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Structured two-stage population model with migration between multiple locations in a periodic environment

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Structured two-stage population model with migration between multiple locations in a periodic environment

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Content STRUCTURED TWO-STAGE POPULATION MODEL WITH MIGRATION BETWEEN MULTIPLE LOCATIONS IN A PERIODIC ENVIRONMENT by Selenne Hayde Garcia-Torres A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) August 2013 Copyright 2013 Selenne Hayde Garcia-Torres Dedication To Raul, Evan, and baby. i Acknowledgments I would like to thank my committee members: Cymra Haskell and Firdaus Udwa- dia. A special thanks goes to Dr. Cymra Haskell who made many valuable sugges- tions for improving my dissertation. My sincerest thanks to my advisor, Robert Sacker, for his guidance, patience and time. I was fortunate to have many friends during my graduate school years that I now consider family. Xin, Kasia, Alyona, and Jinlin thank you for all of the memories, in class, studying and outside of school. A special thanks to Jinlin for all of her help during our rst years of graduate school. I would also like to thank the Women in Science and Engineering (WiSE) program of USC and the Mathematics Department for the fellowship and help I received when my son was born. I thank my parents, Alex and Georgina, for all of their love and support over my years of study. My brothers, Omar and Cesar were always there for me when I needed a laugh to de-stress. Thanks guys! ii Finally, to my husband, Raul, and my son, Evan, who sacriced so much of their time with me so that I could continue my studies and career. Raul, thank you for all of the support and for keeping me grounded. Evan, you cured all of my stress with your beautiful laugh and smile when I walked through the door. iii Table of Contents Chapter 1: Introduction 1 Chapter 2: Ackleh, Chiquet, and Zhang's Results 6 Chapter 3: Cones, Monotonicity and Convexity 12 3.1 Positive Convex Cones . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Partial Order Relation . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Concave, Sublinear, and Monotone Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4.1 Irreducible Matrices . . . . . . . . . . . . . . . . . . . . . . 28 3.4.2 Primitive Matrices . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 4: Structured Population Model with Migration 36 4.1 Revised Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Statement of Main Result: n Locations and p-periodicity . . . . . . 37 4.3 Beverton-Holt Functions . . . . . . . . . . . . . . . . . . . . . . . . 40 4.4 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chapter 5: Further Interpretations of the Model 55 5.1 Weaker Condition for 2 Locations and Constant Birth and Migration Rates . . . . . . . . . . . . . . . . . 55 5.1.1 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . 61 5.1.2 Expanding Theorem 5.1 . . . . . . . . . . . . . . . . . . . . 67 5.2 A Look at the Directed Graph of Locations . . . . . . . . . . . . . . 69 5.3 Two Locations: One Population with i b i + i > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Bibliography 77 iv Chapter 1 Introduction We study a multidimensional nonlinear discrete system which describes the dynam- ics of a structured two-stage population model with migration. A structured pop- ulation is one where there are consistent dierences among the members of the population as a function of some attribute such as age, size, or physiological condi- tion as they develop. Here we partition the population by reproductive maturity. We take populations of the same species in somewhat adjacent locations and con- sider migration between the locations. Dynamical systems are systems that evolve over time according to some xed rule. If time is discrete the rule is described by mappings. In the study of dynamical systems one is interested in the changes the system goes through over time. That is, given an initial point in the state space we wish to determine all future states. The study of long time behavior of solutions of dierence equations is the study of iterations of mappings F : X! X. Here, we are interested in the existence and stability of a xed point of the mappingF , or in the context of population models, stability of the population. 1 We describe this multidimensional population model by using mappings that are monotone and concave with respect to a cone in the space. We do so in order to study the p-periodic system, or the case where we have p varying seasons in the model. Monotone maps have received much attention in the eld of dynamical systems because of their applications to mathematical models and nding xed points. However, applications of monotone maps to the study of periodic systems in higher dimensions are rare. Elaydi and Sacker [5] show that a concave increasing map from the positive reals to the positive reals satisfying certain conditions has a globally attracting exponentially asymptotically stable xed point. In particular, they show that this class of functions forms a semigroup under composition, thus obtaining a condition under which a periodic dierence equation will have a periodic solution having the same stability properties. The notions of semigroup and periodic dierence equation go hand in hand since whenever the functions F i ; i = 0; 1;:::;p 1; F n+p =F n belong to a semigroup, then nding a xed point of F (x) =F p1 F p2 :::F 1 F 0 (x) 2 is equivalent to nding a p-periodic solution to the p-periodic dierence equation x n+1 =F n (x n ): For example, Sacker [9] showed that mappings F : R + ! R + that are either concave increasing or convex decreasing and have non-negative Schwarzian, form a semigroup. In Gaut, Goldring, Grogan, Haskell and Sacker [6] conditions are given that guarantee the Sigmoid Beverton-Holt maps x 7! ax 1 +x ; for certain values of a and form a semigroup. Our motivation was to generalize the result obtained by Elaydi and Sacker to higher dimensions. We work with this model because it allows us to generalize Elaydi and Sacker's work to a class of monotone concave maps in higher dimen- sions, in particular, maps composed with a linear transformation matrix. This work involves a balance between theory and application and the crucial interplay between them. We utilize the theory to address a specic model and utilize the application to gain necessary insight to advance the theory. In Chapter 2 we describe the model rst proposed by authors Ackleh, Chiquet, and Zang in [1]. We summarize the results they obtained for the case where the birth and dispersal rates are constant. In Chapter 3 we develop the theory of 3 n-dimensional monotone concave maps and show that such a map composed with a linear transformation matrix forms a semigroup under composition. We further advance the theory in Chapter 4.3 for vector-valued mappings whose components are Beverton-Holt functions. We show that a mappingF written as a composition of several pairs of matrix-vector mappings as mentioned above satisfy the condi- tions of the Cone Limit Set Trichotomy Theorem [7, 8]. This theorem states that orbits of a monotone map from a cone to itself satisfying certain conditions behave in one of the following ways: the orbits are unbounded, each orbit converges to the origin, or each orbit converges to a unique nonzero xed point in the interior of the cone. We prove that a simple condition is sucient to obtain a unique globally asymptotically stable nontrivial xed point. In Chapters 4 and 5 we apply this work to dierent scenarios of the population model. In section 4.1 we propose a variation of Acklheh et. al.'s model which allows us to write its matrix-vector form as we described above: a transformation matrix composed with a vector-valued mapping whose components are Beverton- Holt functions. In section 4.2 we state our result for the existence and uniqueness of a globally attracting nontrivial asymptotically stable periodic state for the model with n locations and p periods. Here nontrivial means none of the populations in the various patches goes extinct. The result is proved in Section 4.4 after Section 4.3 described above. We show that a sucient condition for this occurrence is that the product of the inherent survivorship fraction of the juveniles and the intrinsic 4 birth rate of each patch over each season is greater than 1 (Theorem 4.1). In section 5.1 we study the case of two locations and constant migration and birth rates and show that a weaker condition over each location also gives us a globally attracting asymptotically stable steady state. We study the case where some migration rates are zero for locations greater than 2 in Section 5.2. Finally in Section 5.3 we study the case when only one location is thriving in the 2 location model. 