non -negative cone Population dynamics motivates the study of linear models whose coefficient matrices are non-negative or positive.

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1 LECTURE 3 Linear/Nonnegaive Marix Models x ( = Px ( A= m m marix, x= m vecor Linear sysems of difference equaions arise in several difference conexs: Linear approximaions (linearizaion Perurbaion analysis As model equaions in and of hemselves Populaion dynamics moivaes he sudy of linear models whose coefficien marices are non-negaive or posiive. P= ( pij, i, j m P or P> means p or p > for all i, j f( x = Px R R = ij m m maps non -negaive cone The soluion of he marix, difference equaion x ( = Px (, Z x( = x remains in he non-negaive cone, which we say is (forward invarian. ij Models make simplifying assumpions. Homogeneiies Demographic homogeneiy Temporal homogeneiy Spaial homogeneiy We ll consider relaxing he firs wo assumpions. Demographic inhomogeneiy Differences among individuals include: Age Size Gender Geneics Behavioral : A he exremes: Classical models (mos oal populaion size is he sae variable (all individuals reaed as idenical Individual based models Each individual is a sae variable In beween: Srucured models Sae variables are classes of individuals (individuals wihin classes are idenical

2 A Classic Example: (Leslie Age Srucured Marix Models x x i vecor of age classes x i number of individuals in age class i,,...,m For example, age classes could be in decades: x pre-eens, x eens, x 3 wenies, For i =,3,, m xi( = τ i, i xi( where τ ii,- is he fracion ha survives from age i o age i. For he las age class i = m x ( = τ x ( τ x ( m m, m m mm m For he i = age class, we have o consider newborns If fi is he per capia offspring produced by age class i per uni ime, hen m = i = i i x ( f x ( where In marix form : f f f 3 f m L x ( = Lx ( 3 m,m mm A more general marix model allows for ransiions among all classes for newborns o fall ino more han one class. x ( = Px ( P= F T, F = f, T = τ ( ij ( ij f ij per capia j-class newborns from i-class individuals or L f f f m f m 3 m,m mm ij fracion of j-class individuals ha move o i-class m ij, ij i The asympoic dynamics of soluions depend on he eigenvalues of he ransiion marix A. Does non-negaiviy ell us anyhing abou eigenvalues?

3 D Example a b Consider P = >. c d Could be a Leslie age-srucured model: Eigenvalues : L ( ( 4 λ = a d ± a d bc ± Some algebra shows : λ < λ b Eigenvecors : v± = λ± a More algebra shows v > and v is no posiive (second componen is negaive f f Asympoic dynamics of he equaion Similarly, Noe x ( = Px ( x ( = Px x = cv c v x( x c > = Px = cav cav x( = cλ v c λ v and x( = Px( = cλ v c λ v x ( = cλ v cλ v λ λ x ( = cλ v cλ v > lim x ( = (unbounded growh < lim x ( = (exincion Moreover, x ( cλ v c λ v cv c( λ / λ v = = x ( c v c v cv c λ / λ v Therefore, λ λ x ( v lim = x ( v ( Regardless of he populaion growh, he normalized (class disribuion sabilizes! Wha made all his work mahemaically? The posiive x marix has a posiive, simple, sricly dominan eigenvalue wih an associaed posiive eigenvecor no oher independen eigenvecor is non-negaive Perron s Theorem A posiive m x m marix has a posiive, simple, sricly dominan eigenvalue wih an associaed posiive eigenvecor no oher independen eigenvecor is non-negaive 3

4 As in he x case, we have he resul---- Suppose P > is a posiive m x m marix, λ > is is sricly dominan (simple eigenvalue, and v > is an associaed eigenvecor. Then he soluion of saisfies x ( = Px (, x lim x ( = if λ < lim x ( = if λ > x ( v In eiher case : lim = x ( v When does a non - negaive marix P a b = c d have a posiive eigenvalue & eigenvecor and no oher independen eigenvecor non-negaive? No boh b and c can equal. Assume WLOG ha b >. A Drawback: In applicaions P > is rarely posiive! An invesigaion of he formulas λ ( a d ( d a 4 bc = b v = ( d a ( d a 4bc shows a necessary and sufficien condiion is bc >, A permuaion marix M is he ideniy I wih is rows reordered. MP is he marix P wih is rows reordered PM is he marix P wih is columns reordered MPM is he marix P wih is rows and columns reordered (by he same permuaion i.e., boh off diagonal enries are nonzero. Such a marix is called irreducible. 4

