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Spatial structure and population dynamics in an insect epidemic ecosystem Clark, William C. 1979

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SPATIAL STRUCTURE AND POPULATION DYNAMICS IN AN INSECT EPIDEMIC ECOSYSTEM  by William Cummin Clark B.Sc, Yale*University, 1971 A thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy The Faculty of Graduate Studies Department of Zoology  We accept this thesis as conforming to the required standard  The University of British Columbia June 1979 (c) William Cummin Clark, 1979  In  presenting  an  advanced  the I  Library  further  for  degree shall  agree  scholarly  by  his  of  this  written  this  thesis  in  at  University  the  make  that  it  purposes  for  freely  permission may  representatives. thesis  partial  be  It  financial  fulfilment  of  of  Columbia,  British  available for  extensive  granted  by  the  is understood gain  for  shall  Head  be  requirements  reference copying  that  not  the  of  this  copying  or  allowed  without  William C. Clark  The  Zoology  of  University  of  British  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  5 October 1979  Columbia  that  study. thesis  Department  permission.  Department  agree  and  of my  I  for  or  publication my  ii ABSTRACT A major theme in contemporary ecology is how, and whether, spatial structure considerations such as dispersal and environmental heterogeneity must be invoked for satisfactory explanations of observed population dynamics. This study shows how behavioral and environmental components of spatial structure shape emergent population dynamics in one relatively simple and well studied natural ecosystem:  the spruce budworm (Choristoneura fumiferana, Lepidoptera:  Tortricidae), and the boreal forests of eastern Canada which it periodically defoliates. The study begins by partitioning spatial structure considerations.'into components of local structure, dispersal processes, and local uniqueness. Existing data on the budworm-forest ecosystem are then synthesized into an array of mathematical models reflecting possible relationships among those components. Numerical simulation techniques are used to explore these models. Subsequent qualitative analyses simplify the simulations, focussing on equilibrium isoclines for population growth and the associated phase portraits of system dynamics. The principle methodological contribution of the study is its explicit inclusion of dispersal effects in these qualitative analyses. Model predictions are compared with a variety of real world experiments and observations to assess the limits of present understanding. The study concludes that spatial population dynamics in the budworm ecosystem can be explained largely in terms of local structure relationships, modified to account for the local effects of insect immigration and emigration. Detailed knowledge of dispersal distance or direction distributions is unnecessary.  Population dynamics are shown to be sensitive to (a) the propor-  tion of moths dispersing, and its modification by habitat conditions or moth density; (b.) the survival of passively dispersing young larvae, and its dependence on forest structure; (c) the relationship between foliage quality/quan-  tity and budworm feeding; and (d) predation by vertebrates on larvae at low insect densities. Priorities for future research are identified, and some retrospective doubts concerning the utility of the population dynamics perspective in ecology are expressed.  iv TABLE OF CONTENTS  List  of  tables  List  of  figures  vi i  Acknowledgement  xi -  PART I INTRODUCTION Chapter 1  Introduction  PART  II  LOCAL STRUCTURE Chapter 2  Local  Structure:  Description  19  Chapter 3  Local  Structure:  Analysis  35  PART DISPERSAL  III PROCESSES Processes  55  Chapter 4  An O v e r v i e w o f D i s p e r s a l  Chapter 5  The E x o d u s  Response:  Description  60  Chapter 6  The E x o d u s  Response:  Analysis  85  Chapter 7  The D i s p l a c e m e n t  Chapter 8  The S e t t l i n g  Response  Response  110 140  PART IV INTERACTIONS Chapter 9  Interactions Dispersal  o f L o c a l S t r u c t u r e and Processes  158  v PART V SPATIAL  STRUCTURE  Chapter  10  D e f i n i n g , B o u n d i n g , and O r c h e s t r a t i n g the S p a t i a l Modeling E f f o r t  197  Chapter  11  Describing Spatial  214  Patterns  of Population  Dynamics Chapter  12  The R o l e o f D i s p e r s a l  256  Chapter  13  The R o l e o f L o c a l  286  Uniqueness  PART VI SUMMARY AND CONCLUSIONS Chapter  14  How t h e M o d e l s Work  351  Chapter  15  Back t o N a t u r e  391  BIBLIOGRAPHY  432  APPENDICES Appendix  A  S c a l e A n a l y s i s o f Budworm P a t t e r n s  458  Appendix  B  D e r i v a t i o n of the M u l t i p l e F l i g h t Displacement D i s t r i b u t i o n  482  Appendix  C  Technical  494  Appendix  D  Glossary  D a t a on C o m p u t e r of  Terms  Simulations  497  L I S T OF TABLES  2-•1  Summary o f l o c a l  structure  variables  32  5-•1  Local  estimates  f o r FFLY  75  7-•1  Flight  8-•1  Parameterization  9-•1  The  population  speed e s t i m a t e s  sensitivity  f o r budworm  of the s e t t l i n g  128  moths  response  o f cf> t o a l t e r n a t i v e  154  function  181  hypotheses  12-•1  Summary o f r e s u l t s :  13- •1  Wind d i r e c t i o n  13- •2  Wind d i r e c t i o n f r e q u e n c i e s  (flight  altitude,  lowland)  322  13- •3  Wind d i r e c t i o n f r e q u e n c i e s  (flight  altitude,  highland)  323  13- •4  Driving  13- •5  Wind d i r e c t i o n f r e q u e n c i e s  13- •6  External  13- •7  Nz(t)  C- •1  frequencies  variables  immigrant  boundary  Technical  exodus  for Basic  and s e t t l i n g (surface,  lowland)  321  324  Model (flight  contribution  altitude,  potential  boundary  zone)  325 326 327  values  d a t a on c o m p u t e r  275  simulations  simulations  495  vii L I S T OF FIGURES  1^1  The e p i d e m i c p a t t e r n  i n time  15  1-2  The e p i d e m i c p a t t e r n  i n space  16  1- 3  Spatial  2- 1  T h e p r o c e s s c y c l e f o r budworm a n d f o r e s t  34  3- 1  Local  47  3-2  Recruitment  3-3  Family  3-4  E q u i l i b r i u m m a n i f o l d f o r budworm l a r v a e  51  3-5  E q u i l i b r i u m manifold with outbreak  52  3-6  Equilibrium manifolds with  5-1  Exodus T y p e s :  5-2  Fecundity  5-3  Oviposition  5-4.  Pupal  5-5.  Oviposition-fecundity  relationships:  5-6  Oviposition-fecundity  relationships  5- 7  Exodus T y p e s :  6- 1  Recruitment  6-2  E q u i l i b r i u m m a n i f o l d f o r c o n s t a n t exodus  102  6-3  Equilibrium manifolds f o r four  103  6-4  Superimposed m a n i f o l d s f o r f o u r  6-5  Equilibrium manifolds f o r alternative  6-6  S i m p l i f i e d exodus  6-7 6-8  .  .  budworm d y n a m i c s i n New B r u n s w i c k  model  dynamics,  zero migration  function with  three e q u i l i b r i a  of recruitment functions  structural  17  49 50  trajectory  immigration  53  forms  76  as a f u n c t i o n o f l a r v a l  density  schedule  77 78  e m e r g e n c e a n d moth e x o d u s  schedules lab results  79 80 82  p a r a m e t r i c forms  83  f u n c t i o n w i t h c o n s t a n t exodus  exodus Types exodus  structures  Types FFLY p a r a m e t e r s  101  104 105 106  Exodus as a s o u r c e o f c o l o n i s t s  107  Local  109  model  d y n a m i c s , Type B exodus  vi i i 7-1  Temporal distribution of female moth take-offs  1.29.  7-2  Radar observations of moth flight altitude distributions  130  7-3  Single flight duration distribution: interval data  132  7-4  Single flight duration distribution: smoothed data  133  7-5  Cumulation single flight and duration distribution  134  7-6  Flight altitude distribution  135  7-7  Wind velocity distribution at flight altitude  136  7-8  Single flight displacement distribution  138  8-1  Targetting relationships  155  8-2  Cover type distribution for New Brunswick  156  8-3  Descriptive equation for settling success  157  9-1  Displacement distribution from a point source  182  9-2  Source and Sink site definitions  184  9-3  Displacement distribution from larger sources  185  9-4  Equilibrium manifolds for budworm egg densities  187  9-5  E(SAR.)  188 189  9-6 9-7  N*  190  9-8  <>j = 1 contours for various SAR values  192  9-9  General <J>/K contours for Type B exodus  193  9-10  General <J>/K contours for Type C exodus  194  9-11  <>| = 1 contours for various displacement hypotheses  195  10-1  Location and configuration of the modeled area  213  11-1  Spatial population dynamics of the Basic Model  235  11-2  Spatially averaged population dynamics of the Basic Model  238  11-3  Space-time transect of the Basic Model  240  11- 4  Observed outbreak  intervals  11- 5  "Epidemic  curves"  f o r budworm i n  11- 6  "Epidemic  curves"  f o r budworm i n New B r u n s w i c k  11- 7  Frequency d i s t r i b u t i o n of h i s t o r i c a l  11- 8  T y p i c a l s a m p l i n g l o c a t i o n s f o r New B r u n s w i c k egg s u r v e y  11- 9  Spatial autocorrelation budworm d e n s i t i e s  11- 10  Historical  11- 11  Q u a n t i t a t i v e egg d e n s i t i e s f o r v a r i o u s  11- 12  Q u a n t i t a t i v e egg d e n s i t i e s f o r s i n g l e sample  locations  255  12- •1  Zero d i s p e r s a l ,  predictions  276  12- •2  S h o r t d i s p l a c e m e n t : " " model  12- •3  M u l t i p l e f l i g h t and v e r y s h o r t distributions  flight  12- •4  Very s h o r t  predictions  12- •5  Multiple  12- •6  Type B e x o d u s ,  12- •7  Type C ( 5 0 ) e x o d u s ,  Passive s e t t l i n g :  12- •8  Type C ( 3 5 ) e x o d u s ,  Active  12- •9  Relative  loading  12- •10  Type B e x o d u s ,  13-•1  Local  13-•2  Mean w i n d  13-•3  F r e q u e n c y d i s t r i b u t i o n o f mean w i n d v e c t o r s  13-•4  Wind s h i f t w i t h  13-•5  A conceptual  13-•6  Critical  spatial  population  displacement:  uniqueness  for  velocities  budworm  model  model  regions  254  model  displacement  predictions predictions  model model  scales  278  279  predictions predictions  rate  280 281 282 283 284  predictions  i n New B r u n s w i c k  airflow  285  329 331  f o r low s t a t i o n s  altitude of a i r f l o w  251  252  rose f o r low s t a t i o n s  model  250  277  model  settling:  249  i n New B r u n s w i c k  predictions  s e t t l i n g : model  factors  247  observed  dynamics  Passive s e t t l i n g :  Active  245  propagation  uniqueness:  displacement:  immigrant  Canada  estimates  zero local  flight  243  332 333  convergence  and f r e q u e n c i e s o f  convergence  334 335  13- -7  Boundary zones f o r the modeled  13-- 8  Q.j  13--9  EQ. , z 1 >z  13-- 1 0  Historical  13--11  EN. , contours z i >z  13-- 1 2  Basic convergence  phenomena:  13-- 1 3  Zero convergence:  model  13-- 1 4  Strong  13--15  Zero boundary  13--16  Boundary-driven  13- •17  P r e v a i l i n g winds  (highland  13--18  P r e v a i l i n g winds  ( s i n g l e mean w i n d r o s e ) :  1 4 - -1  ,  A  region  336  contours  337  contours  338  d e n s i t i e s o f boundary  migrants  339  computed f o r 1 9 5 3  convergence:  model  predictions  model  342  predictions  outbreaks:  343  model  predictions  344  model  predictions  345  only):  model  predictions model  predictions  explanation  346 347  386  1 4 - •2  Qualitative  equilibrium manifolds  14- •3  Qualitative  predator-prey  1 4 - •4  Isocline  trajectories  1 4 - •5  Relative  survivorship  ,A- 1  Spatial  autocorrelation  A- 2  Spatial  a c r s u b d i v i d e d by r e l a t i v e  A- 3  Spatial  acr for  A- 4  Spatial  a c r p r e d i c t e d by B a s i c Managed  A - •5  Spatial  a c r p r e d i c t e d by B a s i c M o d e l  B- 1  Multiple  flight  341  predictions  immigration:  A framework f o r  340  f o r b r a n c h e s a n d budworm  isoclines  387 388 389  and d e f o l i a t i o n  f o r observed  390  recruitment  orientation  subregions  oviposition  475 476 478  rate  Model  calculations  480 481  493  xi ACKNOWLEDGEMENTS The work reported here developed is part of a larger study, jointly supported by the International Institute for Applied Systems Analysis, the Maritimes Forest Research Center of the Canadian Forestry Service, and the Institute of Animal Resource Ecology at the University of British Columbia. D.O. Greenbank, CA. Miller, R. Rainey, and G. Schaefer of the Budwrom Dispersal Project gave freely of research facilities, unpublished data, critical commentary, and hospitality. I have particularly benefited from discussions with Werner Baltenzweiler, Ray Hilborn, Rhondda Jones, Don Ludwig, Jack McLeod, Judy Myers, Randall Peterman, Chris Sanders, and Bill Wellington. Neil Gilbert provided critical questions and invaluable statistical assistance. Warren Klein programmed, ran, and filed the bulk of the computer simulations with a blend of ability, efficiency, and tolerance which leaves me deeply in his debt. Buzz Hoi ling, Judy Myers, Carl Walters, and Bill Wellington read the entire manuscript: their suggestions have improved it substantially. Production assistance along the way was provided by Ulrike Hilborn, Vicky Hsiung, and Mary McGechaen. To all, my thanks. Finally,< three individuals have so infused by professional life these last^ many years that it would be pointless to try to itemize their contributions: to Carl Walters, Buzz Holling, and a much missed Dixon Jones, I can only remain grateful for the education.  xii  On n'a pas besoin d'esperer pour entreprendre, ni de reussir pour perseverer. - William (the Silent) of Orange  1  CHAPTER 1  INTRODUCTION  •1.1  Spatial  structure  and p o p u l a t i o n  dynamics  1.2  Objectives  1.3  T h e s p r u c e budworm a n d i t s e c o s y s t e m  1.4  P a s t and present  and d e s i g n o f t h e s t u d y  research  i n ecology  2 1.1  Spatial  Structure  Early attempts local  spatial  Birch  averages,  prompted  with The  tions  of  Hutchinson  amplified  neity  elicited  Huffaker  The  and e n v i r o n m e n t  (1951,  1953),  of others  spatial  to  little  (1958),  argue f o r  i n the  1976),  laboratory  studies  (Bailey,  "experiments" of  (Krebs,  experiments 1975;  further  Andrewartha  than per-  and  i n c o r p o r a t i o n of  envi-  organisms  e c o s y s t e m and p o p u l a t i o n  dynamics.  the experimental  l a s t twenty years  (e.g.  MacArthur, support  the  and N o r t o n - G r i f f i t h s ,  adaptations  in  demonstra-  have been c o n f i r m e d  for  and  1969;  Pimentel  spatial  1966;  1963),  1974a,  i n nature  1979;  spatial  et a l .  Steele,  view that  Paine,  b;  M y e r s and  Levin,  1978a).  1977;  kingdoms  (reviews  Kennedy,  1975;  in Baker, Emlen,  Natural  is a critical  1940;  F e n n e r and R a t c l i f f e ,  structure  Campbell,  and m a t h e m a t i c a l  structure (Dodd,  heteroge-  directly  de-  Huffaker, 1965).  1964; Finally,  i s mirrored i n the s o p h i s t i c a t e d  d i s p e r s a l r e v e a l e d by p h y s i o l o g i c a l and b e h a v i o r a l  Strickler,  Wynne-Edwards,  has b e e n d e m o n s t r a t e d  et a l .  1972;  eds.,  importance of  t h e p l a n t and a n i m a l  to  It  p o p u l a t i o n dynamics observed  the e v o l u t i o n a r y  But  the  this  e c o s y s t e m dynamics t o d i s p e r s a l and s p a t i a l  f i e l d manipulations  Sinclair  shortcomings of  and t h e movement  response p r i o r to  but  rather  repeatedly. s e n s i t i v i t y of  terminant  conveniently  of  i s now a c c e p t e d e c o l o g i c a l dogma.  through  i n t e r a c t e d as  Skellam (1951),  arrangement  s p a t i a l l y s t r u c t u r e d view of  Their writings  Ecology  e m p i r i c a l and t h e o r e t i c a l  heterogeneity,  a broader,  in  v a r i a t i o n c o n c e p t u a l i z e d as a t e m p o r a l  ( 1 9 5 4 ) and a m i n o r i t y  ronmental  1977;  Organisms  phenomenon.  spective  Dynamics  t o e x p l a i n e c o l o g i c a l p o p u l a t i o n dynamics were e s s e n t i a l l y  in character.  homogenized  and P o p u l a t i o n  1978;  1975;  Dingle,  ed.,  studies across 1978;  Myers and K r e b s ,  Harper,  1971;  1962).  believe.that  u n d e r s t a n d how i t  spatial  does s o .  structure affects  Contemporary  ecosystem dynamics  ecology lacks  anything  i s not  approaching a  to  • .3 • general,  w e l l - t e s t e d theory  population of  dynamics.  environmental  growth,  spatial  of  s t r u c t u r e and  o f i t s consequences f o r  We c a n n o t  to the environmental  significance.  heterogeneity,  competition,  i n terms  and p e r s i s t e n c e .  strategies  evolutionary  spatial  We c a n n o t a s s e s s t h e s i g n i f i c a n c e o f p a r t i c u l a r  heterogeneity  stability,  dispersal  o f t h e r e l a t i o n s h i p between  reproduction,  which presumably  We c e r t a i n l y do n o t u n d e r s t a n d "local"  that characterize  to y i e l d o u r most  alternative define  how a n i m a l  r e l a t i o n s h i p s of  and t h e l i k e i n t e r a c t  d i s t r i b u t i o n and abundance  population  systematically relate  patterns  and t h e c l a s s i c a l  patterns  their  movement,  predation,  the dynamic immediate  patterns  perspective  on t h e e c o l o g i c a l w o r l d . Attempts extremely al.,  difficult  1975),  ecology,  t o move f r o m l o c a l l y t o s p a t i a l l y s t r u c t u r e d i n f i e l d s as d i v e r s e as geography  epidemiology  (Bailey,  o u r most immediate d i f f i c u l t y  vant e m p i r i c a l and t h e o r e t i c a l this  1975),  ultimately  devolves  into  and e c o n o m i c s  and m e t e o r o l o g y  (Monin,  i s lack of i n t e g r a t i o n  research. problems  theories  And a s S t e e l e  of data.  have (Cliff  1972).  between  (1975)  has  proved et In  the r e l e -  emphasized,  Consider the following  partic-  ulars. There are e x c e l l e n t b e h a v i o r a l the s t r u c t u r e  of t h e i r  on t h e s t r u c t u r e temporary  spatial  environments,  of theoretical theories  s t u d i e s r e l a t i n g movement o f o r g a n i s m s  b u t t h e s e h a v e had e s s e n t i a l l y no i m p a c t  models  in spatial  population  are l a r g e l y arrogations  from  dynamics.  s t a t i s t i c s w h i c h c o n t a i n no e m p i r i c a l e c o l o g i c a l i n f o r m a t i o n .  sent  tendency  can o n l y  be a f r u i t l e s s r e e n a c t m e n t  infatuation  with  understanding certainly da,ta.  e v e r more b a r o q u e  the Lotka-Volterra  spatial  require  variants  of mathematical equations.  structure-population  The p r e -  o f t h e s e b a s i c models  ecology's  Future progress  fifty-year towards  dynamics r e l a t i o n s h i p s w i l l  i n f u s i o n of the mathematical  theory  Con-  p h y s i c s and mathemat-  ical  to construct  to  with  empirical  almost behavioral  The the  reverse  population  tionships  predictable spective,  1952)  detriment  of  aspects  spatial  distributions,  use o f  sorts  geographers dynamics scales  spatial  of  more b e h a v i o r a l  predicted, patterns  s t r u c t u r e combined. of  the  Yet  to  think  reveal  l i m i t a t i o n s of The  i n terms our  of  boundary  conditions)  vioral,  mathematical,  evaluating  and d a t a  into  line,  literature  t h a t on a l l  critical  And  their  what  statis-  other  asno zoo-  population spatial  perturbations  in ecology.  We  e x p l i c i t l y designed them f r o m  (along with  to  their  i n t e r c a l a t i o n of spatial  structure  f i e l d experimentation  and w o u l d mark t h e e m e r g e n c e o f experimental  spatial  models.  differentiate  effective  i n a program o f  on  e c o l o g i s t s and  opportunistic  such experiments  r e s e a r c h i n e c o l o g y as a m a t u r e  a  per-  understanding  perspectives  and s t a t i s t i c a l a p p r o a c h e s  successful implementation  for  s p e c i f y t h e k i n d s and  and t o  would r e q u i r e  the  From t h i s  decades.  d i a g n o s t i c experiments  s p e c i f i c a t i o n of  tingent  .b-ring.;theory  for  suggest novel  theories,  to  t h e o r y makes v i r t u a l l y  that plant  to  rela-  Under such  rather  The  exceeds  present  data  required for  are l i k e l y to  key  i n the c h a r a c t e r i z a t i o n of  probably  at  interaction.  t i m e now i s p a s t when a r b i t r a r y ,  structure  competitors.  but  and o b s e r v e d .  scale-heterogeneity  f i e l d observations  define  issues.  data,  dynamics  i s encountered  spatial  must b e g i n the  to  resolution.  structure  t h e o r i s t s h a v e made l i t t l e e f f o r t  spatial  ture  degree of  have been c o l l e c t i n g and a n a l y z i n g  Finally,  Their  for  situation  postulated,  spatial  the  overall  fail  framework  have h i s t o r i c a l l y p r e v a i l e d ,  structure/population  d e s c r i p t i o n of  pects of  of  of  theoretical  w h i c h d a t a a r e n e c e s s a r y and s u f f i c i e n t f o r  An a n a l o g o u s  tical  o r an a p p r o p r i a t e  need i s n o t  definition  no e f f e c t i v e  s t u d i e s themselves  reductionist tendencies  the  of  With  l e v e l , behavioral  (Smith  circumstances  is also true.  discipline.  to  their conbehaproblems would  spatial  struc  The  gap b e t w e e n w h a t we h a v e and w h a t we n e e d i n s p a t i a l  research is  is substantial.  No s i n g l e a p p r o a c h ,  l i k e l y to monopolize  future  structure  including that  a d v a n c e s when t h e y  do o c c u r .  pursued  here,  What i s  viri  t u a l l y certain i s that traditionally illuminating  those advances w i l l  d i s t i n c t research areas recent studies  u n p u b l i s h e d ; and S t e e l e a n d and h i g h l i g h t present of  the  stage of  observe  know when t h e i r diagnostic  its  they  theories  Even t h e b e s t s u c h s t u d i e s brought  a theory  of  theless,  they  of for  1.2  spatial  spatial  spatial  structure  structure,  one way  and t h e r e b y I report  provide  analysis  to  note  that  had s u g g e s t e d  forest-spruce  budworm  nature  of  environmental  an e x c e p t i o n a l l y that  spatial  heterogeneities,  in  at  their  that  have  to  utilize  "edges." resolved  c o u l d be c a l l e d  in ecosystems.  None-  t h e d a t a and t h e  theory  and j u s t i f i c a t i o n  Study  that  spatial  rich  structure  harvest  structure  (Choristoneura  For  body o f  to  population  fumiferana)  s y s t e m and t h e  i n v e s t i g a t i o n are d i s c u s s e d i n the next s e c t i o n . sufficient  the  here.  concerns the r e l a t i o n s h i p of  The  occurs  and d e s i g n o r  both m o t i v a t i o n  This  North America.  dynamics  integrate  and D e s i g n o f  i n the boreal  the  to  Wiens,  in sufficient detail  r a i s e d as many i s s u e s as t h e y  Objectives  eastern  s t r u c t u r e as i t  inadequate;  and p o p u l a t i o n  a,b;  most  case studies at  i l l u m i n a t e s y s t e m and t h e o r y have  the  essential characteristics  consequences are  1977,  the  p r e c i s e l y such c o u p l i n g ,  comprehensive The  Some o f  Jones,  involve  research.  dynamics  have d e m o n s t r a t e d  study  R.  us o n l y m a r g i n a l l y c l o s e r t o a n y t h i n g  the case study which  dynamics of  detailed,  or techniques to  (e.g.,  1977)  confront  population  perturbations  and h a v e  area  structure  such s t u d i e s are t h a t  nature;  for  e x p l i c i t coupling of  I outlined earlier.  Henderson,  potential spatial  in this  involve  history  present previous  of  purposes research  complex its it  is  and  considerations—insect dispersal,  pattern,  and t h e  l i k e - - c o u l d be e s s e n t i a l  to  understanding  contemporary  Finally,  spatial  an e x t e n s i v e  population  standard a g a i n s t which to context,  I defined  (a)  to  hypotheses (b)  the  develop  evaluate  to employ  a formal  that  and r e l e v a n c e o f  ments  for  (1976b)  tionships  in  Local develop.  to  for  the  cesses for  "Local"  Within  objectives: for  integrating  hypotheses, for  this  existing  i n t h e budworm  system;  dynamics  and f o r  con-  evaluating  an u n d e r s t a n d i n g  of  spatial  field  experi-  system;  hypotheses.  spatial  sets the  framework  proposed  structure-population  phenomena  s t a g e on w h i c h l a r g e  are formally  in space.  For  by  dynamics  competition,  interact  predation,  l o c a l l y determines  I address  local  these  (1963):  i n c l u d e most o f  processes of  p a r a s i t i s m , and so o n .  t h e number o f  d i s p e r s a l , and t h e  scale space-time  t h o s e w h i c h c a n be t r e a t e d  budworm  i n t e r a c t i o n s d i s c u s s e d by M o r r i s  export through  abroad.  of  rela-  ecology.  structure  reproduction,  theories.  issues r e f l e c t s a general  a n a l y s i s of  those  an i n v a l u a b l e e m p i r i c a l  deducing the population  e x i s t i n g data  major  h i s t o r i c a l record  relationships  structure  a  many o f  r e s e a r c h p r i o r i t i e s and s p e c i f y c r i t i c a l  these  m i x e d o r homogeneous forest  Further,  investigate  framework  d i s t i n g u i s h i n g among c o m p e t i n g  My a p p r o a c h Levin  spatial  for  i n t h e budworm  identify  system.  alternative  structure  framework  the adequacy  dynamics  to  provided  structural  alternative  to  the  following s p e c i f i c research  sequences of  (c)  of  and i n t e n s i v e  dynamics  and d a t a on s p a t i a l  population  control  r e s e a r c h p r o g r a m was u n d e r w a y  suggestions. observed  a n d management  effect  i n t e r a c t i o n s of  the  of  patterns as w e l l  the  growth,  feeding,  How t h e s e  insects potentially importing  budworm  budworm-  other  available  insects  system i n Part  pro-  II  from of  this  report. Dispersal  processes l i n k  may be v i e w e d a s c o n t i n u o u s  local  s i t e s in space.  or d i s c r e t e .  In e i t h e r  In g e n e r a l  this  c a s e , depending  linkage on  the  details  of ecosystem s t r u c t u r e ,  variations Levin,  i n the s p a t i a l  1978a).  For  the  earlier  by W a t t  budworm  system s p a t i a l  into  and a n a l y z e  processes  in Part  the  structure  Local referred  are considered i n Part  gradations  amount o f  local  uniqueness  patterns  are everywhere  uniqueness  of  spatial  identical.  in dispersal  the  t h e s e may h a v e t h r o u g h  population. Levin's  (1976b)  i n my a n a l y s e s budworm  I explore  of  system.  spatial To  these  general  "bounding"  effort  to  local  framework  the  from d i s a p p e a r i n g i n a haze of organized  the  the  local  the  interactions  local  structure processes.  structure  Jones  structure  1974) (1974)  spatial  and  A  classic  i s induced  system,  sets the  of  can  when d i s p e r s a l  however,  it  attention.  of  I  relationships  those  i s mandatory.  each of  the  follow in  the  relationships  and r e d u c t i o n i s t d e t a i l ,  problem  of  V.  broad o u t l i n e s which  features  issues are addressed e x p l i c i t l y f o r  is  i n s e c t movement,  issues in Part  dynamics  by  shows t h a t a  d i l u t i o n and c o n c e n t r a t i o n  uniqueness  the  such  processes  relationships  dynamics,_even  budworm  computational  s i m p l i f y a n d bound  of  processes  system.  (Whittaker  i n the patterns  essential  of  divided  the e c o l o g i c a l  place i n the  structure-population  prevent  studies  p r o c e s s e s w h i c h has r e c e i v e d t h e m o s t  R e s e a r c h has f o c u s e d on i n h o m o g e n e i t i e s and t h e e f f e c t s  t h a t any o f  population In  in their  these d i s p e r s a l  parameters.  i n the  adopted  linked via dispersal  polyclimax  habitat  procedure  each g r i d ,  ;  IV.  from p l a c e to  in physical  local  1978;  ecosystem i s  regarding  between  r e f l e c t s the f a c t  i n Okubo,  and s u b j e c t t o  hypotheses  vegetational  (1974)  Within  g r i d s are  Interactions  a b o v e may d i f f e r  r e s u l t i n complex  local  III.  o c c u r s where  processes  Individual  the  budworm-forest  s i t e s or g r i d s .  alternative  Uniqueness  to  example  small  local  and P e t e r m a n The  damp o r a m p l i f y  reviews  I follow  be s p a t i a l l y homogeneous  I develop  spatial  and W a l t e r s  defined above.  local  (mathematical  structure.  a l a r g e number o f  analysis  pattern  present a n a l y s i s ,  (1964),  are considered to  d i s p e r s a l may e i t h e r  Local  an  Such Structure,  Dispersal  P r o c e s s e s , and L o c a l  respectively. own s t u d y ,  But b e f o r e  it will  1.3  proceeding to  be u s e f u l  budworm s y s t e m a n d t h e  Uniqueness  to  previous  related forms,  Virginia Yukon  to  Labrador  usually a rare  i s found  the p a r t i c u l a r c o n s i d e r a t i o n s of  my  1967;  i n s e c t , with  in the eastern part  been made upon  It  is a native  host t r e e s ,  throughout  of  Davidson  populations inadequate  the range,  balsam f i r  (Abies  and a p p r o a c h e s  rapidly.  The m o s t r e c e n t e x p a n d e d  million  hectares  100%  Clem.  (Lepidoptera:  fir-spruce-pine forests into  by budworm moths  behavior of  r e s o u r c e s , and w e a t h e r .  b a c k e d by c o n t e m p o r a r y (Blais, It  1968).  five years  Mortality  stands.  than  (FIDS,  c o n s i s t s of  v a l s and l a s t i n g ( i n terms o f 1975).  budworm of  to  1969-1974).  preferred  visible defoliation)  from 5-11  dismay  and s p a t i a l  outbreaks  r e c u r r i n g at  is  nearly  Flight  Tree r i n g  p a t t e r n w h i c h emerges  and  spread  i n d i v i d u a l moths  of  is  factors  down  the  Outbreaks  D a t a on t h e t e m p o r a l  i r r e g u l a r outbreaks  and  Particularly  to  two m i l l i o n  system are e x t e n s i v e .  broad temporal  with  spruce (Picea glauca)  r e c o r d s , a l l o w documentation  The  given l o c a t i o n ( M i l l e r ,  and w h i t e  from l e s s  kilometers.  the budworm-forest  The  h e l d i n c h e c k by a v a r i e t y  i s i m p l i c a t e d i n the s p r e a d : of  1967).  of  from  the Mackenzie V a l l e y  and P r e n t i c e ,  i n d e n s e , mature  in only  t e n s o r even hundreds  '•"ure 1 - 1 .  it.  these c o n t r o l s p e r i o d i c a l l y break  balsamea)  extensive  1704  the  species which, along  t h e s y s t e m e x p e r i e n c e s e p i d e m i c budworm o u t b r e a k s .  fly  p e r s p e c t i v e on  Choristoneura fumiferana  and w e s t a c r o s s C a n a d a  w h i c h may i n c l u d e p r e d a t o r s ,  persal  III,  Ecosystem  forest.  (Freeman and S t e h r ,  fifty  II,  i s t h e m o s t w i d e l y s p r e a d and d e s t r u c t i v e d e f o l i a t i n g i n s e c t  North American boreal  closely  V,  s t u d i e s w h i c h have  The e a s t e r n s p r u c e b u d w o r m ,  the  and  e s t a b l i s h a more g e n e r a l  The S p r u c e Budworm and i t s  Tortricidae)  analyses in Parts  measurements, back  to  i s shown i n  20-90 year years  Fi.g-  inter-  in a  9 Synoptic  maps o f o u t b r e a k s  beginning of t h i s century 1947;  Brown,  1970).  