5 Chapter 2 Ackleh, Chiquet, and Zhang's Results Consider the nonlinear discrete-time model proposed by Ackleh et. al. [1], which describes the dynamics of a two-stage population model with migration between two locations. The population is partitioned by reproductive maturity- juveniles and adults. The number of juveniles at time t in location i for i = 1; 2 is denoted by J i (t): Similarly, A i (t) denotes the number of adults. J 1 (t + 1) = b 1 (t)A 1 (t) J 2 (t + 1) = b 2 (t)A 2 (t) (2.1) A 1 (t + 1) = 1 J 1 (t) 1 + 1 J 1 (t) + 1 (1m 12 (t)) A 1 (t) 1 + 1 A 1 (t) + 2 m 21 (t) A 2 (t) 1 + 2 A 2 (t) A 2 (t + 1) = 2 J 2 (t) 1 + 2 J 2 (t) + 1 m 12 (t) A 1 (t) 1 + 1 A 1 (t) + 2 (1m 21 (t)) A 2 (t) 1 + 2 A 2 (t) where ( J 1 (0); A 1 (0); J 2 (0); A 2 (0) )2R 2 + n (0; 0)R 2 + n (0; 0): 6 Ackleh et. al. arrive at this model by assuming that there is no competition between the juveniles and adults and that only the adults migrate. These condi- tions are consistent with frog populations; tadpoles stay in their place of birth until reaching adulthood and do not compete with adults for resources. The intrinsic birth rates are denoted as b 1 (t) and b 2 (t) in locations 1 and 2, respectively. Let m ij (t) denote the migration rate at timet from locationi to locationj fori;j = 1; 2 and i6= j. Hence, it is assumed that 0 m 12 (t);m 21 (t) 1. The terms 1+ J(t) and 1+A(t) represent the fraction of juveniles and adults, respectively, at time t that survive one unit of time. Thus, since i and i are the inherent survivorship fraction for each group, we assume 0< i ; i < 1 for i = 1; 2. Furthermore, we let the time variablet be such that after one unit of time the juveniles become adults. Therefore 1+ J(t) represents the fraction of juveniles that survive and join the adult group at timet + 1. In the following we will summarize the results obtained in [1]. The system is rewritten as X(t + 1) =C(X(t))X(t); where X(t) = (J 1 (t);J 2 (t);A 1 (t);A 2 (t)) T , and C has the form 7 C(X(t)) = 0 B B B B B B B B B B @ 0 0 b 1 0 0 0 0 b 2 1 1+ 1 J 1 (t) 0 1 (1m 12 (t)) 1+ 1 A 1 (t) 2 m 21 (t) 1+ 2 A 2 (t) 0 2 1+ 2 J 2 (t) 1 m 12 (t) 1+ 1 A 1 (t) 2 (1m 21 (t)) 1+ 2 A 2 (t) 1 C C C C C C C C C C A : (2.2) The authors rst consider constant birth and dispersion. Therefore m ij (t) = m ij and b i (t) = b i are positive constants, for i;j = 1; 2;i6= j. Evaluated at the trivial steady state E 0 = (0; 0; 0; 0) T , C becomes a constant matrix C(E 0 ) = 0 B B B B B B B B B B @ 0 0 b 1 0 0 0 0 b 2 1 0 1 (1m 12 ) 2 m 21 0 2 1 m 12 2 (1m 21 ) 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ 0 0 b 1 0 0 0 0 b 2 0 0 0 0 0 0 0 0 1 C C C C C C C C C C A + 0 B B B B B B B B B B @ 0 0 0 0 0 0 0 0 1 0 1 (1m 12 ) 2 m 21 0 2 1 m 12 2 (1m 21 ) 1 C C C C C C C C C C A _ = F +G 8 Using a result from another publication based on net reproductive value by authors Cushing and Yicang [4], the net reproductive is dened as the positive dominant eigenvalue of F (IG) 1 . The third and fourth rows of the matrix F (IG) 1 are zeroes. Therefore the eigenvalues of F (IG) 1 are zero and the eigenvalues of the submatrix F (IG) 1 [1; 2 : 1; 2] _ =M given by M 11 = b 1 1 (1 2 +m 21 2 ) (1 1 )(1 2 ) +m 12 1 (1 2 ) +m 21 2 (1 1 ) ; M 12 = b 1 m 21 2 2 (1 1 )(1 2 ) +m 12 1 (1 2 ) +m 21 2 (1 1 ) ; M 21 = b 2 m 12 1 1 (1 1 )(1 2 ) +m 12 1 (1 2 ) +m 21 2 (1 1 ) ; M 22 = b 2 2 (1 1 +m 12 1 ) (1 1 )(1 2 ) +m 12 1 (1 2 ) +m 21 2 (1 1 ) : Let D 1 = det(M) = b 1 b 2 1 2 (1 1 )(1 2 ) +m 12 1 (1 2 ) +m 21 2 (1 1 ) and D 2 = tr(M) det(M) = b 1 1 [1 2 (1m 21 )] +b 2 2 [1 1 (1m 12 )]b 1 b 2 1 2 (1 1 )(1 2 ) +m 12 1 (1 2 ) +m 21 2 (1 1 ) : 9 Then the net reproductive number, in terms of D 1 and D 2 , is given by R 0 = D 2 +D 1 + p (D 2 +D 1 ) 2 4D 1 2 : Ackleh et. al. prove a stability result for a nontrivial xed point formulated in [1]. Theorem 2.1. Consider system (2.1) with b i (t) =b i ;m 12 (t) =m 12 ; and m 21 (t) = m 21 positive constants for i = 1; 2 and t = 0; 1; 2:::. If R 0 > 1, then there exists a unique nontrivial interior steady state which is globally attracting. The authors also show that if R 0 < 1 then the trivial steady state E 0 = (0; 0; 0; 0) T is globally attracting. In another section they consider the case where (2.1) varies with period = 2 and obtain results by nding the net reproductive number for that case. We are interested in nding simpler sucient conditions for which there exists a unique nontrivial steady state. We take into account the following issues raised by this model: (1) all of the eggs are assumed to hatch (2) due to the complex formulation of the conditions, a generalization to a larger number of locations seems out of reach (3) for the same reasons given in 2 it is not clear how to generalize to a consid- eration of periods greater than 2. 10 We will be addressing these issues in Chapter 4 by considering a variation of the model described here. 11 Chapter 3 Cones, Monotonicity and Convexity 3.1 Positive Convex Cones Let X be a Banach space and denoteR + =fa2Rja 0g. We say KX is a positive convex cone if the following are satised: i) K is closed, nonempty, and K6=f0g ii) ax +by2K for all a;b2R + and x;y2K iii) K \ (K) =0. Example 1. There are only two positive convex cones when X =R, K = [0;1) and K = (1; 0]. Example 2. For X =R n each orthant is a positive convex cone. 12 3.2 Partial Order Relation We dene a closed partial order relation on X with respect to the cone K: x K y () yx2K x< K y () x K y and x6=y x K y () yx2 Int(K) The Banach space X together with the above partial order relation gives us a partially ordered set (X; K ). In population model applications we are typically interested in lettingX =R n and thus the interior of an orthant ofR n is nonempty. In order to obtain a better grasp of this partial order relation, we prove a few properties that will be helpful later. Proposition 1. If x K y and y K z, then yx K zx. Proof: By denition of the partial order relation we know zy2K and zx (yx) =zy2K: Therefore, yx K zx. 13 Recall that a Banach space, X, is a topological vector space. Thus, the vector space operations are continuous with respect toT , a topology onX. We associate to each xed z2X the translation operator T z dened by T z (x) =z +x (x2X): Lemma 3.1. The translation operator T z is a homeomorphism from X onto X. Proof: The vector space axioms imply thatT z is one-to-one and onto. Indeed, for each y2 X we let x = yz, hence T z (x) = z + (yz) = y. Also, closure under addition and scalar multiplication imply T z is a mapping from X onto X. Note that the inverse of T z isT z and by continuity of vector space operations we know both mappings are continuous. Therefore T z is a homeomorphism. We thus conclude that every vector topologyT is translation-invariant. Corollary 1. If AX is open, then z +A =fz +aja2Ag is open. We now have the necessary tools to prove the following proposition. Proposition 2. If x2 Int(K) and x K y, then y2 Int(K) 14 Proof: Let x2 Int(K), then there exists an open neighborhood of x, denoted N x , such that x2 N x K. We know yx2 K and Corollary 1 together with the denition of a positive convex cone imply (yx) +N x K is open. Therefore, since Int(K) is the largest open subset of K we obtain y2 (yx) +N x Int(K): 3.3 Concave, Sublinear, and Monotone Mappings Let X and Y be Banach spaces. We say a map F : X! X is monotone if for every x;y2 X with x K y we have F (x) K F (y). For mappings F : X! Y we say F is convex with respect to K if 8x 1 ;x 2 2X and ; 0 with + = 1 we have F (x 1 ) +F (x 2 )F (x 1 +x 2 )2K: 15 or equivalently, F (x 1 +x 2 ) K F (x 1 ) +F (x 2 ): (3.1) Similarly, we say F is concave with respect to K, or K-concave, if F (x 1 +x 2 ) K F (x 1 ) +F (x 2 ): (3.2) This coincides with our usual notions of convexity and concavity for functions f :R!R and K =R + since condition (3.1) is simply f(x 1 +x 2 )f(x 1 ) +f(x 2 ); (convexity) and condition (3.2) is f(x 1 ) +f(x 2 )f(x 1 +x 2 ); (concavity) for x 1 ;x 2 2R and ; > 0 with + = 1. Remark 1. We make a side remark that denitions (3.1) and (3.2) stated above are equivalent to the denitions of convexity and concavity with respect to the epi- graph of a mapping. 16 Let X and Y be Banach spaces with function F : X! Y dened on convex UX. Let K be a positive convex cone in Y , then we dene the K - epigraph of F as epi K (F ) _ =f(x;y)2UYjx2U;yF (x)2KgXY: We say F is convex with respect to K if epi K (F ) is a convex set in XY . Note this denition satises our notion of convexity for functions f :R!R; namely, for X = Y = R and K = R + a function f is convex with respect to the positive real line if the region above the graph of f is convex inR 2 . We sayF is concave with respect toK ifF is convex with respect toK. That is, if epi K (F ) =f(x;y)2UYjx2U;y +F (x)2Kg is a convex set in XY . The following theorem states that the two given denitions are equivalent. Theorem 3.2. Let F :X!Y be dened on convex UX. Then epi K (F ) is a convex set in XY () 8x 1 ;x 2 2U and; 0with + = 1, F (x 1 ) +F (x 2 )F (x 1 +x 2 )2K: 17 Proof: Assume epi K (F ) is a convex set in XY and let x 1 ;x 2 2 U with ; 0 and + = 1. Then for (x 1 ;F (x 1 )); (x 2 ;F (x 2 ))2 epi K (F ) it follows by the convexity of epi K (F ) that (x 1 ;F (x 1 )) +(x 2 ;F (x 2 ))2 epi K (F ) ) (x 1 +x 2 ;F (x 1 ) +F (x 2 ))2 epi K (F ) )F (x 1 ) +F (x 2 )F (x 1 +x 2 )2K: Conversely, we show epi K (F ) is a convex subset of XY . Let ; 0 with + = 1 and (x 1 ;y 1 ); (x 2 ;y 2 )2 epi K (F ). Then, y 1 F (x 1 );y 2 F (x 2 )2K: It follows that, y 1 F (x 1 ) +y 2 F (x 2 )2K: by (ii) of the denition of K. Again by (ii) with a =b = 1 and our hypothesis, we have y 1 F (x 1 ) +y 2 F (x 2 ) +F (x 1 ) +F (x 2 )F (x 1 +x 2 )2K: 18 That is, y 1 +y 2 F (x 1 +x 2 )2K: Thus (x 1 +x 2 ;y 1 +y 2 )2 epi K (F ) and therefore (x 1 ;y 1 ) +(x 2 ;y 2 )2 epi K (F ): The K-epigraph of F is a convex subset of XY . A similar proof shows the K-epigraph ofF is a convex set in XY if and only if8x 1 ;x 2 2U, with ; 0 and + = 1, F (x 1 +x 2 )F (x 1 )F (x 2 )2K: We now dene strict convexity and concavity by placing a stronger condition on (3.1) and (3.2), respectively: A map F :X!Y is strictly convex with respect to K if F (x 1 +x 2 ) K F (x 1 ) +F (x 2 ); (3.3) 19 and strictly concave with respect to K, or strictly K-concave, if F (x 1 +x 2 ) K F (x 1 ) +F (x 2 ) (3.4) for all x 1 ;x 2 2X such that [x 1 ] i 6= [x 2 ] i for all i, and ; > 0 with + = 1. Dene the setS(K) to be the class of all functions from a positive convex cone, KX, to itself that are strictly concave and monotone with respect to K. That is, S(K) : =fF :K!KjF is K-monotone, F (x 1 ) +F (x 2 ) K F (x 1 +x 2 )g for ; 0 and + = 1. We now state and prove a useful theorem that lays the groundwork for our study of periodic