5 DEFINITION: P is reducible if for some permuaion marix M A MPM = B C A and B are square marices Oherwise P is irreducible. In he dynamical model, irreducible means here is flow from each class o all oher classes (in a finie number of seps. Frobenius Theorem A non-negaive, irreducible m x m marix has a posiive, simple, dominan eigenvalue: λ > λ i wih an associaed posiive eigenvecor no oher independen eigenvecor is non-negaive F. R. Ganmacher, The Theory of Marices, Volume II, Chelsea, 96 Berman & Plemmons, Nonnegaive Marices in he Mahemaical Sciences, SIAM, 994 Wha is missing? Sric dominance of he posiive eigenvalue : λ > λ ι This is necessary for convergence of he normalized disribuion : x ( v lim = x ( v DEFINITION: A non-negaive, irreducible marix is primiive if is dominan eigenvalue is sricly dominan. The Fundamenal Theorem of Demography Suppose he m x m marix P is non-negaive & irreducible. Le > be he dominan eigenvalue of P and v > be an associaed posiive eigenvecor. Le x ( be he soluion of x ( = Px ( Then λ λ λ x( = x ( x. > lim x ( = (unbounded growh < lim x ( = (exincion Suppose P is primiive. Then x ( v lim =. x ( v 5

6 How o deermine if a non-negaive marix is irreducible? and primiive? Some useful facs: An irreducible, non-negaive marix is primiive if is race is posiive. A non-negaive marix is irreducible and primiive if and only if n P > for some posiive ineger n. There are many ess for specialized marices. L Example: Leslie Marices f f fm fm τ = τ 3 τmm, τ mm fi, τ mm, < τ ii, for i =,3,, m. Suppose m is an adul class : fm >. Then L is irreducible.. If, in addiion, race L = f τ mm >, hen L is primiive. 3. Bu i is common in models ha f = (juvenile class τ = (maximum age class. m L is primiive if wo consecuive classes are ferile, i. e., if wo consecuive f i are nonzero. Example: Usher Marices f τ f f3 fm fm τ τ fi, < τ i, i τ3 τ33 for i =,3,, m U = τii, τii τi, i τ m, m for i =,,, m τmm, τ mm Commonly used as a size-srucured model: individuals eiher remain in a size class or move one ahead each ime sep.. Suppose m is an adul class : fm >. Then L is irreducible.. If, in addiion, a leas one τ ii >, hen race U > and U is primiive. x ( = Px ( P= F T, F = f, T = τ ( ij ( ij f ij per capia j-class newborns from i-class individuals ij fracion of j-class individuals ha move o i-class m ij, ij i Suppose P > is irreducible and primiive. Then he Fundamenal Theorem of Demography applies and he normalized disribuion sabilizes. 6

7 Problems: How o calculae he dominan eigenvalue λ? Deermine when λ saisfies λ < or λ >? How does λ depend on he model parameers? EXAMPLE: x Leslie Marix L f f, = τ < τ τ ( ( ( 4 λ = f τ f τ fτ f τ Rewrie he inequaliy λ < as ( ( f τ f τ 4 fτ f τ < ( f τ 4( fτ f τ < ( f τ ( f τ 4( fτ f τ < 4 4( f τ ( f τ f fτ < τ Define n= f f τ n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : f Number of offspring when of age τ f f τ Define he number of offspring when of age x he probabiliy of reaching age n= f f τ n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : τ 7

