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Modeling of semelparous/iteroparous polymorphism in Botryllus Schlosseri Omielan, John 1991

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MODELING OF SEMELPAROUS / ITEROPAROUS POLYMORPHISM IN BOTRYLLUS SCHLOSSERI By John Omielan B. Math., University of Waterloo  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS F O R T H E D E G R E E O F MASTER'S OF SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES MATHEMATICS INSTITUTE O F APPLIED MATHEMATICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH COLUMBIA  September 1991 @ John Omielan, 1991  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or by his  or  her representatives.  It is  understood  that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  ATa/fA p  -hv  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  Or-f.Ur  9  /91l  >i .  T\4^ie_  <oT~ Appbttf  At*-^  Abstract  This thesis describes and models the age of first sexual reproduction in Botryllus schlosseri. B. schlosseri is a sessile, colonial fouling organism that lives mainly in the low intertidal zone of temperate waters. In Eel Pond, Woods Hole, Massachusetts, semelparous (reproduces only once) and iteroparous (reproduces several times) morphs apparently co-exist. A survey of previous life-history models is given, but none of them can be used very effectively on B. schlosseri. Instead, a dynamic programming model is presented that models, with a good fit to field data, the age of first sexual reproduction in each morph separately. The model makes several predictions. First, the life history characteristics, particularly the age of first sexual reproduction, are near equilibrium. Secondly, the growth reduction after winter affects the optimal age of sexual reproduction throughout the year, suggesting that colonies can tell the time of year through the water temperature or the photoperiod. Thirdly, the primary cause of variance in the age of first sexual reproduction is the spatial variation in the environment. Finally, iteroparous colonies continue reproducing every generation once they begin, not necessarily due to a physical constraint, but simply because it is optimal for them to do so. Next, the coexistence of the morphs was looked at. An important factor is Botrylloides leachi, a closely related competitor, that in the summer and early fall overgrows semelparous colonies but not iteroparous colonies. However, since B. leachi was only introduced into Eel Pond about 30 years ago, an interesting question is whether the current conditions (i.e. coexistence) is stable or not. A survey was taken of previous models that look at the coexistence of species (the two morphs are treated as separate species). Although these models indicated that the two morphs can co-exist, none of ii  them are accurate enough to be able to confidently determine if the coexistence is stable. Although a dynamic programming model would be precise enough, too much unavailable data would be required.  iii  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  viii  Acknowledgement  x  1 Introduction  1  2  Botryllus schlosseri  2  2.1  Basic Morphology  2  2.2  Semelparous/Iteroparous Polymorphism  5  2.2.1  6  2.3  2.4  Cause of Polymorphism  Growth  11  2.3.1  12  Substratum Size  Self/nonself-recognition and Fusion  13  2.4.1  15  Inbreeding Considerations  2.5  Senescence  15  2.6  Mortality  15  2.6.1  Predation  17  Life-History Correlations  18  2.7.1  Semelparous Colonies  18  2.7.2  Iteroparous Colonies  18  2.7  iv  2.8  3  2.7.3  Semelparous/Iteroparous Colony Differences  19  2.7.4  Fx Correlations with Parents  20  2.7.5  Energy Reallocation  20  Competition  21  2.8.1  Botrylloides Description  22  2.8.2  Competitive interactions  23  2.8.3  Coexistence of Polymorphism  26  A g e o f s e x u a l r e p r o d u c t i o n i n B. schlosseri  3.1  Comparative Studies  27  27  3.2 Genetic Models  28  3.3 Life-History Theory  29  3.3.1  The Characteristic Equation  29  3.3.2  Discrete forms of Characteristic Equation  30  3.3.3  Principle of Allocation  32  3.3.4  Fisher's Reproductive Value  32  3.3.5  Cole's Model and Extensions  33  3.3.6  Stochastic Models  39  3.3.7  Discussion  42  3.4 Dynamic Programming  43  3.4.1  Definition and Assumptions  43  3.4.2  Solution Techniques  44  3.4.3  Advantages and Disadvantages  45  3.5 The Model  47  3.5.1  Background Information and Assumptions  47  3.5.2  Equation and Parameter Values  57  v  3.5.3  The Model Cases  70  3.5.4  Density and Reproduction Amounts  88  3.5.5  Lack of Time Dependent Mortality  91  3.5.6  Conclusions and Possible Enhancements  91  4 Modeling of Semelparous/Iteroparous Coexistence 4.1  4.2  94  Previous Modeling  94  4.1.1  Lotka-Volterra Equations  94  4.1.2  Gause(-Volterra) Principle  95  4.1.3  Interference Competition  96  4.1.4  Spatial Heterogeneity  97  4.1.5  Temporal Variation  97  Dynamic Programming  99  4.2.1  99  Environmentally Stable Strategies (ESS)  5 Discussion  101  Bibliography  102  vi  List of Tables  2.1  Life table of September 1-15 iteroparous colonies  16  3.2  Temperature and generation length schedule  53  3.3  Approximate dates for the end of each generation  54  vii  List of Figures  2.1 Habit sketch of B. schlosseri on glass  4  2.2 Bivariate plots based on field data  7  2.3 Frequency distributions of four life-history traits  8  3.4 The proportion of a cohort that survives from settlement to sexual maturity. 40 3.5 Growth trajectories of eight cohorts  50  3.6 Mean duration of an asexual generation and temperature  51  3.7 Seasonal mean daily settlement of B. schlosseri  52  3.8 Frequency distributions of age at first sexual reproduction  67  3.9 Field data summaries of Fig. 3.8  68  3.10 Iteroparous first sexual reproduction age cases (a) to (f) with k = .001 . .  71  3.11 Semelparous first sexual reproduction age cases (a) to (f) with k = .001 .  72  3.12 Iteroparous first sexual reproduction age cases (g) to (1) with k = .001 . .  73  3.13 Semelparous first sexual reproduction age cases (g) to (1) with k = .001 .  74  3.14 Iteroparous first sexual reproduction age cases (m) to (r) with k = .001 .  75  3.15 Semelparous first sexual reproduction age cases (m) to (r) with k = .001  76  3.16 Iteroparous first sexual reproduction age cases (a) to (f) with k = .002 . .  77  3.17 Semelparous first sexual reproduction age cases (a) to (f) with k = .002 .  78  3.18 Iteroparous first sexual reproduction age cases (g) to (1) with k = .002 . .  79  3.19 Semelparous first sexual reproduction age cases (g) to (1) with k = .002 .  80  3.20 Iteroparous first sexual reproduction age cases (m) to (r) with k = .002 .  81  3.21 Semelparous first sexual reproduction age cases (m) to (r) with k = .002  82  viii  3.22 Iteroparous colony density and rep. amounts  89  3.23 Semelparous colony density and rep. amounts  90  3.24 Age of first sexual reproduction without time dependent mortality  92  ix  Acknowledgement  First, I would like to thank my advisor, Dr. Colin Clark, for his patience with how long it took for me to write a thesis and for my continually bypassing my own deadlines. Next, Dr. Don Ludwig helped by giving me a little nudge when things slowed down too much for me, and also helped me present a lot of graphic information in an intelligble fashion. Dr. Brian Seymour deserves thanks for taking on the task of being the second reader of this thesis on short notice. Last, but definitely not least, I owe most of the background and data in this thesis to the PhD thesis of Dr. Richard Grosberg. Also, Dr. Grosberg has my thanks for giving me some help and advice despite how busy he is at U.C., Davis.  x  Chapter 1 Introduction  Among living organisms, the most common, basic, and important function is reproduction. A fundamental classification is whether the organism reproduces only once (semelparity) or several times (iteroparity). A clonal organism, Botryllus schlosseri, has the unusual distinction of having one genetic morph that is semelparous and one that is iteroparous. The next chapter will discuss the basic biology and life-history characteristics of B. schlosseri in  Eel Pond, Woods Hole, Massachusetts where the two morphs apparently co-  exist. Chapters 3 and 4 attempt to explain some of these characteristics using life-history models. Chapter 3 examines modeling the optimal age of first sexual reproduction for each morph. First, previous models are investigated, but none are sufficiently general to accurately handle B. schlosseri. Instead, a dynamic, state-based model is presented and solved using the dynamic programming technique. Chapter 4 studies the co-existence of the two morphs. Previous models show that co-existence is possible, but none account for all of the important factors affecting possible co-existence. As in chapter 3, a dynamic, state-based model is required. However, problems with the model complexity and lack of required data prevent giving and solving a model. Finally, chapter 5 provides discussion and conclusions about the models discussed.  1  Chapter 2 Botryllus schlosseri  This chapter describes the characteristics of the colonial ascidian Botryllus schlosseri. It discusses the basic biology, two life-history morphs, growth, sexual reproduction, senescence, mortality, and competitive interactions among the two morphs and the closely related species Botrylloides leachi. Botryllus schlosseri (Phylum Chordata; Subphylum Tunicata: Class Ascidiacea: Order Pleurogona: Suborder Stolidobranchia: Family Styelidae) is a fouling organism that lives mainly on hard substrata in the low intertidal zone in protected waters. In more exposed locations, it dwells in secure areas like the undersurface of rocks. The species is limited to temperate waters of the Atlantic Ocean, the Mediterranean, the Adriatic, the Black, and the Baltic Seas (Grosberg 1982, 1987). This thesis concentrates on the studies performed at Eel Pond in Woods Hole, Massachusetts. 2.1  Basic Morphology  Colonies of Botryllus schlosseri are founded by a sexually produced swimming tadpole larva (about 1.2 mm long) which settles onto any hard surface such as rocks, algal blades, floating docks, or even the surface of another ascidian (Grosberg 1982). The larva does not feed, but a metamorphosis occurs producing the first zooid of the colony, the oozooid, which filter feeds using a siphon. Small protrusions appearing on the oozooid become the palleal buds, the primordia of the asexually produced blastozooids (to distinguish them from the founder zooid, the oozooid). The blastozooids bud off still more zooids 2  Chapter 2. Botryllus schlosseri  3  (the primary buds). The primary buds have palleal buds (called the secondary buds) (Grosberg 1988). The blastozooids are morphologically identical to one another and, apart from the presence of gonads, are also almost identical to the oozooid. Due to asexual reproduction (called blastogenesis) occurring synchronously among all zooids in a colony (Milkman 1967; Sabbadin 1971), there is a concurrent maturation of zooids to the next stage (i.e., blastozooids being resorbed, primary buds becoming blastozooids, secondary buds becoming primary buds, and a new set of secondary buds forming). This process is termed takeover. The synchronous cycle is called an asexual cycle or generation (Grosberg 1982). Each generation thus behaves as a single individual since the zooids originate at the same time (Sabbadin 1977). The length of the asexual generation varies inversely with the water temperature (Sabbadin 1955, 1958 cited in Grosberg 1988) except during colony starvation (Grosberg 1982). The oozooid secretes a tunic (or test) which consists of proteins and carbohydrates. Embedded in the tunic, the zooids are connected to each other by a complex vascular system. The zooids use the blood-vascular system to remain in physiological communication. The blood circulation, among other functions, synchronizes the asexual cycle (Sabbadin and Zeniolo 1979; Sabbadin 1977). The vessels terminate along the periphery of the colony in distally expanded dead-end vessels termed the ampullae. The ampullae serve several purposes: (1) they regulate blood pressure; (2) they allow the colony to move along and adhere to the substratum; (3) they play a crucial role in recognition of contacting organisms; (4) they may function to allow or prevent overgrowth of one colony by another (Grosberg 1982). The Botryllus colonies are of irregular shape. They may be over a foot in diameter, although they are usually much smaller. When small or medium colonies have ample space, they grow isometrically with a circular encrusting surface (Brunetti 1974). However, most colonies (Fig. 2.1) are composed of rosette-like systems (called <Cpendants^>)  Chapter 2. Botryllus schlosseri  4  Figure 2.1: Habit sketch of B. schlosseri on glass. This caption and figure is from Fig. 1 in Milkman, 1967. of generally 5-18 blastozooids with each pendant being like a solitary ascidian in form (Milkman 1967; Brunetti 1974). The pendants are connected to each other via the vascular system (Grosberg 1982). This level of organization makes B. schlosseri one of the most highly integrated ascidian colonies (Milkman 1968). After five to ten asexual generations, a colony reaches sexual maturity. The sexual reproductive cycle is synchronized with the asexual cycle. However, in a process called protogyny, ovaries with mature ova appear first with the testes ripening synchronously across the colony several days later. Thus, the rapid fertilization of ova is presumably by  Chapter 2. Botryllus schlosseri  5  sperm brought from the incurrent flow of water (Milkman 1967; Grosberg 1982, 1988). The combination of protogyny and synchronization thereby prevents self-fertilization. Grosberg (1982) contains more detailed descriptions of the colony morphology. In the next section, the presence of two different morphs will be described and discussed. 2.2  Semelparous/Iteroparous Polymorphism  Two distinct clusters of observations are present in bivariate plots (Fig. 2.2) of first sexual reproduction age (in asexual cycles), number of clutches, number of buds per zooid per asexual cycle, and number of embryos per zooid per clutch. The distribution remains weakly bimodal even for the later first reproduction age of members of the last cohort that fail to reproduce until the following spring. The two types are termed semelparous and iteroparous as per the classification of Cole (1954) on the basis of the number of clutches. Semelparous colonies are characterized by: (1) death immediately following the production of a single clutch; (2) early first sexual reproduction age (at the fifth or sixth asexual generation); (3) rapid growth to first sexual reproduction (at rates approaching 4-5 buds per zooid); and (4) high reproductive effort (10-12 embryos per zooid in the clutch). Iteroparous colonies, however: (1) produce at least three clutches before dying; (2) postpone sexual reproduction until they are almost twice the age of semelparous colonies (at least nine asexual generations); (3) grow at about half the rate of semelparous colonies (at rates between 1.5 and 2.5 buds per zooid); and (4) invest roughly 75% less in reproductive effort than semelparous colonies (between two and four embryos per zooid). Only rarely do morphs intermediate in characters appear. The relative abundance of the iteroparous and semelparous phenotypes depends on the season as shown in Fig. 2.3. Early in the summer, most colonies are of the semelparous type while by midsummer the two phenotypes are almost equal and by the end of the settlement season iteroparous  Chapter 2. Botryllus schlosseri  6  type colonies predominate (Grosberg 1982, 1988). 2.2.1  Cause of P o l y m o r p h i s m  The two morphs are interfertile since semelparous and iteroparous colonies have been successfully cross-bred (Grosberg 1982). This interfertility plus the seasonal change of abundance suggests that the polymorphism is of a phenotypic nature. Two potential sources of environmentally induced variation are: 1. The parental environment - does the parental environment influence the phenotypes ofFi's? 2. The Fi's environment - is the phenotype determined solely by the progeny's environment during development? Likewise, three potential contributions to genetically determined variation are: 1. the maternal genotype 2. the paternal genotype 3. the F i genotype To determine the relative influence of phenotypic and genotypic factors, Grosberg (1982) performed several laboratory experiments. These experiments involved a series of denned single-pair crosses among several strains of Botryllus schlosseri with parents and progeny reared under a variety of environmental conditions. Seven traits were recorded for each of the parental strains and their progeny: (1) first sexual reproduction age (in days and asexual generations); (2) size at first sexual reproduction; (3) number of clutches; (4) mean number of embryos per zooid per asexual generation (before sexual  Chapter 2. Botryllus schlosseri  o  *  Nl  cr UJ a. cn O >cr  CO  S  ' ro  Q.  CM " IO  ^ cn CD  £  CD  cr  UJ  UJ  2  5  o  <D  * CM  TT  rO  CM " CM  CM  CM  —  O  o o> co  N0iionaoad3a  O  K>  .  o  — O CT> CD f~ Nouanaoad3H tsi IV 39V  CM  aiooz/sans  isi IV 33V  T  cn ui x CJ  CJ  => _1 u  * 3  => CJ  CJ  C CJ U C D 2  C Q 2  cr  U J co  m  I  X  x o  O  ZOOl  >  o  " rt  Q  PER  .  2 § . <J>  _o  BUDS  II  z  2  Z  i  CM  1  —  1  1  1  1  — r  O C7> CO f*Nononao«d3a tsi IV 33V  CO  o  to  aiooz/sans  o  CM  CT>  CD  lO  ai00Z/S0AMSlAl3  Figure 2.2: Bivariate plots based on field data showing the relationships among 1) age at first reproduction, 2) number of clutches, 3) number of buds per zooid per asexual cycle (growth rate), and 4) number of embryos per zooid per clutch. Numbers in parentheses represent overstrikes. This caption and figure is taken from Fig. 2 in Grosberg, 1988.  Chapter 2. Botryllus schlosseri  AGE  AT  NUMBER OF  lit  CLUTCHES  REPRODUCTION  I - IS J U N E O S  16- 3 0 JUNE  1-15 JULY  16-31  JULY  0 S  16-31 AUGUST  | BUDS/ZOOID [  I 1 L_I  | EMeffYOS/ZOOID 1  IIA  OS  i  IL  jhl  05  g~|  jtUCL  •n ml J]  - IS AUGUST  8  05  lulu  1-15 SEPTEMBER 0 5  JOJH  JUL | 1  M40I  1980  n>«SI  1979  n»334  I960  i n 397  1979  n i 194  I960  n.2IS  p  H^n^, 1 lull  0 5  I9T9  1979  n«l28 !  1980  1979  (t'170  I960  n • 233  1979  n . 101  I960  n • 144  1979  n • 120  I960  n • 117  n-177  n  16-30 SEPTEMBER 0 5  u3  CO  O  OJ  f«-  *~  cn  ^  *o  g~| CNJ  v  iO  O  «r>  irt  T in  pg •  Q  ~  m rfi *> V  <0 m &  cv*  (M Q  o ^  o o tfl S  o o  ~  » «  5  1979 1980  n . 100  (i • III  o ^ o  Figure 2.3: Frequency distributions of four life-history traits: 1) age atfirstsexual reproduction (asexual cycles), 2) number of clutches, 3) number of buds per zooid per asexual cycle (growth rate), and 4) number of embryos per zooid per clutch (reproductive effort). The distributions are based on colonies that recruited in the field during the intervals shown in a-h. Solid bars represent data from 1979 cohorts; open bars represent data from 1980 cohorts. This caption andfigureis taken from Fig. 1 in Grosberg, 1988.  