systems. Theorem 3.3.S(K) forms a semigroup under composition. Proof: Let F;G2S(K) and assume x 1 K x 2 then by K-monotonicity of F and G we have F (G(x 1 )) K F (G(x 2 )) so that FG is K-monotone. Since G is strictly K-concave we have G(x 1 +x 2 ) K G(x 1 ) +G(x 2 ): 20 Then by K-monotonicity and strict K-concavity of F we obtain F (G(x 1 +x 2 )) K F (G(x 1 ) +G(x 2 )) K F (G(x 1 ) +F (G(x 2 )): Proposition 1 implies F (G(x 1 +x 2 ))F (G(x 1 )F (G(x 2 )) K F (G(x 1 ) +G(x 2 ))F (G(x 1 )F (G(x 2 )) where F (G(x 1 ) +G(x 2 ))F (G(x 1 )F (G(x 2 ))2 Int(K). Thus, invoking Proposition 2 we know F (G(x 1 +x 2 ))F (G(x 1 )F (G(x 2 ))2 Int (K): Therefore, FG is strictly convex with respect to K and FG2S(K): We state the denition of a strongly sublinear map, which appears in the state- ment of the Cone Limit Set Trichotomy Theorem in Section 4.2, and study the dierences between a sublinear map and a concave map. We say a mapF :K!K is strongly sublinear if 0<< 1 and x2 Int(K) imply F (x) K F (x): 21 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 x y 4x 1+ x Figure 3.1: 4x 1+x is strongly sublinear In order to gain a better understanding of strongly sublinear maps and how they dier from concave and convex maps with respect to a positive convex cone we look at functions from R + to R + and let K =R + . Note that constant functions are both concave and convex however, only constant functions f(x) =c wherec is positive are strongly sublinear. Functions f :R + !R + dened as f(x) = cx +b where c> 0 are also concave and convex however, such functions are not strongly sublinear unless f(0)> 0, i.e. b> 0: We may look at the graph of a function and conclude whether or not it is strongly sublinear if all lines from the origin to the graph lie underneath the graph; see Figure 3.1 and Figure 3.2. In population dynamics we are interested in functions that are positive, concave, and increasing. The following proposition shows us that strictly concave mappings are strongly sublinear. 22 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x y ³ x 2 ´ 2 + 1 2 Figure 3.2: x 2 2 + 1 2 is not strongly sublinear Proposition 3. Let X be a Banach Space, K X a positive convex cone and F :K!K strictly concave with respect to K, then F is strongly sublinear. Proof: Let F be strictly K-concave, then F (x 1 +x 2 ) K F (x 1 ) +F (x 2 ) for x 1 ;x 2 2 K such that [x 1 ] i 6= [x 2 ] i for all i, and ; > 0 with + = 1. Consider x 1 2 Int(K) and x 2 =0, then F (x 1 ) K F (x 1 ) +F (0) K F (x 1 ): 23 Therefore F (x 1 ) K F (x 1 ) where 2 (0; 1), hence F is strongly sublinear. As described in Chapter 1 we are working with mappings that are strictly K- concave and K-monotone because they form a semigroup under composition. We use the semigroup property to study ap-periodic system. The denition of strongly sublinear mappings is discussed here because it is a condition that must be satised in order to apply the Cone Limit Set Trichotomy Theorem (page 46). Thus, although the class of strictly K-concave mappings belongs to the class of strongly sublinear mappings we are not working with this larger set because the composition of two strongly sublinear mappings is not necessarily strongly sublinear, as the following example illustrates. Example 3. Let f;g :R + !R + be dened as f(x) = 1 x + 1 100 g(x) =e x Both f and g are strongly sublinear. However, (fg)(x) =e 1 x+ 1 100 is not strongly sublinear. See Figure 3.3. We will consider the composition of p strictly K-concave and K-monotone maps from the cone to itself. By Theorem 3.3 we know that the composition is strictlyK-concave. Then we apply Proposition 3 to show that the composition is 24 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 3.3: e 1 x+ 1 100 strongly sublinear and thus satisfy a condition of the Cone Limit Set Trichotomy Theorem (page 46). 3.4 Matrix Operations We use the standard notation M mn (R) to denote the set of mn matrices with elements in R and M n (R) the set of nn matrices. Let A = [a ij ]2 M mn (R), we say A is nonnegative and denote A 0 if a ij 0 for all i = 1;:::;m and j = 1;:::;n: Similarly, A is positive or A > 0 if a ij > 0 for all i;j. Thus we say A > B if AB > 0 for A;B2 M mn (R). For the rest of the paper we will be considering A2M mn (R + ), or A 0. 25 Let KR n and dene the setM(K) as M(K) =fM2L(R n ;R n )jK and Int(K) are invariant under Mg: We observe the following property satised byS(K) and the setM(K): Theorem 3.4. If F2S(K) and M2M(K) where K is a positive convex cone in R n , then MF2S(K). i.e. the setS(K) is closed under left composition with elements ofM(K). Proof: Let M 2 M(K) and F 2 S(K), then MF is a mapping from K to K and for x 1 ;x 2 2 K with x 1 K x 2 we have that F (x 1 ) K F (x 2 ) by K-monotonicity of F . By denition of the partial order this implies F (x 2 ) F (x 1 )2K and asM is a linear map that leavesK invariant we have (MF )(x 2 ) (MF )(x 1 )2 K. Therefore MF is K-monotone. Since F is strictly K-concave (3.4) holds, and by invariance of Int(K) under M it follows that (MF )(x 1 +x 2 )(MF )(x 1 )(MF )(x 2 )2 Int(K) so that MF is strictly K-concave. Therefore MF2S(K). 26 The following theorem will be useful in the next chapter in order to show that the map describing our model satises another condition of the Cone Limit Set Trichotomy Theorem. Theorem 3.5. Let the cone K be R n + and H a function from K to K for which H 0 is a positive matrix for all x2 K, i.e. H 0 (x) > 0. Then for x > K 0 we have H(x) K 0, i.e. H maps the boundary of the cone K into the interior. Proof: Let `(t) = (1t)x 0 +tx where x> K x 0 and H(x 0 )2K. Then, H(x)H(x 0 ) = H(`(1))H(`(0)) = Z 1 0 DH(`(s))ds = Z 1 0 DH(`(s)) d ds `(s)ds = Z 1 0 DH(`(s))(xx 0 )ds: Thus H(x) = Z 1 0 DH(`(s))(xx 0 )ds +H(x 0 ) Therefore, sinceH(x 0 ) K 0 andH 0 (x)> 0 we concludeH(x) K 0 forx> K 0 27 3.4.1 Irreducible Matrices A permutation matrix is a square binary matrix with exactly one 1 in each row and column and 0's elsewhere. Note that we can obtain an nn permutation matrix P by interchanging rows or columns of the nn identity matrix. When annn matrix A is multiplied on the right by a permutation matrix P, the result AP, is a permutation of the columns of A. Similarly, PA is a permutation of the rows of A. An nn nonnegative matrix A is said to be reducible if there exists a permu- tation matrix P such that PAP T = 2 6 6 4 A 11 A 12 0 A 22 3 7 7 5 where A 11 and A 22 are square submatrices. We say the matrix is irreducible if it is not reducible. Another method of identifying an irreducible matrix, A = [a ij ]2 M n (R + ), is by examining its corresponding directed graph, G A . A directed graph is a set of nodes,p 1 ;:::;p n , connected by edges with a direction associated to them. We call such edges directed arrows and denote ! p i p j to be the directed arrow from node p i to nodep j . We say a directed graph is strongly connected if for any pair of distinct nodes p i and p j there exists a directed path, ! p i p i 1 ! p i 1 p i 2 ::: ! p i k p j , from p i to p j . A 28 directed graph is totally connected if for any pair of distinct nodes p i andp j there exists a pair of unique directed arrows, ! p i p j and ! p j p i . We construct the corresponding directed graph G A , of a matrixA, by drawing a directed arrow ! p i p j if a ij 6= 0 for i;j = 1;:::;n. Thus the graph G A is strongly connected if there exist indices k 1 ;:::;k m such that a ik 1 a k 1 k 2 :::a kmj > 0: It is clear by the denition of positivity that any matrix A > 0 is irreducible. The corresponding directed graph of A> 0 is totally connected. Example 4. Consider the following matrix A = 2 6 6 6 6 6 6 4 0 1 2 0 0 1 5 0 0 3 7 7 7 7 7 7 5 (3.5) thus, we plot three pointsp 1 ;p 2 ;p 3 on the plane. We see thata 12 > 0, thus we draw a directed arrow from p 1 to p 2 . Continuing to draw the directed arrows we have G A as shown in Figure 3.4. From the gure we can see that there is a directed path from p i to p j where i;j = 1; 2; 3 and thus the directed graph corresponding to A is strongly connected but not totally connected. An isomorphism of graphs G and H is a bijection between the node sets of G and H f :V (G)!V (H) 29 r p 1 r p 3 r p 2 1 Figure 3.4: directed graph for (3.5) such that any two nodesa andb ofG are connected inG if and only iff(a) andf(b) are connected in H while preserving direction. If an isomorphism exists between two graphs, then we say the graphs are graph isomorphic and write G =H. Note that the directed graph, G A , of a matrix A = [a ij ]2 M n (R + ) depends only ona ij > 0, not on its value. Thus it suces to work with binary matrices. We associate to the nonnegative matrix A a binary matrix A = [ a ij ] where a ij = 0 if a ij = 0 and a ij = 1 if a ij > 0. Binary addition and multiplication of two nn square binary matrices K = [k ij ] and L = [l ij ] is dened in the usual sense: K +L = [k ij +l ij ] and KL = [ P n r=1 k ir l rj ], with the exception that 1 + 1 = 1. Note that for A;B2M n (R + ) and C =AB we have C = A B. 30 Identifying a givennn matrix as reducible or irreducible by applying the above denition would take much computation as there are a total of n! permutation matrices. The following theorem tells us we need only examine its directed graph. Theorem3.6. A matrixA2M n (R + ) is irreducible if and only if its corresponding directed graph G A is strongly connected. Proof: Assume A2 M n (R + ) is reducible. Then there exists a permutation matrix P such that PAP T = 2 6 6 4 A 11 A 12 0 A 22 3 7 7 5 : Say A 11 2 M k (R + ) and A 22 2 M nk (R + ) so that 0 is an nkk matrix. We examineG PAP T and thus consider the pointsp 1 ;p 2 ;:::;p n . We will show that there is no directed path fromp n top 1 . Note that because of the location and dimension of our 0 matrix the points p k+1 ;p k+2 ;:::;p n have no directed arrow to the points p 1 ;p 2 ;:::;p k : The only directed arrows fromp k+1 ;:::;p n , if any, are to themselves. Thus, although p n may have a directed arrow to any of the points p k+1 ;:::;p n1 there is no way to get to p 1 from p n . We have therefore shown G PAP T is not strongly connected. Since A is similar to PAP T , G PAP T is isomorphic to G A and thus G A is not strongly connected. We have shown if the corresponding directed graph is strongly connected then the matrix is irreducible. 