8 Define f f τ f τ τ n= f f τ he number of offspring when of age x he probabiliy of reaching age x he probabiliy of surviving one more ime uni n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : τ Define n= f f τ n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : f f τ f τ τ f τ τ τ he number of offspring when of age x he probabiliy of reaching age x he probabiliy of surviving wo more ime unis Define n= f f τ n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : τ f f τ f τ τ f τ τ f τ τ 3 he number of offspring when of age x he probabiliy of reaching age x he probabiliy of surviving hree more ime unis Define n= f f τ n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : τ f f τ f τ τ f τ τ f τ τ 3 = expeced number of offspring during lifeime = ne reproducive number 8

9 Define n= f f τ n < λ < Similar calculaions show n > λ > n = λ = n has a biological inerpreaion : f f τ f τ τ f τ τ f τ τ 3 3 ( = f fτ τ τ τ = f fτ τ τ n = ne reproducive number General Definiion of he Ne Reproducive Number The eigenvalues of a marix ( T = τ, τ, τ ij ij ij i= saisfy λ <. We assume all eigenvalues saisfy λ <. Tha is o say, he specral radius ρ(t of T saisfies ρ(t <. Then he inverse exiss. m ( 3 I T I T T T = - > DEFINITION: If F( I T has a dominan eigenvalue n, hen n is called he ne reproducive number of P = F T. THEOREM: Suppose P = F T is irreducible and le λ > denoe is dominan eigenvalue. Suppose ( F, ρ( T <, F I T has a dominan eigenvalue n >. Then one of he following holds λ < n<, < n< λ, n= λ =. This is a sligh improvemen of a heorem of JC & Z. Yicang proved by Li & Schneider (J. Mah. Biol. 44, no. 5 (, AN EXAMPLE: he X Leslie marix (revisied L f f, τ = < τ τ f f =, = ( = τ < F T ρ T τ τ f f F( I T = τ τ f fτ f = τ τ Has a sricly dominan eigenvalue n= f f τ τ. 9

10 ij Biological Inerpreaion of n e e em e e e m 3 I T = = I T T T em em emm ( e = he fracion of ime an j - class newborn spends in class i Suppose here is only one newborn class: n * * ( F I T = f f f m F = n = f e f e f e m m For an inerpreaion of n when here is more han one newborn class see he monograph erraa. Selec a disribuion of newborns (among he classes. Calculae he expeced per capia number of offspring from each newborn class and ake he minimum. n is he maximum of hese minima aken over all possible disribuions of newborns. (The max and min can be inerchanged. n and λ can be used o sudy asympoic dynamics of populaions One advanage for n is ha formulas ofen exis ha relae i o he enries in he marices F and T. This is almos never he case for λ. Examples appear in he Monograph. f τ f f3 fm fm τ τ Usher Marix : τ3 τ33 (for size srucured populaions U = τ m, m τmm, τ mm Assume τ ii <. Then m i τ j, j n= f i ( where τ = i= j= τ jj An Applicaion Hudson River sriped bass (P. H. Leslie Populaion is divided ino 6 age classes: -yr-old (eggs, -yr-old (larvae -yr-old,, 4-yr-old (juveniles 5-yr-old,, 5-yr-old (aduls Ages 3 5 are subjec o fishing

11 The Leslie marix Can calculae λ numerically (using a compuer. n= f f τ fτ τ f m τ τ τ m m 3 3 3, ( τ τ τ f ( τ τ τ = f ,5 =.999 < λ < populaion is going exinc (I urns ou ha λ = Formulas for n can make parameer sudies possible. Suppose fishing moraliy could be managed in such a way ha he wo oldes ages are no aken. Suppose his resuls in an increased survivorship of s for hese older aduls. Quesion: wha survivorship s, if any, will keep he populaion from going exinc? ( τ τ τ ( τ τ τ ( τ τ f ( τ τ τ f5 ( ττ3 τ4,3τ5,4 f6 ( ττ3 τ4,3τ5,4τ6,5 ( τ τ f ( τ τ τ f5 ( ττ3 τ4,3s f6 ( ττ3 τ4,3s n= f f ,5 = f ,3 = f ,3 3 n= s s 3 n> if s > No possible.