Chapter 2. Botryllus schlosseri  9  reproduction); (5) the mean number of embryos per zooid per clutch; (6) colony life span (in days and asexual generations); (7) the total reproductive output of each colony. Crosses between like phenotypes produced progeny with phenotypes like the parents and crosses between semelparous and iteroparous parents produced roughly equal fractions of Fi's with high and low reproductive effort. There were only two exceptions to this over the five replicates of semelparous colonies (1-5) andfivereplicates of iteroparous colones (6-10): (1) in crosses among semelparous colonies with strain 4 as one of the parents, an occasional iteroparous progeny appeared, and likewise in crosses among iteroparous colonies involving strain 9 an occasional semelparous progeny appeared; (2) crosses between strains 4 and 9 produced the only 6 intermediate Fi phenotypes detected. To determine if the life-history type of a particular genotype persisted in a variety of constant controlled conditions, Grosberg (1982) performed a factorial experiment in which he manipulated two factors: temperature and density of food. There were three levels of each factor, thus nine treatments in all. The three temperatures were 15, 20, and 25 °C which encompasses the in situ temperature range during the reproductive season of Botryllus schlosseri. Colonies were fed a commercially manufactured synthetic diet (Marine Invertebrate Food). The food seemed acceptable since at the middle and high feeding levels used, colonies grew as rapidly in the laboratory as in the field. The three feeding levels were: (a) low (lx) - 0.1 ml of Marine Invertebrate Food per 1000 zooids every other day, (b) medium (2x) - double the low level, and (c) high (3x) - triple the low level. Replicates offivesemelparous andfiveiteroparous colonies were used in each of the nine combinations of temperature and food levels. Despite the range of the conditions, none of the genotypes changed phenotypes. The analysis of variance shows that regardless of the culturing conditions, all semelparous colonies remain distinct from all iteroparous genotypes for all seven life-history traits mentioned above. In addition, under the constant laboratory conditions, no significant differences arose among genotypes  Chapter 2. Botryllus schlosseri  10  (nested within life-history type). A limited number of Fi backcrosses and F2 crosses were made at 20 °C and feeding level 2x. Fi backcrosses between semelparous progeny and parents uniformly produce semelparous progeny and likewise for iteroparous colonies. Similarly, crosses among phenotypically similar Fi's yield F2's that are phenotypically equivalent to their parental Fi's. Crosses between unlike Fi phenotypes produce mixed clutches with, on average, roughly equal numbers of semelparous and iteroparous progeny. Matings among the rare intermediate phenotype Fi's and between these Fi's and their parents (strains 4 (semelparous) and 9 (iteroparous)), do not result in an increased frequency of intermediate phenotypes, at least not over the span of the few generations of crosses performed. The factorial experiment of temperature and food levels described above implies that the polymorphism is not due to the Fi's environment, and the back-cross results imply that the polymorphism is not due to the parents' environment. On the other hand, all three potential contributions to genetically determined variation have an effect. It is possible that a single locus (or a few tightly linked loci) pleiotropically controls the life-history polymorphism through developmental switching or a regulatory locus. However, even if such a locus does exist, the Fi data show that it does not obey simple Mendelian inheritance. Another possible cause for this genetic effect is gene dosage whereby just simply the number of genes, regardless of their location, determines the phenotype (Grosberg 1982). Then the determination of semelparity and iteroparity may depend on the number of certain genes at several to many loci. Whether above or below a certain threshold, the phenotypic expression of the genotype is canalized so that a variety of sub-threshold combinations produce nearly identical phenotypes (Rendel 1967). This argument is consistent with some of the life-history inheritance patterns among B. schlosseri. For instance, if most semelparous and iteroparous colonies have dosages well above or below threshold, then most progeny from within life-history type crosses will  Chapter 2. Botryllus schlosseri  11  also be clearly above or below threshold. This is not unreasonable due to the predominance of semelparous morphs during the spring and of iteroparous during the fall making mid-summer the only time when there would likely be a significant number of crosses. Most of the resulting progeny from the crosses would be above or below the threshold, although some may be very close. Strains 4 and 9 (mentioned in the experimental results above) may represent these marginal individuals. When a colony that is near threshold dosage is mated with a colony that is well away from threshold, one would predict that relatively more progeny would be phenotypically like the parent away from the threshold. This would explain the lower proportions of like phenotypes produced from crosses involving strains 4 and 9. It would also account for the F backcross results. However, x  the presence of intermediate phenotypes remains difficult to explain (Grosberg 1982). Assuming that the intermediate phenotypes result from being essentially at the threshold level, one would expect a larger proportion of F intermediate phenotypes when the 2  Fi intermediate phenotypes were mated among themselves or with their parents. This, however, was not the case in Grosberg's experiments. In the next sections, the various life-history parameters will be discussed, starting with colony growth. 2.3  Growth  Colonies can increase their size in two ways. They may secrete tunic or they may produce zooids at a rate greater than the zooids are lost. Secretion of tunic only increases the surface area of the colony. Most of the growth is due to the addition of blastozooids. Thus, "growth rate" is best described in terms of the number of zooids per blastozooid per asexual generation. Various researchers (e.g. Brunetti 1974; Boyd et al. 1986) have noted a large variability in growth rate, although this could be due mostly  Chapter 2. Botryllus schlosseri  12  to the different life-history types. During the 'change of generation', some buds could be resorbed together with their parent so the total number of zooids of a colony can either increase, remain constant or decrease (Brunetti 1980). However, growth-rate data (Grosberg 1982; Yamaguchi 1975; Brunetti 1974) suggest that growth rates per individual zooid is constant, resulting in exponential colony growth, until the start of sexual maturity. Nonetheless, Brunetti and Copello (1978) found that the maximum growth is between the 2nd and 4th asexual generation with the rate then falling until it stabilizes around a constant value. Although they do not present any values, I assume that this effect is minimal. The abundance of the colonies and the exponential growth means that during the summer and fall the colonies effectively cover all surfaces. By mid-July in Eel Pond, there is virtually no open space on hard substrata, with Botryllus schlosseri coverage at 50-100% (Grave 1933; Grosberg 1982). This means that often colonies will lack substratum space and/or encounter another colony. The next section discusses the semelparous/iteroparous responses to this situation. 2.3.1  S u b s t r a t u m Size  Grosberg (1982) manipulated the substratum size to study its influence on growth and sexual reproduction. For semelparous colonies, substratum size affected neither first sexual reproduction age nor reproductive effort. With insufficient substratum space, the colony will throw itself into convolutions until it reaches a size of « 200 zooids before sexually reproducing. However, when the colonies are so convoluted, they may pry themselves off the substratum and eventually die. A larger substratum results in only a minimal increase in size at sexual maturity. Sexual maturity size for iteroparous colonies, however, increases in roughly direct proportion to substratum size, except that colonies will initiate sexual reproduction when they are  800 zooids even if they have  not reached the bounds of the plate (Harvell and Grosberg 1988). Notably, the age of  Chapter 2. Botryllus schlosseri  13  first sexual reproduction is not affected by the substratum size. Instead, the confined iteroparous colonies make significantly more embryos per zooid than unconfined colonies (Grosberg 1982). Semelparous and iteroparous colonies encountering conspecifics behave in an analogous manner to insufficient substratum space (Brunetti 1974; Grosberg 1982; Harvell and Grosberg 1988). Despite the general lack of space (or perhaps because of it), larvae tend to settle near the parents as discussed in the next section. L o c a l S e t t l i n g of Colonies  Grosberg (1987) found minimal dispersal, with greater than 80% of the marked recruits located within 25 cm of the parental colony. He also studied fertilization and hatching success as a function of distance between mating colonies. Across all distances studied, between 90% and 95% of the ova were fertilized. Individuals separated by <1 m had the greatest success but this declined only slightly for more distant matings. However, considering the subset of fertilized ova that hatched into normal larvae, the success rate remained over 95% for distances <1 m but dropped to 80% for colonies separated by 3 m or more. 2.4  Self/nonself-recognition a n d F u s i o n  Related colonies not only settle near one another, but sometimes fuse together. Tanaka and Watanabe (1973) (cited in Scofield 1983) have shown that the fusion/rejection reaction is controlled by contact responses between the ampullae. In addition, Taneda and Watanabe (1982a,b,c) (cited in Scofield 1983) firmly established that humoral and cellular elements in the blood are the actual allorecognition elements that control the fusion/rejection reaction. The histocompatability system that controls fusion and rejection is controlled by a single Mendelian locus with many codominant alleles such that two  Chapter 2. Botryllus schlosseri  14  colonies sharing at least one allele can fuse (Grosberg and Quinn 1986, Scofield 1982). Grosberg and Quinn (1986) found through their studies on the Marine Biological Laboratory supply dock at Eel Pond that only 4.2% of the colonies fused, showing that B. schlosseri has many different alleles. Fusion provides several possible advantages. It increases colony size. A larger size may increase survivorship, decrease first sexual reproduction age and affect growth rate (Grosberg and Quinn 1986). Finally, germ cells or stem cells of the germinal line are exchanged between the fused colonies. Zooids of successive generations mature these germ cells even if one or more of the component colonies subsequently perishes (Sabbadin 1977). Rinkevich and Weissman (1987) found that the offspring suffered deleterious effects by fusing with the parents or even just contacting them. The progeny which remained in contact with the parental colony suffered growth failure and poor survivorship, but those progeny that later disconnected from the parental colony both survived and grew. They suggested that colonies nonetheless cosettle due to interspecific competition such as from Botrylloides leachi. The costs due to loss of growth potential by settling close could be less than that due to increased mortality from interspecific competition when the colonies settle further away. Despite the apparent connections between gregarious settlement and self/nonself recognition, these two phenomena (or strategies) may have evolved independently and are still in the process of adaptation. However, Rinkevich and Weissman (1987) might have misinterpreted the results. Due to the transfer of germ cells, the genome of the progeny would be preserved in the parental colony. The subsequent resorption benefits the progeny since they do not have to make a somatic contribution (Grosberg, personal communication). The germ cell transfer could also prove beneficial by maintaining the genetic variability within communities of reduced size (Sabbadin 1979) as discussed in the next section.  Chapter 2. Botryllus schlosseri  2.4.1  15  Inbreeding Considerations  Since one possible consequence of close kin association is increased inbreeding (Grosberg and Quinn 1986), an important function of self/nonself-recognition is avoidance of inbreeding depression. When two colonies fuse, the vascular systems join. The common circulation results in synchronization of asexual and sexual cycles which, due to protogyny, prevents the fused colonies from interbreeding. Outbreeding is thus favoured since the probability of fusion increases with the relatedness of the colonies (Sabbadin 1977). 2.5  Senescence  After iteroparous colonies reproduce several times (3-4 generations) they go into a period of senescence. This senescence seems to consist of essentially a reduction of the renewing capacity of the tunic, and is probably related to the intensity of sexual reproduction (Brunetti and Copello 1978). However, no obvious reduction of the blastogenic potential occurs over time (Sabbadin 1979). 2.6  Mortality  During the first week after settlement, juvenile mortality can approach 90% (Brunetti 1974; Grosberg 1982). However, apart from after overwintering, the proportion that survived fromfirstrecruitment to sexual maturity ranged from 7.7% to 21.8% over the period 1979-1980 at Eel Pond. Thus, most of the mortality is during the colony's first week. Most of the colonies that recruited at the end of the season (September 15) did not sexually reproduce until the following spring. These overwintering colonies had a survivorship in excess of 90% in both 1979 and 1980 (Grosberg 1982, 1988). As the iteroparous colony produces more clutches, its mortality rate increases (Table 2.1). However, sexual reproduction does not represent the end of the life cycle but actually  Chapter 2. Botryllus schlosseri  16  Table 2.1: Life table on the cohort of iteroparous colonies recruited in the field during 1-15 September 1979. Iteroparous colonies do not, in general, reproduce sexually before the ninth asexual cycle. Hence, survivorship data prior to the ninth asexual cycle are omitted. Estimates of survivorship do not include planktonic mortality of the larvae. Decreasing age-specific fecundity results from decreasing colony size with age and not from decreasing reproductive effort. The above caption and the table below are taken from Table 2, Grosberg 1988. Age (x) (number of Survivorship (/ ) asexual cycles) (proportion of cohort) 9 0.207 10 0.201 11 0.186 12 0.173 13 0.110 14 0.053 0.021 15 x  Fecundity (m ) (mean clutch size) 1,596 1,932 2,318 1,957 1,843 1,639 1,704 x  represents a difficult period when elevated metabolic activity influenced by environmental conditions, especially temperature, causes the increased mortality (Brunetti 1974). Semelparous colonies die after the production of a single, large clutch (9-12 embryos). Actually, the colony will often cease functioning before the embryos are mature. Since this occurs after the eggs are already fully provisioned with yolk, the parental colony is then merely a receptacle. Although allocating all of the colony's energy to the embryos could cause the mortality, the parental mortality also could be due to enlargening embryos mechanically interfering with respiration and somatic activities (Grosberg 1982). To determine the mortality causes, Grosberg (1982) performed a series of embryo manipulation experiments. He manipulated ten clonal replicates of three semelparous and three iteroparous colonies, with ten clonal replicates of each strain used as controls. The first manipulation involved an early embryo stage removal of embryos from all zooids. In the second manipulation, all the embryos were removed as before but each was replaced with  Chapter 2. Botryllus schlosseri  17  a glass bead of similar size. In the third manipulation, performed only on iteroparous colonies, the clutch sizes were augmented, with glass beads, to the normal semelparous clutch size. 80% of semelparous colonies from which embryos had been removed survived to produce a second clutch compared to only 3.3% (1 out of 30) for the unmanipulated control colonies. However, the replacement of embryos with glass beads only allowed fewer than 7% to live to make another clutch. This data, especially the glass bead replacement experiment, is consistent with the hypothesis that mechanical impairment is the primary cause of death among semelparous colonies. The enhancement of survival of colonies after embryo removal shows that the surgical procedure did not have gross deleterious effects. For iteroparous colonies, there were no conspicuous differences among the control, embryo removal, and glass bead replacement manipulations. However, the augmentation of embryos with glass beads caused the death of 90% of the colonies. This provides further evidence that the number of embryos per zooid is the primary cause of semelparous colony mortality rather than the energetic investment. Furthermore, after the removal of an entire clutch of nearly mature embryos, during the next generation the semelparous colonies revives and produces a clutch of nearly the same size as the one removed. 2.6.1  Predation  B. schlosseri colonies also perish from predation. Predators include the snail Mitrella lunata, bacteria and probably some flatworms and nematodes (Milkman 1967). Bancroft (1903) (cited in Millar 1971) found that crabs eagerly attacked colonies of Botryllus schlosseri. Further predators include Cycloporus papillosus and a number of nudibranch molluscs (Goniodoris nodosus (Montagu), G. castanea Alder and Hancock, and Ancula cristata (Alder)) (Millar 1971). In Eel Pond, the sessile stage of jellyfish occasionally  Chapter 2. Botryllus schlosseri  18  heavily consumes the larvae, but it does not have an appreciable effect on the recruitment rate. Also, there is some minimal predation from nudibranches and fish grazing (Grosberg, personal communication). 2.7  Life-History Correlations  Despite a lack of significant differences among genotypes within life-history morphs, the life-history type had a highly significant effect on all the dependent variables (Grosberg 1988). Thus, the two morphs (semelparous and iteroparous) each have distinct life-history characteristic values. The next sections discuss both the intra and inter life-history morph correlations. 2.7.1  Semelparous Colonies  In semelparous colonies, first sexual reproduction age is positively correlated with per zooid fecundity and negatively correlated with the growth rate. Also, a marginally significant negative correlation exists between the growth rate and the total reproductive effort. These correlations imply that semelparous colonies that grow slowly will postpone reproduction and have a slightly higher reproductive effort (Grosberg 1982). This pattern is similar to that of the iteroparous/semelparous colony comparison (see section 2.2). 2.7.2  Iteroparous Colonies  Iteroparous colonies have negative correlations between the number of clutches and both the growth rate and the reproductive effort per clutch. Also, colonies which make more clutches invest less in each one. Thus, for iteroparous colonies the reproductive "cost" of high growth rate is paid for by producing fewer, rather than smaller, clutches. In contrast,  Chapter 2. Botryllus schlosseri  19  among the semelparous colonies, the negative correlation between growth rate and total reproductive effort suggests that rapid growth exacts its toll by a lower reproductive effort (Grosberg 1982). 