31 To explore the other direction we rst note that the (i;j) element ofA m , which we denote by a (m) ij , is given by n X k 1 ;k 2 ;:::;k m1 =1 a ik 1 a k 1 k 2 ... a k m2 k m1 a k m1 j : (3.6) Thus, if a (m) ij > 0 then the product a ik 1 a k 1 k 2 ... a k m2 k m1 a k m1 j > 0 for at least one collection of indices k 1 ;k 2 ;:::;k m1 2f1;:::;ng. Assume the directed graph G A is not strongly connected. Then there exist indicesi;j such thata (m) ij = 0 for allm. We will construct the permutation matrix P such that [PAP T ] n1 =a ij . Let e k be the unit column vector with 1 in the kth position and zeroes elsewhere. Place the column vector e n in column i and e 1 in column j in the nn matrix P . Fill in the rest of the columns so that there is exactly one 1 in each row and column and zeroes elsewhere. Then note that the rst row of P is e T j and the nth row is e T i . Thus the rst column of P T is e j and the nth column is e i . Since PA permutes the rows of A we have that [PA] nj =a ij Then, since (PA)P T permutes the columns of PA we have that [PAP T ] n1 =a ij : 32 R R R c R c A11 A12 A22 0 Figure 3.5: PAP T For notational convenience we will denote [PAP T ] rs =a rs so that a (m) n1 = 0 for all m. Let R =f kj a (m) nk = 0 8mg. That is, if we consider points p 1 ;:::;p n in the plane, the set R is the set of all indices k such that their does not exist a directed path from p n to p k . We will show that the matrix PAP T has the form as shown in Figure 3.5. CASE 1: AssumeR c is nonempty. We claim that for alll2R c andk2R we have a ik = 0. Assume towards contradiction that for some l2 R c and k2 R we have a lk 6= 0. Then there exists a positive integer m l such that a (m l ) nl > 0. Thus a (m l ) nl a lk > 0. Therefore, since a (m l +1) nk =a (m l ) n1 a 1k +::: +a (m l ) nn a nk and all terms are nonnegative we have that a (m l +1) nk > 0 which contradicts the denition of R. 33 CASE 2: Assume R c is empty. Then a nk = 0 for k = 1;:::;n. Therefore A 11 is an n 1n 1 block matrix, A 22 = 0, and the size of the zero matrix in PAP T is 1n 1. Note Theorem 3.6 and Figure 3.4 tells us our previous 3 3 matrix example (3.5) is irreducible. 3.4.2 Primitive Matrices A nonnegative square matrixA is primitive if there exists a natural numberm such that A m > 0. It is easy to see that any primitive matrix is irreducible. However, not all irreducible matrices are primitive. Take for example, A = 0 B B @ 0 1 1 0 1 C C A Its powers alternate between the identity matrix and itself. However, an irreducible matrix is primitive when a simple condition is met. Lemma 3.7. If a nonnegative irreducible matrix A has at least one positive diag- onal element, then A is primitive. 34 Proof: Let A = [a ij ]2M n (R + ) and consider nodes p 1 ;:::;p n . Assume with out loss of generality that a 11 > 0. Denote n ij = 1;:::;n 1 to be the minimum number of directed arrows needed to get from node p i to node p j . Let N = max ij fn i1 +n 1j g: We claim that A N > 0. Note that a (n ij ) ij > 0 and A N = A n i1 A N(n i1 +n 1j ) A n 1j : For any i;j = 1;:::n we have that a (n i1 ) i1 a (N(n i1 +n 1j )) 11 a (n 1j ) 1j > 0: Therefore a (N) ij > 0 for all i;j. 35 Chapter 4 Structured Population Model with Migration 4.1 Revised Model We describe a variation of the model of Ackleh et. al. found in [1], where a species of frogs migrates between two ponds. Recall the issues raised by the model in Chapter 2. In order to address the rst issue - all of the eggs are assumed to hatch - we consider model (2.1) in a slightly dierent form: J 1 (t + 1) = b 1 (t) A 1 (t) 1 + 1 A 1 (t) (4.1) A 1 (t + 1) = 1 J 1 (t) 1 + 1 J 1 (t) + 1 (1m 12 (t)) A 1 (t) 1 + 1 A 1 (t) + 2 m 21 (t) A 2 (t) 1 + 2 A 2 (t) J 2 (t + 1) = b 2 (t) A 2 (t) 1 + 2 A 2 (t) A 2 (t + 1) = 2 J 2 (t) 1 + 2 J 2 (t) + 1 m 12 (t) A 1 (t) 1 + 1 A 1 (t) + 2 (1m 21 (t)) A 2 (t) 1 + 2 A 2 (t) where ( J 1 (0);A 1 (0);J 2 (0);A 2 (0) ) 2R 2 + n (0; 0)R 2 + n (0; 0): 36 The presence of the 1+A terms in the denominator of juvenile equationsJ 1 (t+1) andJ 2 (t+1) introduce a depensatory eect; for large numbers of eggs, the amount of eggs that hatch and reach the tadpole stage decreases. Thus we have taken into account that not all eggs produce juveniles. In the next section we will write the model for more than two locations and state our main result forn locations andp varying seasons. We show in Section 4.3 that the matrix-vector form of model (4.1) belongs to the semigroupS(K) discussed in Chapter 3. There we state a theorem whose simple conditions guarantee a unique nontrivial globally asymptotically stable steady state for our population model. We apply this theorem to dierent scenarios of our population (Section 4.4 and Chapter 5) model and show sucient conditions on the birth, migration, and survival rates that guarantee a nontrivial xed point. 4.2 Statement of Main Result: n Locations and p-periodicity We extend model (4.1) to include more locations, n> 2. We let X(t) = (J 1 (t) A 1 (t) J 2 (t) ::: A n1 (t) J n (t) A n (t)) T 37 and write the system in matrix vector form for n locations and a given time t as: X(t + 1) = L(t)B(X(t)) (4.2) = f t (X(t)) (4.3) where the composition (product) of the 2n 2n matrix L(t) and vector-valued mapping B(X(t)) is given by L(t)B(X(t)) = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ 0 b 1 (t) 0 0 ::: 0 0 1 1 (1 1 ) 0 2 m 21 (t) ::: 0 n m n1 (t) 0 0 0 b 2 (t) ::: 0 0 0 1 m 12 (t) 2 2 (1 2 ) ::: 0 n m n2 (t) . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ::: 0 b n (t) 0 1 m 1n (t) 0 2 m 2n (t) ::: n n (1 n ) 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ J 1 (t) 1+ 1 J 1 (t) A 1 (t) 1+ 1 A 1 (t) J 2 (t) 1+ 2 J 2 (t) A 2 (t) 1+ 1 A 2 (t) . . . Jn(t) 1+ nJn(t) An(t) 1+nAn(t) 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C C A 38 and k = n X j=1 j6=k m kj (t). We dene the mapping F :R 2n + !R 2n + to be: F = f p f p1 :::f 0 = L(p)B(X(p))L(p 1)B(X(p 1)):::L(0)B(X(0)): (4.4) In Section 4.3 we develop the theory necessary to show existence of xed points of a map such as (4.4). In Section 4.4 we apply the theory of Section 4.3 and prove the following theorem: Theorem 4.1. Consider the system given in (4.4) with 0<m ij (t)< 1 for i;j = 1;:::;n, i6=j and for all t = 0;:::;p. If jdet(F 0 (0))j =j 1 b 1 (p)::: n b n (p)::: 1 b 1 (0)::: n b n (0)j> 1 then there exists a unique globally asymptotically stable steady state in the interior ofR 2n + . 39 4.3 Beverton-Holt Functions In this section we examine a vector-valued function whose components are Beverton-Holt functions. We show this function is monotone and strictly concave with respect to the standard positive convex cone of nonnegative coordinates, K =f(x 1 ;x 2 ;:::;x n )2R n jx 1 ;x 2 ;:::;x n 0g =R n + : We denote the set of vector-valued functions whose components are Beverton- Holt functions asBH, dened as BH : =fB :R n + !R n + jB(x) i =b i (x i ); i = 1;:::;n for some 1 ;:::; n 2R + nf0gg where b :R + !R + is dened as b () = 1 + : We noteB isK-monotone. Indeed, forx;y2R n + withx K y we haveyx2K, i.e. y i x i 0. Since 1+ is an increasing function we have b i (x i ) = x i 1 + i x i y i 1 + i y i =b i (y i ) 8i = 1;:::;n: Therefore, B(x) K B(y) and B is K-monotone. 40 In order to show B is strictly K-concave, we apply the denition and show B(x +y)B(x)B(y)2 Int(K) for x;y2 R n + such that x i 6= y i for i = 1;:::;n, and ; 0 with + = 1. Because of our choice of positive convex cone K, it suces to show the inequality holds component wise. By denition of B(x) we show for all i = 1;:::;n x i +y i 1 + i (x i +y i ) > x i 1 + i x i + y i 1 + i y i : Or equivalently, x i +y i 1 + i x i + i y i > x i +y i + i x i y i 1 + i x i + i y i + 2 i x i y i : (4.5) Proof: x 2 i +y 2 i > x 2 i +y 2 i (x i y i ) 2 = x 2 i +y 2 i x 2 i y 2 i + 2x i y i = (1)x 2 i + (1)y 2 i + 2x i y i = 2 x 2 i + 2 y 2 i + 2x i y i 41 Multiplying both sides of the above inequality by i > 0 we obtain i x 2 i + i y 2 i > 2 i x 2 i + 2 i y 2 i + 2 i x i y i : Adding common terms to both sides of the inequality gives us i x 2 i + i y 2 i +x i + 2 i x 2 i y i +y i + 2 i x i y 2 i + i x i y i + i x i y i > 2 i x 2 i + 2 i y 2 i + 2 i x i y i +x i + 2 i x 2 i y i +y i + 2 i x i y 2 i + i x i y i : Which factors into (x i +y i )(1 + i x i + i y i + 2 i x i y i )> (x i +y i + i x i y i )(1 + i x i + i y i ) Hence we obtain equation (4.5). We also note B(x) is bounded. Indeed, we have for all i = 1;:::;n i x i 1 + i x i ) x i 1 + i x i 1 i ; where i > 0: Therefore, for all x2R n + we have kB(x)k = s x 1 1 + 1 x 1 2 + ... + x n 1 + n x n 2 s 1 1 2 + ... + 1 n 2 _ =M: 42 We denote as the set of all nonnegative irreducible matrices with at least one positive diagonal element. That is, =fL2M n (R + )j L irreducible and l ii > 0 for some i = 1;:::;ng: Consider a collection B 1 ;B 2 ;:::;B p 2BH and L 1 ;L 2 ;:::;L p 2 such that L 1 = L 2 =::: = L p , i.e. the binary forms of the matrices are equal. Let x2R n + and dene F(x) =L p B p L p1 B p1 :::L 1 B 1 (x): (4.6) Note that mapping (4.4) has this form. We have shown above that B j 2S(K) and since each L j 2M n (R + ) is irreducible we also know that L j 2M(K) for j = 1;:::;p. Thus, by Theorem 3.4 we know that each L j B j 2S(K) for j = 1;:::;p. Finally, we have thatF2S(K) by Theorem 3.3. Note thatD(L j B j )(x) =L j DB j (x) whereDB j (x) is annn diagonal matrix with positive diagonal entries. Indeed, B 0 j (x) =DB j (x) = diag 