yearlings a. 0 ----- adults 0 0 a. C"') C\I //-- ---.../.,..--- T'"" '... -...... 2 4 6 8 10 year Fig 2.3 Simulatin f the stable ppulatin mdel btained by adjusting P1 30

LO Q) 0 N... "en c: 1a ::J a. a. 0 ('I) pults yearlings adults C\I... 'f, /------------------- :'...,/ 2 4 6 8 10 year Fig 2.4 Simulatin f the stable ppulatin mdel btained by adjusting P2 31

LO 0 Q) V N.c;; c: 0 pults 1a... yearlings ~ 0-0 ----- 0 0 adults 0- ('I) C\I..- '.,,,,.,.....t 1\'. "...... '\ II \\. /"""-_--------------- j\ / -' \J 5 10 15 20 year Fig 2.5 Simulatin f the stable ppulatin mdel btained by adjusting P3 32

LO pults yearlings adults Q) 0 N ~.Ci) c: 15 ~ a. 0 0 a. ('I) C\I,... /"- / "-, ~--------------,. /,~. ', ' '.. '/.: /'.., ',:~..J.,...r,............ 2 4 year 6 8 10 Fig 2.6 Simulatin f the stable ppulatin mdel btained by adjusting F3 33

4. The effect f harvest Harvest will influence the parameters in yearlings and adults stages directly and in pults stage indirectly. We assume that nly the survival prbabilities f yearlings and adults are affected by the harvest level which is defined as the percentage f the ttal ppulatin. Only fall hunting seasns are cnsidered because the mdel is fr female ppulatins and it's usually nly males can be hunted in the spring seasns. Data frm the bk named "The Wild Turkey" (Dicksn 1992) suggests pssible reductins in P2 and P 3 due t harvest (Table 2.5). Table 2.5 The survival rates f adult hens and yearling hens fr different fall harvest level Survival rates Fall harvest level female adults female yearlings ( % f Ttal ppulatin) (P 3 ) (P 2 ) 5 0.610 0.596 10 0.579 0.550 15 0.551 0.508 20 0.523 0.466 I changed the parameters f the riginal matrix based n this pssible effect f harvest and tried t figure ut the trend. Accrding t the different harvest level, P 2 and P 3 are replaced by the values in the table. Then the dminant eigenvalue is decreasing with the increase f harvest level (Table 2.6). 34

Table 2.6 The eigenvalues f different turkey systems based n the riginal Leslie matrix accrding t different harvest levels Harvest level Al A 23 5 1.146-0.268 ± 0.388i 10 1.114-0.268 ± 0.365i 15 1.085-0.267 ± 0.342i 20 1.056-0.267 ± 0.317i These results are surprising and suggest sme inadequacy f ur mdel because when the 20 percent f ttal ppulatin is remved by hunting ne might expect the ppulatin size shuld be smaller and smaller, that is, the dminant eigenvalue shuld be smaller than 1. The result didn't tell us that. Maybe it is because the mdel is t simple r there is smething wrng with ur parameter estimatin. As it has been mentined befre, the survival and fecundity rates will vary frm time t time depending n many factrs, like weather cnditin. Amng thse parameters, the survival rate f pults, PI' will change much widely than the thers. S PI is chsen t be adjusted t d sme mre analysis. If PI changed frm 0.445 t 0.330, the result f simulatin will be as fllws (Table 2.7). 35

Table 2.7 The eigenvalues f turkey ppulatins based n the mdified matrix by changing P I accrding t different levels f harvest Harvest level Al A 2,3 5 1.0549-0.2224 ± 0.3597i 10 1.0241-0.2226 ± 0.3405i 15 0.9957-0.2223 ± 0.3209i 20 0.9669-0.2219 ± 0.2992i This result perhaps lks mre plausible than the previus ne. The dminant eigenvalue starts t be belw than 1 when the harvest level is 15, 15 percent turkeys f the whle ppulatin have been hunted. Future wrk This turkey mdel is very simple. The turkey ppulatin is a cmplicated system. There are t few parameters in the mdel t prperly describe the cmplexity f a real turkey ppulatin. The matrix is extensible and mre parameters can be added t the mdel. Interactins between thse parameters culd be cnsidered r the whle ppulatin may be divided int mre stages. Fr example, if the time unit is six mnths, each stage can be divided int tw grups. Then the new mdel will have twice as many as stages the ld mdel has and the parameter estimates will be mre reliable because f less envirnmental variatin. The Leslie matrix A wuld be extended t be 6*6. 36

In a real turkey ppulatin, the number f each grup will affect the survival and fecundity rates f the ppulatin. The influence f density may nt be ignred. If a mdel takes accunt f the effect f density dependence n the survival and fecundity rates, then the mdel will be a nnlinear mdel. Anther way t make the mdel pwerful is t allw stchasticity. The Missuri Mdel in the bk (Dicksn 1992) used a stchastic mdel. In that mdel, if the data f thse parameters which can be estimated frm a radi-marked sample f wild turkeys fr specific years is inputed, then the parameters are randmly generated frm a unifrm distributin within the 95 percent cnfidence interval arund the mean f each parameter value fr the data years. This makes the mdel stchastic. Anther type f future wrk wuld be t cnsider survival as different functins f harvest rate. The functins culd be cnstructed t reflect additivity f hunting and natural mrtality as sme degree f cmpensatin (Le. as hunting mrtality increases perhaps natural mrtality decreases t sme extent). This is a very cmplex tpic beynd the scpe f this prject. 37

LITERATURE Caughley, G. 1977. Analysis f vertebrate ppulatins. Jhn Wiley and Sns Ltd, New Yrk, N.Y. 234pp. Cruse, DT., L. B. Crwder, and H. Caswell. 1987. A stage-based ppulatin mdel fr lggerhead sea turtles and implicatins fr cnservatin. Eclgy 68: 1412-1423. Cullen, M.R. 1985. Linear mdels in bilgy. Ellis Hrwd Ltd, England. 213pp. Dicksn, J.G. 1992. The wild turkey: bilgy and management. Stackple Bks. Harrisburg, PA.463pp. Ducet, P, and PB. Siep. 1992. Mathematical mdeling in the life sciences. Ellis Hrwd Ltd, England. 490pp. Grenendael, J.van, H. de Krn, and H. Caswell. 1988. Prjectin matrices in ppulatin bilgy. Trends in Eclgy and Evlutin 3, 264-269. Lefkvitch, L. P. 1965. The study f ppulatin grwth in rganisms gruped by stages. Bimetrics 21, 1-18. Leslie, P.H. 1945. On the use f matrices in certain ppulatin mathematics. Bimetrika 33: 183-212. Luenberger, D.G. 1979. Intrductin t dynamic systems: thery, mdels and applicatins. Jhn Wiley & Sns, New Yrk, N.Y. 446pp. Pielu, E. C. 1977. Mathematical eclgy. Jhn Wiley & Sns, Inc., New Yrk, N Y. 385pp. Ple, R. W. 1974. An intrductin t quantitative eclgy. McGraw-Hill, Inc., New Yrk, N.Y. 532pp. 38

Suchy, W.J., W.R. Clark, and T.W. Little. 1983. Influence f simulated harvest n Iwa wild turkey ppulatins. Prc. Iwa Acad. Sci. 90:98-102. Usher, M.B. 1969. A matrix mdel fr frest management. Bimetrics 25, 309-315. Yu, Y. 1990. A Leslie mdel, threshld functin and uncertainty fr chemical cntrl f crn earwrm. Ph.D. Thesis. Nrth Carlina State University. Raleigh, N.C. 148pp. 39