2.7.3  S e m e l p a r o u s / I t e r o p a r o u s C o l o n y Differences  From the feeding level experiment described in section 2.2.1, middle (2x) and high (3x) effects are not significantly different from each other. However, semelparous and iteroparous colonies react differently to the lack of food at feeding level 1 x. The following compares the lx feeding level effects to that of feeding levels 2x and 3x, with semelparous values first and iteroparous values in brackets: first sexual reproduction age: +60% (+35%); size at first sexual reproduction: +10% (—1%); buds per zooid: —45% (—20%); clutch size (number of embryos per zooid): 0% (—35%); clutch number: 0% (-15%); fecundity: +10% (-45%); and longevity: +60% (+22%) (Grosberg 1982). When starved, semelparous colonies will grow much more slowly but will wait until they have reached their normal size and can produce their normal clutch size. Iteroparous colonies also grow to their normal size, but relatively quickly thereby sacrificing some of their potential sexual reproductive output (see section 2.7.5 about energy reallocation between growth and reproduction). Analysis of semelparous and iteroparous colony differences from field studies reveals a positive correlation between growth rate and reproductive effort. This appears to imply that no reproductive price is paid for rapid growth. However, semelparous colonies are smaller at first reproduction and reproduce only once thus resulting in lower total reproductive output (Grosberg 1982).  Chapter 2. Botryllus schlosseri  2.7.4  20  Fi Correlations with Parents  The maternal and paternal variance components for parental life-history type and genotype are very similar. This suggests both that F i dependent variables are genetically determined and that the maternal and paternal contributions to the phenotypes of progeny are roughly equivalent. In addition to the absence of any parental male by female interactions, this shows that both parents in a cross additively determine the phenotype of the resultant progeny (Grosberg 1982). 2.7.5  Energy Reallocation  To determine how energy can be reallocated, Grosberg (1982) performed a series of experiments where he prevented colonies from allocating resources to either growth or sexual reproduction. During the sexual reproduction process prior to ovulation, the parent supplies the developing oocyte with a substantial quantity of yolk. In the first experiment, Grosberg removed oocytes before the provisioning begins to prevent significant parental energetic investment in sexual reproduction. To test for allocation from somatic growth to sexual reproduction, in the second experiment, Grosberg severed all but one secondary bud from each primary bud. During thefirstexperiment, the number of buds maturing one and two asexual generations later was recorded. For semelparous colonies, no significant difference existed between the experimental colony and control colonies before they sexually reproduced. However, control colonies die after sexual reproduction whereas the manipulated colonies live as long as their oocytes are removed each generation. Thus, it is possible that the oocyte removal procedure of itself inhibits growth in semelparous colonies. However, after oocyte removal, iteroparous colonies had significantly enhanced growth compared to control colonies (18% one generation later and 50% two generations later). This implies that  Chapter 2. Botryllus schlosseri  21  the experimental procedure has at most a minimal negative impact on colony growth. Thus, this experiment indicates that iteroparous colonies can reallocate resources from yolk provisioning to growth while semelparous colonies cannot. During the second experiment, the resulting fecundity was noted and analyzed. While iteroparous colonies increased fecundity in the two asexual generations following the bud removal, semelparous colonies showed no fecundity increase. These two experiments together imply that semelparous colonies cannot reallocate their energetic resources between growth and sexual reproduction, whereas iteroparous colonies can reallocate their energy. 2.8  Competition  As mentioned in section 2.2, over the summer semelparous colony density decreases while iteroparous colony density increases. Semelparous and iteroparous colony fecundity does not change significantly and thus cannot be the cause. Instead, overwintering iteroparous colonies reproducing and increased semelparous colony mortality are partly responsible for the seasonal change. The increasing semelparity mortality occurs before the additional recruitment of iteroparous colonies, so larger inter-morph competition is not the mortality cause. In fact, semelparous/iteroparous density change is mediated primarily by the density of a closely related colonial ascidian, Botrylloides leachi. Botrylloides leachi is competitively dominant to semelparous B. schlosseri but not the iteroparous morph (Grosberg 1982, 1988). As such, semelparous colony mortality increases as the Botrylloides leachi density increases over the summer. Semelparous colonies predominated both in tidal ponds without Botrylloides leachi and in Eel Pond before the introduction of Botrylloides leachi about thirty years ago (Grosberg 1981, 1982).  Chapter 2. Botryllus schlosseri  2.8.1  22  Botrylloides D e s c r i p t i o n  Botrylloides is so closely related to Botryllus that it has often been relegated to the status of a subgenus (Berrill 1947). The minor differences include its gonad arrangement, larval incubation and some digestive tube morphological aspects (Brunetti et al. 1980). Botrylloides has a central stalk with adhesive organs. This stalk is surrounded by a ring of eight epidermal ampullae that aid in maintaining colonial circulation (Bancroft 1899) and form the permanent organ of attachment on the substratum (Berrill 1947). In the Venetian lagoon, sexual reproduction occurs mainly during the 2 months from the second half of May to the first half of July. Assuming that water temperature provides the controlling influence, it appears that sexual reproduction occurs in the temperature range 17°C to 25°C (Brunetti 1976). At Woods Hole, this corresponds roughly to the period of July to September. The colonies are always in an encrusting form with well developed colonies even forming surface folds that resemble a brain surface. The colonies assume a spherical shape with no cases observed of pendant type growth present in B. schlosseri (Brunetti 1976). As the temperature falls below 10°C, B. leachi goes into hibernation. First, there is a large reduction in general activity in feeding and blastogenic development. Not all the buds reach the stage of filtering zooid. Many in fact degenerate together with the parent zooids until there are no functionalfilteringzooids left. With only small rounded ampullae remaining, the colonies resemble a carpet (Brunetti 1976). Botrylloides has many predators in common with Botryllus including the posobranch gastropod Erato voluta (Montagu), nudibranch molluscs such as Goniodoris nodosus (Montagu), G. castanea Alder and Hancock and Ancula cristata, andfinallyCycloporus  Chapter 2. Botryllus schlosseri  23  papillosus. However, although crabs enjoy Botryllus, they quickly learn to avoid Botrylloides (Millar 1971). 2.8.2  C o m p e t i t i v e interactions  Myers (1988) describes the competitive effects of other organisms on Botrylloides at Woods Hole. The presence of Bugula decreases the Botrylloides growth rate. This is most probably caused by food competition because Bugula cannot outcompete Botrylloides for space. The solitary tunicate Mogula spp. overgrows Botrylloides thereby causing a high mortality rate. Bugula has a strong positive settlement effect on Mogula. As mentioned previously, Botrylloides competitively interacts with B. schlosseri. Grosberg (1982) performed a series of experiments to investigate the effect of Botrylloides on Botryllus survival. He used five experimental plates mixed with five control plates. He performed weekly removals of all newly recruited Botrylloides leachi on the experimental plates. For Botryllus colonies that died, he noted the type, extent (in terms of length of contacting margins), and competitive outcomes of contacts involving the B. schlosseri colonies. In a second experiment, Grosberg performed a natural, though controlled, experiment in Green Pond. This pond, located about 10 km northeast of Eel Pond, is similar, but not identical, to the Eel Pond. Most importantly, it does not have Botrylloides. Grosberg placed five clonal replicates each of four strains of semelparous colonies and of four strains of iteroparous colonies at both Eel Pond and Green Pond. This was done on June 1, 1981 before any significant Botrylloides recruitment at Eel Pond and again on July 24, 1981 during the first large Botrylloides recruitment at Eel Pond. In thefinalexperiment, clonal replicates of four iteroparous and four semelparous strains were placed in symmetrical pairwise competition with 3 colonies of Botrylloides randomly collected from the field. All possible pairwise competitions among the four iteroparous and four semelparous strains were performed. In all the above experiments,  Chapter 2. Botryllus schlosseri  24  a colony with greater than 50% of its area covered by another colony was considered overgrown. In the first experiment, the proportion of semelparous Botryllus colonies on the control plates began decreasing at the end of July, reaching only 10% at the end of August. On the experimental plates, in the absence of Botrylloides, the semelparous proportion did not begin declining until the end of August. This one month delay is about the life span of a semelparous colony. After mid-October, on both the experimental and control plates, the semelparous colony proportion began to rise. At this time, the growth rates, and hence contact frequencies, of all colonial ascidians have declined resulting in few semelparous colonies being overgrown by Botrylloides. In the second experiment, in Eel Pond, the semelparous colonies initiated on June 1 had a significantly higher survival rate to sexual reproduction than did the colonies initiated on July 24. However, in Green Pond, the semelparous colonies initiated on both dates had a similar survival success rate. On the other hand, at Eel Pond, the iteroparous colonies initiated in June had a significantly lower survival rate to first sexual reproduction than the July initiates. In Green Pond, iteroparous colonies from both dates had a low survival rate. The above results suggest that the Botrylloides presence hinders semelparous survival but aids iteroparous colony survival. In the third experiment, all Botrylloides/semelparous contests resulted in Botrylloides overgrowing the semelparous colony before it could sexually reproduce. In contrast, in only one of the twelve Botrylloides / iteroparous competitions did the Botrylloides overgrow the iteroparous colony, with the remaining eleven resulting in a mutual cessation of growth along the interspecific contact region. Most other pairings of dissimilar genotypes also resulted in growth ceasing at the contact margin. The two exceptions involved the same semelparous colony overgrowing two different iteroparous colonies. In summary, the above data shows that Botrylloides consistently overgrows semelparous, but  Chapter 2. Botryllus schlosseri  25  not iteroparous, colonies. Also, intraspecific contacts usually result only in cessation of growth rather than death of one of the interacting colonies. The data above strongly support the hypothesis that the increased semelparous colony mortality is primarily caused by the increased competition from the seasonal increase of Botrylloides leachi (Grosberg 1982). On the other hand, the increase of iteroparous colonies probably results from the decrease in intraspecific competition after the semelparous colony decline. The cause of the morph specific difference in Botrylloides competition appears to be related to the ampullar density. Contacts between botryllid ascidians first involve epidermal contact and then interactions among the ampullae. Botrylloides overgrowth of semelparous colonies involves the interdigitating of the ampullae followed by the Botrylloides ampullae pulling their colony edge over the now underlying Botryllus colony. This could be due to the Botrylloides colony possessing an ampullar density often double or triple that of the semelparous colony. Iteroparous colonies, on the other hand, have an ampullar density about double that of the semelparous colonies and thus similar to that of Botrylloides. This similar ampullar density could be why Botrylloides is incapable of extending its ampullae over those of iteroparous colonies. However, this remains tentative at best until pertinent experiments are performed (Grosberg 1982). Also, at least part of the reason for the iteroparous Botryllus/Botrylloides outcome could be Botrylloides recognizing, through some sort of chemical recognition at the ampullae, the iteroparous colonies as being the same species. However, this would not be evolutionarily stable since a random mutation in a semelparous colony to produce these chemicals should quickly spread (Grosberg, personal communication).  Chapter 2. Botryllus schlosseri  2.8.3  26  Coexistence of P o l y m o r p h i s m  Semelparous colonies continue to coexist with iteroparous colonies despite a severe drop in semelparous colony numbers in the late summer. However, few semelparous colonies die during the winter when shrinkage and death of colonies disrupts the competitive process. The resulting empty space becomes filled with semelparous progeny since overwintering semelparous colonies reproduce about 4 weeks before iteroparous colonies. Finally, substantial numbers of Botryllus likely arrive in Eel Pond on boat hulls (Grosberg 1982). Considering the predominance of semelparous morphs in local areas, the majority of the hitchhikers would be semelparous morphs. The next two chapters will use mathematical models to examine Botryllus schlosseri. Chapter 3 will study various life-history characteristics of B. schlosseri while Chapter 4 will investigate how the competitive interactions between the two Botryllus morphs and Botrylloides can result in coexistence.  Chapter 3  A g e of sexual r e p r o d u c t i o n i n B. schlosseri  The previous chapter discussed correlations among life history traits (growth, reproduction, mortality) of semelparous and iteroparous B. schlosseri. However, understanding the underlying causes for the correlations requires a model. This chapter first describes the progression of modeling techniques and then presents a model to study the age of first sexual reproduction in both morphs of B. schlosseri. 3.1  C o m p a r a t i v e Studies  The comparative approach involves broad comparisons among similar groups of taxa (preferably conspecific populations) in different environments (Stearns 1976, Grosberg 1982). Where applicable, the comparative approach provides the best type of explanation for the basic life-history traits (Stearns 1976). However, an implicit assumption is that the environment is the primary cause of any differences in life-history traits. Also, different species or populations often have different genetic histories which can have a significant effect on the life-history traits (Grosberg 1982). Since both morphs of B. schlosseri live together in the same environment, the basic premise of the technique does not hold.  27  Chapter 3. Age of sexual reproduction in B. schlosseri  3.2  28  Genetic Models  Genetic models, as opposed to correlational studies, attempt to directly model the genetic influences on life-history traits. Many genes acting in combination likely determine lifehistory traits (Grosberg 1982, Stearns 1977). However, most genetic models examine only one locus. Three, apparently intractable, reasons disallow generalization to the multi-loci model required. First, two or more loci can have wildly nonlinear interactions (Wright 1968 cited in Stearns 1977). Second, the selective value of an allele can depend on the frequency of alleles at other loci (Wright 1968 cited in Stearns 1977). Finally, two sorts of evidence indicate that changes in only a few regulatory genes, rather than many structural changes, can establish large differences between species: (a) men and chimpanzees differ strongly in morphology, ecology, behavior, and life-history traits, yet the electrophoretically detectable structural loci are more similar between chimpanzee and men than between pairs of sibling fruit flies or mammals (King and Wilson 1975 cited in Stearns 1977); (b) serum albumins of frogs and placental mammals have evolved at the about the same rate while the chromosome number of mammals has changed 20 times faster than that of frogs, paralleling the much greater morphological diversity of mammals (Wilson et al. 1974 cited in Stearns 1977). The above complications will likely keep multi-loci genetic models unrealistic for a long time, perhaps forever (Stearns 1977). As discussed in section (2.2.1), no simple mode of genetic inheritancefitsthe data, thus making genetic modeling difficult. An alternative technique, biometrical (or quantitative) genetics, provides a direct, although imprecise, method to study life-history. The biometrical approach compares the phenotypic resemblance among individuals of known relatedness. This allows the separation of the observed phenotypic variance into genetic and non-genetic (environmental) components. However, the technique has three major difficulties: (1) it does  Chapter 3. Age of sexual reproduction in B. schlosseri  29  not specify the genetics of inheritance and allows easy interpretation only of the additive effects of alleles at different loci; (2) it assumes that phenotypic distributions satisfy a normal distribution which is not true for threshold characters; (3) its results are specific to the time, population, and environmental conditions in the study. 3.3  Life-History Theory  Yet another approach is life-history theory. It directly ignores the genetics but instead models a basic tenet of evolution: survival of the fittest where fitness is a measure of the expected number of descendents (Bell 1980; Stearns 1976). 3.3.1  The Characteristic Equation  Consider a population formed from the descendents of one organism. The growth rate of that population will depend on its age structure and the probabilities of giving birth and of dying at different ages. Assume a stable age distribution has been reached. Let l = the probability of surviving to age x, and b dx = the number of female progeny of x  x  females of age x to x-\-dx. Newborns at time t have parents of age k that were themselves born at time t — k and survived to age k. Thus, letting n (t) = the number of newborn 0  at time t, we have (3.1) If further we assume that the number of newborn progeny changes exponentially in time, then n {t) = Q  Ct \ r  (3.2)  where r = the intrinsic growth rate of the population. Fisher (1930), considering the number of genes instead of the number of individuals, used the symbol m which he called the Malthusian parameter. However, for consistency, I will use r throughout this thesis.  