1 (1 + j 1 x 1 ) 2 ;:::; 1 (1 + jn x n ) 2 : 43 Let x 0 2 R n + and denote x (j) = L j B j :::L 1 B 1 (x 0 ) for j = 1;:::p. Then DF(x)j x=x 0 has the form F 0 (x 0 ) =L p B 0 p (x (p1) )L p1 B 0 p1 (x (p2) ):::L 1 B 0 1 (x 0 ): (4.7) The following Lemma will be useful in order show that the positivity condition of the Cone Limit Set Trichotomy Theorem (page 46) is satised. Lemma 4.2. There exists a positive integer r such that (F 0 (x)) r > 0. Proof: By Lemma 3.7 we know that each L j for j = 1;:::;p is primitive. Since their corresponding binary matrices are equal, there exists a positive integer m such that L m 1 ;L m 2 ;:::;L m p > 0: Since we are considering the multiplication of nonnegative matrices it suces to work with binary matrices. Let L : = L 1 = L 2 =::: = L p and since each B 0 j (x (j1) ) for j = 1;:::;p where x (0) = x 0 is a diagonal matrix with positive diagonal entries, its binary form is the nn identity matrix, I n . Therefore F 0 (x) =LI n LI n :::LI n =L p : 44 Recall the denition of multiplication of binary matrices (page 30). Then ifpm we let r = 1 since F 0 =L p =L m > 0: If p<m we let r =kp where k is the smallest positive integer such that kp>m. We have shown there exists a positive integer r such that (F 0 (x)) r > 0. Let r> 0 be as in Lemma 4.2. Note that (F 0 (x)) r = F 0 (x)F 0 (x):::F 0 (x) = (L p B p :::L 1 B 1 (x)) 0 ::: (L p B p :::L 1 B 1 (x)) 0 = (L p B 0 p (x (p1) ):::L 1 B 0 1 (x))::: (L p B 0 p (x (p1) ):::L 1 B 0 1 (x)) = L p e B 0 p L p1 e B 0 p1 :::L 1 e B 0 1 :::L p e B 0 p L p1 e B 0 p1 :::L 1 e B 0 1 where e B 0 j for j = 1; ... ;p are diagonal matrices with positive diagonal elements. Also, (F r ) 0 (x) = (FF:::F) 0 (x) = (L p B p :::L 1 B 1 :::L p B p :::L 1 B 1 ) 0 (x) = L p b B 0 rp L p1 b B 0 r p1 :::L 1 b B 0 r 1 :::L p b B 0 1p :::L 1 b B 0 1 1 45 Hence, the binary forms of (F 0 (x)) r and (F r ) 0 (x) are equal. Therefore (F r ) 0 (x)> 0 for all x 2 R n + . Applying Theorem 3.5 with H(x) = F r (x) we have that F r (x) 0 for x> K 0. We state the following denitions can be found in [7]. For any given x;y2X, a Banach space with cone K, such that x < K y we say [x;y] =fz2 Xj x K z K yg is the order interval with endpoints x and y. A mapping F :X!X is order compact if it maps each order interval into a precompact set, i.e. its closure is compact. The orbit of x 0 is the set of pointsfx 0 ;x 1 ;:::g where x j+1 = F (x j ). The following theorem can be found in [7]. An earlier nite dimensional version can be found in [8]. Theorem 4.3 (Cone Limit Set Trichotomy). Let X be a Banach space with cone K. Assume F : K ! K is continuous and monotone and has the following properties for some r 1: (a) F r is strongly sublinear (b) F r 0 for all x> 0 (c) F r is order compact Then precisely one of the following holds: (i) each nonzero orbit is order unbounded (ii) each orbit converges to 0, the unique xed point of F . (iii) each nonzero orbit converges to q 0, the unique nonzero xed point of F . 46 In the application of our population model we wish to show that conclusion (iii) prevails. This implies a balance in the populations where no given location becomes extinct. We have shown in our above discussion that our functionF(x) given by (4.6) isK-monotone, strictlyK-concave and satises condition (b). Now, by invoking Theorem 3.3 we know thatF r 2S(K) dened on page 20. Therefore, by Proposition 3 we know thatF r is strongly sublinear. The order compactness follows from the continuity ofF from a nite dimensional space to itself. We may at once exclude conclusion (i) since we know our map is bounded. Thus it is left to specify conditions about our irreducible nonnegative matricesL j so that the origin is not an attracting xed point of our map. Lemma 4.4. LetF(x) be given as in (4.6). ThenF 0 (0) =L p L p1 :::L 1 . Proof: We have shown that F 0 (x 0 ) =L p B 0 p (x (p1) )L p1 B 0 p1 (x (p2) ):::L 1 B 0 1 (x 0 ) where x (j) = L j B j :::L 1 B 1 (x 0 ). We rst note that the origin 0 is the trivial xed point ofF(x). Since 0 (j) =0 we have F 0 (0) =L p B 0 p (0)L p1 B 0 p1 (0):::L 1 B 0 1 (0): 47 Since B 0 j (x) = diag 1 (1 + j 1 x 1 ) 2 ; ... ; 1 (1 + jn x n ) 2 it follows that B 0 j (0) =I for j = 1;:::;p. Therefore we have shown F 0 (0) =L p L p1 :::L 1 Corollary 2.F 0 (0) is an irreducible matrix. Proof: By the argument in Lemma 4.2 we know that there exists a natural number m such that L m 1 ;L m 2 ;:::;L m p > 0: Furthermore, there exists a natural number r (r = 1 if pm) such that (L p L p1 :::L 1 ) r > 0: Therefore,L p :::L 2 L 1 is primitive and hence,F 0 (0) =L p :::L 2 L 1 is irreducible. By the Perron-Frobenius Theorem we know a nonnegative irreducible matrixA has a simple positive eigenvalue equal to its spectral radius and the corresponding eigenvector is positive. Also, A cannot have two linearly independent nonnegative eigenvectors. Recall if at least one eigenvalue of the Jacobian matrix evaluated at a xed point has absolute value larger than one then it is an unstable xed 48 point. Thus since the Jacobian matrix of our system evaluated at the origin is a nonnegative irreducible matrix, requiring the spectral radius to be larger than one will give us the result we seek. Theorem 4.5. LetF(x) be given as in (4.6). If the xed point 0 is hyperbolic and (F 0 (0)) > 1 the mappingF(x) has a unique globally asymptotically stable xed point in the interior of the cone K =R n + . Proof: By Lemma 4.4 and Corollary 2 we know that F 0 (0) =L p L p1 :::L 2 L 1 is an irreduciblenn matrix. Let 1 ; 2 ;:::; n be the eigenvalues ofF 0 (0) with (F 0 (0)) = 1 > 1 and arranged into the two sets: U =f 1 ;:::; l g wherej i j> 1 S =f l+1 ;:::; n g wherej i j< 1: Let E u and E s be the eigenspaces spanned by the generalized eigenvectors corre- sponding to U and S, respectively. ThenR n =E s E u . The sets E s and E u are the stable and unstable subspaces of 0, respectively. 49 The local stable manifold of 0 in an open neighborhood G is dened as W s (0;G)W s (0) =fx 0 2GjO(x 0 )G and lim t!1 F n (x 0 ) =0g: We look at negative orbits to dene the unstable manifold. Let x(0) = x 0 and F (x(t 1)) =x(t) for t2Z + . Then the principal negative orbit of a point x 0 is dened asO (x 0 ) =x(t). The local unstable manifold for0 inG is dened to be the set W u (0;G)W u (0) =fx 0 2Gj9O (x 0 ) and lim t!1 x(t) =0g: The Stable Manifold Theorem [3] states that in an open neighborhood G of 0 the unstable manifold W u (0) is of the same dimension as E u such that E u is tangent toW u (0) at0 and ifx(0)2W u (0), then there exists a principal negative solution x(t) with lim t!1 x(t) =0. By the Perron-Frobenius Theorem we know that the corresponding eigenvector v 1 of 1 is positive, i.e. v 1 K 0, and v 1 is the only linearly independent eigenvector lying in K = R n + : Since we assumed 1 > 1 we have that v 1 E u . Thus, we take x(0)2W u (0) such that x(0)2K and we have a nonzero orbit, in K as our map leavesK invariant, which is tending away from the origin. Therefore by the Cone Limit Set Trichotomy Theorem, each nonzero orbit converges to a 50 unique nonzero xed point q of F such that q2 Int(K). In the next section and chapter we show sucient conditions under which the Jacobian of the mapping (4.4) evaluated at the origin has spectral radius larger than 1. 4.4 Proof of Theorem 4.1 Recall our mapping describing the n locations and p varying seasons model (4.4). We rst take note that the determinant of each of the matricesL(t) fort = 1;:::;p is quite simple to compute. We use the property of determinants which states that if columnj of a matrixA is written as the sum of two column vectors v +w, then the determinant ofA is the sum of the determinants obtained from A by replacing columnj with the vectorsv andw respectively. At each step we expand a column j = 2k as a sum of two column vectors. When writing the determinant as the sum of two determinants we see that one determinant is equal to zero as its matrix has a row of zeroes. Take for example, the case of only two locations: det(L(t)) = 0 b 1 (t) 0 0 1 1 (1m 12 (t)) 0 2 m 21 (t) 0 0 0 b 2 (t) 0 1 m 12 (t) 2 2 (1m 21 (t)) 51 = 0 b 1 (t) 0 0 1 1 0 2 m 21 (t) 0 0 0 b 2 (t) 0 0 2 2 (1m 21 (t)) + 0 0 0 0 1 1 m 12 (t) 0 2 m 21 (t) 0 0 0 b 2 (t) 0 1 m 12 (t) 2 2 (1m 21 (t)) = 0 b 1 (t) 0 0 1 1 0 2 m 21 (t) 0 0 0 b 2 (t) 0 0 2 2 (1m 21 (t)) + 0 = 0 b 1 (t) 0 0 1 1 0 0 0 0 0 b 2 (t) 0 0 2 2 + 0 b 1 (t) 0 0 1 1 0 2 m 21 (t) 0 0 0 0 0 0 2 2 m 21 (t) = 0 b 1 (t) 0 0 1 1 0 0 0 0 0 b 2 (t) 0 0 2 2 = 1 b 1 (t) 2 b 2 (t) This method for nding the determinant of a matrix L(t) works nicely when gen- eralizing to n locations. We see that the determinant of the matrix with positive 52 migration rates is equal to the determinant of the matrix with no dispersal between locations. Therefore we have for n locations: det(L(t)) = 0 b 1 (t) 0 0 ::: 0 0 1 1 (1 1 ) 0 2 m 21 (t) ::: 0 n m n1 (t) 0 0 0 b 2 (t) ::: 0 0 0 1 m 12 (t) 2 2 (1 2 ) ::: 0 n m n2 (t) . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ::: 0 b n (t) 0 1 m 1n (t) 0 2 m 2n (t) ::: n n (1 n ) = 0 b 1 (t) 0 0 ::: 0 0 1 1 0 0 ::: 0 0 0 0 0 b 2 (t) ::: 0 0 0 0 2 2 ::: 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ::: 0 b n (t) 0 0 0 0 ::: n n = (1) n 1 b 1 (t) 2 b 2 (t)::: n b n (t) = 1;t 2;t ::: 2n;t 53 where 1;t ; 2;t ;:::; 2n;t are the eigenvalues of the matrix L(t), for a time t and n locations. We know the spectral radius of an irreducible matrix is equal to its positive dominant eigenvalue. Hence, by assuming jdet(F 0 (0))j =jdet(L(p):::L(0))j =jdet(L(p))::: det(L(0))j> 1 we have that there exists a i;t = (F 0 (0)) that is greater than 1 for some i = 1;:::; 2n and t = 0;:::;p. Therefore, by Theorem 4.5 we proved Theorem 4.1. That is, j 1 b 1 (p)::: n b n (p)::: 1 b 1 (0)::: n b n (0)j> 1 is a sucient condition for a unique steady state q2 Int(K), K = R 2n + , that is globally asymptotically stable for ourn locations andp varying seasons population model (4.4). 