shown  complete  undergone  the  t h e most  against  budworm  The s p r a y  o f t h e program  (Blais,  years  the unprecedented  begun  to k i l l  have  a seven m i l l i o n  The c o n f u s e d s p a t i a l  food  tively  different  natural  system.  in  V.  1975).  Clark,  o f budworm  Jones  hectare  But p r o t e c t i n g Insect  and temporal  These pattern  effects  population  (Baskerville,  a thousand  consequently much In  yearly  locations  recent  estimates  distributed  in Figure  preservation,  pattern  that  d y n a m i c s must a c c o u n t f o r  1-3. system  and i s q u a l i t a -  c h a r a c t e r i s t i c of the  of behavior  i s s u f f i c i e n t to note  of  1976).  of t h i s management-perturbed  forms  for  c h r o n i c i n f e s t a t i o n has  data a r e summarized  from the p e r i o d i c , w a v e - l i k e  it  foliage  populations  i n the 1950s,  of foliage  t h e s e and o t h e r  i n the e a r l y  d e n s i t i e s throughout  of the forest  a c t i v i t i e s began  area.  t h e f o l i a g e and  a n d Hoi 1 i n g , i n p r e s s ) .  been t a k e n a t r o u g h l y  F o r t h e moment  i n 1 9 5 2 , and have  w h i c h was a c h i e v e d a t l e a s t  s t r e s s r e s u l t i n g from t h i s  I explore  some o f t h e w o r s t  Large s c a l e i n s e c t i c i d e  i n New B r u n s w i c k  f o r t h e budworm.  1974;  shows t h e c h r o n i c i n f e s t a t i o n  theory  a goal  s i g n i f i c a n t portions  insect density  Part  s e q u e n c e f o r t h e 1938  p r o g r a m was d e s i g n e d t o p r o t e c t  S i n c e budworm management  over  spread,  and c o m p i l e d t h e most  patterns.  and i n s t e a d r e m a i n e d a t m o d e r a t e  t h e managed a r e a  of  began  ( M i l l e r and K e t e l l a ,  also protected  did not starve  (DeGryse,  of spatial  has e x p e r i e n c e d  i n t e n s i v e management,  l i m i t tree mortality,  trees  sources  i n the absence of i n s e c t i c i d e a p p l i c a t i o n s - - i s  records of space-time population  continued s i n c e .  years  The s y n o p t i c  P r o v i n c e o f New B r u n s w i c k  spraying operations  thereby  patterns  since the  1-2.  Canadian  outbreaks,  variation.  l a s t to develop  i n Figure The  local  North America  been c o m p i l e d f r o m v a r i o u s  T h e s e maps show w a v e - l i k e  s u p e r i m p o s e d on c o m p l e x outbreak—the  have  occurring in eastern  i n more  an a d e q u a t e both.  detail spatial  10 1.4  P a s t and P r e s e n t My s t u d y  Research  d r a w s on t h i r t y  years  o f f i e l d r e s e a r c h on t h e s p r u c e  ecosystem i n e a s t e r n North America.  Investigations  d y n a m i c s h a v e been c a r r i e d o u t s i n c e t h e m i d - 1 9 4 0 s , of  the Canadian  Green  River  direction  Forestry  Project  o f R::F.  Service  i n the Canadian  Morris.  dynamics. ed.,  Their  P i o n e e r i n g w o r k was done by t h e  River  stages  r e s u l t s were p u b l i s h e d as an e x t e n s i v e  1963) w h i c h p r o v i d e s  the l a t e  tions  a starting point  1950's  break  towards  Short  f o r t h e work  throughout  permitted  and used i t  to  monograph  (Morris,  presented  here.  much o f t h e s p e c i e s '  further  laboratories.  summaries- a r e g i v e n  importance of s p a t i a l  structure  range  s t u d i e s on e n d e m i c  popula-  Much o f t h i s work was  in Prebble  (1975)  and o u t -  a n d B e l y e a :.  i n t h e budworm e c o s y s t e m was a p p r e -  f r o m t h e b e g i n n i n g o f t h e CFS s t u d i e s .  and l a b o r a t o r y  a n a l y s e s had e l u c i d a t e d b o t h  A program o f f i e l d  observations  t h e b r o a d o u t l i n e s a n d many d e t a i l s  t h e d i s p e r s a l p r o c e s s by t h e t i m e t h e M o r r i s m o n o g r a p h was p u b l i s h e d i n  1963  (Wellington  Greenbank, however, Its  1960's  life  (1975).  The ciated  a dynamic  the evaluation of s t r a t e g i e s f o r population control  containment.  et a l .  of  and e a r l y  a t a number o f CFS a n d u n i v e r s i t y  directed  group developed  f o r analysis of the population's  The d e c l i n e o f budworm p o p u l a t i o n s in  p r i m a r i l y by s c i e n t i s t s  o f e p i d e m i c budworm p o p u l a t i o n s  key p r o c e s s e s and l i f e  population  P r o v i n c e o f New B r u n s w i c k u n d e r t h e  The G r e e n  t a b l e approach to the study identify  (CFS).  o f budworm  budworm  and H e n s o n ,  1954, 1956, 1957,  1947;  1963; M i l l e r ,  s u f f i c i e n t l y advanced to t r e a t  e f f e c t s w e r e subsumed w i t h  mortality  rate  Subsequent landings  Wellington,  1948, 1954; Henson,  1963).  Understanding  d i s p e r s a l as a dynamic  was n o t ,  process.  o t h e r s as a c o n s t a n t p r e r e p r o d u c t i v e  i n the p r e d i c t i v e models o f t h e time  (Morris,  f i e l d s t u d i e s were d e s i g n e d t o o b s e r v e  throughout  1950, 1951;  t h e s e a s o n o f moth a c t i v i t y .  ed.  adult  1963).  flight-take-offs  Coupled with analyses  and of  •11 spatial  population  systematically  behavior  findings  of  desert  moths  (Greenbank,  i n t e r a c t i o n s between  moth  phenomena  Kingdom.  technology  by C F S '  flight  might  Maritimes  combined l o c a l  in Africa.  Forest  expertise  developed  R a d a r was e m p l o y e d  Research from  through  Doppler  navigation  perature visual  observation  Schaefer,  and R a i n e y  are s t i l l  largely  described  here.  A major  thrust  t h e s e new s p a t i a l  (unpublished).  structure  t h e budworm s y s t e m . synthetic  Morris  An o v e r v i e w  o f my w o r k  individual  of the p r o j e c t The v a s t  model  v e r t i c a l w i n d and t e m The e a r l i e r  during take-off  i s given  quantities  by  program and  Greenbank,  o f data  collected  has been t h e i n c o r p o r a t i o n a n d e v a l u a t i o n  data w i t h i n appropriate  numerical  I h a v e made e x t e n s i v e  simulation  use o f  of models  several  efforts.  and h i s c o l l e a g u e s summarized  a key-factor  moths  meteoro-  b u t w e r e made a v a i l a b l e t o me f o r t h e s t u d i e s  I n so d o i n g ,  modeling  providing  quantitative  Ground based  t h e moth d i s t r i b u t i o n s .  and s a m p l i n g o f  unpublished,  1976).  airborne  (Schaefer,  a l l o w e d l o c a t i o n and  (Rainey,  the radar work,  p r o f i l e s t o compare w i t h  l a n d i n g was i n t e n s i f i e d .  previous  phenomena  budworm  studies  to monitor  Airborne  equipment  Canada  on t h e  p r o f i l e s o f d i s p e r s e r s i n space and t i m e  s t u d i e s complemented  be  Dispersal  a number o f i n s t i t u t i o n s  and r e s e a r c h p e r s p e c t i v e s  of wind convergence  1973).  redistribution of dispersers.  and i n v o l v i n g The p r o j e c t  more  reassess-  density  logical  in  and f o r c e d a f u n d a m e n t a l  km) m e t e o r o l o g i c a l  the ultimate  locust migration  description  of  d i s p e r s a l operated  and d e r i v e  1976).  of  that  1970s, coordinated  i n New B r u n s w i c k ,  system with  that  l e d t o e s t a b l i s h m e n t o f t h e S p r u c e Budworm  i n the early  and t h e U n i t e d  suggested  i n i t i a t i o n and p r o p a g a t i o n  e v i d e n c e was g r o w i n g  i n determining  These  Centre  i n outbreak  and m e s o s c a l e ( c a . 2-200  important  Project  t h i s work  than o r i g i n a l l y supposed,  ment o f i t s r o l e Furthermore,  trends,  b a s e d on v a r i a b l e  their  analyses of epidemic  life-stage survivorship  rates  budworm (Morris,  12 ed.,  1963).  Neither  considered. model, field  (1964,  sity to  (cf.  by 4 - m i l e  "local"  values  on i m p r o v e d  dispersal  process.  1972,  Forest  University  of this  Canadian  Forest  that  proportion overall  (CFS)  Institute  i n New B r u n s w i c k .  spatial  265 g r i d a r e a s .  were a g a i n d e r i v e d  basic studies.  was made t o i n t r o d u c e plicit  i f p r e l i m i n a r y model  and P e t e r m a n , various  realistic  1974;  management  the o v e r a l l  budworm  scored several  s p r a y i n g would t u r n  1973).  This  in forest  Ecology  into  of  (IRE) budworm-  of the  however, cover,  an  (Walters explore  and weakness  intensive  chronic infestations  effort  and an e x -  i n space  i t s approximations,  province  dynamics  was u s e d t o  of strength  s u c c e s s e s , among them t h a t  budworm e p i d e m i c s  time,  M F R C - I R E model  Despite  from t h e  population  to l i n k the g r i d s  and t o a s s e s s a r e a s  research program.  predictive  uniqueness  of dispersal  Stander, options  local  This  the  from t h e M a r i -  s i m u l a t i o n model  Local  that  things)  Walters  Resource  den-  remained  (among o t h e r  A 4 . 5 m i l l i o n ha p o r t i o n  was m o d e l e d by s u b d i v i s i o n i n t o from M o r r i s '  and s t r e s s e d  under C . J .  o f Animal  by d i s p e r -  was s e n s i t i v e  s c i e n t i s t s , mainly  (MFRC) a n d a g r o u p  into  population  p o p u l a t i o n model of  budworm  represen-  neighbors  of the local  "dispersal" term,  realistic  against  divided  as an a p p r o x i m a t e  system behavior  understanding  Service  t o p r o d u c e a more  models o f t h e  10,000 square m i l e f o r e s t ,  gradient  of B r i t i s h Columbia's  interactions  spatial  and l i n k e d t o i t s n e a r e s t  quantitative  Research Centre  collaborated forest  to develop  t e s t s and a p p l i c a t i o n s o f t h e budworm  dependent  times  model,  of the  1977).  g r i d was t r e a t e d  Watt d e m o n s t r a t e d  the r e a l t i v e  In  Each  explicitly  i t p o s e d f o r t e s t i n g t h e model  a r b i t r a r i l y set at a constant  gradient.  specific  grids.  d i s p e r s a l were  o m i s s i o n as a major weakness  S t e e l e and F r o s t ,  1 9 6 8 ) was t h e f i r s t  of Morris'  rates  nor a d u l t  identified this  He s i m u l a t e d a h y p o t h e t i c a l  625 4 - m i l e tation  variation  p a r t i c u l a r l y the problems  observations  system.  sal  The a u t h o r s  noting  Watt  spatial  in  t h e model  insecticide (cf.  Blais,  13 1974).  Walters  overall  behavior  better  spatial  and Peterman  (1974)  again noted  to i t s dispersal assumptions,  structure  data  before  u s e d d i r e c t l y a s a management  and emphasized  t h e model  I spent a year with  evaluation  context  mentioned  forest  managers,  s e r v e as a d a p t a b l e findings  earlier.  tools  with  Drs. C S ,  adapting  (1974)  and exchanges  was e x p a n d e d  from the I n t e r n a t i o n a l  in Vienna.  and Peterman  a s e r i e s o f workshops Brunswick  (IIASA)  t h e IIASA g r o u p ,  d e s c r i b e d by W a l t e r s  for  rigorously  or  tool.  a team o f p o l i c y and systems a n a l y s t s Analysis  t h e need  c o u l d be t e s t e d  The M F R C - I R E c o l l a b o r a t i v e m o d e l i n g e f f o r t  A p p l i e d Systems  the s e n s i t i v i t y of the model's  i n 1974 t o Institute  Hoi 1 i n g , " D . D .  the o r i g i n a l  Jones and  f o r use i n t h e p o l i c y d e s i g n and  involving  we o r g a n i z e d  t h e MFRC e n t o m o l o g i s t s  o f d e v e l o p i n g an a r r a y  f o r r e s e a r c h and p o l i c y development.  of that p o l i c y design e f f o r t  for  M F R C - I R E model  Over t h e n e x t t h r e e y e a r s ,  the goal  include  a r e summarized  a n d New  o f models  to  The b r o a d  elsewhere  (Clark,  Jones,  and H o i 1 i n g , i n p r e s s ) . My a n a l y s i s o f t h e r e l a t i o n s h i p s among d i s p e r s a l , and p o p u l a t i o n context Watt  of this  dynamics  i n t h e budworm  careful  analysis of s p a t i a l  m o d e l s w e r e t o be i m p r o v e d . the  k i n d and q u a n t i t y  feasible with  f o r the f i r s t  budworm  context  Walters  and Peterman  The o n g o i n g  The w o r k r e p o r t e d Dispersal  Project  (1974)  Budworm D i s p e r s a l  The f i n d i n g s  time.  Additionally,  i f their Project  research  of  system  provided  analysis  i t was c l e a r f r o m e a r l y  population  the  argued t h a t a very  o f d i s p e r s a l d a t a w h i c h made s u c h d e t a i l e d  researchers that a broader  and emerging  program.  r e l a t i o n s h i p s w o u l d be r e q u i r e d  was r e q u i r e d f o r a s s e s s m e n t a n d e v a l u a t i o n  findings  heterogeneity,  s y s t e m has b e e n c a r r i e d o u t w i t h i n  ongoing MFRC-IRE-IIASA c o o p e r a t i v e  (1964) a n d , p a r t i c u l a r l y ,  spatial  dynamics/forest  workshops management  o f t h e s e new d i s p e r s a l  options.  here developed  as a c o n t i n u i n g d i a l o g u e w i t h  r e s e a r c h e r s a n d New B r u n s w i c k  forest  management  Budworm  personnel  .14 over  the period  shaped i n t o analyses  hypotheses,  and t h e i r  d a t a w e r e made a v a i l a b l e , d i s c u s s e d ,  implications explored  as w e l l  and o t h e r  exchanges  The m o d e l i n g a n d a n a l y t i c a l  as t h e c o n c l u s i o n s r e a c h e d ,  t h e New B r u n s w i c k  v i a t h e models and  The p r o c e s s was an i t e r a t i v e o n e ,  the s e r i e s o f extended, workshops  and IIASA d e s c r i b e d above.  developed, to  Unpublished  I describe in later chapters.  focused through IRE,  1974-1977.  a t MFRC,  t o o l s which  were  h a v e now been t r a n s f e r r e d  back  r e s e a r c h e r s , and t h e p r o c e s s c o n t i n u e s  there.  1-1: The pattern in time. Representative historical pattern of spruce.budworm outbreaks, synthesized from the work of Blais (1968). There have been four major outbreaks since 1770. The density measure of budworm is per typical balsam fir branch (one square meter df foliage).  Fig.  1-2:  The p a t t e r n i n s p a c e . M a x i m a l e x t e n t o f budworm i n f e s t a t i o n s i n e a s t e r n North America f o r 1938-1948, the l a s t outbreak sequence b e f o r e e x t e n s i v e i n s e c t i c i d e s p r a y i n g was i n t r o d u c e d . Redrawn f r o m Brown ( 1 9 7 0 ) .  17  68*  69 48'  v/  (  •  AREA MEANS  8 46*  g ~  1952 - 1977 (YEAR)  65*  66*  67  #  V V V V V v  V s  V  r  if  45*  Fig.  1-3:  Budworm d e n s i t i e s i n New B r u n s w i c k u n d e r h i s t o r i c a l management c o n d i t i o n s . Each g r i d r e p r e s e n t s a 50 x 50 km square. Mean e g g d e n s i t i e s (n = 2 0 ) a r e shown on t h e o r d i n a t e as t h e n a t u r a l l o g a r i t h m o f (one p l u s egg d e n s i t y per square meter o f f o l i a g e ) . Time i s g i v e n on t h e a b s c i s s a f o r y e a r s 1952 t o 1 9 7 7 . Mean d e n s i t i e s f o r a l l areas combined a r e g i v e n i n t h e i s o l a t e d grid-. M a r g i n a l numbers show l o c a t i o n o f g r i d s i n d e g r e e s n o r t h l a t i t u d e and d e g r e e s west l o n g i t u d e . .  PART  II  LOCAL STRUCTURE  Chapter 2  Local  Structure:  Description  Chapter 3  Local  Structure:  Analysis  19  CHAPTER 2  LOCAL STRUCTURE: DESCRIPTION  2.1  Local structure in the budworm system  2.2  Bounding the local structure model  2.3  The forest submodel  2.4  The budworm submodel  2.5  Summary of variables, units, and typical values  20-, 2.1 Local Structure in the Budworm System In the general sense defined by Levin (1976b), "local structure" encompasses those relationships among organisms and their environments which can be described without recourse to spatial dimension. The term is obviously a relative one, reflecting the questions and degree of resolution appropriate for any given investigation. For the budworm system models described in Part I, and for my own spatial studies, it has been convenient to include as "local structures" relationships such as growth, feeding, predation, parasitism, and short range larval dispersal. Large scale variationupatterns in. theorelat'ionships, and the various :  functions pertaining to long range adult dispersal have been excluded. This partitioning reflects technical problems of field research, and consequent differences in the degree of understanding available for "small" and "large" scale phenomena. Morris (ed., 1963) and his co-workers necessarily concentrated their initial budworm field- studies at relatively small spatial scales. As a result, their monograph reflects a rather detailed understanding of the "local" phenomena noted above, but very little of large scale spatial effects. Subsequent research has improved understandingat both scales, but not dramatically shifted the imbalance. One consequence of this situation is that models of local structure in the budworm system are now relatively advanced, and a number have been pubo  lished (Morris, ed., 1963; Walters and Peterman, 1974; Jones, 1977; Stedinger, 1977). For my spatial structure studies, I take this local structure as given, and confine my efforts to exploration of its role as the "stage" on which essentially spatial phenomena develop. I employ the broader spatial analysis as a framework for testing and evaluating the general applicability of certain predictions made on the basis of local hypotheses, but stop short  21. of proposing alternative hypotheses at the local structure level. This approach leaves a number of interesting questions unexplored, but has been necessary to keep the spatial study per se focused and manageable. The particular local structure hypothesis I employ is a slight modification of the model which Jones, Hoi 1ing and I developed in the context of our policy design studies on the New Brunswick budworm system. Though hardly the final word in local structure of the system (see Stedinger 1977 for an alternative view) it has been adopted as the best currently available state-of-theart synthesis by a variety of budworm research"laboratories and management agencies.throughout North America. Technical details are given in Jones (1977). In the remainder of this Chapter I describe those elements of the local structure model essential to an understanding of my spatial analyses. I concentrate on the "bounding" decisions made regarding variables included, the temporal and spatial scales represented, and attitude towards causal resolution adopted in the local model. Population dynamic patterns predicted by this model are analyzed in Chapter 3. There I also introduce the "recruitment curve" and "equilibrium manifold" compressions of the local structure which will serve as a basis for the full spatial analyses of Parts III through V. 2.2 Bounding the Local Structure Model Any useful model is a caricature of reality. The question is not whether, but what to leave out of the analysis. The general philosophy of model bounding adopted in the overall budworm policy design study is discussed at length in Clark, Jones, and Hoi 1ing (in press). The essential argument developed there is for a rigorously parsimonious selection of variables, and a "functional component" or "process oriented" approach to their causal interaction. Both of these themes are carried through in the present modeling effort.  For the local structure model considered in this and the following Chapter, explicit bounding decisions were required concerning the treatment of causation, the extent of temporal and spatial scales, and the variables and relationships included in the analysis. Detailed arguments are given in Yorque et al (in prep.). A summary of the conclusions follows. Causation Given the overall goals outlined in Part I, the models I employ must . have a causal structure which permits exploration of alternative ecological hypotheses, and suggests generalization and comparison to other ecological situations. Classical descriptive statistical approaches cannot meet these requirements, nor can more modern but essentially phenomenological methodologies which seek to define the model (generating process) from the input-output behavior of the system. The difficulties of such approaches are aggravated in cases such as that of budworm, where natural time and space scales of the system are so large as to make dense sampling of its behavior impossible. In the alternative approach pursued in the local modeling work and the dispersal analysis of Part III, attention focuses on those basic processes or functional relationships (e.g. predation, competition, feeding, reproduction, dispersal) which are the fundamental "building blocks" (Gilbert et al, 1976; Steele and Frost, 1977; Clark and Hoi ling, in press) out of which ecological theory is built. These processes are disaggregated into their component parts (Holling 1959, 1965), parameterized through reference to extensive local data on animal behavior and ecology (here, from Morris, ed.,1963), and recombined to yield hypotheses of (local) system structure and predictions of (local) system behavior. Parameters of such theories are directly interpretable as attributes of animals, subject to natural selection and experimental manipulation, and they are plausibly comparable to similar parameters in other systems. This "process" approach to causal analysis is no substitute for appropriate  23 d a t a on s y s t e m s t r u c t u r e , And  it  but  it  does s e r v e t o m o b i l i z e  timal  conditions  Temporal  invariably  and S p a t i a l  Forest  for  at  1968;  order of  age s t r u c t u r e ,  outbreaks  effectively  encountered  (Blais,  p e r i o d on t h e  those data  in  under the  see C h a p t e r  40 y e a r s  for  11.4)  the  New B r u n s w i c k  least that  long.  For  the  local  structure  i.e.  Seasonal  on t h e o r d e r o f  captures time of  events  Criteria  100 y e a r s .  budworm,  each y e a r  bounding  are  spatial  treated  extent  model  is that  the beginning of without manner  i m p l i e d i n the  this  e x p l i c i t reference i n which the  pattern,  age c l a s s  present  The The  Forest  local  local  V. of  that  spatial  l e a s t two  The  only  "local"  "local"  size  resolution year—the  local  the  agencies.  model.  full  spatial  c o n s t r a i n t on  the  a s i t was i n t r o d u c e d  phenomena Jones  "homogenizes"  a n d budworm l a r v a l  " p a t c h e s " on t h e o r d e r o f  past  out-  t h e management  i m p l i c i t l y i n the  dimension.  model  of  of  c a n be  (1977)  small  treated  notes  that  scale tree  km on  the  species  dispersal limits 2-20  at  its  edge.  Submodel  principal tree  species of  and b l a c k s p r u c e  (Picea  and a v a r i e t y  hardwoods  of  s u s c e p t i b i l i t y to  the  g l a u c a and P .  long term i n t e r a c t i o n of tial  to  i.e.  at  and r e s o l u t i o n f o r  notion  heterogeneities,  d i r e c t a p p l i c a t i o n to  2.3  Chapter';  unit  system.  a n a l y s i s we c h o o s e  t h e s y s t e m i s one  and t h e o p e r a t i o n a l  s t r u c t u r e a n a l y s i s are discussed i n Part local  op-  out-  "memory"  The minimum t i m e  the e s s e n t i a l c h a r a c t e r of  within  for  budworm  d i s c u s s e d i n the next s e c t i o n , r e t a i n s  break  generation  than  suggest a natural  which would d i s p l a y these dynamics through  which s t i l l  less  practice.  a time horizon cycles -  are.relevant.  Scales  Tree r i n g data break  can h e l p to d e f i n e which data  (Loucks,  system are b i r c h (Betula mariana),  1962).  t h e i r own w h i c h  budworm.  Fir  balsam f i r  These  is highly  (Abies  have a l i t t l e  is evidently  sp.),  dependent  s u s c e p t i b l e to  white  balsamea), understood,  on  differen-  damage,  white  spruce moderately so, black spruce only slightly, and birch and hardwoods not at all.  Since my focus in this study is the budworm rather than the  forest, the susceptible tree species (fir, white spruce) are grouped together as "host species" in the model., and the others are eliminated from dynamic consideration. The fixed proportion of the land area in any locality. which is covered with host species turns out to be an important parameter of the system, and is termed PHOST in the model. Dynamics of- the host species and their interactions with budworm and man are strongly dependent on the age structure of the forest. The model subdivides host species into 75 distinct one year age classes (H , a = 1,2,...,75), each a constituting a state variable of the model. In practice, it is convenient to define (H ) as the proportion of the total land area in a patch of host a forest which is covered in host trees of age (a). Note that in this definition  75 • I H. = 1.0, {f PHOST). . a=l a  No particular spatial arrangement among age classes of host species or between host and nonhost areas is assumed. That is, the local model does not . distinguish between the extremes of homogeneous mixing and totally segregated age and species land types. Parameters used in the model define an implicit spatial distribution for New Brunswick that appear reasonable for "local" areas of 2-20 km on edge. Branch surface area For budworm, one very important property of the host forest is its branch surface area. This constitutes the basic habitat or real estate of the. insect and, as I show later, in large part drives the system's dynamics.  Branch surface area is essentially the quantity of host species perceived by budworm. Technically, it is the quantity obtained by circumscribing a polygon around each branch in a given patch of host forest, and then summing the polygon areas. The resulting branch surface area is converted to a branch surface area density by dividing the total polygon (branch) area by. the (land) area of the host forest patch over which it was measured. In practice, branch surface area density is treated as an age-specific property of the host trees. A unit area of land covered by host species of a single age  (a)  will have a branch surface area density of (a-.). Values of a  (a ) for New Brunswick host forests are given in Jones' (1977) Figure 5. Mean Q  branch surface area density characterizing a mixed age host forest patch is 75 a=l  a  ,  a  where (H ) is the above noted state variable value describing the proportion of the host patch land area covered by host trees of age (a). Budworm utilize only the branch area on trees of age >21, so that the surface area relevant to budworm is 75  SA = j o H a=22 a  a  .  For most purposes it is convenient to use a relative, dimensionless measure defined as  SAR = SA/(a ) k  ,  where (a^) is constant value selected so that SAR will normally range from  26 0-1.  In the remainder of this report, I use the terms "SAR" and "branch  density" interchangeably to mean the relative branch area density of host species relevant to budworm as defined above. Foliage The second host forest property of interest to budworm is foliage. In the local structure model, foliage is the density of green needles supported by a unit of branch surface area. Foliage is consumed by budworm larvae and provides oviposition sites for adult moths. It also has an "age class" structure of its own to reflect partial retention of the "evergreen" needles from year to year. Though eight year classes of foliage are present in nature, satisfactory results are obtained in the model when only two aggregate classes are identified as state variables. "New" or "current" foliage (Fl in the model) represents newborn needles of the present year. "Old" foliage (F2) includes all needles of age 2-8 years and, as a state variable, serves to retain the host forest's "memory" of previous years', defoliation by budworm. It is often convenient to express these foliage classes together as "total foliage" (FT = Fl + F2). All foliage densities are expressed in arbitrary "foliage units", where one foliage unit is defined as the quantity of new foliage supported by a unit of branch area (SA) in the absence of budworminduced defoliation. Ratios of new to old foliage observed in nature then bound variable values as follows: 0 < Fl  <  1.0  0 < F2 < 2.8 0 < FT < 3.8  .  27 Budworm prefer to consume new foliage and survive better when doing so (Jones, 1977; Equations A.1-A.4, A.7).  Losses of new foliage and old foliage to bud-  worm differentially affect new foliage production and tree mortality in the following year (Jones, 1977; Equations A.32-A.38; A.44-A.50). 2.4 The Budworm Submodel Principal life stages of budworm were modeled using a "dynamic life table" (sensu Gilbert et al, 1976) treatment of the structural elements described ih Morris (ed., 1963) field studies. Following Morris (ed., 1963), no distinc1  tion was made between different possible genetic "types" of budworm. Stehr (1955) identified certain sex-linked, sex-limited characters in budworm, but did not relate these to ecological or behavioral parameters. Campbell (1962) did hypothesize ecologically significant differences among subpopulations of budworm, but failed to distinguish between ontogenetic and genetic origins of those differences. Holling, in an unpublished reanalysis of Campbell's data, judged that the postulated differences—even if allowed to have a genetic basis-- were very unlikely to affect substantially the population dynamics of the budworm system. The work I report here supports this view. Ontogenetic differences, in contrast, receive considerable attention in my analysis (see Part III). These can be studied without introducing additional state varir ables to the local structure formulation and, in the interest of parsimony and the absence of compelling arguments to the contrary, I confine my attention to such nongenetic differences in all the studies reported here. As Wellington (1964) has emphasized, these can provide a faster-than-genetic mechanism for adjustment to changing ecological conditions. Each year of the life table begins with eggs newly deposited on host tree branches. Since branches are the relevant habitat, budworm densities are expressed as numbers per unit branch area (SA). The following notation  2s;  distinguishes such densities for the different life stages referred to later in this essay:  .  NE : egg density NL : large larval density (instars III-VI) NA : adult moth density NF : adult female moth density .  Egg survival is treated as a constant.  Small larval disperse on the  wind, using silken thread for buoyancy. This dispersal generally involves distances of a few kilometers or less (Mason and McManus, 1978), "mixing" the system at corresponding scales. Survival at this stage requires that the passively dispersing larvae encounter suitable needles. It is thus heavily dependent on the SAR and FT properties of the host forest defined above (Jones 1977, Equations A.28-A.31; see also my Chapter 14). Large  larval survival rates are one principal mediator of budworm popula-  tion regulation (Morris, ed. 1963). Feeding, competition, parasitism, predation, and weather each exert potentially significant influences. All important feeding is done by large larvae. Per capita feeding rates, and the consequent impact on forest and budworm survival, are functions of both foliage and larval (i.e. competitor) densities. (Jones 1977, Equations A.1-A.5). Parasitism in the model inflicts relatively Tow mortality rates and even these become insignificant at epidemic density budworm populations (Jones 1977, Equation A.6).  Parasites are not treated as dynamic variables because,  under normal conditions, their numerical response is too slow to raise net parasitism rates before the outbreak collapse is already well underway. This simplification may not be justified under sustained semi-outbreak conditions  29 resulting from human management (see Part V), and predictions of the model under such conditions must be weighed accordingly. Large larvae are consumed by a number of vertebrate predators. Net impact of predation on large larval survival peaks at intermediate budworm densities, falling at low densities due to switching or search inefficiencies, and at high densities due to saturation of the functional response'(Hoiling, unpubl.; Clark, Jones, and Hoi 1ing, in press; Ludwig, Jones, and Holling, 1978). Numerical response, effects are minimal and can be represented implicitly in the model (.Jones 1977, Equations A.8-A.12), avoiding the necessity i  of including predators as dynamic state variables. Weather-related variability in budworm survival is concentrated at the large larval stage (Jones 1977, Equation A.7).  Unpublished studies by  Professor M.B Fiering of Harvard showed no statistically significant temporal autocorrelation in the historical weather patterns of New Brunswick, when weather was indexed according to biologically meaningful criteria employed by budworm researchers (Morris, ed. 1963). Weather effects therefore function as a simple stochastic driving variable. They are implemented in the model via a suitable "synthetic weather trace" of the sort used in hydrologic stream flow simulations (Fiering., 1967). Pupal and (predispersal) adult survival, sex ratio of adults, and fecundity all reflect the large larval feeding histories noted above (Jones 1977, Equations A.14-A.19). Adult dispersal occurs next in the annual cycle of budworm, redistributing the eggs produced in each local patch of host forest through space. A major focus of my own research has been the development of alternative hypotheses regarding adult dispersal., and the evaluation of these hypotheses with respect to overall spatial structure-population .dynamics relationships in the budworm system. Such issues move beyond the present local structure analysis and  30 occupy me for the remainder of this study, beginning in Part III.  For pur-  poses of my local structure analysis, I therefore simply assume that no adult dispersal occurs: for any local patch, all eggs produced by female moths at the end of one generation become the "newly deposited" eggs with which that local patch begins its next generation.. Because a "local" patch is by definition spatially homogeneous, this zero dispersal assumption is equivalent to a total dispersal assumption which mixes eggs throughout the patch while conserving absolute egg number. Note that only in this aspect of the local structure model do I depart from Jones' (1977) basic treatment. The local structure model I explore below is derived from Jones' by replacing his Equations A.20-A.26 with the single equation N = e^. g  2.5 Summary of Variables, Units and Typical Values In concluding this brief review of the budworm system local structure model, some attention must be paid to the question of measurement units. Whenever possible, the need for such essentially arbitrary decisions has been eliminated by adopting relative expressions such as PHOST, Ha and SAR, (see above).  In Part IV of this essay I take the process further by developing  dimensionless parameters to express certain critical aspects of system behavior... But in order to employ field data for parameterization and validation of the models, some standard measurement units must be adopted. These units and their appropriate handling become all the more important when the arbitrarily "local" model presently considered is given absolute spatial dimension and linked to other models for the full spatial analysis of Parts  IVTV.,  The key absolute unit of the local model is that in which the branch surface area polygons of Chapter 2.3 are measured. The dimension of this unit is necessarily (length)2, and a metric system unit of (meters)2 would be natural. In practice, all relevant field measures and all previous studies 2 have treated branch surface area in English units of "(feet) x 10" : in  31  common usage "ten-square-feet" or "tsf" (Morris, 1955). Since  1 tsf = 10.(ft) ~ 0.929(m) , 2  2  related measures differ by only 1% or so, a trivial amount relative to the order of sampling error encountered in this system. For most purposes, 2  "1 tsf" can be read for (l.m ) and vice versa. Throughout this study T maintain the traditional "tsf" unit in order to facilitate communication with field workers, to impart direct comparability with previous studies, and to eliminate one ubiquitous source of raw data conversion errors. Branch surface area polygons are therefore quoted in tsf units. Insect and foliage densities "per unit branch area" become.:numbers or units per tsf. The branch surface area densities become tsf's of branch area per unit land area (say, hectares) of host forest, and so on. In Table 2-1 I summarize the local structure variables introduced in this Chapter, along with the explicit units adopted for their expression. I include characteristic numerical values encountered in nature, which provide a convenient reference point for subsequent discussions.  }  Table 2-1f Summary df Local Structure Variables Variable Name Forest PHOST  Description  Units  .  Typical or Reference Values  Proportion of local land area covered by bw host species  area host species land area all land  New Brunswick mean about 0.40  ~a (a=l,2,...,75)  Prop, of local host species land area covered in host trees age (a)  area host species land in age class (a ) area host species land in all age classes  for stable age structure, which does not occur in nature, H = H, ='1/75, (a,b = 1,2,...,75) a p  Surface Area  branch surface area unit  tsf ("ten square feet").  1 tsf is the area of about 3 average balsam fir branches  a (a=l,2,... ,75)  age specific branch surface area density  tsf ha. host species land in age class (a)  H  a  o 2 0 = 43,200 a 4 0 = 59,300 a = 66,700 &0  SA  aggregate branch surface area density  tsf ha. host species land  Normally 0 < SA < 60,000; for stable age structure, SA~~ 53,000  SAR  relative aggregate branch surface area density  tsf/59,300 ha. host species land  Normally 0 < SAR < 1; for stable age structure, SAR ~ 0.9  Foliage f .u.  foliage unit  1 f.u. is the amount of new foliage on one tsf of branch area in absence of budworm  Fl  new foliage density  f.u./tsf  0< Fl< 1.0  F2  old foliage density  f.u./tsf  0< F2< 2.8  FT  total foliage density  J  f.u./tsf  0 < FT < 3.8 absolute bound 3.6<FT< 3.8 "light defoliation" 3.1.< FT.1'3.6 "moderate defoliation" FT ± 3.1 "severe defoliation", tree death may follow (see Part V).  Budworm NE  egg density  number/tsf  3000 epidemic  NL  large larval density  number/tsf  200 epidemic  NA  adult density  number/tsf  60 epidemic  NF  female density  number/tsf  30 epidemic  34  Fig.  2-1:  The p r o c e s s c y c l e f o r t h e b u d w o r m - f o r e s t s y s t e m . The inner r i n g represents the f o r e s t c y c l e , the o u t e r r i n g t h e budworm c y c l e . Ellipses indicate insect l i f e stages; a r r o w s show c a u s a l r e l a t i o n s h i p s among p r o c e s s e s a n d i n s e c t l i f e stage d e n s i t i e s . A f t e r Jones (1977).  35  CHAPTER 3  LOCAL STRUCTURE: ANALYSIS  3.1  Computing predictions of the local model  3.2  Recruitment function analysis  3.3  Equilibrium manifold analysis: derivation  3.4  Equi1ibrium manifold analysis: interpretation  3 6; 3.1  Computihg Pfedictions of the•:Local Model Once system structure has been described, there remains the problem of  computing its consequences as predictive theory.  Because the theory has been  assembled according to criteria of ecological relevance rather than mathematical elegance, it is usually necessary to perform these computations via a numerical simulation model. A sample simulation solution of the local structure model is given in Figure 3T1. Budworm densities are expressed as NL (large larvae/tsf), and forest condition by the aggregate variables for branch (SAR) and foliage (FT) density.  Initial conditions for this run reflect a young forest in endemic  condition. Note the general pattern of regular outbreaks at 40-odd year intervals (weather effects are constant rather than stochastically varying in this model run), lasting for 6-8 years on the local site. Foliage losses accumulate rapidly as budworm numbers rise, and are immediately followed by loss of branch density as the forest dies back to a young age structure. In comparison to behavior observed at large scales in real world budworm systems (see Morris, ed. 1963; Part V of this report), the predicted pattern is much too extreme. The model's larval densities are too high by a factor of two, the predicted forest mortality (through SAR) too extensive, and the associated defoliation too acute. These are all classic symptoms of tight, homogeneous coupling in preypredator models (Jones, 1974), a fact which is hardly surprising given the explicitly local structure of the present system. The predicted temporal frequency behavior of the local structure model, though more in line with historical observation, turns out to be highly sensitive to assumptions made about very low density (e.g. NL« 10"^) survival of budworm. With arbitrarily different assumptions than those made by Jones (1977) or, for that matter., by Morris (ed., 1963), predicted frequency behavior changes substantially. Endemic budworm may asreadily go' extinct. Such local  extinctions may occur in nature (Morris 1955; Greenbank 1956, 1957). But the important point here is that correspondence—or lack thereof—between.-.time series predictions of local structure models and time series observations of :  spatially structured nature may be largely devoid of meaning (cf. Steele 1975). There is clearly no possibility of "validation" in such comparisons. More important, there is little to suggest what the local structure will mean when integrated in more realistic and spatially structured theories to which I turn in subsequent Parts of this investigation. For both purposes, we need understanding as well as predictions, which is to say that we need to know why the model generates the solutions it does. Numerical simulation!is indeed the principal tool for generating solutions for realistic ecological models, but it is no more attractive for its inevitability,  To compute results of the sort shown in Figure 3-1 is often expensive,  always ambiguous, and occasionally wildly misleading. Solutions depend on initial conditions and complex joint parameter distributions in ways which virtually no amount of blindman's buff exploration (sensu HoTling-.and Ewing, 1971) can fully disentangle.  I adopt a somewhat more general procedure here, seeking  to "compress" rather than "simplify" the realistic structural models (sensu Hoi ling, ed. 1978, Chapter 6). The basic idea of this compression exercise is to employ the simulation model over single time steps only ^.calculating net generational rates of change and, particularly, net equilibrium values for selected variables as explicit functions of other structural variables and parameters of the model. These relationships are then plotted as  recruitment  functions''(cf .^Ri cker-1954) and;, equi 1 ibraumt. manifolds, (Thorn975),'which,. in turn, provide graphical insights into the "how" of the theory Vs- predictions. I summarize recruitment function and equilibrium manifold analyses of the budworm local structure model in the following two sections of this Chapter. These provide an essential foundation on which I ^base subsequent explorations  38  and explanations of local structure-dispersal interactions in. Parts III and IV. The model compression approach, used.here was introduced into the ecological literature by Jones  (1975,  and in Holling et al  1978,  Chapter  6)  whose original  papers should be consulted for both conceptual and technical background. 3.2  Recruitment Function Analysis Consider the local structure model for budworm outlined in Figure 2-1,  still assuming that no adult dispersal occurs. Once the parameters are fixed, specifying any combination of particular values for the state variables and weather in one year is sufficient to determine budworm populations for the next year. The local structure simulation model can compute this density very rapidly for a wide range of interesting variable and parameter values, because only a single iteration is required for each case. The results of one such set of calculations are presented as a single recruitment function ,in Figure, 3-2.  The. curve shows generationalcnate  of change for local budworm population size as a function of budworm density (NL).  Local structure (model) parameters are representative for New Brunswick  (see Jones, 1977, for numerical values), foliage density (FT) is held constant at its undefoliated level, .branch density (SAR) is that for a medium aged forest, and the weather input is set at its mean historical level. The theory embodied in the local structure model implies three budworm density equilibria under these conditions, one at each of the points where the recruitment curve crosses the line where log  (NL^/NL^JH  log(R) =  0.  The mid-  density equilibrium (NL°) is unstable for budworm, because densities slightly larger lead to continued increase (log R>0) and densities slightly smaller to continued decrease (logR<0).  The high and low density equilibria ( N L and +  NL , respectively) are stable by an analogous argument. It is intuitively plausible, and turns out to be correct, to identify the lower stable equilibrium  3 9'(.NL ') with endemic system condition, the upper stable equilibrium (NL ) with +  the epidemic condition, and the middle unstable equilibrium  (Hi .) 0  with a thresh-  old density of budworm which, once exceeded, allows the system to move to its upper epidemic level. This latter "release" point was originally suggested by Morris (ed., 1963) for the budworm system, and was explored by Holling (1959) and Takahashi (1964) in a more general context. The shape of the recruitment function directly-reflects action of . the key ecological processes included in the local structure model. It can be shown by selectively adding and deleting processes from the model and recomputing the curves that a basic survival rate, relatively independent of budworm density, is set by weather, a variety of constant factors, and forest density. The reduction of survival at high budworm densities is largely due to competition for food. Very low density survival is reduced by parasitism. The obvious "dip" in the recruitment curve at medium-low densities is in large part attributable to vertebrate predation on the insects (Peterman, Clark, and Holling, in press). The recruitment function also suggests how the moth immigration explored in later Chapters may alter the population dynamics implicit in the purely local structure. In the world represented by Figure 3-2, once the system reaches its endemic equilibrium at (NL ), it will stay there. Immigrants from outside the system can change this if they arrive in quantities which are sufficient to boost the local budworm density to a level greater than the escape threshold (NL°). This occurs "instantaneously" if the equivalent of (NL°- NL~) immigrants arrive in a single pulse. It happens more slowly but no less surely if the effective arrival rate is greater than the (locally determined) negative log (R) occurring between NL" and NL°. Parts IV and V of this report explore such immigration effects in detail. Unfortunately, the local structures and their dynamics are not in reality so simple as those of Figure 3-2, and a more complete picture of the "recruitment structure" of the local model must first  40 •  be obtained. The single surface of Figure 3-2 reflects budworm. recruitment rates in a world where everything but budworm remained constant. But it is abundantly clear in the time series predictions of Figure 3-1 that foliage levels (FT) and branch density (SAR) co-vary with budworm. The recruitment function formulation of local structure loses its attractiveness if we attempt to handle such joint variation simultaneously. But for the budworm system it is possible to make use of the fact that important variables operate at radically different time scales. In particular, though branch density (SAR) changes as the forest ages, this aging process occurs much more slowly than the associated budworm (or foliage) dynamics. It is therefore informative to proceed on the assumption that branch density will change very little if at all during the single budworm generation implied by a given recruitment function value. It follows that recruitment functions can be calculated meaningfully for each of a sequence of individual branch densities, without worrying about short term feedback of the recruitment rates (or associated budworm densities) on branch area. Plotted in the same format as Figure 3-2, budworm recruitment curves for forests of higher and lower branch densities reflect higher and lower recruitment rates, respectively. Figure 3-3 shows one such family of recruitment curves. The system dynamics originally displayed,as time series predictions in Figure 3-1 can be understood through Figure 3-3 in the following manner. Assume that the cycle "starts" with medium aged forest originally encountered in Figure 3-2 and marked as curve 'A' in Figure 3-3.  Low density  budworm populations willV.be drawn to the stable endemic equilibrium (NL~). As the forest ages, the relevant recruitment curve moves upwards towards 'B. 1  Since forest aging is slow relative to budworm population change, (NL") "tracks" the growing forest to higher and higher endemic larval densities.  Simultaneously,  the escape threshold density (NL°) is moving down. These two equilibria coalesce  41;: and vanish at some branch density between 'A' and 'B', leaving only the epidemic equilibrium (Nl-t) as a stable attractor of budworm densities. A budworm outbreak necessarily develops, during which the forest may age a bit more towards its maximum densities around 'C. If these were the whole story, the system would remain at the epidemic "equilibrium" forever.  In fact (and in the theory), such high insect  densities rapidly kill the forest, reducing its average age to that shown by curve 'D. This time it is the upper stable equilibrium that coalesces with 1  the escape threshold and disappears, leaving the system back at a low, endemic budworm density (which may or may not represent extinction). At such low densities forest recovery is possible, with the system gradually returning through 'E' to 'A, A recruitment curve for forest—as opposed to budworm—density 1  has been computed and shows this explicitly. Although explicit information on rates of density change between equilibria is available from the recruitment curves, attention in the above discussion focused on the evolution of the "equilibrium" points on those curves. This realization allows a further graphical compression to be carried out in which only the equilibria themselves are represented. The result is the "manifold of equilibria" discussed in the next section. 3.3  Equi1ibriurn Mahifbld Analysis: Derivation Equilibrium budworm densities for all the branch density conditions  spanned in by the recruitment functions of Fig. 3-3 are represented explicitly in the single equilibrium manifold of Figure 3-4.  I emphasize that this mani-  fold is calculated directly from the local structure simulation model in the same way as the recruitment functions discussed above: parameter weather, and foliage are held fixed as before.  values,  For each value of branch sur-  face area (SAR) and initial budworm density (NL ) a budworm generational f  recruitment rate log (-|/N'.) is calculated. The manifold is simply a graphN  t +  t  ical presentation of the SAR, NL^ combinations (the latter plotted logarithmically) for which those resulting recruitment rates reflect budworm equilibria of R = 1, log(R) =0. (Despite the similarity of presentational formats, absolutely none of the assumptions or inferences of mathematical catastrophe theory (Thorn, 1975) are involved in the "compressions" employed here.) As was the case for the original family of recruitment functions in. Figure 3-3, the manifold exhibits from zero to three budworm equilibria for each value of SAR.  Since slope information is lost in the manifold representa-  tion, the unstable equilibria (NL°) are marked here with a dashed line. For the single branch density (SAR) value represented earlier by the recruitment function of Figure 3-2, the arrows show the pattern of attraction experienced by nonequilibrium values of (NL): densities are drawn towards their local stable equilibria (NL~ or NL ), with the unstable escape threshold (NL°) +  acting as a "watershed" or boundary. The general shape of the budworm equilibrium manifold is typical of a wide range of other ecological systems, ranging from man-malaria to wolf-, caribou interactions, and including a large number of well studied insect systems (Southwood 1975; Jones and Walters 1976; Haber, Walters, and Cowan 19.76; Clark 1976; Peterman 1977; May 1977; Levin 1978b). As noted in my previous discussion of the recruitment functions, the upper equilibrium surface is defined by competition for food at high densities. The existence of the intermediate unstable surface (NL°) is critical to many of the spatial structure arguments advanced in Parts III and IV of this report. A similar topological property is implicated in most of the interesting theoretical results reviewed by Levin (1978a). The phenomenon can arise from any of several "autocatalytic" or Allee-type mechanisms which impart an increasing  survivorship rate with increasing population size (0kubp,1974; Segel and ;  Jackson, 1972; Segel and Levin 1976). In budworm and many of the other situations cited above it is due to negative second derivative of parasite/predator functional responses over certain ranges of prey (budworm) densities (Holling 1959, 1965, 1973; Oaten and Murdoch 1975). A lower equilibrium (NL~) will always exist at zero density. In the present case, a non-zero,.stable lower equilibrium also may arise due to search inefficiencies or switching of budworm predators at low densities.  Whether that  lower stable equilibrium is significantly different from zero at low branch densities remains, as noted above, problematical. In any event, eventual merging of the lower stable and unstable equilibria represents the fact that basic forest (branch) density has become so high that predation can no longer compensate for increasing survival of the budworm population.  This, once  again, is Ho.lling's (1959)~and-Takahashi's (1964) "escape;" 3.4  Equilibrium Manifold Analysis: Interpretation It is instructive to review the outbreak sequence originally described  with the recruitment functions in terms of the equivalent local structure equilibrium manifold.  Refer to Figure 3-5.  When forest branch density is  reduced below SAR by fire, logging, or budworm attack, only the lower stable 1  equilibrium surface NL"(SAR),which may-equal zero, exists.' Induced fluctuations are drawn to that surface. Once the forest ages sufficiently to support a branch density SAR>SAR", only the upper stable surface, NL(SAR), exists. +  Any positive density will be drawn to that surface, independent of immigration, and.l budworm outbreak ensues. This upper surface, though stable for budworm, is unstable for the forest. Epidemic densities kill sufficient trees to decrease SAR below SAR'. sarily collapses.  The upper equilibrium vanishes, and the outbreak neces-  44  I't is in forests with branch densities SAR <SAR<SAR" that immigration 1  can trigger outbreaks, as opposed to merely altering frequency behavior of the system. As the maturing forest achieves branch densities in this range, budworm tend to remain in endemic condition, attracted to the lower stable surface NL~(SAR). Now, however, random fluctuation or immigration of budworm densities equivalent to NL°(SAR)-NL"(SAR) = *A(SAR) is sufficient to raise local population densities above the escape threshold and into the outbreak domain. The necessary perturbation A(SAR) decreases as SAR increases. The qualitative significance of such immigration can be appreciated by recalculating the equilibrium manifolds which 'result when the local structure model is subject to arbitrary rates of external immigration. Results are plotted in Figure 3-6. For an "influx" rate of zero, the manifold is that of the basic local structure model already presented in Figures 3-4 and 3-5. For higher rates of "influx", two changes occur in the manifolds. The ambiguous nature of the basic model's lower equilibrium surface at low branch densities is immediately resolved: endemic equilibrium densities are now clearly positive, supported by the colonization of exotic budworm. More significantly, it is obvious that with rising net immigration rates, SAR" decreases dramatically and less dense (younger) forests support "spontaneous" outbreaks. SAR' remains reasonably constant, since immigration rates are not sufficiently high to affect NL . +  It follows that the range of branch densities (SAR" - SAR') over which  three equilibria exist declines rapidly with-]increasing immigration. This range has been reduced essentially to zero by the highest immigration rates shown in the illustration.  It is clear from Figure 3-6 that immigrants and  branch density can interact to determine whether and how an outbreak will occur in a given local area. Budworm immigrants will generally be dispersing moths from other localities, some distance away and already in outbreak. The general  4.5 spatial structure problem suggested by Figure 3-6 is thus one of spatial outbreak propagation, and the interactions of local manifold structure and dispersal processes which this involves. After developing the necessary dispersal process hypotheses in Part III, I return to the manifold compressions to analyze this propagation issue in Part IV. As a first step in this analysis, it will be necessary to explore the "rules" which real world budworm moths use in determining whether to abandon their breeding site and disperse. Once again, the local manifold compression sets the stage for effective exploration of this ultimately spatial question. The local model equi1ibriurn manifold of Figure 3-5 highlights two potentially conflicting functions of animal movement which concern me throughout the remainder of this report. Epidemic insect densities limit the duration of favorable growth conditions by killing their food source. A. budworm "family" remaining on the local breeding ground of its ancestors through several years of outbreak is more than likely doomed. There would appear to be a strong selective pressure to migrate in search of better habitat under such conditions, a circumstance remarked upon in both the earliest and later work on "adaptive" dispersal (Andrewartha and Birch 1954; Southwood, 1977). But compare the situation in a forest of intermediate branch density, i.e. SAR < SAR<SAR". Here the forest is capable of supporting a high rate 1  of budworm increase, but only if local population densities are boosted above the escape threshold NL°(SAR).  Local dynamics cannot accomplish this. And  the tendency to disperse which would seem adaptive at high densities now has opposite effect. What is adaptively "needed" in these endemic situations is a local anti-dispersal or even aggregative phenomenon: one which collects budworm from a large area of low density, extinction-liable populations and concentrates them sufficiently to trigger an outbreak with its attendent selective advantage for outbreak founders. Such aggregative adaptations are  46.'  in fact common in insect systems (Clark, Jones, and Holling, 1978) and are often presented as key determinants of their spatial population dynamic behavior (Taylor et-al. ,1977). I explore their role in the budworm-forest system in Chapter 13. An evolutionarily sophisticated budworm might be expected to have developed a variable dispersal response, tuned to make the best of both endemic and outbreak conditions. To what extent real world data suggest such a design, and to what extent the possible alternatives yield different population dynamics consequences, are the first subjects addressed in the next Part of this essay.  47- •  Fig.3-1: Local. modeIdynamics under zero migration. Predictions with no emigration or immigration (Type 0 exodus). Variables defined in Table 2-1.  FOLIAGE DENSITY (FT)  SURFACE AREA (SAR)  BUDWORM DENSITY (NL)  00,  4-9 .  CD O  L0G(  NL  )  Fig. 3-2: Recruitment function with three equilibria computed from the local model. NL" is endemic stable, NL is epidemic stable, NL° is boundary unstable. Abscissa is (NL.), R is (NL /NL ). Logs are base 10. +  t+1  t  Fig. 3-3:  Family of recruitment functions for various branch surface areas. Higher curves are for larger surface areas. See text, fig. 3-2.  51 3  2 +  CD CD I  0+  -1  BRANCH DENSITY (SflR)  Fig. 3-4: Equilibrium manifold for budworm larvae, computed from local model with zero migration. Ordinate is equilibrium density for large larvae (log base 10). Abscissa is branch density. NL , NL , as in fig. 3-2. Unstable equilibria (NL°) represented as dashed line. +  -  3  BRANCH DENSITY (SflR)  Fig. 3-5: Equilibrium manifold with outbreak trajectory. Manifold as in fig. 3-4. A (SAR) is amount of fluctuation or immigration needed to trigger outbreak. SAR is minimum branch density for which NL+ exists. SAR" is maximum branch density for which NL°, NL exist. 1  -  53"  Fig. 3-6:  Equilibrium.manifold with immigration. "Front" of box is same manifold as fig. 3-4, calculated with zero immigration. Other manifolds are recalculated under increasing amounts of immigration.  54  PART III DISPERSAL PROCESSES  Chapter 4  An Overview of Dispersal Processes  Chapter 5  The Exodus Response: Description  Chapter 6  The Exodus Response: Analysis  Chapter 7  The Displacement Response  Chapter 8  The Settling Response  55  CHAPTER 4  AN OVERVIEW OF THE DISPERSAL PROCESS  4.1  An overview of the dispersal process  4.i An Overview of the Dispersal Process Insect dispersal is now widely recognized as a distinct behavioral process through which organisms interact with spatial variation in their environments. As with other ecological behaviors, it can be analyzed via a functional components approach of the sort developed for predation studies by Hoi ling (1966). Reviews by Kennedy (1975, 1961), Southwood (1962), Dingle (1974, 1972), Johnson (1969, 1966), Schneider (1962), and Williams (1957) identify three basic components of flight dispersal, although terminology varies: the exodus response determines who leaves the local habitat, and under what conditions they do so; the displacement response determines the distance and direction moved following exodus; the settling response determines the "trivial" or "appetitive" search flight undertaken on termination of displacement activities and locates the animal in its new habitat. I analyze budworm moth dispersal with respect to each of these components in the Chapters which follow. There exists no previous history of modeling and analysis for these dispersal processes which is in any way comparable to Morris' (ed., 1963) extensive local structure studies. For each dispersal component, I therefore begin by considering its alternative possible functional forms, drawing where appropriate on relevant studies of other insects. Next, I mobilize available data and conjecture to parameterize the specified funcrn;:. tions for budworm. From this analysis I define a plausible range of alternative hypotheses for each component, and compare these with others proposed in earlier analyses and modeling efforts. A preliminary synthesis of the basic dispersal components in terms of- their interactions withRthe local structure of Part II is developed in Part IV. To keep the initial argument tractable, I confine discussion in Parts III and IV to the analysis of budworm moth dispersal, under spatially uniform meteorological conditions, reserving heterogeneity considerations for treatment in Part V.  ".  57 Long range dispersal.of budworm is carried out through flight of adult moths. The adults feed little if at all during their one or two weeks of life, and the significance of dispersal lies in its ability to "redistribute populations adaptively and periodically beyond the breeding place" (Johnson, 1966, p. 233)'. The broad elements of adult budworm dispersal may be summarized as folrr lows, drawing from the published and unpublished work of Greenbank, Sanders, Rainey, and Schaefer (Greenbank, 1954, 1956, 1957, 1963, 1973; Greenbank, Schaefer and Rainey, unpublished; Sanders, Wallace, and.Lucuik 1978; .Sanders, 1975; Sanders and Lucuik, 1975; Rainey, 1976; Schaefer, 1976). Emergence from pupae is governed by degree-day accumulations above a threshold. At a given location in New Brunswick 50% of the females will emerge during a peak three day period, with 90% or more emerging during the same week. Moths of both sexes live about ten days. Climatological differences across New Brunswick result in a phasing difference of seven days between peak emergence in the warm lowlands and cooler highlands. Adult flight dispersal in budworm is an essentially female phenomenon. Females carry the eggs, and it is dispersal-mediated redistribution of eggs, not parents, which concern us here. Females fly only after mating; no virgin has ever been found in active dispersal flight. Mate location is not a serious problem: females emerge two.or three days after males and "call" them with pheromones effective over a range of 30m.  This is substantially more c\  than the mean intermoth distance at even the lowest endemic densities. Most females are therefore mated on the first evening following emergence. Since male behavior tends to maintain local concentrations near the site of emergence, these matings are more than likely to be between animals with similar ecological histories. Multiple matings, and some long range (though perhaps accidental) male dispersal occur and some genetic mixing is therefore possible. The effects of such mixing are probably slight and, in any event, are  58 explicitly outside the limits of this study. I therefore confine all discussion in subsequent Chapters to the dispersal processes of female moths and to the eggs they carry. Females deposit eggs in the afternoon, beginning a minimum of 24 hours after emergence. Oviposition rate peaks quickly and then declines with time over the remainder of the animal's life.  Dispersal.of females takes place in  the evening, proximately triggered by falling light intensity. As opposed to the "flitting" activity of local movement within the canopy, true dispersal begins with an abrupt takeoff-and steep, deliberate climb into the wind. The climb proceeds until the boundary layer (Taylor, 1958) is breached, at which point the moths turn downwind and are displaced in the direction of the prevailing air flow. Takeoff time and downwind orientation are highly synchronized among individuals. On very calm nights with thick boundary layers, the moths may give up their ascent and return to local stands. Heavy rain and c: cold temperatures may also inhibit or prevent dispersal. A key question is to what extent some oviposition precedes dispersal, and how this is affected by local conditions. I consider this topic in my discussion of the exodus response (see Chapters 5 and 6, and Part V ). Flight displacement is influenced by the wind fields at flight altitudes, and by the insect's airspeed and duration of flight. Detailed radar and meteorological studies show that the exodus climb leads dispersing moths to cruising altitudes commonly in the 100 to 300m,.range. The vertical distribution is strongly influenced by temperature profiles and (perhaps) wind flow phenomena associated with atmospheric temperature inversions. Typical single flight displacements fall between 20 and 50km, with extremes of more than 300km recorded. Prevailing wind patterns for the appropriate season and hours give a net displacement bias to the northeast, but local effects and fine details of timing can alter this average picture considerably (see Chapters 7,-9 and Part V ).  59  Normal settling behavior involves a seemingly purposive plunge from cruising altitude down into the boundary layer. Few moths are recovered from inhospitable habitats, and it is likely that some degree of host selection is involved in settling. Density of foliage in the invaded habitat may affect subsequent oviposition success and/or the tendency to re-emigrate the following night. Few explicit causes of presettling female (or egg) mortality have been discussed in the literature, although it is clear that moths land on water and die (see- Chapters 8, 9.)'. Several meteorologically induced variants on normal displacement and settling occur. The literature documents many ihstances.:of"massive locallmoth'; depositions, usually associated with the passage of cold fronts and/or thunderstorms. Little quantitative study of the frequency and extent of such depositions has been carried out.  In addition to such frontal effects, a number of  local and zonal wind convergence phenomena have been suggested as agents of moth displacement and deposition. These are thought to exert a potentially important influence on budworm population dynamics and have been extensively studied in recent years. explored in Part V .  Certain broad implications of these studies are  60  CHAPTER 5  THE EXODUS RESPONSE: DESCRIPTION  5.1  Framework for the analysis  5.2  Independent variables of the exodus response  5.3  Oviposition before flight: FOVP  5.4  Proportion of females flying: FFLY  5.5  Direct estimates of FMIG  5.6  Summary  61  5.1, Framework for the Analysis  My analysis of local interactions in Part II suggests that a variable exodus response would be advantageous to budworm. The analyses of subsequent chapters show that this variability is indeed a critical determinant of population dynamic patterns. The same view emerges from general theoretical studies by Gadgil (1971), who shows that the sensitivity of exodus to habitat quality should be a critical determinant of fitness in environments heterogeneous in space or time. Finally, in an empirical review of insect migration strategies, Southwood (1962) concludes that "irruptive" species in general exhibit a facultative, habitats-sensitive exodus behavior.* Many mechanisms capable of maintaining a variable exodus response in insect populations have been documented. These include genetic (e.g., Dingle, 1968, 1974; Caldwell and Hegman, 1969), ontogenetic (e.g., Wellington, 1957, 1960, 1964, 1965; Leonard, 1970a,b; Edwards, 1969), and short term behavioral (e.g., Dixon, 1959; Chandler, 1969; R. Jones, 1977a,b) effects. Several of these mechanisms may operate together in a particular species and, except in very carefully studied situations, the detailed causal constitution of the exodus response is far from clear (Gould, 1977; Caldwell, 1974; Brinkhurst, 1963; Guthrie, 1959). The proportion of the local budworm population which emigrates from its breeding ground is determined by two factors: (1) the proportion of female moths which disperse at all, and (2) the proportion of their total fecundity oviposited before flight. Early field  observations suggested that the  *Gadgil, Southwood and most other workers with the exception of Myers (1976) have misconstrued the situation faced by animals, like budworm, which destroy their own habitats. This does not seriously affect their general conclusions,"but does alter some specific classifications.  proportion of the population undergoing exodus varies during the course  62  of a budworm epidemic, and may correlate with various measures of habitat and/or budworm "condition" (Wellington & Henson, 1950; Blais, 1953.)'. Local interactions (Part II) determine budworm and habitat conditions up to the time of moth dispersal. The exodus response specifies how these, in turn, determine the exodus rate. Adopting FORTRAN terminology consistent with the local interaction model (Jones, 1977), I formulate the several exodus response problems as follows:" FMIG (x,y) = FFLY(x) • [ 1 - FOVP(y)]  ,  (Eq. 5-1)  where FMIG (x,y) 5. function determining the fraction of total local fecundity emigrating; FFLY (x)'  = function determining .the fraction of local females emigrating;  FOVP (y)  = function determining the fraction of total fecundity oviposited before flight by those females wftich do fly;  x, y  = independent variables reflecting habitat and/or budworm conditions.  The different exodus hypotheses historically proposed for budworm are essentially structural variants of Equation 5*4.  Tn general, either or  both of FMIG.'s component functions may be constant, i.e., independent of values of variables x and/or y. The four resulting structural alternatives are depicted in Figure 5-1  v  (A)  Type A exodus response represents constant FFbY- and FOVP functions,  and therefore yields a constant value of FMIG. Type A exodus is equivalent to that Implied by passive diffusion processes (Skellam, 1973). Because of its comparative mathematical tractabillty, it is adopted in  most theoretical studies of dispersal (levin, 1976a; Okubo, 1978), Watt (1964, 1968) employed a variant of Type A exodus in his original spatial simulation o f budworm population dynamics. (B) In Type B exodus response, the FFLY function varies but FOVP is constant. Greenbank, Schaefer and Rainey (unpubl.) have hypothesized a Type B response for budworm. (Cj Type C exodus^ reflects constant FFLY but variable FOVP functions. This is the exodus structure originally hypothesized by the MFRC-IRE-irASA modeling group, and derives largely from the ontogenetic arguments of Wellington and Henson (1947), Henson (1950j, Blais (1953), Campbell (1962) and Greenbank (1963). These hold that; smaller, less fecund and presumably starved females, are less likely to oviposit before flight than their larger comrades. The budworm model analyses of Walters and Peterman (1974), Baskervnie (1976) and Clark, Jones and Holling (.1978) are all based on Type C exodus response formulations. (D) Type D exodus response Incorporates variability in both FFLY and FOVP. In practice, it is a hybrid form o f Types B and C. Greenbank (1973) and Sanders (Sanders and Luculk, 1975; pers.  comnK.)  have raised  the possibility of Type D exodus in budworm, but no previous studies have evaluated the resulting hypothesis in a population dynamics context. It will also be useful to introduce comparable terminology for the zero migration case treated in the local structure analysis of Part II. I refer to zero.migration as "Type 0" exodus below. Preferences among existing budworm exodus hypotheses result largely from conflicting interpretations of the same data. In the next four sections, I assemble and analyze available data to identify the basic structural exodus. form(s). most consistent with, observation and to define parametric structures and values for the representation of specific hypotheses.  5.2  Independent yariab.1 es; of the'.exodus response The empirical exodus studies I discuss in this section utilze a wide  range of "independent variables" (the x and y terms of Eq. 5-1), including budworm densities at the larval, pupal and adult stages, foliage density before and after feeding, and fecundity.  The choice of variables  usually reflects technical convenience more than firm causal hypotheses. Since all these variables are correlated in normal Budworm population dynamics, the most expedient measure Is often satisfactory for predictive purposes as well. This is not necessarily the case, however, and an important function of analysis is to determine under what, if any, conditions exodus predictions are sensitive to the particular choice of independent variables. A structural analysis of local interaction models for budworm (Part II; Jones, 1977j Stedinger, 1977; Morris, ed., 1963) separates the potential independent variables for exodus into two groups. The first is essentially ontogenetic In character, encompassing larval density, prefeedlng foliage densities, and fecundity.  The second group  consists of pupal and adult densities and forest defoliation.. In present models, all of the latter group are derived from larval and foliage densities, but additionally reflect.Independent effects of bird predation, weather, and/or insecticide spraying (see Fig. 2-1 ;)•'. ,It follows that an exodus response actually governed by group one variables cannot be unconditionally predicted by group two variables, and vice versa,  (The details of this argument are implicit in Eqs. A.01-A.18  of Jones, 1977]) Within either of the structurally defined groups of variables defined above, relationships are simpler.. For predictive purposes, the pupal, total adult, and female densities defined in the MFRC-IRE-IIASA models are Interchangeable: conditions which determine any .one are  necessary: and sufficient, to determine all the others (Eqs. A.13, AJ4,  65 A.15,  A.l8 of Jones,: 1977), and more than sufficient to determine defoliation (Eq: ,A.4 of Jones, 1977). Relationships within the first group merit more attention. Fecundity is an easily measured indicator of the individual female budworms physio1  logical condition, and is therefore a natural choice as an Independent variable In ontogenetic studies. Lab and field studies on budworm have demonstrated that well fed, uncrowded larvae produce large pupae and large moths"with high, fecundities. Crowding and food stress have the opposite effect. Data from Blais (1953), Campbell (1962), Miller (in Morris, ed., 1963) and Sanders and. Luculk (1975) yield the relationships summarized in Figure 5-2 and employed in the local structure model (Eqs. A.16-A.17..:of Jones, 1977)., ... Fecundity..is:accurately-  by regression against pupal or  adult size (Miller, 1963; Outram, 1973; Harvey, In Sanders and Lucuik, 1975; Thomas, 1978). Tt Is therefore possible to estimate original fecundities before, after or during the dispersal process from collections of individual moths.. No other Index of larval ontogeny has remotely comparable sampling properties,, and fecundity is consequently the independent variable of choice in most of the studies 1 review here. It is important to emphasize, however, that although, fecundity is the only indicator of ontogeny.readily available to the entomologist, it is not the only one available to the insect. The subtle role of ontogenetically altered hormone levels in regulating various life history phenomena (Wigglesworth, 1954) has been shown to apply to dispersal processes as well (Southwood, 1961 i Kennedy and.Stroyan, 1959; Dingle, 1974). The causal role of any ontogenetic variable In budworm exodus remains one alternative hypothesis, not a premise, for the present study.  66 5.3 Oviposition before flight: FOVP As for most flight dispersing insects, both: emigration and oviposition occur early in the adult period (Johnson, 1966", 1969). The relative timing of the two processes determines FOVP and constitutes a life history property of potential adaptive significance and sensitivty (Southwood, 1962, 1977; Dingle, 1972, 1974; Johnson, 1966,. 1969; and. Gould, 1977). Budworm oviposition schedules are.reported by Campbell (1962), Outram (1971) and Sanders and Luculk. (1975).. Field and laboratory results are consistent, yielding the oviposition. schedule shown in Figure 5-3. Direct studies of ovlpositlon-flight sequencing are rare but important. Field observation by Greenbank. (1973) and laboratory experiments by Sanders and Luculk. (1975) suggest that the average female is relatively inactive until the evening of the day following emergence. She is therefore likely to oviposit her first major egg.batch (about half of her fecundity, cf Fig. 5-3) at her breeding site. Field studies by the Budworm Dispersal Project at Chlpman in 1975 monitored pupal emergence and female exodus from the same stands of trees through'the dispersal season (Greenbank, pers. comm.; Schaefer, unpubl ..).„. 1 plot these data as scaled cumulative proportions in Figure 5-4.  Exodus lags emergence by a mean of 20 hours  for the period of significant activity.. Due to phasing, problems in the daily counts, this is rather less than the 30 hour lag of Sanders and Lucuik's (1975) laboratory results. Tt is nonetheless consistent with a second rather than a first or third night exodus, and again suggests that the first major egg batch, is oviposited locally. Wellington and Henson (1547) and Henson (1950) note that fully gravid budworm moths do not fly until.ovipositing some of their eggs. Blais (1953) argued that this might,be true for large moths produced under endemic conditions, but that small, less fecund moths produced  under conditions of food stress "fly in an upward direction soon after emergence" (Blais, 1953, p. 448). This pair of observations led Campbell (1962 and in Morris, ed. 1963) and Outram (1971, 1973) to explore an ontogenetic hypothesis linking food quality, fecundity, wind loading,, and dispersal tendency. Their studies confirm the relationships of Figure 5-2, but fail to show hypothesized trends in either reduced wing loading or higher dispersal rates in small females. Sanders and Lucuik (1975) discuss laboratory studies of the relationship between fecundity and oviposition-flight relationships (FOVP) in budworm. Unfortunately, their experimental design could not distinguish "trivial" (sensu Southwood, 1962) from true exodus flight. FOVP estimates from these data are therefore lower bounds to actual values expected in the field. Sanders and Lucuik found that the proportion of female moths ovipositing before flight increased with'increasing total fecundity (Fig. 5-5a). However, the proportion of their total fecundity oviposited before flight decreased with total fecundity (Fig.. 5-5b). The proportion of the population fecundity oviposited before "flight" is the product of these two opposing functions. This is plotted in Figure 5-5c and shows a fecundity-independent and constant value of about 0.17 as the minimum population FOVP. New Brunswick researchers have made extensive field collections of moths in exodus, dispersal and settling flight (Greenbank, 1957; Morris, 1963; Outram, 1973; Thomas, 1978; and Greenbank, Schaefer and Rainey, unpubl.). For each female, they counted eggs still carried and estimated original fecundity from body size regressions. I plot available data as FOVP versus original fecundity in Figure 5-6.  Sanders and Lucuik's (1975) lab results  are superimposed as probable lower limits. Records published by the Forest Insect and Disease Survey (FIDS 1950 - 1977) and Webb, Blais and Nash (1961) show that most of the populations included in Figure 6T6 originated in regions  68 with moderate to severe defoliation (FT < 3.5). More detailed information is not available. Many of the populations were also sprayed with insecticides during larval development. These data are therefore not sufficient to test possible relationships between FOVP and insect density or foliage condition. Note, however, that no field sampled population has a mean FOVP < 0.5. Furthermore, all populations collected in exodus have a mean FOVP < 0.6.~ Higher values come only from immigrant populations which may have oviposited after landing. Many of the immigrants represented in Figure 5-6 were captured in light traps which are known to traumatize moths and. induce substantial premature oviposition on trap walls (McLeod, pers. comm.). The field data therefore are consistent with the earlier hypothesis based on oviposition and emigration schedules : most females which disperse at all fly the evening of their second day, after oviposition of the first half of their fecundity and independent of the moth size or site conditions. 5.4 Proportion of females flying: FFLY Early views of insect flight dispersal treated the process as an involuntary extension of local appetitive search (Williams, 1957). This was the initial interpretation applied to budworm (Greenbank, 1957, 1963). Southwood (1962) considered budworm in his review of insect migration strategies. Drawing mainly from Greenbank's descriptions, he represented budworm as a species for which increased dispersal tendency under outbreak conditions resulted fortuitously from increased "jostling" activity and extended local searches for scarce oviposition sites. Subsequent work on a number of species has given rise to a much more purposive, adaptive view of dispersal (Kennedy, 1975, 1961; Dingle, 1974, 1972; Johnson, 1966, 1969). Recent behavioral studies by Greenbank and his colleagues (Greenbank, 1973) firmly establish budworm among the purposive dispersers. The comparative  morphometryc analyses of Outram (1973) further suggest that budworm is a well designed flight machine in comparison with other dispersing Lepidoptera. Laboratory studies are as yet of little use for quantifying budworm FFLY rates, due to their previously mentioned inability to differentiate trivial from true migratory flight. Nonetheless, Sanders and Lucuik's (1975) work shows that essentially 100% of all females engage in some sort of flight, regardless of fecundity. Field counts of observed exodus flights have been compared with local pupal densities to estimate FFLY under a range of population and habitat conditions (Greenbank, Schaefer, and Rainey, unpubl.). A number of technical problems, including the possible re-emigration of immigrant females, contribute to a high variability in the results. The range of FFLY for the ten siteyear combinations available is zero to more than one (reflecting invasion and re-emigration, or sampling error). However, no moths at all are observed to exodus at very low endemic densities, although males and mated females are observed in local "trivial" flight. The relevant data (Greenbank, 1973; Greenbank, Schaefer, and Rainey, unpubl.; Sanders, unpubl.) are summarized in Table 5-1. Current defoliation in the zero-exodus sites (1-4 in the Table) was light to nondetectable (FT >_ 3.55). In areas (5 - 10) from which substantial exodus was observed, current defoliation was above 70% (FT <_ 3.10). No trend of FFLY with defoliation or population density is detectable in the data for these latter sites. The ground observations are confirmed by helicopter and aircraft sampling. Greenbank (1973) found no moths at typical dispersal altitudes above lightly infested areas resembling those described in Table 5-^1. A large "removal" experiment conducted in 1975 provides additional evidence on FFLY (Miller, Greenbank and Ketella, 1978). A 4000 ha. block in west central New Brunswick was repeatedly treated with larval insecticides, reducing  70 resident pupal populations to  6%..of  the unsprayed control area. Local female  density on the treated block averaged 0.86 females/tsf. Exodus counts showed a very low proportion of resident females ever flying, with FFLY rates six to seven times lower than those in the untreated area. This supports the general picture developed earlier of suppressed exodus tendency in areas with low moth density. It also suggests that it is the density of females, rather than that of the early larval stages, which determines FFLY. Further evidence for reduced flight tendency at very low densities is given in the following section. In summary, the immediate stimulus for budworm exodus flight cannot be definitively identified from the data presented here. Actual physical "josr tling" of the sort discussed by Southwood (1962) seems unlikely at the low densities indicated in table 5-1. Furthermore, female moths exhibit no preexodus "searching" flight, and therefore, cannot physically survey the density of other females, or oviposition sites (Greenbank, 1973). However, a density response in adult movement tendency is mediated by chemicals rather than physical contact in such diverse groups as ichneumonid parasites (Price, 1970, 1972) and locusts (Nolte, et al., 1970). Sanders (in press) has raised the possibility that sex pheromone might serve a similar stimulating function for female budworm. Ontogenetic factors associated with larval development could conceivably be implicated in FFLY causation, but this seems less likely than in the determination of FOVP: Figure 5-2 shows that fecundity and, presumably, other nutrition-based indicators of ontogeny, change little if at all over the range of low densities covered by Table 5-1. 5.5 Direct estimates of FMIG Greenbank (1963) used 61 plot-year data sets from the Green River Project to compare actual postdispersal egg counts with counts expected in the absence of dispersal. (The expected values, were derived from measured pupal densities  ?  71  plus the pupal size versus fecundity regressions noted above.) The ratio of observed to expected egg densities represents the net gain or loss rate due to dispersal, with emigration and immigration potentially confounded. For the Green River studies, however, data sets involving mass invasions can be identified and analyzed separately (Greenbank, 1963). The remaining observations (representing 50 of the original 61 data sets) predominantly reflect emigration losses. Any hidden immigration will inflate the observed egg counts, leading to underestimation of the fraction migrating (FMIG). Greenbank's observations are summarized in Figures 14.1-14.3 of Morris (ed., 1963) and cover a range of moth densities from less than one to more than twenty females per square meter of branch area. At very low moth densities (< l./m ) expected counts are less than or equal to those observed, implying FMIG = 0. At higher densities, the mean FMIG = 0.51, with 80% of the plot-year combinations yielding FMIG between 0.25 and 0.75.  There is  no trend of the mean with increasing moth density above the one moth/m threshold. MacDonald (in Morris, ed., 1963) conducted a similar analysis of data from areas subjected to insecticide spraying. Once again, at very low moth densities, plots experienced no change or net gains through migration, again suggesting FMIG = 0 (Fig. 26.1 in Morris, 1963). At higher densities, as in the unsprayed case, a fairly tight regression about a constant FMIG rate was obtained. (The numerical values are not directly applicable due to differences in the sampling procedure.) 5.6 Summary Previous research on the exodus response in budworm has not been directed at critical and comprehensive identification of the variables, functional forms or even parameters relevant to the ecological processes involved. It  is therefore not surprising that despite the substantial investment of time and research implied in the studies reviewed here, no strong inductive case can presently be made for any particular exodus hypothesis. The series of workshops which led to the MFRC-IRE-IIASA model reached a Type C exodus consensus, based on extensive field experience of a large number of entomologists. That hypothesis holds that all females always disperse (FFLY = 1 . 0 ) , but only after depositing an ontogenetically determined fraction of their eggs on the breeding site, i.e., FOVP = f (fecundity). The evidence assembled in this Chapter does not require such an ontogeneti dependence in FOVP. Results from the comparison of flight vs. oviposition schedules and from direct examination of moths captured in exodus both suggest a mean FOVP value of about 0 . 5 under a wide range of conditions. On the other hand, very few data are available for the high density, low fecundity conditions under which the postulated dependence would take effect. The MFRC-IRE-IIASA hypothesis of constant FFLY = 1 . 0 is supported by most of the evidence I have reviewed here. In particular, under non-endemic conditions, the direct FMIG observations give a mean value of 0 . 5 over a large range of insect densities (Chapter 5 . 5 ) . Rearranging Eq. 5 - 1 , this requires FFLY  (x)  = 0.5/(1  - FOVP (y)) .  (Eq.  5-2)  Since FFLY and FOVP are individually bounded between zero and one, this in turn implies .FFLY. >_ 0 . 5 and FOVP <_ 0 . 5 .  The actual data of Fig. 5 - 6 give  a range of FOVP values such that 0 . 5 <_ FOVP £ 0 . 6 .  The only consistent Values  for all three variables under non-endemic conditions are therefore FMIG = 0 . 5 , FOVP = 0 . 5 , FFLY = 1 . 0 .  Blais'  (1953)  further suggestion that high density,  low fecundity conditions may give rise simultaneously to increase in FFLY and decrease in FOVP is inconsistent with the constraint of Eq. 5 - 2 .  The tendency of moths to fly under endemic conditions is more problematical. Few relevant data of any kind exist. The suggestion of an FFLY threshold in Table 5-1 is based on the failure to observe exodus of a.very low density of moths and thus not particularly reassuring. However, the near zero FMIG values occurring in the Green River data reflect positive observations. They require (via Eq. 5.-1) that either FFLY equal zero or FOVP equal one at such densities. The former possibility is consistent with other data reviewed here, while the latter is not. Finally, note that an FFLY threshold of about one female/tsf is suggested by both the Green River FMIG data and the direct FFLY observations (Table 5-1) of twenty years later. This concurrence is unlikely to be fortuitous, and militates strongly for the inclusion of a variable FFLY hypothesis among those to be explored in my subsequent spatial structure analyses. Considering the arguments advanced above, I propose in. Fig. 5-7 some plausible parameterizations for each structural "Type" of the basic exodus response components defined in Eq. 5-1. The Type C values reflect those employed in the MFRC-IRE-IIASA model, described in Jones (1977). The Type B response is that most strongly suggested by evidence reviewed here. The Type A values are derived by eliminating the low density FFLY threshold of the Type B formulations. Type D incorporates that threshold in the basic Type C formulation. The equivalent algebraic treatment is given in the computer program of the full spatial simulation model. All of the particular parameter values implied in Fig. 5-7 are subject to an unknown amount of estimation error and natural variation. These uncertainties cannot be further reduced without recourse to new experimental studies and data. My goal here is to assess their present significance by showing how the alternative structural forms and parameterizations differ in  their effects on overall patterns of spatial population dynamics. This evaluation then serves as a basis for simplifying and structuring the subsequent analysis and for discussing priorities of future research. Full evaluation of the alternative exodus hypotheses must await specification of the other dispersal components and their incorporation in a spatial population dynamics model (Part V). But to interpret that full spatial analysis, it is first necessary to understand the properties of local sites as sources and sinks of dispersers. In both of these cases, it is a local interaction model including the exodus response, rather than the zero migration case of Part II, which provides the relevent framework for analysis. It is therefore important to determine whether, and how, the various exodus alternatives affect the previous zero-migration picture of local site behavior. To do this via a classical "sensitivity analysis" (Waide and Webster, 1976; Tomovic, 1963) is unrewarding, due to the large number of parameters, the high uncertainties and the essential nonlinear ties characterizing the system. Needed is what Overton (1977) has called "succinct behavioral summaries" of the effects of biologically meaningful classes of parametric and structural change. The recruitment functions and equilibrium manifolds introduced in Chapter 3 provide a powerful method for obtaining such summaries (Clark and Hoi 1ing,  in press; Moiling, et al.,. 1978). In the following Chapter, I use  them to analyze the impact of various exodus hypotheses on the local structure model.  Table 5-1f Local population estimates for sites with observed proportion of dispersing females (FFLY)  Site  Local Densities pertsf Branch Area Female Moths Pupae Larvae (NF) (NP) (NL)  Female Moths/ Tree (approx)'  FFLY  1  .3  .05  .02  1.5  0  2  2.5  .5  .20  10.0  0  3  10.6  1.9  .81  4  13.3  2.4  1 .02  77:0  5-10  >40.0  >7.2  >3.06  >230.0  61.0  0  0  >.25  76  FFLY CONSTANT  VARIABLE  TYPE A  TYPE B  TYPE C  TYPE D  FOVP  Fig. 5 - 1 : Exodus types as function of FFLY and FOVP structural components; see text.  77  Fig.  5-2:  F e c u n d i t y as a f u n c t i o n o f l a r v a l d e n s i t y . Different curves are f o r v a r i o u s l e v e l s o f t o t a l f o l i a g e ( F T ) .  78  EXPONENTIAL DECAY (50 %/DAY)  1.0-  D(t)  .6.4\  .2 _  J  I  \  I  3  _1  L  5  DAY (t)  Fig. 5-3: Oviposition schedule for budworm moths as a function of days since emergence. D(t) is proportion of total egg complement oviposited on day (t) based on lab studies of Sanders and Lucuik (1975). Curve is theoretical function of 50%/day exponential decay rate, beginning on day (1).  79  1.0,  TIME (hrs)  Fig. 5-4: Pupal emergence and moth exodus schedules. Scaled cumulative values for Chipman 1975, adjusted to common zero. Horizontal distance between the curves estimates the number of hours by which exodus follows emergence; see text.  80  Fig. 5-5: Oviposition-fecundity relationships for laboratory data of Sanders and Lucuik (1975). Abscissa is fecundity, in eggs/ female moth, (a) gives proportion of moths which oviposited at all before flight, (b) gives the proportion of total fecundity oviposited before flight by those moths identified in (a), (c) is the product of (a) x (b), giving the proportion of total fecundity oviposited by the moth population before flight. These figures are minimum estimates for the field situation; see text.  82  Fig. 5-6: Oviposition-fecundity relationships for all data. Abscissa is fecundity in eggs/female moth. Left ordinate is fraction of total fecundity oviposited before flight; right is fraction carried on the flight. Roman numerals show expected cumulative fraction oviposited I, II,... days after emergence, based on lab results of fig. 5-3. (A) points are for moths trapped during take-off (Greenbank, 1973; Greenbank et al., unpublished); (•) points are for moths trapped as immigrants (source as above); (•) points are for moths with uncertain trapping histories (Greenbank, 1957); (•) are lab results from fig. 5-5b, representing only those females who oviposited before flight; the solid line therefore gives one lower limit expected for field results; (o) are lab results from fig. 5-5c, representing means for all females; the line therefore gives absolute lower limit expected for field results.  83  Fig. 5-7: Exodus Types, by parametric forms used in the models. Four Types (A-D) are those structurally identified in fig. 5-1. For each Type, the right frame defines the fraction of total fecundity oviposited before flight by those moths who fly, as a function of mean moth fecundity. The center frame defines the fraction of the moth population flying as a function of female moth density. The left frame is the product of the other two, giving the fraction of the total egg complement generated on the site which migrates away in exodus flight. Fecundity is plotted on the abscissa; where FFLY is variable, limiting values and an intermediate curve are represented. Parameter values derived from data in text.  TYPE  A  85  CHAPTER 6  THE EXODUS RESPONSE: ANALYSIS  6.1  Exodus rates, recruitment functions and equilibria  6.2  Equilibrium manifolds for the alternative exodus hypotheses  6.3  Equilibrium manifolds under variable FFLY  6.4  Equilibrium manifolds under variable FOVP  6.5  Implications for population dynamics: the endemic domain  6.6  Implications for population dynamics: the epidemic domain  6.7  Summary  86 6.1 Exodus Rates, Recruitment Functions and Equilibria The exodus response partitions each generation's egg production into one fraction which emigrates and one which remains on site and recruits to the next generation.  The FMIG term of Equation 5-1 determines the relative  fraction in each category for particular conditions. The larvae-larvae generation cycle adopted in the local structure analysis of Part II is such that FMIG effects occur "after" all other insect density dependent interactions in the generation cycle have taken place. As a result, the recruitment rates of the zero migration ("Type 0" exodus) structure are simply related and those for the structure when exodus but not immigration effects are included. Using 'z' and 'e' subscripts to indicate the zero migration and exodus cases, respectively: log[R (NL(t))] = log[NL (t+l)] - log[NI_ (t)], z  z  z  log[R (NL(t))] .= log[NL (t+l)] + log [l-FMIG(x,y)] - log[NI_ (t)] , e  z  z  and therefore log[R (NL(t))] = log[R (NL(t))] + log[l-FMIG(x,y)] e  z  .  (Eq.6-1)  Equation 6-1 shows the.critical.role-of exodus as a local. mortality agent which reduces-the generation recruitment rate indirect proportion to the applicable FMIG value. This is illustrated in Figure 6-1 for a simple case of Type A exodus (viz., constant FMIG = 0.5, cf. Fig. 5-7a). Due to the form of Equation 6-1, the recruitment function for Type A exodus (R ) is just g  a rescaled version of the zero migration function (R ). £  Equation 6-1 also shows that exodus'i,eqtnld;bci<a (i.e. conditions such that log[R (NL(t))] = 0) will occur under conditions where previous values of log[R (NL(t))] were numerically equal to -log[l-FMIG(x,y)]. z  The change in  the equilibrium densities themselves (i.e. NL , NL°, NL") is not similarly +  87 regular, due to the curvilinear nature of the recruitment function. These equilibria are most easily and generally calculated by introducing appropriate parameterizations of the FMIG function (Eq. 5-1) into the zeromigration model of Part II. This yields an exodus model which can be used to compute appropriate local dynamics, recruitment rates, and equilibria as before. Details of structure and sequence for the calculations are given in Jones-(1977). Figure 6-2 shows an equilibrium manifold for the constant (Type A) exodus rate hypothesis of Figure 5-7a, and compares the analogous zero-migration manifold originally presented as Figure 3-4.  The following general effects  of increased FMIG are apparent: (1)  For a given branch density, stable equilibrium densities for budworm (i.e. NL , NL ) decrease. Unstable equilibrium densities +  (NL°) increase. The width of the endemic "pit" (A = NL.. - NL") 0  consequently increases. Larger fluctuations or immigration pulses are therefore needed to move the system from its endemic to its epidemic attractor. (2)  There is an increase in the minimum branch density necessary to sustain non-zero endemic budworm populations (i.e. SAR "); to in1  duce spontaneous epidemics (SAR"); and to permit immigration or fluctuation induced epidemics (SAR ). It therefore requires a 1  denser, and generally older, forest to provide budworm opportunities for persistence and outbreak. All of these effects are intuitively consistent with the view of exodus as a local mortality factor..  88 6.2 Equi1ibrium Manifolds for the Alternative Exodus Hypotheses It was convenient in the preceding section to explore exodus response effects in terms of a constant value for FMIG (Type A exodus).  I shall now  consider how the more realistic variations of FMIG with budworm density and/ or habitat conditions (Type B-D exodus) affect local structure in the budworm ecosystem. In particular, I show how the four structural classes of exodus presented in Figure 5-1 can be further simplified in terms of their implications for system behavior. The four specific exodus hypotheses defined in Figure 5-7 provide useful reference points. Their equilibrium manifolds are plotted individually in Figure 6-3.  These are superimposed in Figure 6-4, along with the manifold for  Type 0 (zero migration) exodus. Once again, only the plane of the manifold for full foliage (FT = 3.8) is shown. This is the appropriate portion of the manifold for analyzing the outbreak initiation and endemic situation which are my major concern here. Except as explicitly noted, all of the following arguments apply equally to the FT < 3.8 region, subject to rescaling of the sort implied in Part II. These figures succinctly capture many of the results which would be sought in a standard "sensitivity analysis, ' and do so from a global rather 1  than a local perspective. As demonstrated earlier in the analysis of Figure 6-2, the general effect of an increased FMIG is to shift the equilibrium manifolds "to the right'." The zero migration (Type 0) case thus bounds possible manifolds "on the left'.'.' The absolute "right" bound is theoretically 1  at infinity, for the case of 100% exodus. In terms of the budworm hypotheses reviewed here, the constant 50% exodus rate of the Type A hypothesis described in Figure 5-7a approximates a realistic limiting case. When the exodus response varies with budworm density and/or habitat *•  ~  .  .  .  .  .  89 conditions (Type B, C, D exodus), the familiar shape,of the Type A (and zero migration) manifolds may be altered in ways discussed in the following section. 6.3 Equilibriurn Manifolds under Variable FFLY Consider first the variable FFLY component characterizing Type B and D exodus, as illustrated in Figure 5T7. The manifolds for these exodus hypotheses (refer to Fig. 6-4) exactly follow the Type 0 (zero migration) manifold up to a critical larval density of NL". Above this density they break away 1  from Type 0, only to intersect and follow Type C and A manifolds above a second critical density NL". In other words, with the parameterizations of Figure 5-7 the equilibrium properties of local structures incorporating Type B or Type D exodus hypotheses are both equivalent to previously analyzed zero migration systems for equilibrium densities below NL". Similarly, equi1  librium properties of the Type B and D hypotheses are equivalent to those for Type A and C respectively at equilibrium densities above NL". Only for intero  mediate equilibrium densities (NL" <NL<NL") do the variable FFLY hypotheses 1  take on unique equilibrium properties of their own. The manifold configurations described above are easily explained in terms of the FFLY structures shown in Figure 5-7. Recall that the variable FFLY component consists of a low density threshold (NF'") below which no dispersal occurs and a high density threshold (NF") above which FFLY takes on a value of unity and further changes in FMIG must occur via changes in FOVP. (The piecewise-1inear as opposed to curvilinear structure of the FFLY relationship is clearly artificial,.as is the zero-one nature of the extremes. But these conventions are convenient and data are not good enough to warrant more sophisticated treatment.) It is easy to show via equations of the local structure model (Jones 1977) that NL" and NL" are exactly the larval densities which 1  will produce NF" and NF" females when foliage and forest density, weather 1  90 effects, and management activities are specified. That is, the "breakpoints" of the FFLY functions are directly reflected in "breakpoints" of the related equilibrium manifolds. Combining the two previous paragraphs yields valuable insights for the analysis of exodus effects on local system structure. The existence of a plausibly variable FFLY component gives rise to an equilibrium manifold for which the configuration at relatively low densities approaches that of the zero migration case, and at relatively high densities approaches that defined under hypotheses for^a fixed FFLY. At either extreme, analyses based on these simpler exodus assumptions will suffice for understanding the variable FFLY case. The densities over which the transition occurs from zero to high migration manifolds are determined by the parameters of the FFLY function. For the "best-guess" values implicit in Figure 5-7 and used in generating the manifolds of Figures 6-3 and 6-4, it is clear that virtually the entire stable endemic (NL~) and unstable escape (NL°) surfaces of the variable FFLY (Type B and D) manifolds are identical to the zero migration (Type 0) manifolds. The necessary conditions for outbreak initiation and propagation [i-e. NL°(SAR)  A(SAR)  =.  - N L ' ( S A R ) ] are thus essentially the same as those reviewed in the  local structure analysis of Part II. On the other hand, the entire stable epidemic (NL ) surface is strongly affected by emigration losses. It drops +  far below the upper epidemic surface of the zero migration manifold, eventually to parallel the surface of the appropriate fixed migration case. It follows that the consequences of outbreak initiation under variable FFLY may be radically different from the zero migration case. The above noted qualitative effects of variable FFLY on the local structure of the budworm system are not sensitive to plausible quantitative changes in FFLY structure or parameter values. This is ascertained by calculating and  91  comparing equilibrium manifolds for alternative FFLY formulations suggested by the data analysis of Chapter 5.4.  As one example, the most interesting  issue in this regard is the value of NF'", i.e. the female moth density at which the FFLY function first begins to rise. The data of Table 5-1 show that NF", if meaningful at all, is almost certainly less than 3 females/tsf. 1  If the qualitative character of the resulting manifold is strongly influenced by the exact values which might be taken by NF" below this level, then ade1  quate analysis of the system would require an extremely difficult program of field studies to quantify low density FFLY. Calculating the manifolds for Type B exodus under several values 0 <_NF"' ^4 (see Fig. 6-5), it is clear that such precision is not necessary. Configurations for NF" = 0 to 1  NF" = 1 (the best-guess case of Fig. 5-7) are virtually indistinguishable, 1  and only minor effects occur as NF" rises to the unrealistic value of 4 1  females/tsf. I have shown similar insensitivities to hold for plausible values of NF" and the slope of the FFLY function. 6.4 Equilibrium Manifolds under Variable FOVP A parallel analysis explicates the role of a variable FOVP component to the exodus response. The relevant hypotheses are represented by the Type C and D functions of Figure 5-7.  The relevant manifolds of Figure 6-4 exhibit  no obvious FOVP effect analogous to the FELY "breakpoints" discussed in the previous section. In fact, if we plot a manifold for Type A exodus with constant FMIG = 0.35, it is essentially indistinguishable from the Type C manifold of Figure 6-4.  The somewhat surprising implication is that variable  FOVP of the sort hypothesized in Figure 5-7 does not yield equilibrium structures significantly different from those of appropriate constant migration hypotheses.  Closer examination of the relevant manifolds shows why this is the case. From Figure 5-7, FOVP begins to fall (and FMIG therefore to rise) when fecundities drop below some critical level FEC. (All arguments implicating fecundity apply equally if female weight or size are used as independent variables of the FOVP function. I documented earlier the tight correlations among these characteristics which occur under field conditions.) I showed in Chapter 5 (see especially Fig. 5-2) that fecundity in the local structure model was solely a function of larval (NL) and foliage (FT) density, independent of forest density or management effects. For the full foliage case explored in the present manifolds, the critical fecundity FEC is therefore associated with a critical larval density NL above which reduction of FOVP occurs. For 1  the Type C and Type D hypotheses shown in Figure 5-7, FEC =150 eggs/female and therefore, via the equations implicit in Figure 5-2, NL' = 230 larvae/tsf. If the exact numerical values of the equilibria plotted in Figure 6-4 are examined, the manifolds for variable FOVP (Type C and D.) do indeed begin to diverge from the comparable fixed FOVP manifold at exactly NL'.  The diver-  gence is very slow and slight however, particularly when viewed in the logarithmic plots used here. The postulated saturation of FOVP variability apparent in Figure 5-7 occurs at such high larval densities as to have even less effect on the plotted manifolds. Changes in the parameter values of the FOVP function change both the larval density at which FOVP reductions begin to affect the manifold structure, and the rate at which they do so. Note however that my data analyses of Chapter 5.3 showed the threshold fecundity value of the MFRC-IRE-IIASA exodus hypothesis (FEC =. 150) to be an upper plausible estimate, with more likely values in the neighborhood of FEC <_ 120 eggs/female.  Lower fecundities occur  at higher larval densities (Fig. 5-2). The effect of realistic (parametric) variation in FEC is therefore to move the NL' critical larval density higher,  and the point of divergence between variable and fixed FOVP exodus further "right" along the upper stable (NL ) manifold surface. The net result is +  thus to reduce even further the distinction between variable and fixed FOVP exodus hypotheses. A similar argument holds regarding the slope of the variable FOVP function at fecundities below the threshold level FEC. A less extreme response than that hypothesized in Figure 5-7 is not ruled out by available data, but again hastens convergence with the fixed FOVP.(Type A) manifold. A much steeper slope in FOVP is inconsistent with the date of Chapter 5.3 unless the threshold value FEC is simultaneously increased, in which case the net result is as previously described. Note that changes in the constant FOVP rate for FEC > _ FEC simply shift the relevant portion of the manifold left and right, as in the Type A case. 6.5 Implications for Population Dynamics: The Endemic Domain My analysis of the exodus manifolds is best summarized by. focusing separately on its implications for the budworm system's endemic and epidemic attractor domains. The manifolds of Figure 6-4 show two qualitatively distinct configurations of the lower stable equilibrium (NL~) and the unstable boundary (NL°) which characterize the endemic domain. The first reflects Type 0, Type B, and Type D exodus hypotheses, all of which have identical manifolds over the entire range of Type D's NL~ and NL° values. The second reflects Type A and Type C hypotheses, which are identical for equivalent values of Type A's total fraction migrating (FMIG) and Type Cs maximum fraction oviposited before flight (FOVP). In terms of the structural classification of Figure 5-1, the qualitative character of budworm's endemic domain is therefore determined by the presence or absence of a variable fraction flying (FFLY) response, and is  94 independent of the character of the FOVP response. Additionally, the endemic domain of variable FFLY hypotheses is essentially equivalent to that of the zero migration case analyzed in Part II. The preceding analysis of manifold structure with respect to FFLY and FOVP variation explains why these groupings emerge. It also shows that they are insensitive to parametric variation within the range of uncertainty suggested by the data. The essence of the argument consists of showing how the critical budworm densities (NL , NL", and NL") defined earlier are related 1  1  to the densities defining the upper boundary of the endemic domain (i.e. the unstable equilibria, NL°). The recruitment curves of Part II and Figure 6-1 show that the maximum value of NL° possible in the zero or constant migration case is about NL° ^75 [log(NL°) - 1.9], i.e. the density at which the maximum recruitment rate occurs. Since exodus functions as a local mortality factor, it cannot act to increase this value, no matter what structure or parameterization of the FMIG function is adopted. It follows that processes which affect the system only at larval densities NL > 75 cannot alter its endemic-release equilibrium structure. From Figure 5-2 it is clear that fecundity mediated processes (and, via the correlations discussed in Chapter 5, others of ontogenetic character) do not vary appreciably at densities NL < 75.  In fact the FOVP variation pos-  tulated in the Type C (MFRC-IRE-IIASA) and Type D hypotheses of Figure 5-7 occurs at densities NL >. 230, so far from the endemic domain as to render the variable vs. fixed FOVP distinction completely irrelevent there. It is only the basic, low density FOVP rate which affects endemic domain behavior. For outbreak initiation studies, Type A and C exodus may therefore be treated synonymously, as may Type B and D. This simplification of the original four exodus types of Figure 5-1 is summarized in Figure 6-6.  The variable FFLY response hypothesized for Type B and D exodus serves to eliminate dispersal flight under nearly all circumstances where the system is in its endemic domain. Some dispersal may begin around the NF = 1 (NL-30) threshold, which is indeed below the maximum NL°.= 75 level defined above. But as shown in Figure 6-5, minimal alteration of the NL° surface occurs under any plausible range of variable FFLY parameterizations. For endemic domain and outbreak initiation studies, Type B and D exodus may therefore be treated as-' equivalents-not only of each pother, but also of the zero migration (Type 0) 1  case of Part II. The equivalences summarized in Figure 6-6 will be exploited to  . '  reduce the number of alternative exodus formulations which must be considered independently in the detailed outbreak propagation analyses of Part IV. 6.6  Implications for Population Dynamics: -The,'Epidemic..Domain Analysis of exodus effects on system behavior in the epidemic domain is  more complex and less conclusive. The situation here is essentially dynamic, involving overshoots of the manifold equilibria, significant activity away from the full foliage manifold plane, and strong feedback among insect, foliage, and branch densities. Exodus impact on the endemic domain could be usefully viewed as that of a mortality rate inflicted on the local budworm population. In the epidemic situation, attention shifts to its impact on the rest of the budworm-forest system. Two particular aspects are treated here. First, I consider the role of exodus as a source of migrant moths who may trigger additional epidemics at other spatial locations. Results of this analysis are required for the study of outbreak propagation relationships reported in Part IV. Second, I discuss the complex relationships among budworm density, exodus rates, and forest mortality.  Exodus as a source of colonists To determine how exodus hypotheses differ with respect to the number of eggs dispersed to surrounding areas, I begin with the approach developed in Chapter 6.5.  There I considered the location of the endemic budworm manifold  surface (NL ) with respect to the critical exodus densities NL', NL", and NL"' (Fig. 6-4). Recall that the variable vs. constant fraction flying (FFLY) distinction is relevant only for insect densities NL" < NL < NL", while the 1  same distinction for fraction oviposition (FOVP) is relevant only for NL > NL . 1  It is clear from Figure 6-4 that a budworm population rising from its endemic (NL ) to its epidemic (NL ) equilibrium may pass through all the +  critical exodus densities. Its temporal trajectory, if not its ultimate epidemic equilibrium, is thereby potentially affected by all the structural variations of the exodus response. One useful distinction can nonetheless be made. Outbreaks triggered by random fluctuation or immigration at SAR.values between SAR"' and SAR" (see Fig. 6-2) may rise to very different epidemic levels, depending on the specific character of the exodus hypothesis. But .... the manifolds of Figure 6-4 suggest that spontaneous outbreaks (i.e. those triggered by a rise in SAR above SAR") triggered on the full foliage (FT = 3.8) plane will rise to attractors NL > NL > NL" > NL" regardless of the exodus +  1  1  type. Under such conditions NL and the maximum emigration rates are there+  fore independent of the variable vs. fixed FFLY distinction. On the other hand, they are potentially influenced by both the variable vs. fixed FOVP distinction, and the maximum FOVP rate. The previously noted convergence of all hypothesized budworm manifolds at high SAR values raises a possibility that these latter influences may be insignificant in practice. To test this, I simulated spontaneous outbreak sequences using the local model of Part II,  modified to include the various exodus hypotheses of Figure 5T7. All emigrants were lost to the site and no immigration occurred. For each year of the simulations, I calculated the number of eggs emigrating per tsf unit of branch surface area. Figure 6-7 presents these emigrant densities as cumulative distributions over the five peak years of the outbreak cycle. Dots mark the maximum single year emigrant density for each hypothesis evaluated. Note that these maximum single year values contribute about half of the total outbreak emigration. Several features of Figure 6-7 deserve special mention: (1)  Fixed FFLY hypotheses (Type A and C) show higher total cumulative emigrants,-but these are solely due to additional contributions under endemic conditions.  (2)  Peak emigrant numbers are similar for hypotheses of similar FOVP structure. The variable FOVP of Type C and D leads to the expected increase in this peak, averaging about 30% over fixed FOVP hypotheses.  (3)  Most important, comparing the specific Type B, C, and D budworm hypotheses of Figure 5-7, the peak emigrant numbers are virtually identical.  I conclude that peak emigriation densities under spontaneous outbreak are relatively insensitive to the alternative exodus responses hypothesized for budworm. Once again, this realization will help reduce the analysis of Part IV to a manageable level of complexity. Exodus and the forest There remains for consideration the NL surface of the manifolds for +  intermediate values SAR" > SAR > SAR'". This region may be reached either via fluctuation/immigration-induced outbreak from the endemic surface, or in  98 post outbreak stages as the forest density is reduced by high budworm densities (recall that the NL surface in general is stable for budworm but not +  forest). Figure 6-4. shows radical differences in the manifold configuration of this region, reflecting both the variable FFLY component and the maximum FOVP values of the exodus response. My investigations of system behavior in this region reveal a strong dependence on detailed dynamic relationships among NL, FT, and weather. The equilibrium manifolds are a poor indicator of actual outcomes. At certain parameter values, the "bulge", of the Type B and D budworm manifolds can intersect the manifolds for the forest, signifying a potential stable equilibrium for all variables in the neighborhood of NL ~ 90, SAR ~ 0.6, and FT ~ 3.0. One typical case for the Type B exodus of Figure 5-7 is illustrated in Figure 6-8. Comparing Figure 6-8 to the zero migration (Type 0) dynamics of Figure 3-1, it is clear that the "realistic" epidemic and collapse population dynamics pattern of the local model disappears entirely under the imposition of Type B's variable FFLY exodus "mortalities." Recall that in both simulations all emigrants are lost to the system and no immigrants arrive. This is clearly unrealistic and may be responsible for the odd results of Figure 6-8. But if the stable budworm-forest equilibrium is retained when the variable FFLY exodus hypotheses are incorporated in the more realistic, spatially structured theories of subsequent chapters, then something is clearly wrong. (It is retained. Something is wrong. See Chapter 13.) In the present context, however, the artificiality of a modeled world in which insects can leave but not arrive, and the acute sensitivity of that world's behavior to forest age structure, to weather, and to exogenously imposed, necessarily arbitrary immigration rates, suggest that the limits of an essentially local analysis have been reached. Further understanding of  99 exodus response relation to long term patterns of population dynamics ,1  requires exploration of the other components of dispersal and inclusion of explicit spatial dimensions in the resulting models. I address these issues in the subsequent chapters. 6 . 7 Summary The analyses of this Chapter permit me to simplify the range of alternative exodus hypotheses which must be treated explicitly in the broader spatial structure study. Most significantly, I have shown that endemic, release, and outbreak propagation characteristics of the budworm-forest system are'likely to be critically dependent on the presence or absence of a low density exodus flight threshold (i.e. the FFLY component). In contrast, different plausible oviposition behaviors cannot affect these characteristics. In subsequent chapters dealing with outbreak release o^propagation.;rtitbeipecfoiceg!;pupeexodus 3  hypotheses according to the structural criterion of whether they possess a low density flight threshold (Type B and D), or lack one (Type A and C). For historical reasons, I particularly emphasize the constant FFLY hypothesis proposed by the MFRC-IRE-IIASA researchers (Type C in Fig. 5 - 7 ) and the variable FFLY hypotheses suggested by my present data analysis (Type B in Fig. 5-7).  Type A and D hypotheses are less fully explored. Under suitable param-  eterizations. the former should give results qualitatively similar to Type C, while the latter should correspond generally to Type B. A major goal of the following analysis is to discover whether the differences between variable (Type B/D) and fixed (Type C/A) FFLY revealed in the local analysis are damped or amplified in the full spatial case. The situation posed by epidemic budworm populations is not as easily simplified due to its complex dynamics. Nonetheless, I show through the  100 spatial simulations of Chapter 13 that the structural classification developed for the endemic situation is useful in the dynamic epidemic case as well.  Fig. 6-1: Recruitment function with constant exodus. Basic format and left ordinate as fig. 3-2, for zero migration. Right ordinate for constant exodus rate, as a function of zero exodus relationship and fraction of total eggs leavinq in exodus (FMIG). Note the shifts of equilibrium densities; see text.  102  Fig. 6-2: Equilibrium manifold for constant exodus (Type A, 50%). Zero migration manifold on left is identical to that of the local model, fig. 3-4. Note the shift of critical SAR values; see text and explanation of Fig. 3-4.  TYPE A (50)  .5  .5  BRANCH DENSITY CSflR)  BRANCH DENSITY (SAR)  o  BRANCH DENSITY (SAR  6-3:  .  .5  BRANCH DENSITY (SAR)  Equilibrium manifolds for the four exodus Types defined in fig. 5-7. Conversions as in fig. 3-4.  1  3 Type 0  BRANCH DENSITY  (SAR)  Fig. 6-4: Superimposed manifolds of four exodus Types from fig. 6-4, plus zero migration (Type 0 ) manifold of fig. 6-2. Note that the dashed lines are used only to distinguish the different manifolds, and do not signify unstable equilibria as in previous figures. Critical NL values defined in text.  105 3  2 +  CD O  BRANCH DENSITY  (SAR)  Fig. 6-5: Equilibrium manifolds for alternative FFLY parameters. Basic structure is for Type B exodus of fig. 6-4. Alternatives are for different values of NF*, the density of female moths above which the FFLY response begins to rise, NF* = 1 is value for standard Type B of Fig. 6-4 (see Fig. 5-7 and text).  106  FFLY CONSTANT  VARIABLE  Fig. 6-6: Simplified exodus structures. The original four structures of fig. 5-1 are simplified by recognizing that the. variable vs. constant FFLY distinction dominates population dynamics implications.  107  Fig. 6-7: Exodus as a source of colonists. Figure shows simulation results for density of eggs migrating out of a site during the most active six years of a typical outbreak, under a range of exodus assumptions. Ordinate gives cumulative number of eggs migrating per tsf branch surface. Abscissa is partitional on one year increments. Note that each curve runs for six years. The spacing between and ordering among the curves is arbitrary, the present design being chosen only for ease of presentation. Letters correspond to exodus Types of fig. 5-7. FFLY values as in fig. 5-7; "Basic FOVP" is FOVP at maximum fecundity; remainder of FOVP relationship as in fig. 5-7.  BASIC FOVP= 0.50  BASIC FOVP= 0.65  TIME  (1-YEAR  INCREMENTS)  109  Fig. 6-8: Local model dynamics, Type B exodus. Units as in fig. 3-1.  110  CHAPTER 7  THE DISPLACEMENT RESPONSE  7.1  Displacement versus settling behavior.  7.2  An approach to the displacement analysis  7.3  Flight timing and duration: pre-radar studies  7.4  Flight timing and duration: radar studies  7.5  Flight speed, orientation, and altitude  7.6  Wind velocities and displacement  7.7  A displacement distribution function  Ill  7.1  Pisplacement versus Sett!ing Behavior Many Lepidoptera, including budworm, undergo long-range displacement  in association with large-scale weather systems (Johnson, 1969). As noted earlier, it is now generally accepted that such displacements reflect more or less refined adaptations to "redistribute populations . . . beyond the breeding place" (Johnson, 1966, p. 233). This adaptive perspective is particularly credible in the case of resource depletors such as budworm. In terms of Southwood's (1977) habitat templet, if "here" and "now " is good for such animals, "here" and "later" is bound to be dreadful.  "Elsewhere"  offers at least a temporary.respite from rising competition, parasitism, and starvation pressures. The problem faced by the dispersing organism is how to move from "here" to an "elsewhere" which will better his expected progeny production. Historical records show that budworm outbreak fronts advance 10 to 50 km or more per year (Brown, 1970; FIDS 1938-1977; see Part V). Adaptive displacement should therefore involve distances of at least that order. Like other components of dispersal, displacement involves distinct behavioral adaptations. Studies of insect movement have repeatedly identified two different types of behavior. One is essentially searching, appetitive or "trivial" (sensu Southwood, 1962) in nature, and is terminated by encounter with a suitable vegetative stimulus (e.g., food, mate, habitat). The other has been described as nonappetitive or migratory, and is totally or relatively insensitive to vegetative stimuli. Termination of flight is generally mediated by factors internal to the organism or dictated by the physical environment, e.g., cold or turbulence (Kennedy, 1975; Dingle, 1972, Johnson, 1969; Southwood, 1962, and many references therein). These behaviors may grade into one another (usually in a manner not very different from Sherrington's notions of successive reflex inductions), as illustrated by Kennedy's (1958.) studies of Aphis fabae, Green's (1962, Green and Pointing, 1962) work on the tortricid  112 Rhyacionia buoliana (Schiff.), and R. E. Jones' (1977a) analysis of Pieris rapae. The distinction, however, is critical in terms of the possible approaches for modeling and analysis of displacement behavior. Under appetitive search, movement duration is a function of resource (actually, stimulus) distribution in the neighborhood of the exodus site. Where resource distributions are spatially heterogeneous, duration becomes a nonlinear function of exodus location. Explicit search behavior and/or probability-of-encounter models are then the only, albeit limited and expensive, recourse for analysis (e.g., Fleschner, 1950; Dixon, 1959; Siniff and Jessen, 1969; Chandler, 1969; Murdie and Hasse.ll, 1973; Brunner and Burts, 1975; R. E. Jones, 1977b; Gerri.ts.en and Strick.ler, 1977). But if movement is essentially nonappetltive, then its duration may be independent of local resource distribution and therefore exodus location. The analysis is then greatly simplified, even where duration remains a function of condition of the disperser, or of habitat conditions of the exodus site. Dispersing budworm moths almost certainly exhibit this latter, nonappetitive type of flight behavior, although direct experimental evidence on the matter is unavailable. But it is clear from the budworm research I review below- that the moths do fly for periods up to several hours, passing over suitable and reasonably homogeneous habitat for much of that time. The moths are heavy enough (ca. 50mg wet weight, Outram, 1971) that they cannot remain aloft without sustained flight activity except in unusually severe convective cells.  Visual and radar observations show that displacement termination, when  it occurs, involves an abrupt wing-folded plummet from flight altitude to the ground. This has often been observed to deposit the moths on water, where they drown. For the purposes of this analysis, I use that "plummet" to separate the (nonappetitive) displacement phase from the (possibly appetitive)  113  settling phase of the budworm movement process. I analyze the latter phenomena in the next Chapter, and focus here on the moths' dispersive flight behavior and interaction with large-scale patterns of weather.  7.2 An Approach to the Displacement Analysis Studies on a variety of insects identify three common characteristics: persistent locomotion, depression of vegetative responses, and strai.gh.theni.ng out of the movement track (Kennedy, 1961, 1975; Kennedy and Booth, 1963a,b; Dingle, 1972, 1974). All of these tend to give the disperser a greater net displacement than would be obtained from similar amount of random walk/passive diffusion movement (Skellam, 1973; Clark, Jones, and Holling, 1978). R. E. Jones' (1977a,b) elegant work on the butterfly Pieris rapae shows that, under appropriate conditions, these behaviors can be experimentally decomposed into their component parts, accurately parameterized, and reassembled to predict displacements of individual animals under field conditions. I argue in this chapter that budworm exhibit the characteristic features of "purposive" dispersal , and that the behavioral elements of budworm's displacement response can be usefully approximated on the basis of available data. But the real difficulties in assessing long-range insect flight displacement are meteorological rather than entomological. Exodus behavior carries the insect above the relatively calm "boundary layer" (Taylor, 1958; 1960; 1974) into the more active air above. Here the animal does not (and usually cannot) maintain a ground track independent of wind velocity and direction. Net ground displacement emerges from complex interactions of airflow patterns and insect behavior. Such, interactions have been studied since the pioneering work of Uvarov (1931), Wellington (1945), and Williams (1951). Recent studies are reviewed by Sayer (1965), Johnson (1969), and Rainey (1974). As a result of this work, it is occasionally possible to reconstruct particular episodes  114 of insect-air mass interaction (e.g., Hughes and Nicholas, 1974), and even to predict the spatial redistribution of dispersing insects under specified meteorological conditions (e.g., Rainey, 1976). For the long-term population studies which concern me here however, the high resolution, dynamic treatment of insect-air.mass interactions is neither practical nor desirable. To evaluate the role of moth displacement in establishing these large-scale patterns of population dynamics, I adopt a temporal resolution comparable to that of the local analysis (i.e. one budworm generation) and a spatial scale on the order of the Province of New Brunswick (see Part. V). I therefore seek to develop a net flight season displacement function whereby emigrants originating at a given site in a given year are proportionally redistributed at various distances and directions from their point of exodus. A first approximation to such a function is described in the last section of this chapter. To create it, I begin by adopting a space-time coordinate system fixed with respect to an arbitrary parcel of air inhabited by the dispersing insect. Within this system I analyze the behavioral aspects of budworm displacement, focusing on the duration of flight (Chapters 7.3 and 7.4), and its airspeed and orientation (Chapter 7.5). These data are sufficient to calculate per flight displacement  (distance and direction) distribu-  tions of dispersing moths with respect to the airparcels into which they exodus, I need to obtain net moth displacements relative to the ground, however, and for this I also need to calculate the displacement distribution of airparcels relative to ground locations. I do this in two steps. For present purposes it will be sufficient to evaluate the distribution of airflow velocities in which budworm moth dispersal actually occurs, leaving the complexities of directional distribution for consideration with other spatial heterogeneity issues in Part V . This velocity distribution, discussed in Chapter 7.6, is based on empirical meteorological data, corrected for the  115 season, hours (Chapters 7.3 and 7.4), and altitude (Chapter 7.5) at which dispersal takes place. Combined with the behavioral flight data noted above, it allows calculation of the basic displacement distribution functions of Chapter 7.7.  7.3  Flight Timing and Duration: Pre-Radar Studies I need to estimate both a flight duration distribution, and (for syn-  chronizing with airflow data) the clock time of dispersal flight. The most comprehensive data on these displacement components derive from the radar studies of Schaefer (1976) and his colleagues. But certain ambiguities in the interpretation of those data make an initial discussion of nonradar sources desirable. Th.istle.ton (1975) used a night viewing device to count moth take-offs as a function of time, on each, evening of the 1975 dispersal season, at Chi.pman, New Brunswick.  His raw data, available in the file report noted above, yield  an approximately normal time distribution of take-offs (Fig. 7-1). These t  peaked.sharply at 2200 hrs , with half of the total evening exodus flights beginning in the 20 minutes from.2150 to 2210 hrs. Thistleton also counted descending moths until 2200 hrs, at which time reliable sightings of the rapidly plummeting forms become impractical. In no half-hour period from the onset of exodus at 1930 hrs to termination of the count at 2200 hrs did the number of descending moths constitute more than 7% of the number taking off. Both mean and mode descent rate for the entire observation period is 3%, with no trend in time. From Figure 7-1, I calculate that mean elapsed flight time for moths in the air at 2200 hrs is on the order of 20 minutes. Given the minimal (and nonincreasing) descent rate at 2200 hrs, it therefore seems A11 times quoted are Atlantic Daylight Time (ADT), which is equivalent to GMT minus three hours.  +  116 reasonable to postulate a 20 or 30 minute lower limit to dispersal flight duration. Such "refractory" periods, during which the dispersing organism will not "willingly" cease flight in response to any environmental stimuli, are a characteristic and evidently adaptive feature of dispersal in many insects (Kennedy, 1975). Representative documentation of the behavioral origins of the phenomena are available for aphids (Kennedy et al. ,1961), scolotid beetles (Graham, 1959; 1961; Francia and Graham, 1967; Bennet and Borden, 1971), milkweed bugs (Dingle, 1974), and pierid butterflies (Nielsen, 1961). A physiological upper limit to flight duration is proposed by Outram (1973). He measured fat content of budworm moths at various stages of their flight and oviposition history, and estimated that a minimum of nine hours' active flight would be energetically feasible. An empirical upper limit to flight duration is set by Greenbank (1973) at eight hours, based on the general absence of descending or flying moths in dawn observation surveys. Citing the results of night-time counts from aircraft, he estimates three to seven hours as the most likely figures for mean single flight duration in budworm. Finally, Wellington's (1948) demonstration that female moths are strongly photonegative for a period after initial exodus has led to the suggestion that dawn light may provide the ultimate flight termination stimulus. This cannot be strictly true, since occasional flight lasting past dawn has been observed (Henson, 1950), and post-landing daylight flights are common (see Chapter 8). Nonetheless, rapidly rising light after 0500 hrs may play a role in flight termination, and would suggest a maximum duration of 7 to 8 hours. (Illumination at New Brunswick's Renous radar site first re-attains its 2200 hr value of 0.1 ft.-candle at 0520 hrs the following morning.)  117 7.4  Flight Timing arid Duration: Radar Studies Radar studies of budworm dispersal were begun in 1973, in part to further  refine the crude limiting flight duration estimates quoted above. Ground stations operated from 1973 to 1976, and airborne radar in 1975 and 1976 (Schaefer, 1976; Greenbank, Schaefer, and Rainey, unpublished). Essentially none of the vast quantity of data collected during the radar studies is yet published, or available in useable format. The work reported here employs the raw data sheets kept by radar observers at each of two ground stations operated in 1974 and 1975.  These were compiled by G. Schaefer, I. Norton,  K. Allsopp, and M. J. Farmery of the Ecological Physics Research Group, Cranfield Institute of Technology for the Budworm Dispersal Project, and made available to me by the Projects' coordinator, D. Greenbank. Data were available for 72 location-nights as follows: Juniper Station (1975), 18 nights; Chipman Station (1975), 14 nights; Chipman Station (1974), 18 nights; Renous. Station (1974), 22 nights. Each night's data consist of moth densities per volume of air, estimated at roughly..10 minute intervals throughout the flight period, at each of 7 to 9 altitudes above the radar site (see Schaefer, 1976, for details). I. used these data to calculate time by altitude density profiles for each night. A typical profile is shown in Figure 7-2, along with, the altitude integral representing total airborne densities projected on a unit ground area. Radar density profiles such as those shown in Figure 7-2 necessarily constitute the basis for subsequent flight duration calculations. Note, however, that these data are densities above a point fixed in space. The wind-borne moths are moving with respect to such ground points at velocities on the order of 25 km/hr (see below). The available data therefore do not represent the time history of a single cohort of moths, and cannot be used directly in flight duration calculations. The airborne radar data, should they ever  118  become available, may allow, direct cohort tracking. Some cohort information could have been obtained even from the ground radar had the two available machines been located so that airparcels passing over one would pass over the other some useful time interval later. Unfortunately, the multiple objectives of the radar studies sacrificed such close spacing for more ambitious if less explicit goals. There are circumstances, however, in which crude but meaningful flight duration estimates can be drawn from the existing ground station radar data. If moths initiate and terminate displacement synchronously throughout a given spatial region, then it will not matter where in that region observations of the phenomena are made. If, in addition, the numbers undergoing exodus on a given night are approximately equal throughout the region, then rates of density change above a fixed location can be used to estimate flight duration distributions. The synchrony assumption is not too implausible. Exodus time is largely mediated by light intensity; the relationship is apparently consistent throughout the New Brunswick study area (Greenbank., 1973). As already noted, under "migratory" dispersal behavior of the sort postulated for budworm, duration is largely independent of spatial distribution of resources: synchronized exodus therefore plausibly leads to synchronized landing. The assumption of spatially homogeneous exodus densities is less easy to justify. Note, however, that moths passing over the fixed ground station originate upwind from that station at a radius roughly determined by time since, exodus and net ground speed of the wind blown organism (see below). The homogeneity condition need be met only along the contributing radius. I selected data sets likely to reflect the homogeneity and synchrony conditions by analyzing historical records of insect density (Chapter .11) and nightly wind flow (see below) in the region of the radar stations. Next,  119 I examined the altitude profiles (e.g. Fig. 7-2) of these, candidate, sets. If the synchrony and homogeneity assumptions in; fact apply/to the field situation, then the altitude profile must exhibit certain continuity and conservation properties. In particular, density contours must both, originate and terminate at ground level (or the limits of the observation period), since moths must do the same. Closed contours imply net lateral moth flux across the radar station. The "32" contour of Figure 7-2 provides one example of a high density cloud of moths which exodused locally but disappeared near midnight without losing altitude. This must be interpreted as a "patch" of high moth density; extending less than two hours' fli.gh.t upwind from the radar station, blowing past it in the night, and thus violating the homogeneity assumption.  Such data sets were removed before calculating flight durations  given below. Of the original 72 location-night data sets, I judged 40 suitable for flight duration calculations. I standardized these with, respect to time of peak area density (Fig. 7-2a, point T*). Thistleton's (1975) studies reported earlier defined take-off distributions and justified the assumption of essentially no descent before peak take-off. Combining all data I tallied the proportion of moths aloft at the beginning of a time interval which landed during that Interval for consecutive 30 minute intervals on either side of the exodus peak.. They-translate directly into estimates of the flight duration distribution, but one additional problem needs comment. In combining Thistleton's take-off distribution with the radar data, three extreme assumptions are possible: (I,)  "Independence": The time at which the individual descends is  independent of the time of take-off. (II.)  "First up-firstdown"- : The individuals which descend first  are those which took, off first.  120  (III) "First up-last down": The individuals which descend last are those which took off first; this would reflect a disperser "type" morph in the population. These alternative assumptions cannot be differentiated on the basis of available data, so I tabulated flight duration distributions for each case. Resulting frequency data are presented in Figure 7-3 (those for assumption III almost certainly reflect aliasing due to the crude 30 minute class intervals). Eye-smoothed curves through the points are given in Figure 7-4, and their cumulatives in Figure 7-5.  Regardless of the assumption used, about  half the dispersing population is expected to fly 90 minutes or more, while less than 10% should fly more than 4 hours. All these radar estimates are within the 0.5 to 8 hour limiting values defined earlier, but they give much shorter mean flight times than those originally suggested by Greenbank (1973) from his qualitative, aircraft-based observations. Greenbank (pers. comm.) is now inclined to credit the shorter estimates. 7.5  Flight Speed, Orientation, and Altitude In this section, I evaluate data relevant to how and where the moth moves  during her period of flight duration. Flight Speed Flight speeds of budworm moths relative to the air are available from three independent sources, all of which closely agree. Outram (1971) analyzed the morphometries of wild and laboratory reared budworm moths, and calculated nominal flight speeds from standard aerodynamic formulae (Weis-Fogh, 1976). Relevant results are summarized in Table 7-1, giving a mean of 2.3 m/sec. Greenbank (1973) observed male moths engaged in characteristic positionholding "buzzing" behavior around tree crowns. When wind velocities exceeded  121 2.7 m/sec. these insects darted into the foliage and secured themselves to the tree. Greenbank reasoned that the moths could maintain position against winds up to this velocity by flying into them, but at greater velocities their own flight velocity was exceeded. Schaefer (1976) radar tracked individual female moths in dispersal flight. Weather balloon data and insect orientation information (see below) allowed calculation of flight speed by differencing. Results average '2.5 m/sec. during the majority of the flight period, but dropped as low as 1.0 m/sec. late at night. Results of flight speed research are summarized in Table 7-1. I adopt a mean flight speed of 2.5 m/sec. (9.0 km/hr) in the remainder of this analysis. This is well within the range of velocities found in other migratory moths of comparable size (Johnson, 1969). Flight Orientation Budworm moths orient downwind during dispersal flight. This was noted by ground observers (e.g. Thistleton, 1975), and demonstrated conclusively by the integrated radar and meteorological studies of the Budworm Dispersal Project. Figure 8-12 in Schaefer (1976) shows that budworm orientation tracks altitudinal wind shear very closely. Downwind dispersal orientation has three important implications for this analysis. (1) The common orientation of individual moths tends to keep the exodus cohort together in flight; (2) The lack of turning by individuals yields a net displacement rate, relative to the air, which is equal to the flight speed (random turning would give an expected net displacement of zero);  v.  122 (3) Net displacement relative to the ground becomes a simple addition of flight speed and wind speed at the flight altitude (see below). These features combine to yield the "straightening-out" of the ground track which, as noted earlier, characterizes migratory behavior of insects and results in maximal net displacement per unit time of dispersal. Flight Altitude Although my analysis focuses on horizontal displacement of budworm dispersers, some consideration of its vertical movements is necessary in order to integrate budworm and airparcel  displacement data. Figure 7-2b showed a  characteristic altitudinal distribution of moths in the 50-350 m range. Wind speed and direction are known to vary dramatically with altitude between, say, 0 and 500 m, primarily due to frictional and heat exchange interactions with the earth. This is particularly true for the light breeze, temperature inversion situations which characterize most late summer evenings in the Canadian Maritimes. The budworms' vertical migration behavior within this highly structured airflow affect their net horizontal displacement. Detailed studies of vertical relationships among moth density, air temperature, wind speed and direction were carried out during the Budworm Dispersal Project (Greenbank, Schaefer, and Rainey, unpublished). Present understanding is insufficient to warrant a dynamic treatment of vertical migration in budworm, however, and I confine myself here to characterizing the mean altitude distribution for dispersing budworm. I use the time-by-altitude radar data represented by Figure 7-2 to calculate the mean dispersal altitude for each of the 72 location-nights. "Mean dispersal altitude" (h) is here defined as the altitude such that half of the .time-integrated moth density for the entire night's flight period lies above h. A frequency distribution of h for the 70 useable data sets of the  123  1974  and 1975  and 90% o f  dispersal  the  be u s e d i n t h e represented  h values  next s e c t i o n to  the  this  section,  ground.  spatially ties  As n o t e d  the  earlier,  late  airflows  evening  ( 1 0 0 - 3 0 0 m)  geostrophic  neither  the  daily  winds  ment S e r v i c e  (AES)  displacement. pilot  stations.  velocity frequency  1974  for  furnish  Instead,  indices necessary to  to  be  complete  the  introduced  period  ( 1 0 0 - 3 0 0 m)  i s given  of  in  hours, to  coordinant  the wind system  Part  region.  wind v e l o c i t y  500 m;  relative  New  Brunswick  by t h e  to  to  7-7a.  a secondary  the moth's The  distribution  frequency  peak  at  Environflight  Project's  of  and  ground  Renous of  12  altitude. the main  flight  A bud-  altitudes  has a d o m i n a n t  v e l o c i t i e s of  that  theodolite  an a v e r a g e  principal  level  means  budworm  Chipman  50 m i n c r e m e n t s  flight  station  from s i n g l e  the wind v e l o c i t i e s recorded during at  This  Atmospheric  relevant data  ground  ground  Budworm D i s p e r s a l  roughly  of  complexi-  budworm  that at  Each b a l l o o n p r o v i d e d at  the  hourly  s u c h r e l e a s e s w e r e made a t  season.  at  Canadian  data  wind v e l o c i t y  r e l e a s e d from the  Added  P e t t e r s s e n 1956),.  p r e s s u r e maps n o r t h e  in Figure  velocity  V.  be i n t e r m e d i a t e  say,  (2100-0100..hrs)  5 m/sec w i t h  estimate  air  and d i r e c t i o n e s t i m a t e s distribution  to  study  t h e Wind v e l o c i t y  Fifty-five dispersal  data  the  I obtained  balloons  worm f l i g h t  about  range  throughout  (above,  synoptic  published  the  altitude  m,  will  simplest p o s s i b l e assumption  i s expected  wind r e c o r d s  during  These data  the  dispersal  radar  relevant  175  I adopt  During  of  Mean h i s a b o u t  100 a n d 300 m...  d i s p e r s i n g moths'  are  tracks  the  airflow  and h e t e r o g e n e i t i e s  and t h e  of  specify  I use m e t e o r o l o g i c a l  homogeneous  altitudes  between  7-6.  Displacement  d i s t r i b u t i o n which s h i f t s to  fall  in Figure  analysis.  Wind V e l o c i t i e s and In  i s given  by t h e m e t e o r o l o g i c a l  displacement  7.6  nightly  seasons  10-12  mode m/sec.  124  T h i s peak i s a s s o c i a t e d  with a "low-level  jet"  of laminar flow a i r ,  weaker than but a p p a r e n t l y r e l a t e d t o those d e s c r i b e d by B l a c k a d a r G e r h a r d t (1963),  Taylor  (1965b),  of i n d i v i d u a l instantaneous i t y shows t h a t t h i s j e t  B e r r y and T a y l o r (1968).  C l o s e comparison  tends to l i e above the main d e n s i t y l a y e r o f moths, T h i s view i s s u p p o r t e d by  H e r e , I matched r a d a r and weather b a l l o o n v e r t i c a l  and t a l l i e d o n l y v e l o c i t i e s (i.e.,  much s m a l l e r , but the secondary budworm are not f l y i n g i n the  "jet"  The sample s i z e  peak shows no s i g n s o f a p p e a r i n g :  ( e x c l u d i n g the  w i t h p u b l i s h e d r e c o r d s o f 10 m and g e o s t r o p h i c winds f o r c o a s t a l and western Maine (AES 1953-1975, USDC 1953-1975).  most  jet)  New Brunswick  The p i l o t b a l l o o n  veloc-  i n t e r m e d i a t e to those a t the lower and h i g h e r  alti-  I c o u l d not d e v i s e a good p r e d i c t o r o f 100-300 m winds from the pub-  l i s h e d d a t a , however,  p r o b a b l y due to the d i s p a r a t e  high l e v e l  1974 d i d not p r e s e n t r e c o g n i z e a b l y  stations.  the d i s p e r s a l guide.  is  jet.  I compared t h e s e 1974 p i l o t b a l l o o n v e l o c i t y data  tudes.  profiles  o c c u r r i n g a t the mean a l t i t u d e o f budworm d e n s i t y  Fi) a t the time o f the wind v e l o c i t y measurement.  i t i e s were almost always  (1957),  a l t i t u d e p r o f i l e s f o r moth d e n s i t y and wind v e l o c -  and so does not a f f e c t the b u l k o f d i s p e r s e r s . F i g u r e 7-7b.  generally  p e r i o d i f i t s ground.and g e o s t r o p h i c  l o c a t i o n s o f the low and abnormal winds d u r i n g  velocities  are a  In the d i s p l a c e m e n t c a l c u l a t i o n s o f t h i s Chapter I t h e r e f o r e  s o l i d l i n e frequency  d i s t r i b u t i o n o f F i g u r e 7-7a  d i s t r i b u t i o n , o m i t t i n g the " j e t "  effect.  as a r e p r e s e n t a t i v e  relevant use  the  velocity  125 7.7 A Displacement Distribution Function I now combine the relationships developed in this chapter to yield a distance distribution function of net displacements for single dispersal flights. Define variables as follows: f-j(t)', (i = 1,2,3) is the distribution of single flight durations, in hours, as developed in Chapter 7.4.  Subscripts denote the three alterna-  tive hypotheses stated there. Data are those of Figure 7-4; 'g(v)! is the distribution of wind velocities at flight time and altitude, as developed in Chapter 7.6.  Units are meters per second. Data are those of  Figure 7-7a (solid line); V  is the mean downwind flight velocity from Chapter 7.5., namely a  constant of 2.5 meters per second; 'k is a conversion constant, equal to 3.600 (kilometers x seconds)/ 1  (meters x hours); 'P.j(r) is the calculated proportion of a dispersing population displaced 1  'r' kilometers from their exodus points under flight duration hypothesis ' i . 1  The P.(r) values are most simply obtained by a numerical (FORTRAN) algorithm:  DO 1 t = 1, tmax DO 1 v = 1, vmax x=(v  + y ) * - t * k + 0.5  r = INT (x + 0.5) Pi(r) = f -(t) * g(v) 1 CONTINUE n  establi sh iterations }• compute km index }discretize index }compute P.  126  The resulting function is shown in Figure 7-8.~ Several features'are'noteworthy. First of all, the three alternative flight duration hypotheses result in only moderately different numerical values and virtually identical shapes for their respective P(r) functions. The "Independence" assumption of Hypothesis I yields intermediate values, as expected, and wiUnserve as a focus for my discussion here. (The apparent differences among curves I, II, and III are even.less when their interactions with local structure are considered; see Parts IV and V.) All hypotheses yield a distribution qualitatively similar to that obtained by Gaussian plume models common in pollution dispersion studies (Mason and McManus, 1978): moths are less likely to land near the exodus site than some distance away. The modal distance under Displacement Hypothesis I is about 37 km, with a range of 27-42 for Hypotheses III and II respectively. Roughly 50% of the dispersing population lands within 50 km of the exodus site, and 75% within 75 km.  The spatial scales affected by a single year's production  of the "local" budworm forest system are thus substantial, a fact which I explore in greater detail in Part IV. The radar-based P(r) distributions computed here are in reasonable agreement with earlier estimates based on more qualitative data. Walters and Buckingham, drawing on the empirical experience of Greenbank (in Stander, 1973), hypothesized a P(r) function for New Brunswick budworm with a marked refractory period. The modal distance and 50% cumulative both occurred at about 33 km. They truncated the distribution at 65 km out of computational necessity, and otherwise would have very closely approximated Hypothesis III of my. own calculations. The original MFRC-IRE-IIASA modeling effort (Jones, unpublished),  127  with somewhat more data than were available to Walters and Buckingham, also postulated a Gaussian piume-1ike function, with a mode at 40 km and a long tail analogous to the functions presented here. I show in Part IV that this "tail", reflecting a few budworm flying for long durations in faster winds, can have a major effect.on outbreak propagation behavior. But given that Walters and Buckingham would have included such a tail had their computer permitted, it seems that the sophisticated and expensive radar studies of the Budworm Dispersal Project have largely served to confirm and enforce the gratifyingly accurate previous analyses made by experienced observers of the budworm displacement process.  t  Table 7-1: Flight speed estimates for budworm moths  Source  Outram (1971) (morphometries)  Greenbank (1973) (direct observation) Schaefer (1976) (radar observation)  Description  Flight speed in m/sec (mean and s.d.)  lab reared virgin females ( 2 day old )  2.4 + 0.2  lab reared mated females ( 2 day old )  2.1 + 0.3  wild virgin females  2.3 + 0.2  wild males, buzzing  2.7  \ wild females, dispersing and presumably mated  2.5 + 1.0  129  Fig. 7-1: Temporal distribution of female moth take-off. Eye-fit to data of Thistleton (1975) for 1975 flight season data of Chipman. Time is Atlantic Daylight Time ( GMT-3 hrs).  130  Fig. 7-2: Radar observations of evolution of altitude distribution in moth density above Chipman on the night of 6/7 July 1975. (A) shows the altitude-integrated moth count for a column of air over one hectare of ground area. (B) shows the altitude profile of this count in terms of relative density contours. Divide contour indices by 1.28 x 10^ to obtain absolute moth density per cubic meter of air. (*) indicates the minimum elevation scanned by radar; lower elevations are extrapolated Curves are hand fit to a 100 point time x altitude data grid. Times are Atlantic Daylight Time (GMT-3 hrs).  131  TIME (ADT)  Fig. 7-2  132 • + x  (I) INDEPENDENCE (II) FIRST UP -FIRST DOWN (III) FIRST UP-LAST DOWN  >=3  cr UJ cc u.  o  LU M  CC  o  X  X  X  +  Ob 0  ±  ±1  2  3  +  x x  4  FLIGHT DURATION (HOURS)  Fig'. 7-3: Single flight duration distribution. Interval data computed from radar data for 30 minute periods centered at the indicated duration. Three types of points are for same data under different computational hypotheses (see text).  0  1  2  3  4  5  6  FLIGHT DURATION (HOURS)  Fig. 7-4: Single flight duration distribution. Eye smoothed from interval data of Fig. 7-3. Curve I assumes time of landing independent of time of take off; II assumes moth taking off first land first; III assumes moths taking off first land last.  7  134  1  2  3  4  5  FLIGHT DURATION T (HOURS)  Fig. 7-5: Cumulative single flight duration distribution; from  instantaneous probabilities of fig. 7-4. Three curves as defined in fig. 7-4.  6  135  MEAN N I G H T L Y FLIGHT ALTITUDE (METERS A B O V E GROUND)  Fig. 7-6: Flight altitude distribution—frequency distribution of h, the mean flight altitude (m. above local ground), for 70 nights of 1975 flight season; h is taken from data such as fig. 7-2, and is defined as the altitude above which lies one half of the time integrated moth density for the night. Curves fit by eye.  136 Fig. 7-7: Wind velocity distribution at flight altitude--data from single theodolite pilot balloons released at Chipman and Renous during 1974 flight period; all curves eye fit; A) 55 pilot balloons, velocities estimated at 50 m increments from 100-300 m above ground level; secondary peak is a "low level jet"; B) subset of data from (A), including only data taken at each night's mean flight altitude (h); the "jet" disappears.  138  Fig. 7-8: Single flight displacement distribution, calculated from data of figs. 7-4, 7-7. Three distributions representing three hypotheses defined in fig. 7-4. Abscissa is distance (r) from exodus site. A) Ordinate is proportion of eggs leaving an exodus site which will land (r) kmaway (all directions combined); normalized for 1-km increments. B) Cumulative of curve (A).  139  0.02^  r(km)  Fig. 7-8  140  CHAPTER 8  THE SETTLING RESPONSE  8.1  Overview of the settling analysis  8.2  Descent from flight altitude  8.3  Redistribution following descent  8.4  A descriptive equation for settling success  8.5  Alternative hypotheses for the settling response  141 8.1 Overview of the Settling Analysis A third component of insect dispersal behavior is referred to as "set-; tling" (Johnson, 1966), "stopping" (R. Jones, 1977a), "immigration/deposition" (Greenbank, 1963), or "a return of appetitive response" (Kennedy, 1975). Whatever their terminology, most authors agree that the phenomenon encompasses termination of long-range movement and installation of the disperser in its new habitat. For many insects this installation constitutes an active, sensory search for appropriate vegetative stimuli (Southwood, 1962; Dingle, 1972). But beyond such broad generalizations, the settling response is little understood or studied: in his massive monograph on insect flight dispersal, Johnson (1969) declares the problem to constitute a subject in itself, and leaves it to others—still not in evidence—for review. Research on settling response in budworm is almost nonexistent, and relevant evidence is both largely anecdotal and wholly descriptive. My goal in this chapter is to review that evidence and to develop a simple descriptive model for its inclusion and evaluation in the full population dynamics analysis. The budworm settling response is conveniently divided into two stages: (1) the descent from flight altitude to ground level ("termination"), and (2) the subsequent and relatively local search for oviposition sites ("search"). The crucial issue with regard to the termination stage is whether descent is triggered by perception of "good" habitat below. I assumed that it was not in Chapter 7 so that I could utilize the available radar data in calculations of a flight duration distribution. In the first section of this chapter I show that this assumption is probably warranted, and review counterarguments which have been put forward in the literature.  In the second section I address  142 the "search" issue, suggesting that no matter where moths initially land upon their "termination" descent, subsequent local flight redistributes most of them into potential oviposition habitat. The third and final section presents a simple descriptive modeling framework which encompasses my interpretation of the budworm settling response and a reasonable set of structural alternatives.  8.2  Descent from Flight Altitude Budworm displacement flight generally terminates in an almost vertical  descent from flight elevation to the ground. The important question with regard to settling and its subsequent population effects is whether this descent is a response of the moth to suitable habitat perceived below. Greenbank, Schaefer, and Rainey (unpublished;henceforth "GSR") argue that it is, and that moths therefore descend disproportionately into "good" oviposition sites. I do not find their evidence compelling, and suggest that descent may be taken as essentially random with respect to habitat quality, at least in the.absence of overriding topographic variation. Three relevant observations are well established: The common form of moth descent is a vertical, wing-folded plummet and involves no active flight. Thistleton's..(1975) night-viewing device and Aldis observations showed 85% of descents to be of this sort. The wing-folded descent attitude is the same as that typically assumed by moths at rest, or cooled beyond minimum flight temperature (Wellington, 1945b; Greenbank, 1973). In the absence of strong vertical updrafts, moths in wing-folded position fall and rapidly attain a terminal velocity on the order of 3-4m/sec (Wellington, 1945b; compare Berry and Taylor, 1968; Johnson, 1969- Mason and McManus, 1978). This plummet begins at normal flight altitude, which, in late evening is usually 175-200m (Chapter 7). Moths do not descend to tree-top level, fly  143 laterally, and then plummet to earth upon "close encounter" with suitable habitat. This is documented by the radar time-by-altitude profiles which show very steep late-night slopes of the isodensi'ty lines, with no tendency to plateau at low levels (see Chapter 7, especially Fig. 7-2). The night-viewing device similarly fails to show late night, low altitude concentrations (Thistleton, 1975). Finally, the late night temperature inversions frequently lower canopy and ground temperatures well below the 15°C minimum flight threshold for budworm: no prolonged lateral flight at such temperatures is likely (Sanders, Wallace, Srlucuik, 1978; Taylor, 1963). There is also evidence that the moths "targetting" abilities, if they 1  exist at all, are complicated and imperfect. Radar data frequently show moth flights proceeding across good habitat areas without descending. The literature and word of.mouth budworm apocrypha are replete with accounts of moths descending on water and drowning (Henson, 1950; Greenbank, 1963; GSR, unpubl.). Some of these instances doubtless reflect forced deposition through turbulence or convection, but others seem to represent nothing more nor less than moths drifting over water on the prevailing wind and descending as usual into distinctly suboptimal habitat. How reasonable is it to argue, as do GSR (unpubl.), that moths perceive good habitat from flight altitude and target on it in free-fall descent? Nothing is suggested in their argument regarding mode of such perception, and no relevant evidence exists in the literature. Although behavior studies are constantly uncovering new modes of animal perception (e.g., Strickler, 1975; Keeton, 1974), the possible candidates are limited in the present case. The flying moth is moving quickly with respect to the ground, at a different velocity and direction from air near the canopy. Air-diffused perception of habitat therefore seems highly unlikely. Descent patterns are the same on completely overcast (and essentially zero foot-candle) nights and clear nights  144 (GSR, unpubl.), so reflected light cannot be a necessary condition for the proposed habitat perception. Long-wave radiation is all that obviously remains. Callahan :(:-T97Z.) argues- on neurqphysiologieal grounds that this,may be important, H  but. direct..evidence-'is, lacking (Goldsmith & Bernard, 1974). Even if perception is granted, simple calculations show that serious ballistic problems remain. I noted earlier that a budworm moth in folded-wing free-fall rapidly approaches a terminal velocity of 3-4m/sec. At such velocities it will take 50-70 sec to fall to earth from a mean late night flight altitude of 200m (acceleration to terminal velocity occurs at about g = 9.75m/ 2 sec ; the acceleration period is therefore insignificant relative to the total period of fall).  I showed in Chapter 7 (Fig. 7-7a'-) that lateral wind veloci-  ties encountered during that fall are usually on the order of 3-9m/sec. The moth will therefore land 150-600m •(-.at .least*)' from the point above which descent was initiated. Furthermore, as I show in Part V, the descent will be through decelerating air with a directional shear of (-9°) or so between flight altitude and ground level. Even if the moth could somehow direct her postulated perceptive search to feasible targets a quarter to half kilometer ahead and 9° left of her ground position prior to descent, the unpredictable variation in descent conditions encountered during subsequent free-fall make accurate targetting implausible, at best. In suggesting that descending budworm do target on good habitat, GSR rely on the observation that dawn moth densities on radar fields and airstrips are only 2 or 3% of the densities recorded by radar during the night. This is j-  a rather weak foundation on which to base so strong an argument, for several reasons. First of all, it should be noted that the "clearings" they refer to are of the order of 100-200m or less across their short dimensions. In view of the crude ballistic arguments advanced earlier, these are very small areas in * This is a minimum, since turbulence is not included; see Mason & McManus (1978).  145 terms of the moths' potential targetting (or avoidance) capabilities, even under the most generous of perception assumptions. One would expect such areas to receive a fair share of immigrants, even if moths actively avoid larger areas .of unsuitable habitat. It is well established that even the moths which do .land on fields and airstrips will fly away to neighboring forest in the morning as soon as flight temperature is attained (see Chapter 8.3). The implication in the Greenbank, Schaefer, Rainey argument is that no moths descending onto these habitats during the night could fly off to neighboring forests before dawn, due to cold ground temperatures under the late-night inversion. While it is true that night-time ground temperatures often drop below budworm'sl5°C minimum flight threshold, there is no direct evidence of what moths actually do when dropped into such a temperature layer. In other insects, body heat produced by previous flight or by muscle flexing (Krough and Zuethen, 1941; Taylor,,1963) is often sufficient to enable local flights at subthreshold temperatures, and I suspect this is also true for such big bodied moths as gravid female budworm. Finally, the data on which the GSR 2-3% figure is based are not reported. It appears that the dawn densities were casual inspection counts, mainly on tarmac. The accuracy of absolute moth density estimates from radar is open to serious questions under the best of circumstances (see Communicated Discussion in Schaefer, 1976) and no rigorous calibration experiments have been conducted in relation to the New Brunswick studies. Finally, the steepness of the aerial density profiles represented by Figure 7-2 suggests that radar estimates of descending densities will be extremely sensitive to the spatial homogeneity assumptions discussed in my interpretation of the radar flight duration data in Chapter 7. There is no indication that such factors were taken into account in the GSR calculations. Until their targetting argument is more  146 convincingly advanced, the-preponderance of evidence militates for the simpler assumption that descent occurs independent of habitat quality.  8.3  Redistribution Following Descent Even if initial descent from flight altitude is habitat independent, sub-  sequent flights within the boundary layer may redistribute a more substantial proportion of immigrants into suitable oviposition habitat. Such "searching" flights have been observed and are a potentially important component of the budworm dispersal strategy. Oviposition must occur on live foliage in trees of species and age classes susceptible to budworm attack if the eggs are to survive (see local structure analysis of Part II, and Morris, ed., 1963). Oviposition normally occurs in the afternoon (Sanders and Lucuik, 1975). Immigrant moths therefore have up to 12 hours (some of which may be too cold for activity) in which to redistribute themselves from landing sites to suitable oviposition habitat. Their "targetting" problem is summarized schematically in Figure 8-1. Of all those moths immigrating to a given geographical area, some fraction will find subareas covered by host tree species. Of these, a further fraction will find subareas covered by hosts in susceptible age classes. Finally, the ultimately successful fraction of immigrants will discover oviposition sites on live foliage of the susceptible hosts.  If moths land ran-  domly throughout an area and undergo no net redistribution, the target success at each stage will be roughly proportional to the relative physical area covered by the respective habitat types.  This view is incorporated in the  original MFRC-IRE-IIASA analysis (Jones, 1977), At the other extreme, highly efficient search and redistribution would impart a targetting success approaching unity. In this section I consider the relevant evidence as it relates to these two extreme possibilities.  147 The Province of New Brunswick is a predominantly forested area. The spatial distribution of cover (habitat) types is discussed in Part V.. Provincial averages are given in Figure 8-2. For present purposes, it is sufficient to note that even if moth descent is independent of habitat type, on the order of 40% of the immigrants will be expected to land directly in host species. Another 40% will land in other forest types. Nearly all the remainder will fall into fields, cultivated land, burns, and inhabited areas. (It is this last group of immigrants—the relatively small proportion landing on airstrips, golf courses, or,field station clearings—which furnish most of the above-noted observations on post-descent redistribution.) The following speculation, regarding the fate of moths missing host trees on their initial descent is based upon unpublished observations of a number of budworm researchers, notably David Greenbank, Chris Sanders, B. :W. Flieger, and C. A. Miller. As noted earlier, there is some question as to whether moths landing in open areas during the night can fly off again before daylight. What is not disputed is that essentially all those found in clearings at dawn take off as soon as air temperature warms sufficiently, usually by about 0900 hrs. Winds are normally calm or light at this hour. The moths climb steeply to altitudes of 10-20m, turn towards visible forest, and fly to it in straight, undistracted flight. Once within the forest, moths can be seen to search actively, avoiding hardwoods and seeking out tall, exposed,, undefoliated host species. These are inevitably older trees in the "susceptible" category defined above. These search behaviors have not been quantitatively studied, but their results have. Morris (1955) showed that a disproportionate number of eggs were laid orf\ "sun foliage"—the outer and upper surfaces of the tall exposed hosts sought by flying females (Fig. 30.1 in Morris, ed., 1963). Miller (in Morris, ed., 1963, Table 13.1) found that in continuous forests of host species where  148 young or middle-aged stands adjointed others more mature, eggs were deposited disproportionately on the latter. Miller and Greenbank (in Morris, ed., 1963.) report experimental demonstrations of budworms oviposition preference for 1  normal over defoliated branches, and for host over non-host species. A similar degree of host selectivity is demonstrated in the laboratory studies of Jaynes and Speers (1949).  In the field, it is evident that host selection  is not absolute among softwoods: black spruce and jack pine in particular are occasionally used as oviposition sites, though with generally fatal results for the eggs (Greenbank, pers. comm. ;_Wilson, "1963'). Selection of susceptible hosts from among hardwoods is apparently more effective. Fir and spruce within predominantly hardwood stands develop inordinantly heavy infestations and are often the first trees to die during an outbreak.  This is almost  certainly a result of selective oviposition by females immigrating to the hardwood area and seeking out the few available hosts (Baskerville, pers. comm.). The distance over which post-deposition host searching can be effective is not known. The observations cited above suggest that flights of several hundred to a thousand meters are frequently accomplished.  At flight  speeds established in the previous chapter (2.5m/sec), moths in calm air would cover a kilometer in less than ten minutes. A ten minute search flight is in accord with the casual field observations noted earlier, and substantially less than the maximum flight durations measured by radar or suggested by measured fat reserves (Chapter 7). The general prevalence of host species and their intermixture with other species in most of New Brunswick (Loucks, 1959-1960) suggests that moths descending in forest will usually be within reach of suitable oviposition sites. The possibility of "mistaken" oviposition.on honhost softwoods remains and, from Figure 8-2, might reduce average oviposition success by as much as 13% below the level attainable with perfect softwood discrimination. Locally, the effect could be much greater.  149 Nonforest area constitutes 20% of the province. Some of this land is distributed in patches several kilometers across, particularly in coastal areas and certain inhabited regions of the south. Moths landing in these areas may not find suitable hosts, and those landing on inland or ocean waters certainly will not. "Natural" clearings from fires and blowdowns within the forested area probably pose less of a problem. These have average diameters on the order of 300m or less (Wein and Moore, 1977 ; \Sprugel, 1976), well within the search range suggested above.  8.4  A Descriptive Equation for Settling Success The observations on settling response summarized in this chapter are too  crude to warrant detailed modeling. At best, they suggest potentially relevant variables and some extreme limiting values for the resulting function. The only realistic goal at this stage is to devise a simple formulation through which these approximations can be incorporated in the broader spatial structure-population dynamics analysis. The significance of present uncertainties can then be assessed.within that context, allowing a critical appraisal of both needs for future research and reliability of present models. I adopt here a simple expression of settling success, reflecting the trivial "targetting model" summarized in Figure 8-1.  Independent variables  indicated by the studies cited above are the proportion of total.land area covered by host species (PHOST in the local structure model of Part II, and Jones, 1977); the proportion of host land area covered.by susceptible age classes (HSP in the local model); and the relative density of living foliage on those susceptible trees (FTD in the local model). In keeping with the approach adopted in previous chapters, I assume an arbitrarily small spatial scale for the expression. The implication of this assumption is that spatial distribution of the independent variable values at smaller scales is largely  150 irrelevant to targetting success.  The few available data on budworm moth  search ranges suggest a maximum realistic scale of at least several hundred and perhaps a few thousand meters. The fine scale "patchiness" spectrum of the independent variables is not known, but is probably such that the scale at which my settling analysis applies can be increased to as much as " 10 km. The following component functions are required: TH(PHOST): Targetting success on host land (the proportion of all immigrants arriving in a unit area of land which successfully discover host land). TS(HSP):  targetting success on susceptible age classes (the proportion  of all immigrants arriving in a unit area of host land which successfully discover trees of susceptible age). TF(FTD): targetting success on foliage (the proportion of all immigrants arriving in a unit area of susceptible host land which successfully discover live foliage). TT(PHOST,HSP,FTD) = TH(PHOST) * TS(HSP) * TF(FTD):'. total targetting  success (proportion of. all immigrants arriving in a unit area of land which successully discover susceptible foliage).  (Eq, 8-1)  The structure used for component targetting functions is essentially arbitrary, given the absence of relevant empirical data.  I adopt here a simple  descriptive model based on physical targetting analogy; many other forms are possible. I employ a physical model in which a group of objects are thrown randomly at a target with hit probability 'p'.  Those that miss are collected  and thrown again, and so on through k' rounds or tries. The proportion of 1  1  objects hitting the target (at least once) after 'k' tries is then  T = 1 - (l-p)  k  (Eq. 8-2)  151 The behavior of Equation 8-2 for different values of p (.0 £ P£l,) and k (k > 0.) is shown in Fig. 8-3. The various components of the budworm targetting success equation (Eq. 8-1) can each be described with the basic form of Eq. 8-2.  'p' simply  becomes the relevant independent variable (scaled between zero and one), and 1  k'. is set to various values in some sense indicative of "search power" of 1  the moth for each respective component. A 'k' value of one gives target success directly proportional to the relative density of the relevant independent variable. I thus reformulate Eq. 8-1 in the FORTRAN conventions of themodels ;  to yield my descriptive equation for the settling response:  TH = 1 - (ITPHOST) **  THK  TS = 1 - (1-HSP) ** TSK (Eq. 8-3) ; TF = 1 - (l-FTD/3,8) ** TFK TT = TH * TS.* TF 8.5  .  Alternative Hypotheses for the Settling Response The settling response functions hypothesized in the MFRC-IRE-IIASA models  of Jones (1977) and Clark, Jones, and Hoi 1ing (in press).are equivalent to those of Eq. 8-3 with THK = TSK = 1 and TFK =2.  That is, host species and  susceptible-aged trees are discovered in direct proportion to the land area they cover. Live foliage, which Jones (1977) reasoned to be an essentially three dimensional target as opposed to the simple two dimensions involved in forest cover, was sought somewhat more effectively. These are essentially minimum plausible estimates, and I adopt them here as a limiting hypothesis of."Passive" settling. The arguments considered in earlier sections of this Chapter lead me to suspect that these MFRC-IRE-IIASA targetting parameters are quite conservative  152 estimates of budworm's actual search capabilities. They imply extremely heavy loss rates in young or sparse host forests, and are thus incapable of accounting for the higher than average concentrations of immigrants found in isolated host stands. Various more "Active" search hypotheses implied by higher k. . 1  1  values predict such effects quite nicely. It is both impossible and meaningless to attempt to "fit" specific k.' 1  values to the qualitative settling descriptions of previous sections. But the exact numerical values are of minimal importance in any case. Fig. 8-3 shows that Eq. 8-2 rapidly approaches limiting values as 'k' rises from 1, and that these limits are approached particularly rapidly for high 'p..' For the condi1  tions of outbreak initiation and propagation which most concern me here, high 'p.' values are characteristic for at least two of the independent variables characterizing total targetting success (Eq. 8-3). Foliage under such conditions is essentially undamaged, so that FTD/3,8 ~ 1.0.  And the high SAR  values necessary to allow outbreaks normally arise only where most host trees are of susceptible age (i.e. HSP » 0.5; Jones, 1977). For these two compo- . nents of the settling response, the most important question is therefore simply whether their respective k' values are one or, say, two or more. 1  Values of 'k' higher than 3 yield no additional increases in target success and need not be explicitly evaluated. The third independent variable of Eq. 8-3, i.e. PHOST, typically takes on intermediate values in the neighborhood of 0.4 (see Fig. 8-2). This is just the range where targetting success is most enhanced by higher increments of 'k'. But if evidence for any component of the settling response does suggest limited 'k' values it is surely that for host land targetting. Some moths clearly land beyond reach of host forest, whatever their search capacities within such forest.  153 In full recognition of its arbitrary form and parameterization, I therefore define an alternative "Active" settling hypothesis for which the parameters of Eq. 8-3 are set at THK = 3, TSK = 3 and TFK = 2. This is compared with the-"Passive" MFRC-IRE-IIASA version in Table 8-1. Referring again to Fig. 8-3,.it is clear that radically different total targetting successes can result from the -"Active" and "Passive" settling alternatives. Since targetting success acts as a survival scalar directly on reproductive female-moths, the significance of this large differential may be quite important in determining the ultimate spatial population dynamics of the budworm system. The potential significance of this large uncertainty is the settling response cannot be explored without explicitly integrating the local and spatial perspectives of Parts II and III. It is to the first steps of that integration that I turn now, in Part IV of my analysis.  154  Table 8 - 1 :  Alternative parameterizations for the settling response function of Equation 8 - 3 .  Parameter  ~""~S"ettling Hypothesis . "Passive" "Active"  THK - host land targetting  1  3  TSK - susceptible age class targetti ng  1  3  TFK - foliage targetting  2  2  155 ALL AREA SUBJECT TO IMMIGRATION  ALL HOST SPECIES  SUSCEPTIBLE  I I V E FOLIAGE^  HOST SPECIES  Fig. 8-1: Targetting relationships for the settling response (see text)  156  Fig. 8-2: Cover type distribution New Brunswick, land and freshwater area: Total area is 7.32 million hectares.  157  T  Fig. 8-3: Descriptive equation for settling success. 'T' is proportion of projectiles hitting target at least once in k* tries, where 'p' is probability of success for a single try. 1  i  PART IV INTERACTIONS OF LOCAL STRUCTURE AND DISPERSAL PROCESSES  Chapter 9  Interactions of Local Structure and Dispersal Processes  159  CHAPTER 9  INTERACTION OF LOCAL STRUCTURE AND DISPERSAL PROCESSES  9.1  Overview  9.2  P : The spatial distribution of emigrants  9.3  N : The ratio of necessary to available migrants  9.4  $ : The propagation function  9.5  Summary  i  160 9.1 Overview Before considering questions of spatial arrangement and local uniqueness, it is necessary to explore certain relationships among the local structure and dispersal components already discussed in Parts II and III.  In particular,  it is important to appreciate the spatial scales at which various of these relationships become important. I have explored several approaches to the analysis of local structuredispersal relationships. In this Chapter, I report on the one which has been most useful as a compromise between tractability and relevance. My general focus concerns the determinants of insect outbreak propagation. I confine discussion to events occurring during a single time-interval of the system (i.e. a single year or insect generation), and defer treatment of the dynamic case to the full simulations of Part.V. Within.these constraints, I develop a function which specifies the maximum distance at which dispersers from an outbreak area are sufficiently numerous to trigger outbreaks in surrounding endemic areas in the following year. The function is a complex one, based on the local manifold configurations as developed in Parts II and III. It includes terms reflecting each of the major dispersal elements. The one important factor introduced here for the first time is the spatial extent of the area initially in outbreak. My treatment falls into three parts. First I develop an expression for P' . This expresses the (pre-settling) distribution of immigrants in space as a function of distance from, and size of, the exodus site.  It is based on  the P(r) displacement relationship of Chapter 7-. A term allowing for the radial spread of dispersers is also included. P* is a normalized relative density, for which the actual number of immigrants at each sink location is divided by the total number of emigrants displaced from all active source sites.  161 The second term of this analysis is N . This is a dimensionless number reflecting the proportion of the total quantity of emigrants from source regions which must immigrate to a sink location in order to trigger an outbreak there. N is defined as the ratio N /N . N is based on the manifold structures prec  t  £  sented in Parts.11 and III. It is a function of the "distance" between the lower stable (endemic) surface and the unstable (escape) surface of the relevant manifold, and therefore reflects the quantity of immigrants necessary to flip the system from its endemic to epidemic attractor. N expresses the t  quantity of emigrants generated in a year of outbreak. It essentially incorporates the size of the outbreak area and the slope of the emigrant curves already shown in Figure 6-7. The final term of this analysis is simply a dimensionless ratio of the first two: <j> = P yN*\  This "propagation function" $ gives the immigrant den-  sity actually arriving at various distances from the outbreak center as a fraction of the immigrant density necessary to trigger an outbreak there. For <fi  > 1 an outbreak will be triggered. FoV-'<j>  ><  1 it will not, at least in the cur-  rent generation. By exploring the behavior of <j> and, particularly, of the c>j - 1 surface I show how the considerations introduced in earlier Chapters interact in the incipient stages of outbreak propagation.  The detailed analy-  sis of the P'*, N'-, and § functions follows. 9.2 P*: The Spatial Distribution of Emigrants The P(r) functions of Chapter 7 are dimensionless expressions for the proportion (P) of all moths emigrating from a source site which immigrate at sink sites a distance (r) away. Tn practice, I normalize P(r) over one kilo200  meter increments of (r) such that £  r = 0  P(r) = 1-0 (Fig. 7-8a). Note that P(r)  is implicitly summed over all angles in order to represent a distance distribution, independent of direction.  162 The impact of immigrant moths (actually, eggs) on a forest site is a function of their density distribution in space, rather than their distance distribution per se. But it is easy to show geometrically that if the distance distribution is proportional to P(r), then the density distribution is p.roportibna-1 to P(r-)/r..Spec:if-i-cally, t-he^meankdensity, of ..immigrants which will f  :  occur if moths are distributed via a distance distribution P(r) over an arc of (y) radians, into a band of width (e) centered at radius (r) is P (r) r  ,  =  where k = 1/ye and has units of (length)  (Eq  . 9-1)  .  Budworm's normalized distance distribution P(r) functions from Chapter 7 are replotted in Figure 9-T, along with the related density transform P (r). Normalized values of the latter are appropriate for 1 km intervals, a band width of e = Kkm, and an isotropic dispersal pattern for which y = 2n. The P (r) values plotted in Figure 9-1 reflect the acute numerical sensitivity of Equation 9-1 to small changes in P(r) at low values of r (say, r < 5 km).  Given the rough estimation and smoothing procedures used to estimate  the P(r) functions, no biological significance should be attached to these irregularities. They disappear as the size of the source area is expanded (see below). It is worth noting from Figure 9-1 that the radius (r) at which peak immigrant densities occur (Fig. 9-lb) is considerably less than the modal displacement distance for dispersing moths (Fig. 9-la). In the case of displacement Hypothesis I, these peak radii occur at 23 and 37 km, respectively. Size of Source Site In previous Chapters I avoided explicit consideration of the size of .. "local" sites from which moths disperse. The P(r) and P (r.) functions  163 discussed above essentially reflect point source exodus phenomena, with immigration only quantified at convenient 1 km intervals. I now analyze how the point source picture changes as the exodus site takes on explicit finite dimensions. The problem could be conceptualized assuming infinitely small points sources and continuous space. But the ? functions are not conveniently r  integrable, and the results of the analysis are for use in numerical simulations on a grid system.  In practice it is therefore more natural to adopt a  discrete formulation of the problem. I begin by assuming a spatial surface partitioned into an (X,Y) Cartesian coordinate system of 1 km units. The density form of the displacement distribution function can then be rewritten as a function of the Cartesian coordinates as Pr(x',Y')  .  (Eq. .9-2) _?  Equation 9-2 has units km' . For a unit density of migrants it gives the spatial density distribution among all locations (X ±_-|, Y ± -|), where those coordinates are defined as displacements from the central exodus source site, 'e' may be any increment. In keeping with the normalization increment introduced earlier.I take it here to equal 1 km and omit it for simplicity in the i  i  remainder of the Chapter. Equation 9-2 is defined for all values of (X ,Y ) but, in practice, I take function values at radial distances greater.than 200 km to equal zero (cf. Fig. 9-1). I now consider multiple exodus source sites centered at locations (X , Y „ ) m n' v  in the fixed Cartesian system.  These are conveniently organized into a square  of side dimension (£)km, arbitrarily centered on the Cartesian origin (Fig. 9-2a).  164 From  each  source  exodus  an  i d e n t i c a l  unit  density  of  mi g r a n t s .  These  ' r are  d i s t r i b u t e d  migrants source  i n  i s  P * ( X  The  sink  square  (X^,Yj)  according s i t e s  of  Yj)  r  general  i n =  case  ^  f o r  -  X  at  J  ,  convenient. of  migrants  the  from  ,Y  ( X . , Y . ) .  and  ( X  ) The  m  , Y  n  functions,  and  contribution  ) , t o  a  single  land  from  sink  a  as  im-  s i n g l e  at  denoted  ( X j - Y ^ ) ]  multiple  sources  i n  .  a  square  of  side  £ ( F i g .  9-3c)  i s :  1/2  c a l c u l a t i o n s  Since  Pr(X  location  972b,  1/2  subsequent  s i t y  1  Figure  P  i d e n t i c a l  locations  I =  *  For  at  edge  shown  to  a  entire each  normalized  d i s t r i b u t i o n  displacement  source  to  computation  somewhere  i n  the  function j u s t sink  P  i s  moves  a  region,  more unit  i t  follows  that  +  0  I  i=-co  0  +  0  0  I P . ( V X . J) = 1  •  j=-oo  A l t e r n a t i v e l y  I  (and  I  as  Pr(X'  an  Y1)  e x p l i c i t  =  check  on  the  c a l c u l a t i o n s )  the  i d e n t i t y  1.0  X'=-oo Y ' = - »  is  +  s u f f i c i e n t  00  +  00  any  give  + C . 0 O + 0 O  i=oo j=oo  In  to  J  rase,  for  i/2  1/2  i = - c o j = - < x » m = - > i / . 2 r\=-a/2  - I - M - U - / m  -  .  0  J  den-  165 In any case, for the normalized distribution P * ( V j ) = P* (X Y )/£ Y  £  r  2  j  .  (Eq. 9-4)  P (X.,Y.) can be evaluated numerically for any point source displacement function P ( X ' j Y 1 ) .  In general the P functions will be asymmetric, reflecting r  New Brunswick's prevailing wind biases as discussed in Chapter 13. For present purposes it is sufficient to confine analysis to isotropic (radially symmetric) dispersal, or to the case where all dispersers are concentrated within a given sector (e.g. NW octant), but radially homogeneous within that sector. In either case, it suffices to characterize the resulting P*(X-j>Yj) distribution along a single representative radius as P*(.r)-. The individual values on that radius represent the proportion of the total moth population 2 ? emigrating from a source of size .(z) km , centered at r = o, which lands in ? a sink of 1.0 km centered at radius (r) km (Eqs. 9-3 and 9-4). Results of the numerical evaluations of P'(r) are given in Figure 9-3. Figure 9-3a gives detaiMs for isotropic dispersal under displacement Hypothec  esis I and a = 1,3,5,...,23 km.  Note that the numerical evaluation for  is a simple scalar multiple of the direct results for P (r) as presented in Figure 9-lb. The curves for higher (£) values reflect a "backfilling" effect of multiple sources.  It may be noted in passing that even the slight back-  filling associated with a = 3 is sufficient to damp out the previously noted numerical sensitivities of P (r) for very low (r). The most important and obvious conclusions to be drawn from these results is that substantial source size changes bring about little change in P (r). £  Such source-dependent variation as does occur is almost wholly confined to low radial distances. Below a - 10, the P^ functions are virtually indistinguishable. Above this certain differences become noticeable at low (r) values,  166 but further studies show that these increase approximately as (a) (the source 2 dimension), rather than as (•&) (the source area). The same detailed analyses show that the majority of the low (r) differences among the alternative P^(£ = 1,2,...) functions occur at r- <: «-/-2-.-i. e. within the source region itself. Beyond the edge of the source, P^r) for (rr•>/y£'/2) is insensitive to a. The same general pattern occurs in a comparison of the three displacement hypotheses originally developed in Chapter 7 and transformed to point source density functions in Figure 9-lb. Figures 9-3b to 9-3d give density displacement distributions for each of these functions, plotting only the limiting cases of a = 1 and a = 23 (compare Fig..9-3a). The "backfilling" effect of large sources is again evident, particularly for Hypothesis III which has its maximum P(r) value at a shorter radial distance than either of the other two hypotheses. More significantly, it is again clear that due to the shape of the original displacement function, source size (i) differences in P^(r) manifest themselves almost entirely at r -s .#£2, within the source patch itself. I return to these P (r) density displacement functions for construction £  of the propagation function <j> in Section 9.4 and for determination of grid size for the numerical simulation in Chapter 10. 9.3 N*: The Ratio off Necessary to-Available Migrants The P^(r) functions developed in Chapter 9.2 determine the proportion of 2 all moths emigrating from a source area of side (i) which land at a 1 km sink site (r) km away. I now assess N : a function determining the proportion of all moths emigrating from a source area which must land at a sink site of specified character in order to trigger an outbreak there. The ratio of these two functions will then yield the propagation function <f>.  167  I begin with the problem of assessing the immigrant density (N ) neces£  sary to trigger a local outbreak. Subsequent normalization relative to the total migrants available (N ) is straightforward. t  N : The immigrant density necessary to trigger outbreaks. c  N is essentially a function of each sink site's separation between its c  lower stable and unstable manifold surfaces (Chapters 3 and 6 ) .  I assume here  that sink sites are initially in an endemic condition on their lower equilibrium surfaces, and seek to determine the input of immigrant eggs (E^) necessary to boost them above the corresponding unstable equilibrium. Once there, the site is classed as "in outbreak," though visible forest damage will not appear for two or three years. All factors which alter the manifold will alter the numerical results of these calculations, but not their general form or conclusions. For the sake of definiteness, I deal only with the completely undefoliated (FT = 3 . 8 ) , mean weather case in the numerical illustrations which follow. Other cases--including that in which the lower stable equilibrium is everywhere equal to zero--can be accommodated with one simple recalculation of an (N ) component. ?' c  - ; i w hp-  Since dispersal operates in terms of eggs rather than larvae, it is convenient to recompute the relevant sink site manifolds in terms of equilibrium egg densities. This is done in Figure 9 - 4 for the full foliage, mean weather plane of my bas«ic exodus Type B and C. Only the portion covering the unstable and lower stable manifold surfaces is shown. E(SAR-) represents the additional 1  increment of local density necessary to trigger an outbreak.  (The " i " subscript  indicates "sink," to distinguish it from the SAR of the "source" site, used Q  later in this Chapter.) The E(SAR.j) function is plotted directly in Figure 9 - 5 .  As expected from  earlier study of the manifolds, for sufficiently low values of SAR^. the value of E(SAR) goes to infinity: the site is incapable of supporting an outbreak i  168  regardless of the number of immigrants added. Similarly, at sufficiently  ,  high SAR.J the value E(SAR^) goes to zero: a spontaneous outbreak will occur even if no immigrants are added. Finally, it is evident that the reduced emigrant losses of Type B exodus allow a Type B site to move from endemic to epidemic conditions with a smal1er immigrant input for any given value of SAR... The next step of the analysis is to translate E(SAR|), which expresses necessary number of eggs successfully settling per tsf branch area, into terms comparable to P , i.e. number of eggs arriving over each km of land. Most of this is a simple balancing of units. Since SAR^ has units of "tsf branch area/unit area of host forest," we need only an additional conversion constant C in "unit area host forest/km host forest" to give IT = E(SAR.) *,SAR * C i  (Eq. 9-5) 2  has units of "necessary number of eggs successfully settling/km host forest,";and is plotted in Figure 9-6. The inclusion of a term expressing proportion of sink site land covered in host forest, i.e. PHOST .j, gives N" = N* * PHOST.  .  (Eq. 9-6)  II  N consequently has units of "necessary number of eggs successfully settling/ 2  km of sink site land.!' A comparison of  and -P*(r) units shows that one final term must be  included to convert the "successfully settling" eggs of  into the "arriving"  eggs of P*(r). That term is obviously the targeting success function of Chapter 8. Given the uncertainties in its structure, let alone numerical values, I adopt here a simple total target success term TT^, which may be  169 thought of as the product of the individual terms discussed in Chapter 8. Since TT.. is the ratio of successfully settling/arriving eggs, N = N" * TT," ..... c c i 1  2  N has units of "necessary number of eggs arriving/km sink site land." It £  thus has almost the same units as P£(r), except that the former is expressed as an absolute density, while the latter is a proportion of the total number of emigrants. This discrepancy is remedied through the inclusion of  in  the next section. I summarize the present treatment with the full expression N = E(SAR) * SAR * PH0ST * TT/" * C  (Eq. 9-7)  1  c  i  i  i  N^: The total emigrant density Nj. is the total number of emigrants provided by a source area of side (i) km. The total number of eggs produced per square km of source area is NE' * SAR * PHOST * C O  O  0  where C is the same units conversion constant used in the formulation of N . c  NE^ equals NF * FEC, i.e. the number of female moths/tsf times their mean fecundity. From Chapter 6, we also know FMIG, the fraction of total eggs Q  which actually emigrates. Thus *l *C t = oNE o* SAR„ * PHOST0 * FMIG„ 0 1  .  2  v  (Eq. 9-8) ^ '  The ratio ,N/N =. N .can now.be written as 'follows-;, usying Equation's 9T7" C  t  and 9-8: .  N E(SAR.) * SAR.* PHOST.* C *TT. N = _...=.. . ; . • --. t NE* * 'SAR * PHOST-. * C * FMIG * ll _1  c  1  1  1  1  r  N  0  0  0  0  u  170 The C constants cancel, and the remaining terms group logically to give * E(SAR.) SAR. PH0ST. . TT." . . N = ., *^ * -; , * * -o NE" SAR-• PHOST; FMIG a 1  s  1  O  1  O  O  0  (Eq. 9-9)  •  0  N* has units of "necessary number of eggs arriving per km2 sink site land/ total eggs emigrating from source area." The N equation is valid for any values of its arguments. The most interesting case occurs when the sink site is endemic and the source area is in peak epidemic conditions. The same analysis used to produce Figure 6-6 shows that under spontaneous outbreak conditions NE^, SAR, and FMIG are highly correlated. Their product Q  Q  (NE^ * SAR * FMIG) is therefore confined to a range of 1000-2000 Q  for all  Q  of the exodus hypotheses considered here. Assuming PHOST^ = PH0ST, TT^ = 1, Q  and NE .* SAR * FMIG = 1000, numerical values of N* can be calculated as a Q  Q  Q  function of SAR.1 and I  o  for the two basic exodus types. This is done in  Figure 9-7. Few surprises emerge from the numerical results. The major sensitivity of N is to £ , the size of the outbreak source. SAR. has significant effect O  I  only when N is already quite low, in the 10" to 10" /km  range. I next show  how N interacts with P^ to determine the more interesting propagation function <f). 9:4  The Propagation function The function $ is a propagation index, relating the quantity of immigrants  actually provided a site via P , to the quantity necessary for precipitating an outbreak as reflected by N*. $ is a dimensionless function, formally  171 expressed as n  *  .  NE  1  * SAR * PHOST * FMIG- * i  (Eq. 9-10)  °' E(SAR) * SAR * PHOST * TT^ * 1  N  1  i  i  i  Rearranging terms in functional groups gives a succinct summary of my concerns in this analysis:  4> =  NE'o E(SAR)  'FMIG * P (ao ,r) SAR„ T o * PHOSo ' SAR * PHOST TT. v  i  i  .2  i  local inter-, dispersal compo- habitat quality source site, actions, see nents, see Part see Part V see Part IV Part II III "(Eq. 9.-11) IffJ>>l there are sufficient immigrants to initiate an outbreak in the  specified sink, otherwise not. A bit of numerical experimentation with reasonable values of <> t 's arguments shows that the major effects lie with the spatial dimensions, i.e. the size (a ) of the source area, and the distance (r) of the sink site from that source. Not unexpectedly, the relative surface area of the sink (SAR^) is next in interest. In the graphical solutions of <>j which follow I therefore treat r, & , and SAR^ explicitly, while assuming a constant value for the remaining product terms. In particular, I analyze <> j  = f(r, £ , SAR.) * K. • 0  where  K  1  '  NE' * FMIG„ * SAR o o 1 TT.  (Eq. 9-12) PHOST. o PHOST,  For the moment, assume that K = 10 , based on convenient assumptions of HE'  Q  = 2000, SAR = 1.0, FMIG = 0.5, TT^ = 1, and PH0ST/PH0ST ='l. Figure Q  Q  Q  i  9-8 shows the resulting <>j = 1 contours as a function of (r) and I  Q  for a range  172 of  SAPw  values. Separate graphs are given for exodus Type B and C, due to \.  their different N functions. Both graphs represent the case for isotropic c  dispersal under displacement Hypothesis I. The graphs of Figure 9-8 require careful study if they are not to be misleading. Consider particularly the following: While (r) is a radial distance from the center of the source site, £ is the "diameter" or full side .dimension of.that site. The 0  straight dashed diagonal in the figures therefore demarcates the outer edge of the source region. The contour interval is 0.10 SAR^ units for the exodus Type C figure, but only 0.05 SAR^ for Type B. I showed earlier in Figure 9-6 that dispersal mediated outbreak propagation under Type C exodus was possible over twice the SAR^ range possible with Type B. A comparison of Figure 9-8a and 9-8b additionally illustrates the relative importance for outbreak propagation of different spatial dimensions (r,£ ). Q  For any SAR^ value, there is a minimum source size necessary to initiate outbreaks at even immediately adjacent sink sites. Not surprisingly, this distance falls with increasing SAR^ and the associated fall in N  c>  Typical minimum source sizes for Type C  exodus are on the order of £ =20 km, and for Type B nearer £ = 10 km.  o  0  Once a source size.i'S'-reached sufficient to permit any propagation at all, the maximum distance at which propagation occurs (i.e. the r coordinate of a given <>j = 1 contour) rises rapidly at first, then more slowly. For larger source sites, say £ > 100 km, the <>j = 1 q  contours for all SAR^. values are nearly parallel, rising at a rate just less than the increase in the source size itself. At this stage the propagation phenomenon has lost most of its radial character and  173 functions essentially as a wave front. The relatively constant (r) increment between the <j> = 1 contour and the forward edge of the source area at these large a value sizes is therefore somewhat akin to the •  0  "asymptotic velocity" of simple epidemiological models (Bailey, 1975). The analogy should not be taken very far, but it is all I am able to get in this complex situation.  In any event, the maximal extent of  propagation, measured from the edge of the source area, ranges in Figure 9-8 from about 35 km (Type C, z = 150, SAR^ = .375) to 115 km (Type C, a = 150, SAR = .775). Q  i  The local interactions analysis of Part II showed that on a site of medium to high branch density, budworm numbers require about 4-6 generations to rise from escape densities just above the unstable equilibrium to a full outbreak similar to that postulated for the present source region. At a very crude level, it follows that a mean rate of visible outbreak spread might fall in the range of 35-115 km per 4-6 years or about 10-30 km/yr. This is within a factor of two of the generally accepted historical values cited earlier, and in even better accord with my own detailed data analysis of Chapter 11 (Figs. 11-5, 11-6). This agreement may, of course, be fortuitous. But it is also one of those welcome non-refutations which occasionally arise in this sort of analysis. I return to this issue below after exploring behavior of the <j> function in more detail. Generalizing the analysis for variable K The foregoing contour analysis has been constrained by the need to postulate fixed values for most of the arguments (specifically, the value of K) in Equation 9-12.  Furthermore, plotting only the <j> = 1 contours gives no  indication of the steepness of the <j> surface. A convenient resolution to these  174 difficulties is obtained combining K with c|> in a new function, which is then contoured. 3  Multiplying and dividing Equation 9-12 by 10 gives cf,  = f(r, £ , SAR^ * 10 * K/10 . 3  3  3  = K , and it follows that  Now define K/10 £ = f(r,  A  ,  SAR^ * 10  3  I plot contour values of and 9-10.  .  (Eq. 9-13)  as functions of r, & , and SAR^ in Figures 9-9  CJ>/K  Outbreaks are triggered wherever these contour values are  >,1/K.  (This follows algebraically: if the right hand side of Equation 9-13 is _> 1/K, then cf) >_ 1. By earlier arguments this defines the outbreak threshold.,: In other words, to find the critical outbreak-triggering combinations of r, i , and SAR.J for any value of K in Equation 9-12, we need only locate the contour of value K " = (K/10 )" in Figure 9-9 or 9-10.) 1  3  1  3  When K = 10 as in the examples given earlier, K = 1, <(>/< = cf., and the contour valued 1.0 again defines critical outbreak conditions. If NE^ (the mean egg density available for exodus from the source area) is reduced to . one-fourth of its original value, K = 0.25 and the plotted contour for cf./K = K = 4.may be interpreted as the critical outbreak triggering condition. 1  Figures 9-9 and 9-10 show a geometric progression of $/K contours for selected SAR^. values and both of the Type B and C exodus hypotheses. As before, isotropic dispersal under .displacement Hypothesis I is assumed. The <J)/K = 1 contours for each SAR^ value are identical to those for cf> = 1 given earlier in Figure 9-8. Figures 9-9 and 9-10 should be studied with regard to the same issues I pointed out in discussion of Figure 9-8. Note particularly how the minimum size, the maximum propagation distance, and "front velocity" characteristics  are now clearly functions of K, as well as the r, Jl , and SAR^. values emphasized in the earlier discussions. The following features are also of interest: The contours are plotted with a convenient geometric interval ratio of 4.0.  This should not obscure the fact that the  <J>/K  surface is very steep nea  the source edge, tailing off sharply with (r). This phenomenon reflects the shape of the P(r<) functions. .The results shown in Figure 9-9a are particularly instructive. Here the contour for  <J>/K  = 4.0 barely extends past the source front, and even then  only over a narrow range of source sizes. For < =0.25 this would be equivalent to the critical <> f = 1 contour, representing an essentially stationary outbreak condition. The contours can also be interpreted as the ratio by which the immigration rate under given conditions exceeds or falls short of that necessary for outbreak propagation. Consider first the case where cf> >:'.l. A forest manager faced with such a situation must achieve < $ > < 1 if the sink site in question is not to move into outbreak.  In principal, this can be achieved by suitably  manipulating any of the terms in Equation 9-11.  In practice, the only.realis  tic alternative has been to apply a mortality rate to the incoming eggs.  For  any given local value of cj>, the net achieved mortality on all incoming eggs must be greater than (1 - 1/$) to achieve the desired effect. For given tech nological capabilities of a moth-interception, air-to-air spray strategy such as that proposed for New Brunswick, Figures 9-9 and 9-10 thus provide guides to the budworm-forest conditions and locations for which propagation control via that technology is feasible.  176  Generalizing the analysis for nonisotropic dispersal Finally, use of the  <J>/K  contours allows a partial relaxation of the  isotropic dispersal assumption imposed at the beginning of this Chapter. The approach I have taken here is not a useful one for exploring complex radial inhomogeneities in the displacement process: that is undertaken via numerical simulation in Chapter 13.  I can, however, examine the effect on <>j of concen-  trating all the emigrants from a source into a radial sector of specified extent, so long as I require their homogeneous radial distribution within that sector. More simply, I address the situation where migrants originally distributed isotropically over 2n radians are constricted into a narrower sector of, say, (y) radians. The concentration ratio 2n/y = 9 simply acts as multiplier on the local density of available immigrants along the radius sampled by (r).  Without altering previous arguments, the right hand side of  Equation 9-11 can be multiplied by 0 (0 = 1 for isotropic dispersal), as can the right hand side; of Equation 9-12. Just as with K, this term is then brought into the denominator of the left hand side, giving  T  ± -  g  = f ( r ,  £, q  SAR.J • 1 0  3  .  (Eq.  9-14)  With !<* 0 = K the contours of Figures 9-9 and 9-10 are again directly appli1  ;  cable, reading K for the K given there. The same procedure which previously 1  defined critical contours as those for which <J)/K'  >_ 1/K'  = 1/K9.  <$>/K  >_ 1/K,  now defines them as  The reader can work through the impact of various radial  confinements for himself. By way of a guide to the plausible however, I show in Chapter 13 (Tables 13-2, 13-3) that prevailing winds may confine 50% of dispersing moths to a single quadrant of the airspace. Within that quadrant, K will therefore be reduced to half of its nominal value, but  0 = 2n/0.5n = 4.  177 The critical contour thus occurs at <*>/K' = 1/K9 = 0.5, a value half that of the isotropic dispersal case. For the medium branch density, Type C exodus situation, the radial concentration therefore has outbreak propagation consequences roughly comparable to those obtained by an increase of 0.1 in SAR^, a reduction of 10 km in (r),or an increase of 1.5 times in i . '  o  Alternative displacement hypotheses All of the foregoing analysis has been based upon P functions derived from displacement Hypothesis I. It remains to consider the § value under displacement Hypotheses II and III.  Fortunately, these prove to be so similar  to those derived above as to obviate the need for their explicit discussion here. Figure 9-11 illustrates this with the — = 1 contour for SAR. = .475 under each of the three d'ispT a cement Hypotheses.  As previously established,  Hypothesis I is generally intermediate to II and III.  The most notable dif-  ferences are at very low (r) and (z ) values, often within the source itself, and do not further concern me here. 9.5 Summary This Chapter explains how local structure and dispersal interact within a single year to trigger budworm outbreaks in endemic (sink) sites surrounding an epidemic (source) area. I derive a dimensionless propagation function <j>. This has the property that relationships between source and sink yielding.'. (j> > 1 are necessary and sufficient to result in propagation of the epidemic. The full expression for <f> is given below; terms.have been defined in-the--text, and are summarized in Appendix D : NE.  FMIG * P (A ,r) * 9 Q  E(SAR) i  TT.-1  SAR„ * PHOST o* o l21 0 SAR * PHOST. * Z.' i  (Eq. 9-15)  178 The terms on the right hand side of Equation 9-15 are the determinants of outbreak propagation behavior in the spatial structure theory I developed in Parts II and III. As noted by Smith (1952), one of the most important functions of an analysis such as I have performed here is its indication that each of these terms must be accounted for in any meaningful test of the theory's predictions. Equation 9-15 serves the additional purpose of summarizing relationships among key components of the spatial structure theory. Viewing the equation in functional groupings, the first bracketed term on the left gives the salient local structure characteristics: the ratio of egg density generated on the outbreak (source) site, to additional egg density required to trigger an outbreak on the endemic (sink) site. The second bracketed term