Chapter 3. Age of sexual reproduction in B. schlosseri  30  Charlesworth (1973) showed that in an age-structured, diploid, random-mating population, any nonrecessive mutant which decreases r will be eliminated by selection, while a mutant which increases r will be selected. Thus, maximizing r will also maximize the fitness of a population. Substituting (3.2) into (3.1) yields Ce = rt  /•oo  / Jo  Ce^-^hbkdk,  /•CO  1 = / e- l b dk. Jo rk  k  k  (3.3)  The above equation, called Lotka's equation or the characteristic equation, was first derived by Lotka (1913) (also see Stearns (1976) for the derivation). The equation relates the mortality and fecundity schedule of a population in a stable age distribution with its intrinsic growth rate r. This equation is a basic tenet of most subsequent life history theoretical work (Stearns 1976). 3.3.2  Discrete forms of Characteristic Equation  Although (3.3) uses continuous time, organisms reproduce in discrete time intervals. A discrete form of (3.3), where the polling is done after mortality but before birth, is (Leslie 1948; Schaffer and Rosenzweig 1977) oo  l = £A-(* )£U . + 1  x  (3.4)  x=0  where A = the intrinsic growth rate, B = the number of offspring produced by a i year-old individual which themselves x  survive to the next breeding season, l  x  = the mean survival rate to age x.  Chapter 3. Age of sexual reproduction in B. schlosseri  31  Note that A = e . Also, if c is the probability that an x year-old's new born progeny r  x  survives to the next breeding season, then B = c b . Thus, B is the effective fecundity x  x  x  x  of an x year-old individual (Schaffer 1974). The life-history equation may also be stated in matrix form. Let n,(i) be the number of individuals of age between i and i + 1 alive in the population at the beginning of the breeding season in year t. Define the vector N(t) such that n (t) 0  m(t)  N(t)  (3.5)  ^ n (t) m  where m is the maximum age of the organism. Also, define M(t), the population matrix, as Bo B  x  M =  .••  B -i B m  m  Po  0  .  0  0  0  Pi •  0  0  0  0  . • • Pm-l  (3.6)  0  where p; = U+i/U is the probability that an i year-old survives to the next breeding season. We then have the relation N{t + l) =  MN(t).  (3.7)  Note that the A from (3.4) is the dominant eigenvalue of M (Leslie 1945). Thus, for most M and any initial arbitrary age distribution we have that N(t)  oc  \*Ni as t -> oo,  (3.8)  Chapter 3. Age of sexual reproduction in B. schlosseri  32  where iVj is the associated eigenvector of A (Leslie 1945; Schaffer and Rosenzweig 1977). Existing life-history models assume that disturbances are always sufficiently infrequent that this limiting approximation is valid (Schaffer and Rosenzweig 1977). 3.3.3  Principle of Allocation  With no constraints on Ik and bk in (3.3), the characteristic equation implies that the most fit organism imaginable (i.e., maximum r) starts to reproduce continuously immediately after birth and experiences no mortality. That no such creature exists implies that Ik and bk cannot independently have any value. In real life, resources, or at least their uptake, are limited. Thus, organisms face a tradeoff of allocation of resources among activities (growth, maintenance, reproduction, etc.). The dominant doctrine of theoretical and experimental research into this tradeoff is the "principle of allocation" (Grosberg 1982). The principle of allocation's major prediction is that current reproduction imparts a cost in terms of subsequent survival and/or fecundity (Murphy 1968; Schaffer 1974; Schaffer and Rosenzweig 1977; reviewed in Stearns 1976, 1977, 1980). Energy not allocated to reproduction can presumably be used for other activities (growth, maintenance, predator avoidance, etc.) that enhance future reproduction. 3.3.4  Fisher's Reproductive Value  The tradeoffs limiting reproduction cause the value of reproduction to depend on the age class, as first shown by Fisher (1930). The idea is analogous to that of present value of money invested at compound interest. The reproductive value of a female at age x, denoted by v , is the average number of young the female can expect to have over the x  remainder of its life, discounted back to the present. It is given by (3.9) X  Chapter 3. Age of sexual reproduction in B. schlosseri  33  (Fisher 1930). In discrete form, it is (3.10) where k is the maximum age of a reproductive individual (Leslie 1948). Reproductive value allows the characterization of different age classes as to their "worth" in terms of their contribution to the intrinsic growth rate of the population, r. However, the reproductive value of a female is the present value of her future daughters only with a stable age distribution (Leslie 1948). 3.3.5  Cole's M o d e l a n d Extensions  Cole  Differing reproductive values with age implies that the age of first reproduction and the reproductive effort are important considerations. If the reproductive value decreases quickly enough with age, then the optimal action would be for the organism to sacrifice itself to reproduce the maximum amount possible. Cole (1954) was the first to theoretically study the conditions under which this type of reproduction, called "semelparity" or big-bang reproduction, is superior to reproducing several times, called "iteroparity" or repeated reproduction (Cole 1954, Gadgil and Bossert 1970). For simplicity, in his study, he assumed that semelparity and iteroparity can be represented by annuals and perennials respectively. First, Cole discretized Lotka's equation (3.3) to get 1 = £  (3.11)  e- l b . rx  x  x  x=oc  where a and u are, respectively, the start and end of the reproductive period. Consider an organism that begins producing at the age of ct with a constant mean litter size of b. Also, assume zero mortality from birth to age u so that l — 1 for 1 < x < x  Chapter 3. Age of sexual reproduction in B. schlosseri  34  u>. In the case of semelparity with reproduction at the first time step only, u = a = 1 so (3.11) becomes e = b which gives r  r = In 6.  (3.12)  The most extreme case of iteroparity would be each organism producing b offspring each year for eternity. In this case, a — 1 and u — > oo so (3.11) results in r = ln(6+l).  (3.13)  Comparing (3.12) to (3.13) led Cole (1954) to make the famous statement (called "Cole's paradox"): For an annual species, the absolute gain in intrinsic population growth which could be achieved by changing to the perennial reproduction habit would be exactly equivalent to adding one individual to the average litter size (p.118). Most organisms, especially those with large litters, could presumably easily evolve to produce one more progeny. Thus, why does iteroparity exist? Cole (1954) provided a partial answer to this problem by studying the effect of delayed age at first reproduction on the proportionate increase in r in going from a semelparous to an infinite iteroparous life history. As the age of first reproduction increases, so does the percentage gain from iteroparity (Cole 1954; Fig. 2) implying that iteroparity would be favoured in organisms with longer prereproductive periods. However, in Cole's (1954) analysis some of the assumptions, particularly the lack of mortality and infinite life span, are highly unrealistic. Later researchers expanded this basic model by making the assumptions more tenable or by creating more general models with Cole's model as a special case.  Chapter 3. Age of sexual reproduction in B. schlosseri  35  G a d g i l a n d Bossert  Gadgil and Bossert (1970) criticized Cole's assumption of no mortality. With a large litter size, mortality during the first year of life would be relatively higher than later on. Assuming that an annual organism just basically replaces itself after its first reproduction, they stated that Zi&i ~ 1.  (3.14)  For an annual species, this gives 1 = e- /iib , r  x  r = ln(7i&i).  (3.15)  For a perennial, assuming constant litter size and no mortality beyond the first time step, (3.15) gives l b = l\b\ « 1 for all ages. The discretized characteristic equation (3.11) x  x  yields r = \nilxh + 1).  (3.16)  r ~ ln(2/i6i).  (3.17)  Using (3.14), (3.16) becomes  Thus, Gadgil and Bossert (1970) suggested that Cole's result be modified to read: For an annual species the absolute gain in the Malthusian parameter which could be achieved by changing to the perennial reproductive habit would be approximately equivalent to doubling the average litter size (p. 11). However, Stearns (1976) pointed out that their semelparous (annual) and iteroparous (perennial) organisms are not comparable since the semelparous population is just replacing itself (r = 0) while the iteroparous population is growing rapidly (r = ln2 « 0.69).  Chapter 3. Age of sexual reproduction in B. schlosseri  36  Gadgil and Bossert's argument of the annual organism just barely replacing itself applied to perennials actually leads to the equation oo  l =  ^Kx=l  Finally, if the semelparous organism were to double its average litter size without affecting mortality (highly unlikely in real life), then the basic assumption (3.14) would no longer hold! Gadgil and Bossert also studied the result using computer simulation, finding that in all cases examined the reproductive effort increases with age. However, Fagen (1972) constructed a model where the reproductive effort at first decreased! This shows the potential danger of trying to generalize from computer studies. Bryant Bryant (1971) criticized Gadgil and Bossert (1970) for using a special case (r = 0) to establish a general result. Also, the result in (3.16) shows that the benefit of iteroparity is equivalent to adding one to the effective litter size. This actually makes it a special case of Cole's result. If a general mortality term is used instead of just mortality in the first time period, then the annual species needs to add less than or equal to 1 to the effective litter size to match the iteroparous advantage. Bryant (1971) considered an example with the survivorship being exponentially distributed, l — e~ , where u is the mortality rate. The result for an annual species is ux  x  r = ln(6) — u and for a perennial it is r = \i\(b + 1) — u. Thus, Cole's statement still holds for this example.  Chapter 3. Age of sexual reproduction in B. schlosseri  37  Charnov and Schaffer Charnov and Schaffer (1973) expanded on previous work by considering variation in adult and juvenile mortality. They considered two species: one annual and the other perennial. Both reproduce once at the end of the year and produce B and B progeny, respectively. a  p  Both the annual and perennial offspring have a survivorship the first year of C. The perennial has an adult survivorship of P per year. The relation between the number of individuals next year N(t + 1) and the number of individuals this year N(t) for the annual species is: N(t + 1) = B CN(t)  (3.18)  a  while for the perennial species it is: N(t + l) = B CN(t) + PN(t) p  = {B C + P)N(t).  (3.19)  P  If both populations increased at the same rate, then we would have B C = B C + P or a  B = B + P/C. a  p  P  (3.20)  Cole's (1954) and Bryant's (1971) results are special cases of the above equation with P = C = 1 and P = C < 1 respectively. Thus, Charnov and Schaffer (1973) suggested that Cole's result be modified to read: For an annual species, the absolute gain in intrinsic population growth rate that can be achieved by changing to the perennial habit would be exactly equivalent to adding P/ C individuals to the average litter size. Usually P >• C (for examples, see Deevey 1947 or Caughley 1966 (cited in Charnov and Schaffer 1973)). In these environments, the evolution of iteroparity is likely.  Chapter 3. Age of sexual reproduction in B. schlosseri  38  Charnov and Schaffer (1973) next considered prereproductive time where the age of first reproduction is in year K (K > 1). The resulting equation is §- = l-P/\  = R,  (3.21)  where B{ and B are the iteroparous and semelparous clutch sizes, respectively. Preres  productive time, K, affects R only through its effect on A. Iteroparity is favored by low R. Actually, (3.20) and (3.21) are different forms of the same equation (Young 1981) showing that prereproductive time does not qualitatively change the life-history results. Young Young (1981) created a more general model that included Cole's (1954) and Charnov and Schaffer's (1973) results as special cases. The model allows for prereproductive time, between reproduction time, and adult survivorship to vary independently of one another. Senescence for the iteroparous life type is included by a maximum number of reproductive episodes before death. It is assumed that the intrinsic growth rate (A), and juvenile survivorship and prereproductive development time (PDT) are the same for both genotypes. The yearly juvenile survivorship need not be constant, but the proportion of offspring that survive to reproductive age is assumed to be constant. Finally, Young (1981) assumed a stable age distribution. The resulting model simplifies to:  - -( / r  B  1  p  Bt ~ i - {Pixy  x  z  •  (3 22) { 6  }  where x = maximum number of reproductive episodes before senescence. Z = time between reproductive episodes. Assuming that P/X < 1 and letting Z = 1 and x —» oo in (3.22) results in Charnov and Schaffer's equation (3.21).  Chapter 3. Age of sexual reproduction in B. schlosseri  39  Discussion Cole's model prediction that iteroparity would be favoured in organisms with longer prereproductive times applies to B. schlosseri. However, even the more general assumptions of Young's model do not fit B. schlosseri very well. For example, mortality is assumed to be solely a function of the age of the organism, but, as shown in Fig. 3.4, the mortality rate also depends strongly on the time of year. The next section discusses several models that add the effects of environmental change. 3.3.6  Stochastic Models  Cohen Cohen (1966) constructed a long-term growth rate optimization model that accounted for a randomly varying environment. In particular, he modeled an annual plant. The seeds can germinate the next year, with the resulting plants producing seeds according to environmental conditions, or the seeds can lay dormant, but with a certain portion decaying. The number of seeds present is defined by the recursive relationship S  t+1  = S - S G - D(S - S G) + GY S . t  t  t  t  t  t  (3.23)  where St = the number of seeds at time t, G = the fraction that germinates each year, D = the fraction that decays each year, Y = the average number of seeds per germinated seedling. It is assumed to be a random t  variable depending on environmental conditions but independently of the population density.  Chapter 3. Age of sexual reproduction in B. schlosseri  40  16 - 3 0 J U N E  0-25i  0125-  £  <  O-25-i  '"'  5  J U L Y  6-31  JULY  0-125-  X CO  o z  0-25-  % co  0-125-  > > z  g  tce o  Q.  0-25-  o  X CL  0-125-  1-15  AUGUST  LLl  16-31  11  -15  SEPTEMBER  1979  1980  16-30  1979  AUGUST  SEPTEMBER  I960  Figure 3.4: The proportion of a cohort of field colonies that survives from settlement to sexual maturity. Open bars show semelparous survival and solid bars show iteroparous surival. The caption and figure are from Fig. 5, Grosberg 1988.  Chapter 3. Age of sexual reproduction in B. schlosseri  41  Over a sequence of N steps, this becomes SN = 5oIJ[(l ~  -D) + GYi] ,  (3.24)  ni  t  where n,- is the number of times that a particular Y{ occurred during the N steps. Taking logarithms, dividing by N, and letting N — > oo results in jfe, ^  =  ?  P i l o g [ ( 1  "  G  )  (  1  "  D  )  +  ( 3  -  2 5 )  where P, is the probability associated with Y{ and lim(log SN/N) is the mathematical expectation of the specific growth rate of the seed population. Analysis of (3.25) indicates that a necessary condition for G x < 1 (where G ma  max  is  the value of G giving maximum growth) is Y { \ - & I < \ - D .  (3.26)  Thus, it becomes advantageous to postpone the germination of some fraction of the seeds only when the variance of the yield becomes large enough in relation to the mean yield and the viability of the seeds. If instead of seeds germinating, one considers progeny reproducing, then this analysis shows that iteroparity is favored in a variable environment. The spreading of reproduction over time to enhance the chance of reproducing during a favourable period is known as "bet-hedging" (Cohen 1966). Murphy Using computer simulation, Murphy (1968) found that under conditions of uncertain survival of prereproductives and relatively stable survival of organisms during the reproductive stage, iteroparity is advantageous. This is true both with respect to competition between species and within a freely interbreeding population. Conversely, either high or variable adult mortality encourages early reproduction, high fecundity, and few reproductive periods or even only one.  Chapter 3. Age of sexual reproduction in B. schlosseri  42  Schaffer Schaffer (1974) showed that if, in constant environments, repeated breeding (iteroparity) is the optimal reproductive strategy, environmentally induced variation in reproductive success favors reductions in per capita reproductive output. Conversely, variations in postbreeding survival among adults favor greater investments in breeding. However, the principal issue is the age-specificity of the fluctuations (i.e., do they principally affect immature adults?). 3.3.7  Discussion  Unfortunately, these stochastic models do not allow the specification of an arbitrary temporal variation, thereby seriously limiting their usefulness. The basic result of these models is that iteroparity is favoured in more variable environments. However, since the B. schlosseri semelparous and iteroparous morphs live together, their environments, including any variability, are essentially identical. Also, the range of survival to first reproduction, as shown in Fig. 3.4, is from .04 to .23 for iteroparous morphs and .03 to .24 for semelparous morphs suggesting that the morphs do not experience a large difference in the degree of changes in mortality. Life-history theory is predicated on the assumption that mortality and sexual reproduction are based solely on age. However, for most clonal organisms, including Botryllus schlosseri, this is an inappropriate assumption. Studies have shown that postzygotic age is at best weakly correlated with the development, behaviour, and reproductive success of clonal organisms (Jackson 1985). Partial mortality among genets decouples age and size. Thus, a small individual can be old and likewise a large individual can be young (Grosberg 1982). However, neither age nor size is necessarily an accurate predictor of the start of sexual reproduction. Rather, the onset is determined by a complex interaction  Chapter 3. Age of sexual reproduction in B. schlosseri  43  between intrinsic factors such as age, size, and physical condition and extrinsic factors such as density, food availability, physical disturbance, and season (Harvell and Grosberg 1988). Thus, a stochastic, state-based modeling technique is needed to model the important characteristics of B. schlosseri. Dynamic programming, as described below, is used here. 3.4  Dynamic Programming  3.4.1  Definition and Assumptions  Dynamic programming is a stochastic, state-based modeling approach that allows one to consider dynamic behaviour in organisms. It has several components. First is the specification of the organism's "state". A state variable represents any characteristic of an organism that affects its fitness (Mangel and Clark 1988). Basically, the technique views an animal choosing the behavioural option that will maximize its fitness based on its current state. However, it is not being suggested that an organism actually consciously solves the optimization problem. Rather, the organism is assumed to "breed true" so that by natural selection the behavioural sequence with the greatest fitness will predominate (Houston and McNamara 1988). In its common discrete form (for continuous time dynamic programming, see Mangel 1985; Bertsekas 1976), dynamic programming divides time, t, into discrete units from 1 to T, where T is the final time step in the calculations. The basic concept is the lifetime fitness function F, defined for 1 < t < T as (Mangel and Clark 1988): F(x,t,T) = max E j £ R(X(j), A(j)J,w(j)) + 4(X(T))\X{t) = x  (3.27)  with the maximization taken over possible values of actions A(j), j = t, t + 1,..., T — 1 and  Chapter 3. Age of sexual reproduction in B. schlosseri  44  X(t) = the state (possibly multi-dimensional) of the organism at time t, x = the current state of the organism, A = the set of possible actions of the organism, R = the reward function that denotes the increase in fitness (i.e. current reproduction) from a given action, w = a random variable (such as the environment) that affects the state or reward function,  <f> — the terminal fitness function that denotes the fitness of the organism after the last time step. The equation for updating X for t < T is (Mangel and Clark 1988): X{t + \) = G{X{t),A(t),t,w(t)),  (3.28)  where G includes the state dynamics and constraints as appropriate. 