54 Chapter 5 Further Interpretations of the Model 5.1 Weaker Condition for 2 Locations and Constant Birth and Migration Rates We consider the case of constant birth rates, b 1 (t) =b 1 ;b 2 (t) =b 2 and migration, m 12 (t) =m 12 ;m 21 (t) =m 21 for model (4.1) and write the system in matrix-vector form as F (J 1 (t);A 1 (t);J 2 (t);A 2 (t)) =LB(J 1 (t);A 1 (t);J 2 (t);A 2 (t)) 0 B B B B B B B B B B @ J 1 (t + 1) A 1 (t + 1) J 2 (t + 1) A 2 (t + 1) 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ 0 b 1 0 0 1 1 (1m 12 ) 0 2 m 21 0 0 0 b 2 0 1 m 12 2 2 (1m 21 ) 1 C C C C C C C C C C A 0 B B B B B B B B B B @ J 1 (t) 1+ 1 J 1 (t) A 1 (t) 1+ 1 A 1 (t) J 2 (t) 1+ 2 J 2 (t) A 2 (t) 1+ 2 A 2 (t) 1 C C C C C C C C C C A : (5.1) 55 We wish to show that a sucient condition for having a xed point in the interior ofR 4 + is i b i + i > 1 fori = 1; 2. Clearly, it is a necessary condition as shown in [2] for a population in one location. We repeat the argument here for completeness. Considering our system without migration, that is with m 12 =m 21 = 0, L is a scalar block diagonal matrix. Thus we may investigate the criteria for a nontrivial, i.e. positive, xed point in each patch. F(J(t);A(t)) = 0 B B @ J(t + 1) A(t + 1) 1 C C A = 0 B B @ 0 b 1 C C A 0 B B @ J(t) 1+ J(t) A(t) 1+A(t) 1 C C A =LB(J(t);A(t)) (5.2) The components of a nontrivial steady state (J ;A ) K 0 must satisfy J = bA 1 +A and A = J 1 + J + A 1 +A so that, A = bA 1 +A + bA + A 1 +A and thus 1 = b 1 +A + bA + 1 +A : Therefore, a nontrivial xed point implies b +> 1: (5.3) 56 We next show the converse holds when considering constant migration between two loactions. Namely, Theorem 5.1. Consider the system given in (5.1) with 0 < m 12 ;m 21 < 1. If i b i + i > 1 for i = 1; 2 then there exists a unique nontrivial steady state, i.e. (J 1 ;A 1 ;J 2 ;A 2 ) K 0, which is globally asymptotically stable. We rst state and prove the following lemmas which will be used in the proof of Theorem 5.1. Lemma 5.2. LetA 1 andA 2 bekk nonnegative irreducible matrices such that at least one matrix A i for i = 1; 2 has spectral radius larger than 1. Then the 2k 2k nonnegative matrix A = 0 B B @ A 1 0 0 A 2 1 C C A has spectral radius larger than 1. In general, if A 1 ;A 2 ;:::;A r are nonnegative irreducible block matrices such that at least one A i for i = 1;:::r has spectral radius larger than 1, then the square matrix A = 0 B B B B B B B B B B @ A 1 A 2 0 . . . 0 A r 1 C C C C C C C C C C A has spectral radius larger than 1. 57 Remark 2. It is clear that this conclusion holds true for any such matrix A (even when A 1 and A 2 are not assumed to be irreducible) since the eigenvalues of A are the eigenvalues of the matrices A 1 and A 2 . However, the method of proof that follows is useful for our generalization in Lemma 5.3 Proof: With out loss of generality, we assume A 1 has spectral radius larger than 1. By the Perron-Frobenius theorem, =(A 1 ) is a simple eigenvalue of A 1 and the corresponding eigenvector v is positive. Let ^ v = 0 B B @ v 0 1 C C A a vector inR 2k , then ^ v is an eigenvector of A. Indeed, A^ v = 0 B B @ A 1 0 0 A 2 1 C C A 0 B B @ v 0 1 C C A = 0 B B @ A 1 v 0 1 C C A = 0 B B @ v 0 1 C C A =^ v: Consider u = ^ v k^ vk , then for any positive integer m, A m u = A m ^ v k^ vk = m ^ v k^ vk = m u: Therefore kA m k = max kxk=1 kA m xkkA m uk =k m uk = m : 58 So that (A) = lim m!1 kA m k 1=m > 1: Therefore, the 2k 2k matrix A has spectral radius larger than 1. The same conclusion holds when any or all of the elements in the o diagonal blocks are positive. Lemma 5.3. LetA 1 andA 2 bekk nonnegative irreducible matrices such that at least oneA i fori = 1; 2 has spectral radius larger than 1 andB 1 ;B 2 are nonnegative block matrices. Then the 2k 2k nonnegative matrix A = 0 B B @ A 1 B 1 B 2 A 2 1 C C A has spectral radius larger than 1. In general, if A 1 ;A 2 ;:::;A n are nonnegative irreducible kk matrices such that at least one A i has spectral radius larger than 1 andB ij fori;j = 1;:::;n are nonnegative block matrices, then the square matrix A = 0 B B B B B B B B B B @ A 1 B 12 ::: B 1n B 21 A 2 ::: B 2n . . . . . . . . . . . . B n1 B n2 ::: A n 1 C C C C C C C C C C A 59 has spectral radius larger than 1. Proof: With out loss of generality, we assume A 1 has spectral radius larger than 1 and let;^ v; andu be as in the proof of Lemma 5.2. We show by induction that A m ^ v K m ^ v where K =R 2k + . A^ v = 0 B B @ A 1 B 1 B 2 A 2 1 C C A 0 B B @ v 0 1 C C A = 0 B B @ A 1 v B 2 v 1 C C A = 0 B B @ A 1 v 0 1 C C A + 0 B B @ 0 B 2 v 1 C C A =^ v+ 0 B B @ 0 B 2 v 1 C C A K ^ v: Assume it holds true for m, that is, A m ^ v K m ^ v. Then, A(A m m I)^ v K 0 A m+1 ^ v m A^ v K 0 A m+1 ^ v K m A^ v K m ^ v = m+1 ^ v: Thus, A m u = A m ^ v k^ vk K m ^ v k^ vk = m u: Note that A m u K m u K 0 where K =R 2k + implieskA m ukk m uk. Thus, the rest of the argument follows as in the proof of Lemma 5.2 and we obtain (A)> 1. 60 Assuming that we only need one of the irreducible block diagonal matrices to have spectral radius larger than one is a sucient condition to show the larger matrix has spectral radius larger than one. In the following section we will prove Theorem 5.1 by applying Theorem 4.5. Recall that we assume i b i + i > 1 for i = 1; 2. We show that this is a sucient condition for each diagonal block to have spectral radius larger than one for suciently small migration rates and thus by Lemma 5.3 the 4 4 matrix L as given in (5.1) has spectral radius larger than one. We then proceed with a continuation argument to show that L has spectral radius larger than one for all migration rates 0<m 12 ;m 21 < 1. 5.1.1 Proof of Theorem 5.1 Recall our system with no migration, given in (5.2). Note that F 0 (0) = 0 B B @ 0 b 1 C C A and thus its characteristic polynomial is 2 b: (5.4) 61 Thus we obtain 1 = + p 2 + 4b 2 > 0; 2 = p 2 + 4b 2 < 0: (5.5) By Perron-Frobenius we know (L) = 1 and we see that we arrive at the same condition as shown in (5.3), since requiring 1 > 1 we obtain + p 2 + 4b 2 > 1 p 2 + 4b > 2 > 0 2 + 4b > 4 4 + 2 (5.6) b > 1 b + > 1: Now consider our system (5.1) with 0<m 12 ;m 21 < 1. We let A 1 = 0 B B @ 0 b 1 1 1 (1m 12 ) 1 C C A and A 2 = 0 B B @ 0 b 2 2 2 (1m 21 ) 1 C C A : A simple calculation shows that we need i b i + i (1m ij )> 1 fori;j = 1; 2; i6=j so that (A 1 ) > 1 and (A 2 ) > 1. By our conditions, this is satised for suciently small m 12 and m 21 , say for m 12 2 (0; 1 ) and m 21 2 (0; 2 ). (Note that if i b i are greater than 1, then the inequality holds for any migration rate). Then by 62 1 1 S m 12 β 1 β 2 m 21 b m 21 Figure 5.1: S = (0; 1) (0; 1) Lemma 5.3 we know that for suciently small migration rates m 12 and m 21 , the spectral radius of the matrix corresponding to system (5.1), L, is greater than 1; this corresponds to the dark blue area in Figure 5.1. We now proceed with a continuation argument to show that (L)> 1 for all 0<m 12 ;m 21 < 1: Let S = (0; 1) (0; 1), L = L (m 12 ;m 21 ) = 0 B B B B B B B B B B @ 0 b 1 0 0 1 1 (1m 12 ) 0 2 m 21 0 0 0 b 2 0 1 m 12 2 2 (1m 21 ) 1 C C C C C C C C C C A 63 and g(m 12 ;m 21 ;) = det(L m 12 ;m 21 I) namely, the characteristic polynomial of L m 12 ;m 21 . Note that for (m 12 ;m 21 )2 (0; 1)f0g, the horizontal axis in Figure 5.1, we have L m 12 ;0 = 0 B B B B B B B B B B @ 0 b 1 0 0 1 1 (1m 12 ) 0 0 0 0 0 b 2 0 1 m 12 2 2 1 C C C C C C C C C C A So that g(m 12 ; 0;) = ( 2 2 2 b 2 )( 2 1 (1m 12 ) 1 b 1 ): Therefore as shown in (5.4) - (5.6), since 2 b 2 + 2 > 1, we know the spectral radius of L (m 12 ;0) is greater than 1. Similarly it can be shown that along the boundaryf0g (0; 1) of S the spectral radius of L is also greater than 1. By the argument given above, the spectral radius of L (m 12 ;m 21 ) is greater than 1 for a small half-tube neighborhood about the bottom and left boundaries of S; that is, for (0; 1) (0; 2 ) S and (0; 1 ) (0; 1) S. Let m 12 2 (0; 1 ) and x m 21 = b m 21 2 (0; 1). By the Perron-Frobenius theorem we know (L m 12 ;b m 21 ) is the 64 largest positive eigenvalue of L m 12 ;b m 21 and has multiplicity 1. Therefore, for max (m 12 ;b m 21 ) =(L m 12 ;b m 21 ) we have @ @ g(m 12 ;b m 21 ; max (m 12 ;b m 21 ))> 0: (5.7) Indeed, for each m 12 2 (0; 1 ), g(m 12 ;b m 21 ;) is a monic polynomial of degree 4 in terms of and thus the graph of g(m 12 ;b m 21 ;) rises to the right. This, along with the fact that max (m 12 ;b m 21 ) is a root of multiplicity 1 ofg(m 12 ;b m 21 ;) gives us (5.7). By the Implicit Function Theorem we know d dm 12 = @g @m 12 @g @ : We now gather information about @g @m 12 in order to study what happens to (L m 12 ;b m 21 ) as we let m 12 proceed to 1. @ @m 12 g(m 12 ;b m 21 ;) = @ @m 12 b 1 0 0 1 1 (1m 12 ) 0 2 b m 21 0 0 b 2 0 1 m 12 2 2 (1b m 21 ) 65 = @ @m 12 0 B B B B B B B B B B @ b 1 0 0 1 1 0 2 b m 21 0 0 b 2 0 0 2 2 (1b m 21 ) + 0 0 0 1 1 m 12 0 2 b m 21 0 0 b 2 0 1 m 12 2 2 (1b m 21 ) 1 C C C C C C C C C C A = @ @m 12 0 B B B B B B @ 0 1 m 12 0 2 b m 21 0 b 2 1 m 12 2 2 (1b m 21 ) 1 C C C C C C A = @ @m 12 0 B B B B B B @ 1 m 12 0 2 b m 21 0 b 2 0 2 2 1 C C C C C C A = @ @m 12 1 m 12 ( 2 2 2 b 2 ) 66 Therefore we have @ @m 12 g(m 12 ;b m 21 ;) = 1 ( 2 2 2 b 2 ): Thus, if 0 < e 1 + then @ @m 12 g(m 12 ;b m 21 ; e ) < 0 since 2 b 2 + 2 > 1: We now let m 12 proceed to 1 and claim that max (m 12 ;b m 21 ) is always greater than 1. Indeed, max (m 12 ;b m 21 ) can decrease as m 12 proceeds to 1 but not past 1 + as this would imply d dm 12 > 0, a contradiction. We apply this argument to each b m 21 2 (0; 1) and letm 12 proceed to 1. Therefore, we have shown (L (m 12 ;m 21 ) )> 1 for all (m 12 ;m 21 )2S. Thus we apply Theorem 4.5 whereF(x) is given as in (5.1) (note it has the form of (4.6) where k = 1) and (F 0 (0)) =(L)> 1: We conclude that there exists a unique globally asymptotically stable steady state in the interior of our cone K =R 4 + . 5.1.2 Expanding Theorem 5.1 By applying the general version of Lemma 5.3 our analysis is such that we can take n locations with the assumption that each patch satises the inequality i b i + i > 1 67 and obtain a unique globally asymptotically stable interior xed point for the system with suciently small migration. Theorem 5.4. Assume i b i + i > 1, 0 < m ij < 1 for i;j = 1;:::;n, i6= j and let X(t + 1) =LB(X(t)) (5.8) where X(t) = (J 1 (t) A 1 (t) J 2 (t) ::: A n1 (t) J n (t) A n (t)) T ; LB(X(t)) is given by 0 B B B B B B B B B B B B B B B B B B B B B B @ 0 b 1 0 0 ::: 0 0 1 1 (1 1 ) 0 2 m 21 ::: 0 n m n1 0 0 0 b 2 ::: 0 0 0 1 m 12 2 2 (1 2 ) ::: 0 n m n2 . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ::: 0 b n 0 1 m 1n 0 2 m 2n ::: n n (1 n ) 1 C C C C C C C C C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B B B B B B B B B @ J 1 (t) 1+ 1 J 1 (t) A 1 (t) 1+ 1 A 1 (t) J 2 (t) 1+ 2 J 2 (t) A 2 (t) 1+ 2 A 2 (t) . . . Jn(t) 1+ nJn(t) An(t) 1+nAn(t) 1 C C C C C C C C C C C C C C C C C C C C C C A and where k = n X j=1 j6=k m kj . Then, there exists a unique globally asymptotically stable interior point for suciently small migration rates. 68 5.2 A Look at the Directed Graph of Locations Note that in Theorem 4.1 and in Theorem 5.4 we require the migration rates m ij fori;j = 1;:::;n andi6=j to be greater than zero. However, we will show in this section that for the case of more than two patches not all migration rates need to be greater than zero. In Chapter 4.3 we considered a mappingF of the form F =L p B p :::L 1 B 1 whereB j 2BH, i.e. vector-valued functions whose components are Beverton-Holt functions, and L j 2 for j = 1;:::;k such that L 1 = ::: = L k , i.e. irreducible matrices of the same \form". We showed in Theorem 4.5 that the mapF has a unique asymptotically stable nonzero xed point if (F 0 (0)) > 1. The integral element of this proof is that F 0 (0) =L p L p1 :::L 1 is an irreducible matrix. Thus considering the matrixL in the system described by the single map (5.8) some of the migration rates may be equal to zero provided that the matrix is irreducible. Note that for the case of a p periodic system L(t) given in (4.2) may 69 also have some migration rates equal to zero as long as L(t) is irreducible for t = 0;:::;p and L(p) = ::: = L(0). In the following we will explore what such a situation means in terms of our two-stage migration model by providing a more general lemma and applying it to our model. Consider the matrix A = 0 B B B B B B B B B B B B B B @ A 1 B 12 B 13 ::: B 1n B 21 A 2 B 23 ::: B 2n B 31 B 32 A 3 ::: B 3n . . . . . . . . . . . . . . . B n1 B n2 B n3 ::: A n 1 C C C C C C C C C C C C C C A (5.9) where A i and B ij are nonnegative kk block matrices for i;j = 1;:::;n. We know by Theorem 3.6 that the knkn matrix A is irreducible if and only if its corresponding directed graph comprised of points p 1 ;p 2 ;:::;p nk in the plane 70 is strongly connected. Assume the kk matrices A 1 ;A 2 ;:::;A n are irreducible. Then the directed graphs of each group p 1 ;p 2 ;:::;p k p k+1 ;:::;p 2k . . . p (n2)k+1 ;:::;p (n1)k p (n1)k+1 ;:::;p nk are strongly connected. Therefore we may treat each group of points p (i1)k+1 ;:::;p ik as one single point, l i , for i = 1;:::;n. Indeed, note that a given o diagonalkk block matrixB ij fori;j = 1;:::;n determines the directed arrows from points p (i1)k+1 ;:::;p ik to the points p (j1)k+1 ;:::;p jk . This follows from the position of the matrix B ij in the larger matrix A. That is, the elements ofB ij are on the (i1)k +1 through theik rows ofA and the (j1)k +1 through the jk columns of A. Therefore, if B ij has at least one positive element we draw a directed arrow from the point l i to the point l j . Moreover, we dene a new binary matrix e A = [e a ij ] with size nn such that e a ij = 8 > > < > > : 1 if B ij 6=0 0 if B ij =0 : 71 Thus by the discussion above we have reached the following conclusion: Lemma 5.5. If thekk matricesA 1 ;:::;A n are irreducible and thenn binary matrix e A is irreducible then the nknk matrix A (5.9) is irreducible. Lemma 5.5 tells us that we may determine whether an nknk matrix A is irre- ducible by examining the size-reduced nn constructed matrix if all diagonal kk block matrices ofA are irreducible. In other words, by Theorem 3.6 we draw directed arrows between n distinct points as opposed to nk distinct points. We now consider the 2n 2n matrixL forn> 2 in system 5.8. Note that each 2 2 diagonal block matrix A i = 0 B B @ 0 b i i i (1 i ) 1 C C A is irreducible fori = 1;:::;n and each 22 o diagonal block matrix has the form B ij = 0 B B @ 0 0 0 j m ji 1 C C A for i;j = 1;:::;n. The directed graph of the nn constructed binary matrix e L represents the opposite movement of the adult frogs between the n patches. Therefore, by Lemma 5.5 if the directed graph of the locations is strongly con- nected, then the matrixL is irreducible and Theorem 5.4 still holds for suciently small migration rates. Similarly, Theorem 4.1 still holds for all migration rates 72 Location 3 Location 1 Location 2 Figure 5.2: directed graph of locations 0 m ij < 1 for i;j = 1;:::;n, i6= j provided that the directed graph of the locations is strongly connected and L(0) =::: = L(p 1): Example 5. Consider three locations consisting of two-stage populations of the same species. We assume that the adult population from location i migrate only to location i + 1 for i = 1; 2 and adults from location 3 migrate only to location 1. Then the directed graph of the locations has the form shown in Figure 5.2. It is 73 easy to see that the directed graph of the locations is strongly connected and thus by Lemma 5.5 the matrix L given by L = 0 B B B B B B B B B B B B B B B B B B @ 0 b 1 0 0 0 0 1 1 (1m 12 ) 0 0 0 3 m 31 0 0 0 b 2 0 0 0 1 m 12 2 3 (1m 23 ) 0 0 0 0 0 0 0 b 3 0 0 0 2 m 23 3 3 (1m 31 ) 1 C C C C C C C C C C C C C C C C C C A is irreducible. Therefore, we may still apply Theorem 5.4 for X(t) = LB(X(t)) where n = 3 and Theorem 4.1 for F (X) =L(p)B(X(p)):::L(0)B(X(0)) where L(p) =::: = L(0) and m 13 ;m 32 ;m 21 = 0. 74 5.3 Two Locations: One Population with i b i + i > 1 Consider the case of two locations, constant birth rates, and constant migration rates given in system (5.1). We repeat it here for the reader: F (J 1 (t);A 1 (t);J 2 (t);A 2 (t)) =LB(J 1 (t);A 1 (t);J 2 (t);A 2 (t)) 0 B B B B B B B B B B @ J 1 (t + 1) A 1 (t + 1) J 2 (t + 1) A 2 (t + 1) 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ 0 b 1 0 0 1 1 (1m 12 ) 0 2 m 21 0 0 0 b 2 0 1 m 12 2 2 (1m 21 ) 1 C C C C C C C C C C A 0 B B B B B B B B B B @ J 1 (t) 1+ 1 J 1 (t) A 1 (t) 1+ 1 A 1 (t) J 2 (t) 1+ 2 J 2 (t) A 2 (t) 1+ 2 A 2 (t) 1 C C C C C C C C C C A : Recall that in order to apply Theorem 4.5 to this mapping, the matrix L must be irreducible and have spectral radius larger than 1. Let us now assume that only location one satises the inequality 1 b 1 + 1 > 1. Then we know by the discussion in Subsection 5.1.1 that the diagonal block matrix A 1 = 0 B B @ 0 b 1 1 1 (1m 12 ) 1 C C A 75 has spectral radius larger than one for suciently small migration ratem 12 2 (0; 1). In fact, we need 1 b 1 + 1 (1m 12 )> 1: (5.10) Hence, (A 1 )> 1 provided that 0 < m 12 < [ 1 b 1 + 1 ] 1 1 : (5.11) Therefore, by Lemma 5.3 we know that (L) > 1 as long as (5:11) is satised. Thus, by Theorem 4.5 we obtain a globally asymptotically stable xed point in the interior of the cone K =R 4 + . This tells us that the population in location 2 (with out migration) could be going towards extinction but the thriving adult population from location 1 migrat- ing into location 2 eventually brings a balance to the populations. It is important that the the migration rate from location 1 to location 2 is small enough so that a thriving population is not moving to a location where the general population is not surviving. However, note that if 1 b 1 > 1 then (5.10) is satised for any migration rate 0<m 12 < 1. This implies that the product of the inherent survivorship rate of the juveniles and the birth rate of the adults is larger than 1 in location 1 along with adults migrating in from location 2 is enough to eventually bring balance to both populations even if the migration rate from location 1 to location 2 is large. 76 Bibliography [1] Ackleh, A. S., Chiquet, R. A., and Zhang, P. A Discrete Dispersal Model with Constant and Periodic Environments. Journal of Biological Dynamics, 5(2011) : 563 - 578. [2] Ackleh, A. S., and Jang, S. R. J. A Discrete Two-Stage Population Model: Continuous Versus Seasonal Reproduction. Journal of Dierence Equations and Applications , 13(2007): 261-274. [3] Cushing,J.M. An Introduction to Structured Population Dynamics. SIAM, Philadelphia, 1998. [4] Cushing, J.M., and Yicang, Zhou. The Net Reproductive Value and Stability in Matrix Population Models. Natural Resource Modeling, 8(4)(1994): 297 - 333. [5] Elaydi, S., and Sacker, R. J. Global Stability of Periodic Orbits of Non- autonomous Dierence Equations and Population Biology. Journal of Dif- ferential Equations, 208(11)(2005): 258 - 273. [6] Gaut, G., Goldring, K., Grogan, F., Haskell, C., and Sacker, R. Dierence Equations with the Allee Eect and the Periodic Sigmoid Beverton-Holt Equa- tion Revisited. Journal of Biological Dynamics (6)(2): 1019 - 1033. [7] Hirsch, M. W. and Smith, H. Monotone Maps: A Review. Journal of Dierence Equations and Applications , 11(2005): 379 - 398. [8] Krause, U., and Ranft, P. A Limit Set Trichotomy for Monotone Nonlinear Dynamical Systems. Nonlinear Analysis, Theory, Methods & Applications, 19(4)(1992): 375 - 392. 77 [9] Sacker, R. J. Semigroups of Maps and Periodic Dierence Equations. Journal of Dierence Equations and Applications, 16(1) (2010): 1 - 13. 78