3.4.2  Solution Techniques  From (3.27), we have F(x,T,T) = <f>(x).  (3.29)  Next, consider t = T — 1. Assuming that u>i = w(T — 1) and x = X(T — 1), then (3.28) becomes X{T) = G(x,A,T-l,w ). l  (3.30)  Thus, (3.27) gives F(x,T-l,T)  =maxE {R(x,A,T-l,w ) w  1  + F(G(x,A,T-l,wi),T,T)},  (3.31)  A  where E is the expectation over values of w. By repeating the above procedure, we get w  for< = r - l , r - 2 , . . . , l that F(x, t, T) = max E {R(x, A, t, w) + F(G(x, A, t, w), 14-1, T)}. w  (3.32)  Chapter 3. Age of sexual reproduction in B. schlosseri  45  This is the general dynamic programming equation (Mangel and Clark 1988) that is normally used in actually computing a solution. Note that the solution is calculated iteratively backwards in time. As Kierkegaard (Bertsekas 1987) said "Life can only be understood going backwards, but it must be lived going forwards." Each iteration provides both the maximal future fitness, F(x,t,T),  and the optimal  strategy A* = A*(x,t). The optimal strategy depends on both the current state, x, and the time t. This strategy specification, called feedback control policy (Mangel and Clark 1988), allows the organism to dynamically determine the optimal action. However, the question of whether strategies that use the entire previous state history can be superior to strategies that use just the current state naturally arises. The answer is no (Bertsekas 1978; also see Mangel and Clark 1988 for the case of fixed n previous states). The intuitive reason is that the previous state history is implicitly accounted for in the current state value (Mangel and Clark 1988). The maximization of fitness from time t forward in (3.32) is similar to Fisher's (1930) idea of reproductive value. In fact, the life-history equation is a special case of the dynamic programming technique but without the state dynamics. For example, Mangel (1987) shows how to reformulate the equation for the value of future reproduction from age a, V(a): UI  V(a) =  e ^2(l m /l )era  rt  t  t  a  t=a  to allow the use of dynamic programming. 3.4.3  Advantages and Disadvantages  Dynamic programming is more than just another technique to solve life-history equations since it widely applicable to many different problems in biology (see Mangel and Clark 1988). Being state-based, it uses more biologically relevant parameters than other  Chapter 3. Age of sexual reproduction in B. schlosseri  46  modeling techniques (Mangel and Clark 1988). Also, dynamic programming has the additional flexibility of modeling behavioral responses to environmentalfluctuationsand uncertainty. Integrating the results of actions with qualitatively different effects on fitness into a common currency allows the easy comparison of different behavioural alternatives. Dynamic programming can easily handle difficult constraints like integer or discrete sets (Bertsekas 1987). In the calculation procedure, for each time step and each state value, the optimal action and the correspondingfitnessvalue are calculated. This makes the globally optimal solution readily available. Theflexibilityof dynamic programming, however, may create problems. In an attempt to try modeling nature's intricacies, researchers often devise complex models that are incomprehensible. Unfortunately, it is difficult to prevent people from abusing powerful scientific techniques (Clark 1989). As the dimension of the state space increases, the solutions, both numerical and analytical, of the dynamic optimization problems increase in difficulty. Called the curse of dimensionality, it is an inherent feature of all dynamic optimization problems (Bellman 1957; Mangel and Clark 1988; Houston and McNamara 1988). For example, in the algorithm given in (3.32), thefitnessis calculated for all state variable values and all times. Thus, assuming that each variable has the same number of possible values, the procedure requires an order of T • A • V  N  calculations and V  N  storage. Here T is the  number of time steps, A is the number of actions, V is the number of state variables and N is the number of possible values. Thus, as the number of state variables, V, increases, the number of calculations and storage requirements grows very rapidly (Mangel and Clark 1988). Gradient-based optimization methods allow the use of more state variables than dynamic programming. However, the use of gradient methods entails several disadvantages.  Chapter 3. Age of sexual reproduction in B. schlosseri  47  The major limitation is that the methods do not give feedback control that can incorporate observations of random events as they become available. Also, the optimization model must be computed repeatedly for each different set of initial values. In comparison, with dynamic programming a single computation of the model gives the optimal policies for a range of initial values of the state variables and for all possible values of the random variables, such as the weather. Also, dynamic programming models incorporate fixed costs directly without extra computation (Shoemaker 1981). A potentially limiting factor in dynamic programming is the use of discrete time steps. Note that only a single state transition and a single behavioural decision occurs in each time unit. Nonetheless, even in cases where state transitions occur at random times, it is usually possible to use discrete-time dynamic programming by choosing a sufficiently short time unit (Mangel and Clark 1988). However, the number of calculations increases linearly with T so computational time limitations might limit the reduction of the time step. Often, models that gain realism by adding more parameters or interactions between parameters (dynamic programming does both) end up telling us more and more about less and less. In other words, they become less general (comment by E. Smith in Houston and McNamara 1988, p.150). Nonetheless, any principles revealed by the model can be checked for generality against a variety of data or even with different models.  3.5  3.5.1  The Model  Background Information and Assumptions  In all the previous optimization models listed, the implicit assumption of only a single mode of reproduction is often violated by clonal organisms (Caswell 1985). Most clonal organisms reproduce both sexually and asexually. This raises the question of whether  Chapter 3. Age of sexual reproduction in B. schlosseri  48  the basic unit of selection is the single zooid or the whole colony. Asexual reproduction ensures a common genetic ancestry among all the zooids. It thus acts like growth in nonclonal organisms. Apart from somatic mutations and extrachromosomal factors, all zooids are identical genetically. Urbanek (1973) states that it is widely accepted that somatic mutations have rather minor evolutionary value. Also, mutations do not usually result in instantaneous speciation "because the gametes carrying the new mutation are not reproductively isolated from those carrying the 'parental' growth gene" (Mayr 1963; 1970 p. 251 cited in Urbanek 1973). Also, Boardman (1973) asserts that somatic mutations are probably of negligible phenotypic expression in most groups. Thus, I will ignore the effects of somatic mutations in the model. Botryllus schlosseri rarely propagate clonally. Grosberg (1982, 1988) only saw two colonies fragment among more than 3,000 studied. Thus, the vast majority of colonies start from a sexually produced larva. The basic unit of selection would therefore be the colony. As such, the fitness of a colonial organism is measured, as with non-colonial organisms, by the relative numbers of descendent colonies (or genes) that it contributes to future generations (Harper 1985). Although the age of first sexual reproduction is determined by a complex interaction of many factors, internal and external, the model will consider only the morph type, size, age, and time, since those seem to be the most important factors (Grosberg 1982, 1988). The value of offspring depends on the age of reproduction. For example, early reproduction in a growing population leads to a proportionately greater contribution to population growth (Lewontin 1965 (cited in Grosberg 1982)). Although the population of Botryllus schlosseri is not on average growing over the years, the iteroparous and semelparous morphs have seasonally specific periods of population growth and decline. However, the standard dynamic programming model (see eq. 3.4) considers the fitness being the total expected lifetime reproductive output. In the model, I equatefitnessto  Chapter 3. Age of sexual reproduction in B. schlosseri  49  something similar to Fisher's (1930) reproductive value; I assume that the most fit organism leaves the largest number of descendents at time T in the limiting case as T —• oo. However, this definition is not appropriate in stochastic environments (e.g., Cohen 1966; Stearns 1977; Mangel and Clark 1988; Yoshimura and Clark 1991). I have no data or evidence for significant yearly changes at Eel Pond. Thus, apart from seasonal changes, the model ignores environmental stochasticity. Section (2.1) describes the synchronization of the asexual and sexual cycles. Thus, all times are given in units of generations. As described in section (2.3), the growth rate until sexual reproduction is approximately exponential. However, for both iteroparous and semelparous colonies, the growth rate decreases several generations before sexual reproduction (Fig. 3.5 - for iteroparous colonies, this decrease is most noticeable in the 1 August graph). The model assumes that the colony stores energy for reproduction during the generation just before reproduction, resulting in no growth then. Also, as described in section (2.3), Brunetti and Copello (1978) found that the growth rate fell in later generations until it stabilized around a constant value. Although some, or even all, of this growth decrease could result from production of sexual organs, the model allows the growth rate to change at a fixed age, and then drop at a fixed rate for several generations until it reaches 1.0 at a defined maximum size. The length of a generation varies inversely with the water temperature (Fig. 3.6). At Woods Hole, over the reproductive period, the water temperature roughly increases linearly until August and then linearly decreases (Grosberg, personal communication). Fig. 3.7 shows a May to mid-November growing season. Thus, I assumed the approximate temperature and generation length schedule shown in Table 3.2, resulting in the dates for each generation given in Table 3.3. Mortality depends on many factors, with the most important ones for Botryllus  Chapter 3. Age of sexual reproduction in B. schlosseri lOOOr  1000  2  4  6  8  10 12 14 16 18 2 0 22 24  ASEXUAL  50  13 August  2  4  6  8  10 12 14 16 18 2 0 22 2 4  GENERATION  Figure 3.5: Growth trajectories of eight cohorts (each composed of 10 colonies). The solid circles show trajectories for semelparous colonies; the open circles show trajectories for iteroparous colonies. The caption and figure are taken from Fig. 3, Grosberg 1988.  Chapter 3. Age of sexual reproduction in B . schlosseri  51  R  CVJ  Lo  CM  o o  LLi  cr  I  . o tr LLI  CL  o  O ro  o  CsJ  o  N0llVd3N39 "IVnX3SV U3d SAVQ Figure 3.6: The relationship between the mean duration of an asexual generation and temperature. The error bars denote ranges. The caption and figure are taken from Fig. 2, Grosberg 1982.  Chapter 3. Age of sexual reproduction in B . schlosseri  52  r <  3±Vld/IN3fl31±13S AllVQ NV3*N  Figure 3.7: The seasonal mean daily settlement of B. schlosseri per 100 c m plate. The caption and figure are taken from Fig. 5, Grosberg 1982. 2  Chapter 3. Age of sexual reproduction in B. schlosseri  53  Table 3.2: Temperature and generation length schedule. Month Temp. (°C) May 10 June 15 July 20 Aug. 25 Sep. 20 Oct. 15 Nov. 10  Gen. Length (days) 18 9 6 5 6 9 18  schlosseri being size, age, time, and sexual reproduction. The effect of sexual reproduction on mortality in iteroparous colonies is ignored since I don't have any data to use. I assume that the other three factors act independently, so that total survival is the product of survival from each of these factors. So that time-dependent mortality would be non-negative, I used the mortality schedule at the time of maximum survivorship to define the time dependent mortality. For iteroparous colonies, this is the time period September 1-15 while for semelparous colonies it is September 16-30 (Fig. 3.4). Since per zooid mortality is almost constant, mortality due to all zooids dying from chance events decreases almost exponentially with increasing size (Grosberg, personal communication; Jackson 1985; Harvell and Grosberg 1988). However, large, sexually mature colonies also have a large mortality due to an age effect. Table 2.1 gives the life-history table for iteroparous morphs. The semelparous colonies' life table up to sexual maturity can be obtained from Fig. 3.4. Determining the optimal age of first sexual reproduction requires knowledge of the mortality if the colony does not reproduce at its usual age. However, that information is not known for semelparous colonies since they invariably die after a single reproductive episode. I assume that mortality in older colonies is due to a cumulative effect of metabolism. This is implied for several reasons: (a) high growth rates in iteroparous colonies are correlated with fewer clutches being produced (Grosberg  Chapter 3. Age of sexual reproduction in B. schlosseri  Table 3.3: Approximate dates for the end of each generation. Gen. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  Date End May 18 June 5 June 14 June 23 June 29 July 5 July 11 July 17 July 23 July 29 August 3 August 8 August 13 August 18 August 23 August 28 September 2 September 8 September 14 September 20 September 26 October 2 October 11 October 20 October 29 November 15  54  Chapter 3. Age of sexual reproduction in B. schlosseri  55  1982); (b) the mortality increase in sexually mature colonies is imputed to weakening from abnormally elevated metabolic activity (Brunetti 1974); (c) as shown in Fig. 3.5, overwintering colonies, with a low metabolic rate in early spring, live longer than colonies that do not overwinter; (d) Starving colonies grow more slowly and live longer (Grosberg 1982). Since semelparous colonies grow at almost double the rate of iteroparous colonies, and semelparous colonies first reproduce at about age 6, for age 6 on, I used the combination of 2 iteroparous age-dependent mortality values, starting at age 9, the usual minimum age for sexual reproduction. The numbers are undoubtedly inaccurate, but I believe that the idea of increasing mortality is correct so the modeling results will give useful qualitative information. Mortality of both iteroparous and semelparous planktonic larvae is unknown (Grosberg, personal communication). Thus, without anything else to use, I assume that planktonic survivorship is proportional to the corresponding survival to sexual maturity (Fig. 3.4), with the proportionality constant being such that r « 0. After overwintering, colonies have different mortality and growth patterns. First, about 90% of semelparous and iteroparous colonies survive the winter (Grosberg 1982, 1988). Also, the growth rate is less than one for about 4 generations and then it is slightly greater than one for about 3 generations before the growth rate returns to approximately the pre-winter amount (see Fig. 3.5). A lack of food causes this early spring growth retardation (Grosberg, personal communication). From the Fig. 3.5 data for the September to October cohorts for the first 4 generations after winter, I noticed that, for both semelparous and iteroparous colonies, the rate of size decrease (i.e. size of colony at time t — 1 divided by the size of the colony at time t) is very nearly a linear function of the logarithm of the size of the colony. In particular, I found that y « .167a: + .45 where y is the rate of size decrease and x is the natural logarithm of the colony size in number of zooids. I don't know any theoretical reason why a relationship of this nature should hold, but qualitatively it makes sense since larger colonies (larger x) will, under  Chapter 3. Age of sexual reproduction in B. schlosseri  56  starvation conditions, be able to obtain more food from the meager supply passing by, but less per zooid. Larger colonies decreasing in size more quickly than smaller colonies means that all the colonies' sizes tend towards the equilibrium size (i.e. y = 1). Grosberg (personal communication) has said that after winter, colonies "reset" to the same size basically regardless of age. Although the data in Fig. 3.5 only covers roughly the range of 50 to 400 zooids, without other information, I extrapolated the size reduction equation to all colony sizes. This means that the equilibrium point, y = 1, occurs for a colony of about 27 zooids. Thus, if the equation were to also apply to colonies started during the early spring, then their growth rates, especially for semelparous colonies, would be severely curtailed. However, Fig. 3.5 and 2.3 shows that the growth rate is near normal, although Fig. 2.3 shows that a portion of the June 1-15 colonies had average growth rates (buds/zooid) between 0.5 and 1.5. I can only assume that somehow, such as through good location, the young colonies can get enough to eat. For the fifth to seventh generations of slow growth, I assume, based on the Oct. 1 cohort data in Fig. 3.5, that the growth rate increases by .3 compared to that predicted for the same size in the first four generations. As mentioned previously, I assume that the age dependent mortality depends on the cumulative metabolic rate. Since overwintering colonies live about 6 or 7 generations longer (Grosberg 1982 (Table 5)) and there is about a 7 generation period when colony size is decreasing or not increasing much, I assume that the effective age for age dependent mortality increases in proportion to the metabolic amount over the maintenance amount. Although Grosberg (1982) states that iteroparous colonies continue to sexually reproduce every generation once they begin, this is not necessarily true after overwintering (Grosberg, personal communication). Thus, the model does not require that iteroparous colonies continue reproducing once they start on the likelihood that, apart from after winter, it is optimal for them to do so anyway.  Chapter 3. Age of sexual reproduction in B. schlosseri  57  As stated earlier, the model assumes that there is no growth during the generation before reproduction as the colony stores energy. For overwintering colonies, although they are starving, there is no significant decrease in fecundity (Grosberg 1982 (Table 5)). Instead, the model assumes, on energetic considerations, that the growth rate during a generation of reproduction would be the non-reproduction growth rate divided by the normal growth rate. This would be consistent with the growth stoppage before reproduction in non-overwintering colonies. 3.5.2  Equation and Parameter Values  As stated earlier, the purpose of the model is to predict the optimal age of first sexual reproduction. The dynamic programming technique assumes that this age depends on maximizing the "fitness" F given by: F(s,a,t)  = [1 — m(s,a ,t)]ma,x{F(si,a e  + l,t ), n  rF(s , a + l,t ) + 0ksp{t )F(l, 0, *„)}. 2  n  n  (3.33)  where the first maximand corresponds to the decision "grow" and the second to "reproduce". The variables and parameters are described on the following pages.  Chapter 3. Age of sexual reproduction in B. schlosseri  58  Variable  Interpretation  s  Size of colony in number of zooids.  a  Age of colony in number of asexual generations.  t  Time of the year in asexual generations.  F(s,a,t)  The maximum discounted expected future reproduction of a colony of size 5, age a and at time t. Thus, F(1,0, t ) is the maximum discounted n  expected future reproduction of a new colony at time t . n  m(s, a , t) e  The mortality of the colonies given by the relation 1 - m(s,a ,t) = (1 - m (s))(l - m (a ))(l - m (t)) e  s  a  e  t  (see below and parameter table). a  The effective age. As discussed previously, I assume that this depends  e  on the cumulative metabolic rate above maintenance level. Thus, if the colony has positive growth potential at the current time t, then for the next generation, a increases by (growth potential for size s at time t) e  I (maximum growth potential for size 5 at any time). With negative growth potential, a remains constant. In the computer program, a is e  e  calculated from the colony size at each time step from the colony's start Si,s  2  to the current time. Size of the colony, in number of zooids, if the colony does not or does reproduce (respectively) (see [1] below).  t  The next asexual generation. If t < T (see parameter table), then  n  y  t = t + 1, else t = 1. n  p(t ) n  n  The time dependent survivorship factor of the planktonic larvae. Since I have no data for this, I assume that it is equivalent to 1 — m (t) (see t  parameter table below).  Chapter 3. Age of sexual reproduction in B. schlosseri  Parameter Interpretation  Assumed Values  r  0 (semelparous)  Colony type  1 (iteroparous) Number of embryos per zooid (average 10.10 (semelparous)  j3  from June 15 to Sep. 15 data in Table 3.24 (iteroparous) 5, Grosberg 1982) The basic survival factor of planktonic .001 and .002  k  larvae T  The number of asexual generations in a 26  m (s)  year Size dependent mortality (see [2])  y  s  .75  .25  .01 0 0  0 0 0 0 0 (for semelparous colonies) .61  .37  .13  .02 0  0  0  0  0 0  0  0  0  0 0  0 0 (for iteroparous colonies) m (a ) a  e  Age dependent mortality (see [3])  0  0  0  0  0  0 .09 .40 .81 1.0 (for semelparous colonies) 0  0  0  0  0  0  0  0  0  0  .03  .07  .07  .36 .52  .60 1.0 (for iteroparous colonies)  Chapter 3. Age of sexual reproduction in B. schlosseri  m (t) t  Time dependent mortality (see [4])  60  .02  .02  .02  .03  .04  .05  .07  .11  .15  .17  .18  .18  .18  .22  .26  .22  .18  .13  .08  .03  0  0  0  0  0  0 (for semelparous colonies) .12  .14  .16  .18  .16  .13  .10  .08  .05  .04  .04  .01  .01  0  0  0  0  0  0  0  0  0  0  0  0  0 (for iteroparous colonies) Notes: [1] For times of normal growth (generations 8-26 for all colonies and generations 1-7 for colonies started the same year), s = a s . For overwintering colonies during the first x  four generations, as discussed earlier, y = .167a; + .45 where y = ^- and x = In5. This gives si = s(.1671ns + .45). For generations 5 to 7, the growth rate increases by .3 so we have J =  M  7  ^  a  +  M  +  -3 which gives  3  l  = s  (.osffiffiiss)-  I n  a 1 1 c a s e s  > * = 5  [2] I used the assumption that size dependent mortality decreases exponentially with size. The values listed were obtained from using a = 2.03 and a survival rate of .207 to sexual maturity (Grosberg 1988) for the iteroparous colonies and a = 3.88 and a survival rate of .218 to sexual maturity (Grosberg 1988) for the semelparous colonies. The values given are for the size indices as described in the "Methods" section below.  Chapter 3. Age of sexual reproduction in B. schlosseri  61  [3] Table 2.1 was used to determine the per generation age dependent mortality for iteroparous colonies. Each semelparous age dependent mortality was obtained by combining 2 iteroparous age dependent mortality values on the assumption, as discussed earlier, that the age dependent mortality is due to cumulative metabolic activity with that of semelparous colonies being roughly double that of iteroparous colonies. [4] The mortality to the age of first sexual reproduction relative to its maximum value, as given in Fig. 3.4, is used to determine the time dependent mortality. This mortality is assumed to occur equally over 9 generations for iteroparous colonies and over 6 generations for semelparous colonies. The values for the generations are interpolated from the cohort values given in Fig. 3.4. Equation Analysis  Equation (3.33) has two important features. First, all terms contain the function F. Also, unlike the standard dynamic programming equation (3.27), no final time T is used. Instead, (3.33) uses the implicit limiting behaviour as T —+ oo. But, unless the year-to-year growth rate is exactly 1, limr-xx, F(s,a,t;T) = 0 or oo. Assuming a stable age and size distribution exists (at least on a year-to-year basis), then for T —> oo, the distribution of progeny of a colony of any size and age will approach the stable age and size distribution. The only possible difference among different colonies will be the relative numbers of progeny in each class. In other words, limx-^oo F(s, a, t; T)/F(l, 0, t; T) = L(s,a,t) for some finite function L. Thus, dividing (3.33) by F(1,0,£„) results in: F'(s,a,t)  =  [1 - m(s,a ,t)]max{F'(s ,a + l,t ), e  rF'(s ,a + 2  where F'faa^)  = F(s,a,U)/F(l,0,t ) n  1  l,t )-rPksp(t )} n  n  for h = t,t . n  n  (3.34)  Now, as T -> oo, F' -* L. For  simplicity, I have dropped the prime in the rest of the paper.  Chapter 3. Age of sexual reproduction in B. schlosseri  62  Approximations and Correlations With Previous Work Equation (3.34) is in general too complex for analytic study and must be solved using a computer. However, knowledge about the behaviour of the equations can be gained from studying several simplifications to the equation. First, ignore the after winter size change, time dependence in mortality, and assume a = a. Thus, size can be determined e  from the age so let m (a)m (s) = m(a). This gives (from (3.34)): a  s  F(s,a,t)  [1 — m(a)] max{F(5i, a + 1, t ),  =  n  rF(s ,a  + l,t )  2  n  + pks}.  (3.35)  Assume that the organism first reproduces from some age a to its maximum age A x  (a = A for semelparous colonies). Let p (a) = 1 — m(a). Then the reproductive value x  a  of the organism at age a will be x  F(s,a ,t) x  = ^fcsp (ai){l+p (ai + l)4-p (oi +1)-p (ai + 2) a  0  0  a  + ... + p (a + 1) • p (ax + 2) • • • (A)} a  Since l = nf=iPo(0> t x  n e  x  a  Pa  (3.36)  equation can be rewritten as: A  fiks^2l /l .  F(s,a ,t) = 1  i  ai 1  (3.37)  which is basically the same as Fisher's equation for reproductive value in discrete form (3.10). If the organism does not reproduce at time a but instead grows (at exponential x  rate a) and then reproduces at the next time step, the result is: A  F(x,a t) u  = f3ksap (a ) a  x  £  U/l  ai  (3.38)  t'=ai+l  The age of first sexual reproduction is the maximum a for which the right hand side x  of (3.37) is greater than that of (3.38). Thus, for the iteroparous morph, the resulting  Chapter 3. Age of sexual reproduction in B. schlosseri  !>(<*-!)  63  (3.39)  £ /,//, 1=0!+!  while for the semelparous colonies it becomes: 1 >  (3.40)  ap (ai) a  This shows, interestingly, that the value of 0k does not seem to have an influence. Using the m and m values given in the previous section, for iteroparous colonies with a = 2.03, a  s  (3.39) gives a = 12, while for semelparous colonies with a = 3.88, (3.40) gives a\ = 10. x  The actual values are 11.5 and 6.2 as given in the 2x and 3x feeding levels in Table 4 of Grosberg (1988). While the iteroparous value is quite reasonable, the semelparous value is too high. Equation (3.40) requires that p {ai) < 1/a which means that the mortality a  must become high at about 7 generations. However, this is only an approximation, and there are other considerations, so I will continue using my current assumptions since I don't know the actual mortality values. Consider another approximation. As before, ignore the time dependent mortality.  Let T be some large time to consider and let t be the current time.  n = [F(l,Q,t;T)] ^ ~ ^ 1  T  t  Define  = the geometric average reproduction rate of the organism per  unit time. Assume that this rate applies for all times t. Finally, assume that at age a, s = a . If the organism first reproduces at age a = a , then from equation (3.34) we a  x  have: A  (3.41) Using TJ gives A  (3.42) Dividing both sides by nJ-A-t  gives A  (3.43)  Chapter 3. Age of sexual reproduction in B.  schlosseri  64  T h i s i s a p o l y n o m i a l o f degree A i n 77 w i t h t h e o p t i m a l age o f r e p r o d u c t i o n b e i n g t h e v a l u e o f di t h a t o p t i m i z e s 77. U n l i k e (3.39), t h e v a l u e o f f3k does affect t h e r e s u l t . C o n s i d e r t h e s p e c i a l case o f o n l y o n e r e p r o d u c t i o n ( s e m e l p a r o u s ) . T h e n (3.43) s i m p l i f i e s t o  r) = a(f3kl y' K a  (3.44)  ai  T h u s , t h e v a l u e o f a i s j u s t a s c a l i n g f a c t o r f o r t h e r e p r o d u c t i v e v a l u e a n d does n o t affect the decision o f w h e n t o reproduce!  F o r i t e r o p a r o u s c o l o n i e s w i t h a = 2.03, t h e a g e o f  first s e x u a l r e p r o d u c t i o n i s 13 f o r k — .001 a n d .002. F o r s e m e l p a r o u s c o l o n i e s t h e v a l u e is 8, also f o r b o t h k values. N o t e t h a t f o r b o t h i t e r o p a r o u s a n d s e m e l p a r o u s m o r p h s , a f a c t o r o f 2 c h a n g e i n k does n o t c h a n g e t h e o p t i m a l age o f first s e x u a l r e p r o d u c t i o n , so k does n o t a p p e a r t o b e a n i m p o r t a n t c o n s i d e r a t i o n . A l s o , a l t h o u g h t h e s e m e l p a r o u s v a l u e is c l o s e r t o 6.2 c o m p a r e d t o t h e first s i m p l i f i c a t i o n , i t is s t i l l s i g n i f i c a n t l y l a r g e r .  Methods S i n c e i t e r o p a r o u s c o l o n i e s c a n c o n t a i n u p t o 1000 z o o i d s , t h e c o m p u t e r m e m o r y r e q u i r e d to s o l v e (3.34) d i r e c t l y ( s e v e r a l m e g a b y t e s ) i s t o o l a r g e t o i m p l e m e n t o n m o s t m i c r o c o m p u t e r s . I n s t e a d , s i n c e f o r m o s t of t h e t i m e t h e g r o w t h i s e x p o n e n t i a l a t t h e c o n s t a n t r a t e a , i n (3.34), t h e s i n F is r e p l a c e d b y x g i v i n g  F"(x,a,t)  =  [1 —  m(x, a , t)] e  rF"(x2,a w h e r e x, x\, x  2  max{F"(xi,  a+  1,  t ), n  + l,tn) + Pksp(tn)}  (3.45)  a r e i n d i c a t o r s o f t h e size e q u i v a l e n t t o t h e a g e o f a s i m i l a r l y s i z e d c o l o n y  t h a t h a s n o t r e p r o d u c e d s e x u a l l y o r gone t h r o u g h g r o w t h p h a s e a t r a t e a, w e h a v e s = a  x  a winter.  Thus, during the initial  o r , e q u i v a l e n t l y , x = l o g s. S i n c e o n l y a  fitness  v a l u e s c o r r e s p o n d i n g t o i n t e g r a l values o f x are s t o r e d , o n l y 2 4 v a l u e s o f x a r e r e q u i r e d . F o r c o n v e n i e n c e , I d r o p t h e d o u b l e p r i m e t h r o u g h o u t t h e rest o f t h e thesis.  Chapter 3. Age of sexual reproduction in B. schlosseri  65  Equation (3.45) is solved using an iterative technique. Initial values are assigned to F(x, a, T ) for all x,a. Then, (3.45) is used repeatedly to solve for the previous y  time steps. After each time step is solved, the fitness values are scaled (i.e. divided by F(0,0,t)). When the beginning of the year (t = 1) is reached, during the next iteration the end of the previous year (t — T ) is calculated. After a full year backwards is y  calculated, the values at time t are compared to the values at the same time i, but for the next year. Since the model does not account for year-to-year fluctuations in the environment, at steady state the scaled fitness values on a year to year basis should be the same. Of course, steady state will never be reached in the model, so the model assumes that steady state is effectively reached when the absolute relative difference in the fitness values from one year to the next are all less than a small value (e) and the decision (to reproduce or not to reproduce) does not change between the years. However, convergence is not very fast due to oscillatory behaviour in the fitness values. I used a technique, called "relaxation", to speed convergence. Relaxation, often used influiddynamics equations (e.g. see Mitchell & Griffiths (1980)), is a form of linear interpolation between the previous and current values. If f' is the nth relaxed value n  obtained, and /  n + 1  is the value obtained from the iteration equations using the value f' , n  then we have / ;  +  i = £ + « ( / n + i - / i ) .  (3.46)  where u> is the relaxation parameter. For u > 1, it is called over-relaxation, for u < 1, it is called under-relaxation, and for UJ = 1, it is equivalent to not using relaxation at all. In overwintering organisms, the decrease and then increase in size results in nonintegral values of x. Since I only store fitness values corresponding to integral values of x and the growth is exponential, I use exponential interpolation. For example, let x < g (the size index when the growth rate is reduced). Also, let x,- = int(x) and xj = x — a;,-.  Chapter 3. Age of sexual reproduction in B. schlosseri  Then, F(x,a,t) = F(x a,t) - iF( l  u  x  Xi  66  + l,a,t) f. x  F i e l d D a t a for Comparison Fig. 2.3 shows the frequency distributions of various life-history parameters over the year. Although the data are not separated on a semelparous/iteroparous basis, there is a definite dichotomy in the number of embryos per zooid with no colonies having between 4.0 and 8.0 embryos per zooid. I assume that the colonies with less than 4.0 embryos per zooid are iteroparous and those with more than 8.0 are semelparous. Since Fig. 2.2 shows that in general semelparous and iteroparous have non-overlapping life history parameters, I assumed that the lowest ages of first sexual reproduction are semelparous and the highest ones are iteroparous. Thus, I calculated the approximate frequency distributions as given in Fig. 3.8 with the data summarized in Fig. 3.9. Several patterns become evident. First, the age of first sexual reproduction rises in the July cohort (about generation 8) for iteroparous colonies and in August (about generation 13) for semelparous colonies. Secondly, the age of first sexual reproduction decreases for both morphs in early September (about generation 17). Finally, in late September (about generation 20), the age of first sexual reproduction greatly increases, also for both morphs. Another possible check on the modeling results is correlation with the larval settlement rate (Fig. 3.7). As shown, little settlement occurs at the beginning (May) and end (November) of the growing season. Instead of smooth changes, definite peaks occur. Over the two years of data, these peaks seem to occur at about the same times: mid to late June, late July, late August, early September, and mid October. Also, most of the peaks are about the same size except for the one in late July which is at least 4 or 5 times as large. Unfortunately, the morph specific larval settlement is not given. However, the increase in the proportion of iteroparous colonies in late July (Fig. 3.8) indicates that a lot of the large peak then is likely due to iteroparous settlement.  67  Chapter 3. Age of sexual reproduction in B. schlosseri  Type of Morph uo  (a)  June 1-15  JI  00 UJ  June 16-30  Semelparous 1st Reproduction Age  Iteroparous 1st Reproduction Age  \  iH i i m  08  i l itl  00  r  10  (c) July 1-15  as ao 10  J  1  t < / / C  (dD  Jury 16-31 \  oo  n  IJO  (e) Aug 1-15  as  VP  Il II  ao  JI  (f) Aug. 16-31  as  ao J  1  _01  1JD,  Sep. 1-15  as ao J J Z L  -IZL.  J 2 l  10,  (a) Sep. 16-30 oo J  I  L  r  s  -» s,  j  ^ ^  S -b  'i.  ^  Figure 3.8: Frequency distributions of age at first sexual reproduction for semelparous and iteroparous colonies. This data is adapted from Fig. 2.3. Hashed bars represent data from 1979 cohorts; open bars represent data from 1980 cohorts.  Chapter 3. Age of sexual reproduction in B. schlosseri  10  68  20 T I M E  (a) Iteroparous field data statistical summaries.  (b) Semelparous field data statistical summaries. Figure 3.9: Field data summaries of Fig. 3.8. The x-axis is the time of year that the colonies began. The central line is the mean, the vertical lines are the error bars and the outside dashed lines are the upper and lower bounds.  Chapter 3. Age of sexual reproduction in B. schlosseri  69  Variable Model Parameters In the model, I assume that the value of k is such that the intrinsic growth rate of the population, r, « 0. For the iteroparous morph, Table 2.1 gives the September 1-15 period survivorship and fecundity figures. Fig. 3.8 shows that for that period about 40% started sexual reproduction during generations 9 and 10, 40% started during generations 11 and 12, and finally about 20% started sexual reproduction during generation 13. Thus, in each later generation from 9 to 13, roughly an additional 20% of remaining start sexual reproduction. Therefore, we have [.207(1596)(.2) + .201(1932)(.4) + .186(2318)(.6) + .173(1957)(.8) + .110(1843) + .053(1639) + .021(1704)]*; « 1 which gives k « .0012. For the semelparous morph, from Grosberg (1988), the maximal survival is about .218 with a mean fecundity of 2267 embryos. Thus, (.218)(2267)Jb « 1 which gives k « .0020. This is about 70% higher than the estimated k for the iteroparous morph. Grosberg (1982) states that no reason is evident to believe that larval behaviour differs between semelparous and iteroparous larvae so the k values should be roughly the same. Although the semelparous morph value is likely more accurate, in case the iteroparous morph's k value is lower, I modeled k = .001 and .002 to examine the effect. I modeled 18 cases for both k values and for both semelparous and iteroparous morphs for a total of 72 cases, with the results given in Fig. 3.10-3.21. The 18 cases examine the influence of the following factors: growth rate decrease after winter, growth rate, maximum size, and different numbers of generations at the maximum growth rate and to reach the maximum size. From Grosberg (1982) (Table 5), for iteroparous colonies, the growth rate ranged from 1.90 to 2.14 with an average of 2.03 buds/zooid while for semelparous colonies, the growth rate ranged from 3.23 to 4.16 with an average of 3.88 buds/zooid. The cases cover the minimum, average, and maximum values of a for both morphs. Grosberg (1982) states that semelparous morphs normally only grow to a  Chapter 3. Age of sexual reproduction in B. schlosseri  70  maximum of about 250 zooids and iteroparous to about 800 before sexually reproducing. I used these values as the low maximum size. Fig. 3.5 shows that some semelparous colonies can reach sizes of about 500 zooids and iteroparous morphs can reach sizes of about 1000 zooids. I used these as the medium maximum values. Finally, Fig. 3.5 shows that for the July 15 cohort, a couple of semelparous colonies reached sizes of at least 800 zooids. In case iteroparous colonies do not reach their absolute maximum size even at 1000, I tried using 1300 zooids. These values (800 and 1300) were used as the high maximum size. From Fig. 3.5, the growth rate levels off, on average, at about 200 zooids for semelparous colonies and at about 500 zooids for iteroparous colonies. Thus, the age at which the growth rate changed was calculated as that which was closest to those sizes. The number of generations over which the growth rate linearly decreases to 1.0 is loosely based on reaching maximum size at the latest age that sexual reproduction first occurs. 3.5.3  T h e M o d e l Cases  The 18 cases are summarized below: Cases  Description  (a)-(c)  No after winter growth decrease  (d)-(f)  As (a)-(c) but with after winter growth decrease  (g)-(i)  As (d)-(f) but with low a and varying max. size  G)-(i)  As (g)-(i) but with medium a  (m)-(o)  As (j)-(l) but with a different # of generations for growth rate decrease  (P)-W  As (m)-(o) but with high a  Each case is presented in the figures as 2 superimposed graphs with 8 variables below presented numerically and graphically. On the graphs, the y-axis is age in asexual generations and the x-axis is the time of year, in asexual generations, that the colony started.  Chapter 3. Age of sexual reproduction in B . schlosseri 20  20  15  16 -  71  I  i  / j 10  (cD  (a) 10  Mttv19 Clwigs-rui  20  20  10  30  Qrowttr1D7 K/t\ a - 2 2 1 0 A M 8/-2210 Mn. a - 2 2 1 0  30  Grewm-0011 Mh 1 - 1 6 8  umt-tu.  An&a-ieeo Kta. ai-4210  20  I  - 15  1  j 10 -  j i / j ?\ V-...J....J  '  (G)  10  NurWma  Acta-2.14  20 Oowtt>-274 N*\ a-288 A m a-OSOO Max 1-41000  Alpta-203 Umrt-rta  Qrowth-207 M\ a-9220 AMI 1-10200 Max. a-19700  Attfa-2.14 UMHUL CflQflQS^lA.  Qrow(h-a28 M h 1-208 Awa-6620 k k i U-S0400  Figure 3.10: Iteroparous colony first sexual reproduction age cases (a) to (f) with k = .001 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has no winter effect while the right side has a winter effect, a values increase going down the page.  72  Chapter 3. Age of sexual reproduction in B . schlosseri i  20 -  1  16 -  -  20-  -  16 -  -  10-  -  5 -  30  0 -  1  -  i f\  10 -  i ,  ji  j—•*  5(a) 0 -  It  J""-"  -  <d) I 10  Urfl-na.  1 20 Mh a-oeee A A 8*3888 Max 5.-3688  Nurtoama.  •  " T  10 Urrttna. Oargama.  1  20  30  Orowtfi-283 Mh 8.-1136 Am.a-2630 Mu. a-43670  20  16 -  —  10 -  10  20  Umit-na. GlanQB-na Nuroar-T\a.  Mn a-13200 Ave a-132O0 Max a-13200  ,»W«-418  Qrowth-2110 Mh s-21000 * « (.-21800 Mu. a-21000  30 A**»-3J38 Uirtt-na. CtanQB-oa. Nuntw-na  Atoha-4YIB  UMMUL  CMafrjo-'ia.  QPOW*I-120  Mh «.-132O0 A*a a-20100 MU. J.-614C0  Mh 8.-21600 Atm. «-44C00 Mu. v-80700  Figure 3.11: Semelparous colony first sexual reproduction age cases (a) to (f) with k = .001 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has no winter effect while the right side has a winter effect, a values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri  73  20  fy  j'-  16 ""1  -  /  rj""  ^  10  '  <J> 10  20  AW»-2£H  Qrow#rO209 Mha.-288 Am«-394 Max. a-«31  umt-eoo  Oanga-e  Alpha-US Urrtt-1000 Olaiy-9 NUTtW-6  AW*-1.9 LMM300 Oarga-0 NurtM(-«  30  Afeha-203 Uirtt-1000 CtwQO 6 1  ' I  !  I  I  t»»1h-0242 k*» a-323 Ava. a.-488 Max.a-746  -1 : I  ! I  ]  Mpf»-2D3 Ulrtt-1300 CtWQB'O Hntm-6  I  1  Growth-0.410 Mh a-288 Ax» « ^ 7 0 Mix BHM52  Figure 3.12: Iteroparous colony first sexual reproduction age cases (g) to (1) with k = .001 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has a = 1.9 while the right side has or = 2.03. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri 20 -|  '  '  74  h 20 •  Figure 3.13: Semelparous colony first sexual reproduction age cases (g) to (1) with k = .001 (see sec. 3.5.3 for details). The z-axis is the time of year the colony started; the j/-axis is the age of first sexual reproduction. The left side has a = 3.23 while the right side has a = 3.88. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri  AW*-2JQ3  um-i300  Mh s-288 »*e *-*71 M u »-760  Urrtt-1300  Oaroe-7 Wrtw-7  75  GrowtfK>380 Mh 1.-203 A»o. «-623 Mu. 1-873  Figure 3.14: Iteroparous colony first sexual reproduction age cases (m) to (r) with A: = .001 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has a = 2.03 while the right side has a = 2.14. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri 20  76  20  Figure 3.15: Semelparous colony first sexual reproduction age cases (m) to (r) with k = .001 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has a = 3.88 while the right side has a = 4.16. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri  M(t*--[a IJW-na Chargma NurtMt>na.  '. I  AV«-214 LMMk Oargs^ia t±jrtm-na. I  I  I  _ I  I  Qmn9>-US  !  I  Qrcwttl-6.12 Mh a-2210 At* a-2210 Mu,*-2210  I  Mh S-G22Q A« M u a-0020  I  I  — ! * * 1  ' ' T  77  M(t*-19 Uirfl-tu. CtmgmnM. Mirtar-no. I  Aor«-Z14 Uirtt-na. CfwQa-na. Hnbrra  Z I  1  0nowtf>-24O Mh a-189 Ana, «-1579 Mu&-7Q00  I  GtowtfHSBfl Mh S.-2C6 1 /»*». a-*J7*0 M u 1-00400 I  , I  I  I  Figure 3.16: Iteroparous colony first sexual reproduction age cases (a) to (f) with A: = .002 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has no winter effect while the right side has a winter effect, a values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri 20  78  20  Akta-4.18  LMt-aai  Oowtft-20100 Mh 1-21600 A***-2i800 M u 1.-21000  Urrtt-ru. CtiarQB^*.  Qwtfv-eOCO Mh 1-21600 A M I a-40200 M u *-807O0  Figure 3.17: Semelparous colony first sexual reproduction age cases (a) to (f) with k = .002 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has no winter effect while the right side has a winter effect, a values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri 20 H  1  '  79  h 20  15  15  10-  -  10  -  5  (g)  10 *ena-UB  L»m-eoo  20 QrowtfrO.718 Mh a-68 AM. aH358 Mua-438  —r-  10  30 AWa-203 Unt-600  20  30  QnowW-1.023 Mh S.-142 Ami «-092 Mu.a-431  yy  1  !  If*""" Ij . :—-  :-:-.-i._.  v  ^»  /^  <D  10 LMt-1300 Cftanga Q  At»a-2j03 Urtt-1300 CfiarQa—6 l*nnb«r»«  20  30  Oow<h-1.ei2 Mh »-142  M u k-082  Figure 3.18: Iteroparous colony first sexual reproduction age cases (g) to (1) with k = .002 (see sec. 3.5.3 for details). The ar-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has a = 1.9 while the right side has a = 2.03. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri 20-|  Aljfca-323 Urtt-600 Ct«no»-4 NurtM-<3  Aiffm-323 Urtt-600 Charga 4 Nurbaf-a  "  1  _:  I  Q-owtn-.167 Mh a-070 Ata&-3S9 Max. a.-600  '— : I  Oowth-fl33 Mh s-«T7 Atm. *-«84 Max. a-600  ! ! I  -  I  —  1*1  80  h 20  1  ' ! \ '  I  1  Z ! : I  I  *_J  A*X»-3J38 Umrt-600 Oarxje-4 Nuntw-3  Afc*«-a88 Urrtr-800 Croroa-4 Htttm-d  " ! ! 1*1  I  !_ " _ 1*1  Qrwttv-488 Mh »-3C0 Ami 4-302 Max, a-600  Growth-273 Mn a-617 AK»«-097 I Max. a-800  I  ! ! " _ I  Figure 3.19: Semelparous colony first sexual reproduction age cases (g) to (1) with k = .002 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has a = 3.23 while the right side has a = 3.88. The maximum colony size limit values increase going down the page.  *  Chapter  3. Age of sexual reproduction  in B .  schlosseri  81  20  20 15 -  - 15  10  -  10  -  5  5-  <P)  (m) 10 AlchS-203  QrjwtfrQ87e  ChwxjB-8  Mh a-ee A M . OK316 Max S.-372  om-eoo Hjrtxn-7  AJpha-203 Urrtt-1000 CharxjB-a Nurtw7  Aleha-203 LMM300 Clurxja-a Nut*«-7  ! ii I  I  I  Qw*W3S77 Mh a-142 A*aa-370 Max, a-613  I  Growth-1.123 Mha-142 A»«a-4«a Max. a-680  *  :: — l  I  *  I  -r-  30  20  10  20  umt-eco Charga-7 MntxW  1  Alpta-2.14 Umt-1000 OerQir-7 Mffttr-7  I  Afcht-2.14 Uirfl-1300 Chnioa-7 Mn*ar-7  _! I  Z 'I  : _! I  'H  30  GRM»lh-aS07' Mh a-fl6 Am.«—379 Max. a-482  AtOfa-414  ! I  0mwtt»-1O86 Mn a-9B Avai a-468 Max a-^68  I  Qrowth-1.431 Mh a-SO Aw. 1-62B Max s-673  ! Z I  I  Z  ! L Z I  I  -  Figure 3.20: Iteroparous colony first sexual reproduction age cases (m) to (r) with k = .002 (see sec. 3.5.3 for details). The x-axis is the time of year the colony started; the y-axis is the age of first sexual reproduction. The left side has a = 2.03 while the right side has a = 2.14. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B . schlosseri 20 H  Figure 3.21: k = .002 (see y-axis is the side has a =  1  1  h  82  20  Semelparous colony first sexual reproduction age cases (m) to (r) with sec. 3.5.3 for details). The x-axis is the time of year the colony started; the age of first sexual reproduction. The left side has a = 3.88 while the right 4.16. The maximum colony size limit values increase going down the page.  Chapter 3. Age of sexual reproduction in B. schlosseri  83  The dashed line is the average age of first sexual reproduction from the field data (as given in more detail in Fig. 3.9) and the line with the dots is the model results. Below these graphs, on the left are four input variables and on the right are four output variables. Beside each variable is a box with a central vertical line and an asterisk. The vertical line designates the average or expected value for that variable while the asterisk denotes its actual value. The input variables use a linear scale while the output variables use a logarithmic scale with a factor of 10 difference between the central line and the edge of the box. If the value is outside this range, it is placed at the edge. The input variables are the base growth rate (a), the maximum size limit, the generation # at which the growth rate first changes, and finally the number of generations over which the growth rate decreases linearly to 1.0. For the output variables, the first one is the yearly colony population growth rate (expected value is of course 1.0). Once the age of first sexual reproduction was calculated, forward iteration was used to calculate the densities of colonies, and thus the yearly change in densities (i.e. the yearly population growth rate). The next three output variables are, respectively, the minimum, average, and maximum colony sizes over the year (the central line designates the expected average value of 695.6 zooids for iteroparous colonies and 224.7 zooids for semelparous colonies (Grosberg 1982)). All the computer runs used the small convergence criterion of e = .001 to try ensuring accurate modeling results. For iteroparous cases (a)-(c), I did not use relaxation. For semelparous cases (a)-(c), I used u = .9 since some cases did not converge otherwise. For semelparous case (g) with k = .002, the solution would not converge regardless of the value of u. Finally, I used u> = .8 which gave a maximum difference of .011 after 400 iterations. For all the remaining 65 cases, I used u> = .73 with all of them converging within 400 iterations.  Chapter 3. Age of sexual reproduction in B. schlosseri  84  A n a l y s i s of R e s u l t s  In the following analysis, when I talk about the age of colonies varying over the year, I mean the age of first sexual reproduction varying over the time of year that the colony started. Also, each new set of cases will discuss the differences with the previous set of cases. First, consider k = .001 (Fig. 3.10-3.15). Cases (a)-(c) show very little fluctuation in the age over the year. The rise, dip, rise from field data is not evident for any one of these. Also, the resulting larval settlement curves would be smooth, rather than have large peaks as shown in Fig. 3.7. The annual growth rate is way too high for all except Fig. 3.10(a). Finally, the colony sizes are all too large, especially for the semelparous morphs. Cases (d)-(f), with a growth rate reduction after winter, shows large variations over the year that correspond very roughly to field data. However, for the iteroparous / semelparous morphs, the July / August (respectively) peaks occur about 3-4 generations too soon (about 1 month). Also, the late September dip, then rise, is greatly exaggerated in the iteroparous morph and to a lesser extent in the semelparous morph. This is because the model predicts that colonies started over a fairly wide period in the late summer and early fall will reproduce in the early spring, which is not true. However, the main problem with this model is that allowing the growth to continue unabated results in too large colony sizes, especially for semelparous colonies. Cases (g)-(i), with a size limitation introduced, have ages of first sexual reproduction that correspond much more closely to field data.  With iteroparous colonies, as the  maximum size increases, the age of first sexual reproduction increases but the fall rise in age occurs earlier, up to 5 generations too early. The July peak can be roughly seen a few generations early in Fig. 3.11(h). Since only the first integral value of the age when  Chapter 3. Age of sexual reproduction in B. schlosseri  85  reproducing becomes favoured over not reproducing is shown, the "real" age when this switch occurs would be somewhat lower, Thus, the age shown is the "real" age truncated up. This tends to obscure some of the details of how this age changes over the year. The growth rates are too low and sizes too small, although case (i) has sizes that are close to that expected. For the semelparous colonies, in case (g), the ages follow thefielddata quite closely, with a slight rise around generation 8 and generation 15, with a dip around generation 20 followed by a sharp rise at generation 22. However, the sizes are somewhat low and the growth rate is way too low. Cases (h) and (i) show that, given the chance, the colonies will grow to much larger sizes, but even then the population growth rate is still way too low. Cases (j)-(l) show the results from using the average growth rate. Compared to the left side, the ages are somewhat lower but very similar, but case (1), for both iteroparous and semelparous colonies, has the September rise occurring one generation later. The population growth rate is higher because reaching reproduction size at even slightly lower ages means there is less mortality before reproduction. Cases (m)-(o) show the effect of having a longer period of time after the growth rate is reduced before the maximum size is reached. For iteroparous colonies, the age stayed the same or went down slightly since each generation of growth means one less generation of reproduction so it is better to reproduce if the growth rate is fairly low. On the other hand, the semelparous colonies' age either stayed the same or increased. With only one reproduction episode, the colonies will try to maximize their size until very slow growth and/or high mortality make it unprofitable. For iteroparous colonies, the population growth rate and colony sizes are lower, while for semelparous colonies it is true only for all sizes but the largest, 800 zooids. Finally, cases (p)-(r) show the results for the highest growth rate. The results change  Chapter 3. Age of sexual reproduction in B. schlosseri  86  little except that iteroparous colonies have a more steep decline before the rise in September. The growth rates and sizes are larger but very similar also. Now, compare k = .001 (Fig. 3.10-3.15) to k = .002 (Fig. 3.16-3.21). The ages of first sexual reproduction are, on the whole, very similar. However, for many cases the rise in age before winter is 1 or 2 generations later making it more accurate for iteroparous colonies but less accurate for semelparous colonies. For the iteroparous cases, the population growth rate increases by about a factor of 4 while for the semelparous cases it increases by a range of factors from 8 to 16. For iteroparous cases, the factor of 4 increase is due to about 2 full generations living over the year, while for semelparous colonies, the factor of 8 is due to 3 or 4 full generations living over the year. The larger factor increase (up to 16) is due to the colonies reproducing a generation earlier which avoids the high mortality imposed by the model for later generations. Although with k = .002 the yearly growth rate for iteroparous colonies is close to the expected value of 1.0, for semelparous colonies it is too low for the most realistic cases (i.e. where the maximum size is 250 zooids). In fact, it is too low by a factor of 20 to 40. This is caused by the extra mortality imposed by the model at later ages which makes the densities too low. Assuming about 4 generations per year, this means that the density is too low by a factor of 2 to 2.5. Since k — .002 is fairly accurate for semelparous colonies, this indicates that the mortality values do not rise sharply with age in semelparous colonies. This corresponds to Grosberg (1982) who states that semelparous colonies continued to live as long as he prevented sexual reproduction by removing their oocytes at each asexual generation. On the whole, the model results predict the qualitative, and even quantitative, patterns of age of first sexual reproduction, especially for the iteroparous morph. Since cases (a)-(c) did not show these qualitative features, it implies that the after winter growth rate change is responsible. Thus, since the current conditions are not responsible, the  Chapter 3. Age of sexual reproduction in B. schlosseri  87  colonies can somehow tell the time of year. With the synchronization of the asexual generation length to the water temperature, the current value and changes in the water temperature is the likely method used, although the photoperiod is another possibility. Also, the model allows one to suggest the cause for the changes in the age of first sexual reproduction. The dip in age in early September is for colonies that reproduce in early spring before their size is greatly reduced. The large rise in age in late September is due to the colonies waiting until they regain some of their size after winter before reproducing. The rise in age in July for iteroparous colonies is to avoid reproducing during the period in early September when the age goes down. For both semelparous and iteroparous colonies, k w .002 as expected, although it might be a bit lower for iteroparous colonies. In particular, for iteroparous colonies, the most realistic maximum size is 1300 zooids, while for semelparous colonies it is close to 250 zooids. Thus, although iteroparous colonies do not usually grow to very close or up to their maximum sizes since they want to be able to reproduce for many generations, semelparous colonies always tried to grow to, or at least near, their maximum sizes before reproducing, even despite increased mortality. This would explain why with little substrate semelparous colonies grow into folds despite the extra risk of being torn off (Grosberg 1982). Thus, size is very important for determining the age of reproduction in semelparous colonies, as confirmed by the starvation, substrate, and embryo removal experiments in Grosberg (1982). Finally, except for during the first 7 generations after winter, once an iteroparous colony began sexual reproduction, it continued each generation until death. Thus, Grosberg's (1982, 1988) observation to that effect could be due to it being the optimal thing to do rather than a physical constraint. However, even over the range of parameters studied, the variance in field data shown in Fig. 3.9, particularly for the iteroparous colonies, is not reproduced in the results.  Chapter 3. Age of sexual reproduction in B. schlosseri  88  Possible causes for this are natural individual variation and the model not accounting for spatial heterogeneity since I have no data for it. However, the colonies are sessile so their growth and mortality depends on the suitability of their location which can vary greatly over their habitat. 3.5.4  Density and Reproduction Amounts  As mentioned previously, once the age of first sexual reproduction is found, forward iteration is used to obtain the density and reproduction amounts at each time of the year. Fig. 3.22 shows these values for iteroparous colonies and Fig. 3.23 shows it for semelparous colonies. In Fig. 3.22, there is an internal peak about generation 15 (Aug. 23). However, the major peak is at the end of the year and early part of the next year. Thus, although the ages of first sexual reproduction are fairly accurate compared to the average, the reproduction data is not. Part of the reason is that in real life the age of first sexual reproduction is spread over a range rather than concentrated in the average. Secondly, it is likely that spatial effects make a large difference. As mentioned previously, although overwintering colonies are starving in early spring, it appears that their offspring, to a large extent, do not. Thus, since the progeny tend to locate close to the parental colony (Grosberg 1982), colonies probably only reproduce if the local conditions are favourable. Fig. 3.23 (semelparous colonies) shows 3 peaks at generations 12 (Aug. 8), 19 (Sep. 14) and 25 (Oct. 29). In these cases, the reproduction is in a yearly cycle. Comparing Fig. 3.22 and 3.23 with Fig. 3.7, it is difficult to make any comparisons with any degree of confidence. Density-dependence and the range of ages offirstsexual reproduction are important in determining the amount of reproduction during each time step.  Chapter 3. Age of sexual reproduction in B . schlosseri  0  10  20  30  0  89  10  20  30  Case CD density (on loft) and reproduction (on rlc/it) amoixita The values shown are relative to their maxlrnurna  -i  1  1  r uu i  i  1  r  0 10 20 30 0 10 20 30 Case (r) density (on left) and reprociictton (on rlc/it) amounts The values shown are relative to their maxlmuma  Figure 3.22: Iteroparous colony density and rep. amounts, cases (i), (1), and (r) with k = .002. The x-axis is the time of year the colony started.  Chapter 3. Age of sexual reproduction in B . schlosseri  90  Case (g) density (on left) and reproduction (on right) amounta The values shown are relative to their maximums.  Case (p) density (on left) and reproduction (on right) amounta The values shown are relative to their maximums.  Figure 3.23: Semelparous colony density and rep. amounts, cases (g), (j), and (p) with Jc = .002. The x-axis is the time of year the colony started.  Chapter 3. Age of sexual reproduction in B. schlosseri  3.5.5  91  Lack of Time Dependent Mortality  From the earlier discussion, it appears that the growth rate change after winter is the primary factor controlling the age of first sexual reproduction changes over the year. However, it is possible that it could be due to a combination of the growth rate change after winter and the time dependent mortality. Fig. 3.24 shows the effect of removing the time dependent mortality for several iteroparous and semelparous cases. The ages are almost identical, with the only major difference being that the population growth rate is much higher.  3.5.6  Conclusions and Possible Enhancements  The relatively good agreement between the field data and the model results allows several conclusions.  First, each morph's life history characteristics, particularly the age  of first sexual reproduction, is near equilibrium. The after winter growth rate change causes fluctuations in the age of first sexual reproduction throughout the year for both the semelparous and the iteroparous morphs. Thus, the model suggests that the colonies can tell the time of year. Apart from during the first seven generations for overwintering colonies, iteroparous colonies continue reproducing every generation, as noted by Grosberg (1982), not necessarily due to an energetic or physical constraint, but simply because it is optimal for them to do so. However, although average values of the age of first sexual reproduction are predicted quite accurately, the full range of values (Fig. 3.8, 3.9) cannot be predicted primarily because of a lack of spatial heterogeneity in the model. An interesting enhancement to the model, that could help understand why each morph reproduces in the amount that it does and also why intermediate morphs are so rare, would be to find the optimal amount of reproduction as well. However, the effect of  Chapter 3. Age of sexual reproduction in B . schlosseri  AW»-ug Lmt-i3co ChnrcB-0 Mirtw*  Alpha-203 Urrtt-1300 OhanQB-a MLrrtM-^  AKiw-2,14 LW-1300 CJnngs-7 NUT*«-7  I  Oowth-OTS M»ta-ea4 A M I 1-680 Max.a-74«  |  Oowth-061 Mn a-143 Ave. a-702 Max 3.-362  —:  — I — I  1 I  ;  I  L ! —  L  1_  I  W*-&23 uim-2eo Crwqa-4 Nuixai-O  I  Atpha-a8S Umrt-260 Chvxjo-4 Nujtw-3  :  I  :  t  1 _! I  ! ll  Ataha-4.18 Ufrtt-260 Cftanoa-3 Mnxw-4  92  _* :  I * I  1_  I  <3rowttrfi34 MM 1-174 Ama-203 M n a-318  I  Growtt>-a03 Mh a-197 Avo.a-232 Max 3.-250  ! ! I  *  I  Qrawft-A42 Mh a-172 Awa «—201 Max. a-421  Figure 3.24: Age of first sexual reproduction, using k = .002, without time dependent mortality (see sec. 3.5.5 for details). The x-axis is the time of year the colony started; the ?/-axis is the age of first sexual reproduction. Left side: iteroparous cases (i), (1), (r); right side: semelparous cases (g), (j), (p). a values increase going down the page.  Chapter 3. Age of sexual reproduction in B. schlosseri  93  varying amounts of reproduction on mortality and growth rate of the next generation are not known. Before a successful attempt can be made at this model, more information needs to be known about B. schlosseri, starting with the potential mortality of older semelparous colonies. The next chapter will examine various theories about the coexistence of species that occupy the same niche and their applicability to B. schlosseri. The use of the dynamic programming technique in solving this problem will also be discussed.  Chapter 4  M o d e l i n g of S e m e l p a r o u s / I t e r o p a r o u s C o e x i s t e n c e  As discussed in Chapter 2, the semelparous and iteroparous morphs of Botryllus schlosseri coexist. An interesting question is whether this coexistence, mediated by Botrylloides leachi, is stable (i.e. is in some form of equilibrium) or not. Although the two morphs are actually part of the same species, their population densities fluctuate in such a manner that there is little inter-morph breeding (Grosberg 1982). Also, since most of the young from any inter-breeding are one morph or the other (Grosberg 1982), for the purposes of modeling, the two morphs can be treated as separate species. The next section will discuss some earlier models that looked at two or more species sharing a resource to see how they can be applied to the Botryllus schlosseri coexistence.  4.1 4.1.1  Previous Modeling Lotka-Volterra Equations  In the absence of any restraining influence, the rate of growth of a single population, N, would be proportional to its existing population, i.e.  - dN -  = rN  (4.47)  resulting in an exponential (Malthusian) law of population growth at rate r (Lotka 1932). Of course, for most real populations, although (4.47) might apply to populations at low density levels, as the population grows, the rate of increase, r, would decrease as the population approaches its carrying capacity. In the simplest case, r would be a linear 94  Chapter 4. Modeling of Semelparous/Iteroparous Coexistence  95  function of N so (4.47) would become dN — = N(l dt  - phN)  ro  (4.48)  which is the Verhulst-Pearl law of population growth that has been found to very acceptably fit a number of observed examples of population growth (Lotka 1932). The extension of (4.48) to two populations competing for a common food supply is —77- = rx/Vill-piffc/Vx + at dN = r N [l-p (hN + kN )] dt  fciVa)]  (4.49)  2  2  2  2  1  2  (4.50) (Volterra 1926,1931 (cited in Lotka 1932)). The same constants h, k appearing in both equations mean that both species consume a single food or, if mixed, then the proportion of each food is the same for both species (Lotka 1932). Thus, the two species share each resource identically. This system of equations is analyzed by Lotka (1932) who shows that there are three possible equilibria - an unstable origin, one stable, and one other one unstable. However, all of these equilibria involve one or both species dying out. This implies that two species that are resource limited cannot coexist on one resource. 4.1.2  Gause(-Volterra) Principle  This result was substantiated experimentally by Gause's experiments on competition between Paramecium audatum and P. aurelia (Gause 1934 (cited in Levin 1970)). Later theoretical work by Mac Arthur and Levins (1964 (cited in Levin 1970)) extended this concept (called the Gause(-Volterra) Principle) to show that, in general, there cannot be more species than resources to support them. However, there are many examples in real life, such as that of Botryllus schlosseri, that apparently contradicts the Gause Principle. Some apparent contradictions, on closer  Chapter 4. Modeling of Semelparous/Iteroparous Coexistence  96  examination, have proven to not be so. For example, MacArthur (1958 (cited in Levin 1970)) studied five congeneric species of warbler that at first appeared to have overly similar ecological preferences. However, he showed that their feeding habits were distinct enough for them to be actually occupying distinct ecological niches. This shows that competition can result in adaptive changes among the species rather than the elimination of one species. However, this does not appear to be a factor in the Botryllus schlosseri coexistence. A fundamental reason why Botryllus schlosseri does not obey the Gause Principle is that the underlying model given by (4.49) is too simplistic. The model can be enhanced to make it more realistic in basically two ways: (a) add different types of competition and (b) make the parameters depend on space and/or time. Equations (4.49) involve only exploitation competition, whereby each consumer affects the other by reducing resource abundance. However, interference competition, whereby one species directly interferes with the other species ability to exploit the resources, is also possible. This definitely happens when Botrylloides leachi overgrows a semelparous Botryllus schlosseri colony, or even possibly when two colonies come in contact and stop growing along the line of contact. 4.1.3  Interference C o m p e t i t i o n  Interference competition is less well understood than exploitation competition primarily because it has not been clear how to describe it mathematically (Waltman et al. 1980 (cited in Vance 1984)), although it can still be analyzed qualitatively. The influence of Botrylloides leachi makes the system more of a three-species Lotka-Volterra system. Usually, the system will be transitive in that the most efficient competitor will completely exclude the other two. However, interference competition permits nontransitive relationships since each inter-species interference interaction may be unique (Gilpin 1975), thus  Chapter 4. Modeling of Semelparous/Iteroparous Coexistence  97  possibly permitting stable coexistence. Next, the following paragraphs describe the effect of spatial and/or temporal variation in the model parameters. 4.1.4  Spatial Heterogeneity  Tilman (1980, 1982 (cited in Vance 1984)) and others have suggested that coexistence is possible given the existence of sufficient spatial variability in resource abundance. Chesson and Warner (1981 (cited in Chesson 1985)) state that, with pure spatial variation, species can coexist purely from variation in adult death rates. Of course, the semelparous and iteroparous morphs have different adult death rates. 4.1.5  Temporal Variation  Seasonal (or daily or other periodic) influences make most of the temporal variation in an environment cyclic or periodic in nature. Species survival can depend on this periodicity; namely, two competing species, one of which would be doomed to extinction in a constant environment, can under certain circumstances coexist in a periodic environment in a limit cycle sense. The idea is that at low population densities with plentiful resources for both species, the competition will be low and consequently both species increase (roughly exponentially) as they would in isolation. Usually, this will continue until increased competition causes a decline in the density of the inferior species. However, if some event causes a significant loss in density so that both species return to the same previously low levels, then the cycle is renewed. If this cycle repeats itself periodically, the inferior species might coexist in a periodically fluctuating (limit cycle) fashion (Cushing 1980). This is roughly what happens with Botryllus schlosseri when Botrylloides leachi overgrows the semelparous morph during the summer. However, this seasonal "kill" only affects one of  Chapter 4. Modeling of Semelparous/Iteroparous Coexistence  98  the "species", and also the overwinter mortality/reduction in size has a large influence as shown in the previous chapter. Another important factor is immigration into the pond (Grosberg 1982). Chesson (1985) showed that variation in the ratios of immigration rates will favor coexistence of species. The effect of Botrylloides leachi can be looked at as a physical disturbance that causes disproportionally greater mortality of the superior species (semelparous Botryllus schlosseri). This can prevent the lower ranked species (iteroparous Botryllus schlosseri) from being competitively excluded locally (Paine 1966). The above models suggest, qualitatively, the possibility of stable iteroparous / semelparous coexistence mediated by Botrylloides leachi. However, none of them individually is precise enough to show, with any degree of confidence, that the two morphs are actually stably coexisting. Another important consideration, even more important than in the modeling of the last chapter, is the clonality effect. The following aspects will have a direct bearing on possible coexistence: 1. Clonal organisms are often phenotypically plastic so that the differences in genotypes may be masked by responses to habitat characteristics. This would also likely delay competitive exclusion. 2. Genotypic differences allow the potential for slight niche differentiation between conspecific clones. 3. Seasonal phenomena may provide "balancing selection" whereby two or more genotypes are favored sequentially. Even though the balance may not be exact, it could at least prolong clonal coexistence. (Sebens 1985). Finally, another important consideration is that for sessile organisms the arrangement of neighbours becomes important as individuals cannot move except via  Chapter 4. Modeling of Semelparous/Iteroparous Coexistence  99  dispersal of young (Grosberg 1982). The only modeling technique that I know of that can potentially model the Botryllus schlosseri situation accurately enough is that used in the previous chapter, dynamic programming. 4.2  Dynamic Programming  Dynamic programming, as explained in the previous chapter, assumes that the organism acts in a manner that optimizes its discounted expected future reproduction. In the model of the previous chapter, only the time, size, and age of the colony influenced its behaviour. However, when dealing with the coexistence of species, this decision also depends on the actions of the other members of the same morph and that of the competing morph. The resulting strategy is assumed to result in an equilibrium stable under environmental fluctuations. This strategy is called an environmentally stable strategy (ESS). 4.2.1  E n v i r o n m e n t a l l y Stable Strategies ( E S S )  Unlike the model in the previous chapter, stochasticity and random effects cannot be ignored in an ESS since coexistence depends critically on how the colonies are spatially distributed. A standard ESS predicts a particular strategy to be superior. However, in a stochastic setting, no individual strategy will necessarily always win out over all others for several reasons. Any strategy, no matter how inferior it is on average, could be favoured during a chance run of years (Levin 1984). Also, a strategy's endurance might depend on its density, with certain types of strategies favoured only when they are sufficiently abundant in the population. Although it might have difficulty becoming established among an established morph, it could nonetheless do so sporadically. However, the original morph could also reinvade sporadically, especially among local populations of interbreeding organisms (Yoshimura and Clark 1991).  Chapter 4. Modeling of Semelparous/Iteroparous Coexistence  100  A more basic problem than the potentially complex solutions is the fact that stochastic dynamic programming models are extremely difficult to solve. Even almost trivial problems can involve a lot of work to solve them (Mangel and Clark 1988). A yet more basic problem is that the model requires, among many other things, knowledge of the spatial distribution of the morphs and their progeny and the dependence of the state variables on the population density of all possible phenotypes. Since I do not have this information, I have not tried creating a dynamic programming model to analyze the situation.  Chapter 5 Discussion  The dynamic programming model in Chapter 3 is able to quite accurately predict the average age of first sexual reproduction in Botryllus schlosseri, especially for the iteroparous morph. This indicates that the life-history characteristics of the two morphs of Botryllus schlosseri, individually, are at least near equilibrium. It also suggests that the colonies can tell the time of the year since the after winter growth rate change causes fluctuations in the age of first sexual reproduction throughout the year. Finally, iteroparous colonies continue to reproduce sexually once they begin because it is optimal rather than due to some energetic or physical constraint. However, the full variation in the age of first sexual reproduction cannot be predicted by the model, probably due to spatial heterogeneity not being incorporated. 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