Abstract (if available)

Abstract We study a multidimensional nonlinear discrete system which describes the dynamics of a structured two-stage population model with migration. This model is written as a mapping whose matrix-vector form is a transformation matrix composed with a vector-valued mapping whose components are Beverton-Holt functions. We show this mapping is concave and monotone with respect to the standard positive cone. This mapping belongs to a class of mappings that form a semigroup under composition. Therefore we are able to find a fixed point of the periodic system by placing conditions on the composition of these maps which belong to the semigroup.

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Large-scale inference in multiple Gaussian graphical models

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Creator Garcia-Torres, Selenne Hayde (author)

Core Title Structured two-stage population model with migration between multiple locations in a periodic environment

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 08/05/2013

Defense Date 05/06/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag concave,migration model,monotone,OAI-PMH Harvest,semigroups of maps,structured population model

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sacker, Robert (committee chair), Haskell, Cymra (committee member), Udwadia, Firdaus E. (committee member)

Creator Email garciato@usc.edu,selenne.h.banuelos@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-316756

Unique identifier UC11293418

Identifier etd-GarciaTorr-1964.pdf (filename),usctheses-c3-316756 (legacy record id)

Legacy Identifier etd-GarciaTorr-1964.pdf

Dmrecord 316756

Document Type Dissertation

Format application/pdf (imt)

Rights Garcia-Torres, Selenne Hayde

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA