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UBC Theses and Dissertations

Strategies of cereal rust management : redesign of an agro-ecosystem to alter its stability properties Fleming, Richard Arthur 1982

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STRATEGIES OF CEREAL RUST MANAGEMENT: REDESIGN OF AN AGRO-ECOSYSTEM TO ALTER ITS STABILITY PROPERTIES by RICHARD ARTHUR FLEMING B . S c , T r e n t U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Zoology) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1982 ^ c ^ R i c h a r d A r t h u r F l e m i n g , 1982 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements f< an advanced degree at the Univers i ty of B r i t i s h C o l u m b i a , I a g r e e tha the L ibrary sha l l make i t f ree ly ava i l ab le for r e f e r e n c e and s t u d y . I fur ther agree that permission for extensive copying o f t h i s t h e s i s for scho la r ly purposes may be granted by the Head o f my Department o r by h is representa t ives . It is understood that copying o r p u b l i c a t i o n o f th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i ABSTRACT A major concern i n contemporary e c o l o g y has been the f a i l u r e of management i n many e c o n o m i c a l l y and s o c i a l l y i m p o r t a n t renewable r e s o u r c e systems. In s p i t e of e f f o r t s t o the c o n t r a r y , management has o f t e n w i t n e s s e d the e v o l u t i o n of such systems i n t o b e h a v i o r p a t t e r n s almost d i a m e t r i c a l l y opposed t o t h e i r o r i g i n a l o b j e c t i v e s . A c t u a l l y , i t i s management i n t e r v e n t i o n which o f t e n seems t o have i n i t i a t e d t h i s e v o l u t i o n by d i s t u r b i n g the system's s t a b i l i t y p r o p e r t i e s . T h i s study shows how an a l t e r n a t i v e approach which emphasizes system r e d e s i g n may a l l e v i a t e the .problem of changing s t a b i l i t y p r o p e r t i e s i n some r e l a t i v e l y s i m p l e and w e l l s t u d i e d ecosystems: the c e r e a l r u s t s and the c r o p s which they a t t a c k . The p l a n t d i s e a s e e p i d e m i o l o g i c a l l i t e r a t u r e r e c o r d s a number, of i d e a s about the b e h a v i o r and management of c e r e a l r u s t systems. Many of these i d e a s have been n e i t h e r a d e q u a t e l y f i e l d t e s t e d nor de v e l o p e d i n a c o h e r e n t a n a l y t i c framework. In t h i s s t u d y m a t h e m a t i c a l methods a r e used t o d e t e r m i n e the l o g i c a l consequences of some of the s e h y p o t h e s e s . The a n a l y s i s s u g gests t h a t c e r e a l r u s t systems might indeed be r e d e s i g n e d t o h e l p management meet i t s o b j e c t i v e s . Four a l t e r n a t i v e and m u t u a l l y c o m p a t i b l e p o t e n t i a l s t r a t e g i e s f o r a c h i e v i n g t h i s g o a l a r e c o n s i d e r e d : (1) u s i n g n a t u r a l enemies t o d e l a y d i s e a s e o n s e t , (2) u s i n g m u l t i l i n e s or v a r i e t y m i x t u r e s t o i n h i b i t p l a n t t o p l a n t d i s p e r s a l , (3) cha n g i n g f i e l d geometry t o i n c r e a s e d i s p e r s a l wastage, and (4) employing p o l y g e n i c r e s i s t a n c e t o slow the growth r a t e of the be s t adapted r a c e s . The r e s e a r c h and development r e q u i r e d t o implement any of these s t r a t e g i e s i s b r i e f l y d i s c u s s e d . Recommendations f o r f u t u r e work and comments on t h e promise of p l a n t e p i d e m i o l o g y as an a r e a f o r e c o l o g i c a l r e s e a r c h a r e p r o v i d e d . Acknowledgement TABLE GF CONTENTS PART I INTRODUCTION Chapter 1 I n t r o d u c t i o n PART I I CEREAL RUST DYNAMICS DURING THE DISEASE SEASON Chapter 2 A comparison of D i s e a s e P r o g r e s s E q u a t i o n s f o r C e r e a l Rust Chapter 3 PART I I I GENE-FOR-GENE RELATIONSHIPS S e l e c t i o n P r e s s u r e s and P l a n t Pathogens: Robustness of the Model ChaDter 4 PART IV ALTERNATIVE MANAGEMENT STRATEGIES D i s e a s e C o n t r o l Through Use of M u l t i l i n e s : A T h e o r e t i c a l C o n t r i b u t i o n Chapter 5 The Consequences of P o l y g e n i c D e t e r m i n a t i o n of R e s i s t a n c e and A g g r e s s i v e n e s s i n N o n s p e c i f i c Host: P a r a s i t e R e l a t i o n s h i p s Chapter 6 The E f f e c t of F i e l d Geometry on the Spread of Crop D i s e a s e 119 Chapter 7 The P o t e n t i a l f o r C o n t r o l of C e r e a l Rust by N a t u r a l Enemies 173 PART V CONCLUSIONS Chapter 8 C o n c l u s i o n s : The Broader Context 209 L i t e r a t u r e C i t e d 218 Appendix 239 v i ACKNOWLEDGEMENTS I have b e n e f i t t e d from i n t e r a c t i o n w i t h many people i n d e v e l o p i n g the i d e a s and methodology f o r the t h e s i s , p a r t i c u l a r l y B i l l C l a r k , B i l l F r y , Bean San Goh, Buzz H o l l i n g , Dixon Jones, S i L e v i n , Don Ludwig, L a c h l a n Marsh, Ken Minoque, Ted Munn, Judy Myers, C l a y P e r s o n , R a n d a l l Peterman, Henry T u c k w e l l , C a r l W a l t e r s , and Con Wehrhahn. C h a p t e r s 2 through 7 have been p u b l i s h e d e l s e w h e r e i n s u b s t a n t i a l l y the same form w i t h c o - a u t h o r s : C S . H o l l i n g (Chapter 2 ) , C O . Person (Chapters 4 and 5 ) , and both L.M. Marsh and H.C T u c k w e l l (Chapter 6 ) . C a r l W a l t e r s has g r a c i o u s l y a c c e p t e d and a d m i r a b l y performed the d i f f i c u l t j o b of a c t i n g s u p e r v i s o r i n C S . H o l l i n g ' s 1981-82 absence. Donna C h i n typed the e n t i r e m a n u s c r i p t and a s s i s t e d i n the almost e n d l e s s s e r i e s of a d m i n i s t r a t i v e d e t a i l s and demands. F i n a l l y , p a r t i c u l a r c o n t r i b u t i o n s of f i v e i n d i v i d u a l s were a b s o l u t e l y e s s e n t i a l t o the e x i s t e n c e of t h i s t h e s i s . C S . H o l l i n g magnanimously encouraged me t o take my own r e s e a r c h d i r e c t i o n s even when they d i v e r g e d from h i s ; C O . Person p a t i e n t l y t a u g h t me the b a s i c s of c e r e a l r u s t p o p u l a t i o n b i o l o g y . O f t e n they both s e l f l e s s l y s a c r i f i c e d on b e h a l f of my p e r s o n a l and p r o f e s s i o n a l w e l f a r e . E l s p e t h , M a r i e , and M e r e d i t h have had such a p e r v a s i v e i n f l u e n c e on my work t h a t t h e i r c o n t r i b u t i o n s cannot be i t e m i z e d , I can o n l y remain g r a t e f u l . To a l l , my t h a n k s . P A R T I I N T R O D U C T I O N CHAPTER INTRODUCTI 3 Plant Epidemiology and Ecology "Man suffers from epidemics. His crops suffer from epidemics. Whole nations suffer from epidemics. In f a c t , human suffe r i n g and epidemics of plant disease have gone hand i n hand since the e a r l i e s t history of man. Epidemics of plant disease have influenced man's food, his health, his s o c i a l customs, his economies, and even his a b i l i t y to -wage var." ( H o r s f a l l and Cowling, 1978) The control of epidemics of plant disease l i e s within the j u r i s d i c t i o n of plant disease management and, as Zadoks and Schein (1980)noted, "Epidemiology and plant disease management belong t o -gether as the two sides of a coin; the former has no s o c i a l relevance without the l a t t e r ; the second i s no longer possible without the f i r s t . " They see the ro l e of the epidemiologist as designing a l t e r -native approaches to the control and prevention of epidemics from which decision-makers can choose. Epidemiology provides the means with which speculations about disease control strategies can be r e -fined into testable hypotheses which, a f t e r appropriate t e s t i n g , be-come the elements of disease management programs. Zadoks and Schein (1980) "position epidemiology at the cross-roads of phytopathology, primarily a problem-oriented science, and ecology, p r i m a r i l y a.principle-oriented d i s c i p l i n e . For too long a period, epidemiology has developed within phytopathology, apart from the main stream of ecological thinking.... Long, too long, epidemi-ologi s t s have lingered at d e t a i l s , d e t a i l s of fungus a on host b i n environment c. Such honorable and indispensible studies often tend to lead from one spec i a l question to the next more specialized question. I f a researcher i s caught and encapsulated i n t h i s process, 4 his way toward alternative views may be blocked." Ecology "can serve as a counterpoise and offer an approach to alternatives." The Cereal Rusts In 1979 the Technical Advisory Committee of the Consultative Group on International Agriculture Research- (CGIAR), which operates under the auspices of the World Bank, the U.N.Development Programs, and the F. A. 0., i d e n t i f i e d the management and control of cereal diseases as among the world's most urgent areas for further a g r i c u l -t u r a l research (wortman, 1980). The cereal rusts are unarguabiy the most destructive of a l l groups of cereal diseases. The p a r a s i t i c fungi associated with the various cereal rusts are basidiomycetes i n the order Uredinales. Their l i f e cycle i s complex, involving as many as f i v e kinds of spores. The l i f e cycle often varies from rust to rust i n a given region and from region to region f o r a given rust (ingold, 1973)' The repeating uredial stage, with i t s enormous p o t e n t i a l to spread disease quickly, i s responsible f o r the tremendous y i e l d losses often i n f l i c t e d on cereal crops. Uredospores can often be e f f e c t i v e l y dispersed as f a r as 300 miles i n a matter of days or less (Loegering et_ a l . , 1967) and within a given l o c a l i t y the f r a c t i o n of vulnerable cereal tissue which i s v i s i b l y diseased can increase exponentially at rates of up to .5 per day ( i . e . , double i n less than 1.5 days) (Van der Plank, 1963)• In the great cereal growing regions of the world rusting may. develop to the extent that t h i c k rust coloured clouds of uredospores r i s e into the a i r during harvesting. The cereal rusts are extremely specialized obligate parasites. 5 A given species i s often separated into a number of morphologically s i m i l a r forms, formae speciales, -which occur more or less exclusively on different host genera. Each forma s p e c i a l i s consists of numerous physiologic races which are distinguishable from each other only by the spectrum of host v a r i e t i e s they can attack. Uredospores are haploid and dikaryotlc ( i . e . , two haploid n u c l e i i occupy each uredospore and j o i n t l y determine i t s phenotype as though the uredospore were d i p l o i d ) . Those cereal rusts which have a sexual cycle complete i t on t h e i r alternate ( i . e . , uncultivated) host. Where alternate hosts are absent, mechanisms outside the sexual cycle ( c f . Webster, 197*0 manage to maintain a large (Person et a l . , 1976), though reduced (Roelfs and Groth, 1980), l e v e l of phenotypic v a r i -a b i l i t y . Current Control Methods Current control of cereal rusts i s almost t o t a l l y based on the monoculture of uniformly resistant cereal v a r i e t i e s , unfortunately, t h i s method re s u l t s i n strong selection f o r those physiologic races capable of overcoming the resistance. A positive feedback loop for disease spread ensues: the greater the l o c a l increase of vi r u l e n t races, the greater the number of uredospores dispersed to other l o c a l i t i e s , the greater the number of l o c a l i t i e s i n which further l o c a l increase occurs, the greater the number of other l o c a l i t i e s subject to virul e n t inoculum, etc. This feedback loop i s a d r i v i n g mechanism behind the buildup of r u s t populations t o epidemic l e v e l s (Stakman, 1968). The positive feedback loop f o r disease spread i s intimately 6 connected with a positive feedback loop f o r variety spread ( P r i e s t l e y , 1978). When farmers notice that certain v a r i e t i e s are doing w e l l , they have h i s t o r i c a l l y tended to grow these v a r i e t i e s i n the hope that they w i l l continue to do w e l l . The more successful a certain variety, the more farmers tend to grow i t , the greater i t s acreage, the more v i s i b l e i t s success to farmers who didn't use i t , and so on. However, t h i s increase i n acreage, brought about by the variety's success, i n turn brings about i t s demise by increasing the selection f o r v i r u l e n t races and presenting them with a large area on which to exercise the positive feedback loop f o r disease spread. The eventual epidemic marks the culmination of t h i s p a i r of synchronous positive feedback loops and the i n i t i a t i o n of the next p a i r . This i s the basis of the 'boom and bust' cycle, and as Zadoks (1965) comments, "The greater the success of a resistant variety the more severe the epidemic when resistance breaks down." The use of resistance i n general as a means of disease control faces another problem. I t i s w e l l documented (Frankel, 1977) that the 'primitive' c u l t i v a r s of t r a d i t i o n a l agriculture are being ra p i d l y replaced by higher y i e l d i n g 'modern' c u l t i v a r s i n many remote areas of the world. In what appears to be a c l a s s i c case of option fore-closure (Walters, 1975)> t h i s replacement i s seriously eroding the world's pool of resistance genes ( H o r s f a l l , 1972). Thesis Objectives . Because of the shrinking pool of resistance genes and t h e i r often short-lived effectiveness under current management methods, alternative methods of using resistance and/or other control methods 7 to support resistance w i l l be needed i n the future. In t h i s context Browning (197*0 remarked, "As a prerequisite, we should recognize three things: F i r s t , a sound pest management program must be based on natural or b i o l o g i c a l means of pest management, especially the use of resistance and the encouragement of antagonists, as our f i r s t l i n e of defense. Second, a sound pest management program must not count on one fungicide or one gene or even one type of resistance to do the -whole job of disease control; rather the work load of managing pathogen populations and c o n t r o l l i n g disease must be shared. F i n a l l y , we must study natural ecosystems from which knowledge can be gained t h a t i s r e a d i l y applicable to agro-ecosystems." A number of hypotheses have emerged i n the plant epidemiology l i t e r a t u r e concerning the behaviors of natural and cultivated cereal: cereal rust systems and concerning a variety of possible approaches to c o n t r o l l i n g cereal rusts i n agriculture. A Large number of these hypotheses have been neither adequately f i e l d tested nor developed i n a coherent analytic framework. Here, using the techniques of mathematical analysis, I pursue the l o g i c a l consequences of.some of these ideas which concern the p o t e n t i a l l y f r u i t f u l areas suggested by Browning, above. Occasionally, I show that the consequences of a hypothesis are at odds with the claims made concerning i t . More often, I expand on an old hypothesis, suggest new ones, and recommend methods for f i e l d t e s t i n g where appropriate. Thesis Organization The thesis i s organized around the three broad areas of study: p o p u l a t i o n d y n a m i c s d u r i n g t h e d i s e a s e s e a s o n ( P a r t I I ) , g e n e -f o r - g e n e r e l a t i o n s h i p s a n d t h e i r e v o l u t i o n a r y b e h a v i o u r ( P a r t I I I ) , a n d p o s s i b l e a l t e r n a t i v e a p p r o a c h e s t o c e r e a l r u s t c o n t r o l ( P a r t I V ) . E a c h o f t h e s e p a r t s i s c o m p l e t e l y s e l f - c o n t a i n e d a n d p r o v i d e d w i t h i t s own b r i e f i n t r o d u c t i o n . P a r t V p r o v i d e s t h e o v e r a l l c o n c l u s i o n s o f t h e t h e s i s . T h e s i s M e t h o d s B e f o r e d e l v i n g i n t o t h e p a r t i c u l a r s o f t h e i n d i v i d u a l s t u d i e s i t w i l l be u s e f u l t o o u t l i n e t h e p h i l o s o p h y b e h i n d t h e m e t h o d s o f i n v e s t i g a t i o n ( F o r m o r e d e t a i l s e e F l e m i n g a n d B r u h n , 1 9 8 3 ) . I n t h e t h e s i s , i d e a l i z e d m a t h e m a t i c a l m o d e l s a r e u s e d t o e x p l a i n c e r t a i n a s p e c t s o f c e r e a l r u s t e p i d e m i o l o g y . T h e a s s u m p t i o n s o f t h e s e m o d e l s a r e c o n s c i o u s l y o v e r s i m p l i f i e d . T h i s p e r m i t s t h e i s o l a t i o n o f k e y r e l a t i o n s h i p s i n t h e h o p e t h a t t h e i r e x p l a n a t i o n may be a u s e f u l f i r s t s t e p i n u n d e r s t a n d i n g e n t i r e c e r e a l r u s t s y s t e m s i n w h i c h many f a c t o r s i n t e r a c t ( e . g . F l e m i n g , 1 9 8 0 a ) . T h e p h i l o s o p h y i s a k i n t o t h a t o f l a b o r a t o r y a n d g r o w t h c h a m b e r w o r k . T h e a n a l y s i s o f t h e s e i d e a l i z e d m o d e l s p r o v i d e s s t a n d a r d s o f c o m p a r i s o n a n d a f r a m e w o r k i n w h i c h t o c o n s i d e r m o r e c o m p l e x s y s t e m s ( L e v i n , 1 9 8 0 ) . F o r i n s t a n c e , t h e c l a s s i c a l t h e o r y o f s i n g l e l o c u s M e n d e l i a n p o p u l a t i o n g e n e t i c s a n d t h e i d e a l i z e d H a r d y - W e i n b e r g e q u i l i b r i u m p r o v i d e t h e a l m o s t u n i v e r s a l b a s i s f o r d i s c u s s i n g t h e m o r e c o m p l i c a t e d g e n e t i c m e c h a n i s m s b e h i n d e v o l u t i o n a r y c h a n g e . I n d e a l i n g w i t h m a n a g e m e n t q u e s t i o n s t h e r e i s o f t e n a d e l i b e r a t e s e a r c h f o r t h r e s h o l d s - a n d n o n - l i n e a r i t i e s i n t h e p a t h o s y s t e m d y n a m i c s w h i c h c a n be e x p l o i t e d f o r d i s e a s e c o n t r o l 9 (e.g. F l e m i n g , 1980b). In t h i s r e s p e c t e x p l a n a t o r y models o f t e n p r o v i d e a new and unique p e r s p e c t i v e and because of t h i s , need not be c l o s e l y t i e d t o e x p e r i m e n t a l work i n the way p r e d i c t i v e models must be. F u r t h e r m o r e , as Kranz (1974) n o t e s , e x p e r i m e n t a l t e s t i n g may be i m p a i r e d due t o the l i m i t s of e x p e r i m e n t a l a c c u r a c y or l a c k of a p p r o p r i a t e t e c h n i q u e s or a p p a r a t u s . In a p p l i e d e c o l o g y i n g e n e r a l , the b e s t e x p l a n a t o r y models have become i n f l u e n t i a l d i d a c t i c t o o l s , g u i d i n g r e s e a r c h and d e t e r m i n i n g c o n t r o l s t r a t e g i e s (Conway, 1977). PART I I CEREAL RUST DYNAMICS DURING THE DISEASE SEASON 11 I n t r c d u c t i o n In P a r t I I I f o c u s on the p o p u l a t i o n dynamics of c e r e a l r u s t s i n a g r i c u l t u r a l systems d u r i n g the d i s e a s e season. In Chapter 2 the d e t a i l s of the c e r e a l r u s t uredospore c y c l e , which i s r e s p o n s i b l e f o r c r o p damage, are d e s c r i b e d . A number of s i m p l e d i s e a s e p r o g r e s s models are compared, a new model i s sug g e s t e d , and recommendations f o r f u r t h e r r e s e a r c h are p r o v i d e d . 12 A COMPARISON OF DISEASE PROGRESS EQUATIONS FOR CEREAL RUST CHAPTER 2 13 I n t r o d u c t i o n E v e r s i n c e i t s i n t r o d u c t i o n b y L a r g e ( 1 9 4 5 ) a n d i t s s u b s e q u e n t a d o p t i o n b y V a n d e r P l a n k ( e . g . , 1 9 6 3 ) t h e l o g i s t i c e q u a t i o n h a s d o m i n a t e d p l a n t p a t h o l o g y . I t s s t a t u s a s t h e s t a n d a r d o f c o m p a r i s o n f o r t h e p r o g r e s s o f p o l y c y c l i c d i s e a s e s ( K r a n z , 1 9 7 8 ) i s m a n i f e s t e d i n t h e common p r a c t i c e o f p l o t t i n g t h e l o g i t t r a n s f o r m a t i o n o f d i s e a s e s e v e r i t y ( o r i n c i d e n c e ) , x , a g a i n s t t i m e , t . T h e ' n u l l h y p o t h e s i s ' o f l o g i s t i c g r o w t h , w h i c h p r e d i c t s a l i n e a r r e l a t i o n s h i p b e t w e e n l o g i t x a n d t , o f t e n p r o v i d e s a g o o d f i t t o d i s e a s e p r o g r e s s d a t a ( e . g . , J o w e r t e t a l . , 1 9 7 4 ; V a n d e r P l a n k , 1 9 6 0 , 1 9 6 3 ; W a g g o n e r , 1 9 8 1 ; Z a d o k s , 1971 ) . B u t p r e v i o u s d e r i v a t i o n s o f t h e l o g i s t i c t o d e s c r i b e d i s e a s e p r o g r e s s a r e b a s e d o n c r u d e a n a l o g i e s w i t h b a n k i n g ' s l a w o f c o m p o u n d i n t e r e s t a s m o d i f i e d b y a s a t u r a t i o n f a c t o r w h i c h b e c o m e s i n c r e a s i n g l y i m p o r t a n t a t h i g h e r d i s e a s e s e v e r i t i e s ( e . g . , L a r g e , 1 9 4 5 ; V a n d e r P l a n k , 1 9 6 3 ) . I n c i d e n t a l l y , r e s o r t i n g t o t h e o r e t i c a l e c o l o g y ' s d e r i v a t i o n s o f t h e l o g i s t i c m o d e l i m p l i c i t l y m a k e s t h e d a n g e r o u s a s s u m p t i o n t h a t p a t h o g e n p o p u l a t i o n s i z e ( s p o r e s , m y c e l i u m , p u s t u l e s , e t c . ) c a n be e q u a t e d t o d i s e a s e s e v e r i t y ( v i s i b l y d i s e a s e d h o s t t i s s u e ) . Due t o t h e m e t a p h o r i c a l n a t u r e o f t h e r e l e v a n t d e r i v a t i o n s , i t i s d i f f i c u l t t o e x p l a i n e p i d e m i o l o g i c a l l y w h a t i t m e a n s f o r a d i s e a s e t o p r o g r e s s l o g i s t i c a l l y . N o r a r e b i o l o g i c a l i n t e r p r e t a t i o n s o f v a r i a t i o n s on t h e l o g i s t i c m o d e l ( e . g . , K i y o s a w a , 1 9 7 2 ; W a g g o n e r , 1 9 8 1 ) f o r t h c o m i n g , i n d e e d , t h e l o g i s t i c m o d e l a n d i t s v a r i a t i o n s a r e w i d e l y v i e w e d a s u s e f u l f o r c u r v e f i t t i n g a n d d e s c r i p t i o n b u t i n c a p a b l e o f e x p l a i n i n g 14 the epidemic p r o c e s s i n a b i o l o g i c a l l y m e a n i n g f u l way (Van der P l a n k , 1960; Zadoks, 1972). Here I d e v e l o p a system of s i m u l t a n e o u s d i f f e r e n t i a l e q u a t i o n s t o d e s c r i b e the i n t e r a c t i o n of c o n s t i t u e n t b i o l o g i c a l p r o c e s s e s i n h e r e n t i n the c e r e a l r u s t i n f e c t i o n c y c l e . The e q u a t i o n s are founded on b i o l o g i c a l p r i n c i p l e s and e m p i r i c a l e v i d e n c e . In t h i s m e c h a n i s t i c approach I attempt t o i n c o r p o r a t e the p h y s i c a l form of the component p r o c e s s e s of d i s e a s e p r o g r e s s i n t o the s t r u c t u r e of the e q u a t i o n s . T h i s approach has two advantages over p u r e l y d e s c r i p t i v e m o d e l i n g : F i r s t , the e f f e c t of changes i n component r a t e s on d i s e a s e p r o g r e s s can be d i r e c t l y p r e d i c t e d . Second, independent measurements of what the parameters supposedly r e p r e s e n t can be compared w i t h the e s t i m a t e s r e s u l t i n g from f i t t i n g the models t o d i s e a s e p r o g r e s s d a t a . The system of s i m u l t a n e o u s d i f f e r e n t i a l e q u a t i o n s i s s u b s e q u e n t l y reduced t o a s i n g l e d i f f e r e n t i a l e q u a t i o n which s e r v e s as a b a s i s of comparison f o r the l o g i s t i c and t h r e e o t h e r d i s e a s e p r o g r e s s e q u a t i o n s p r e v a l e n t i n the l i t e r a t u r e . The o b j e c t i v e i s t c compare these e q u a t i o n s w i t h r e s p e c t t o t h e i r e x p l a n a t o r y a b i l i t y , t h e i r d e s c r i p t i v e a c c u r a c y , and the ease w i t h which t h e i r parameters can be c a l c u l a t e d . An a l t e r a t i v e d i s e a s e p r o g r e s s e q u a t i o n appears t o b e s t meet t h e s e c r i t e r i a of c o m p a r i s o n . 15 The Model The i n c r e a s e and spread of r u s t s among c u l t i v a t e d c e r e a l s i n the d i s e a s e season i s l a r g e l y by a s i n g l e spore t y p e , the ured o s p o r e . Uredospores are p r e d o m i n a n t l y wind d i s p e r s e d . A f t e r i m p a c t i n g on s u i t a b l e host t i s s u e s u c c e s s f u l u redospores germinate and m y c e l i a p e n e t r a t e and c o l o n i z e the p l a n t . A f t e r about 1-2 weeks ( e . g . , K a t s u y a and Green, 1 9 6 7 ; Leonard, 1 9 6 9 ) su b e p i d e r m a l u r e d i a become v i s i b l e , s i g n a l i n g the end of the l a t e n t p e r i o d (FBPP, 1 9 7 3 ) and s h o r t l y t h e r e a f t e r , u r e d i a break through the s u r f a c e of the p l a n t . A s p o r u l a t i n g uredium r e l e a s e s v a s t q u a n t i t i e s of ure d o s p o r e s d u r i n g i t s i n f e c t i o u s p e r i o d of 2-3 weeks ( e . g . , C h e s t e r 1 9 4 6 ; Leonard, 1 9 6 9 ) . A f t e r w a r d s the u r e d i a senesce and the t i s s u e which they o c c u p i e d i s e s s e n t i a l l y removed from the e p i d e m i c ; i t i s no l o n g e r i n f e c t i o u s and no l o n g e r c a p a b l e of bei n g i n f e c t e d . C o n s i d e r the p r o g r e s s of c e r e a l r u s t i n a f i e l d of u n i t a r e a . D e f i n i n g v u l n e r a b l e t i s s u e i n the f i e l d as t h a t which i s , or has proven t o be, s u s c e p t i b l e t o r u s t a t t a c k , I s e p a r a t e i t i n t o the f r a c t i o n s which a r e h e a l t h y (H), e x p e r i e n c i n g i n i t i a l c o n t a c t w i t h a uredospore ( I ) , l a t e n t l y i n f e c t e d ( L ) , and v i s i b l y r u s t e d ( x ) . I f s i s the f r a c t i o n of u r e d i a t h a t a re s p o r u l a t i n g , then the p r o p o r t i o n of v u l n e r a b l e t i s s u e which i s i n f e c t i o u s i s s x . In the model I t r y t o d e s c r i b e p r o c e s s e s which govern how the amount of v u l n e r a b l e t i s s u e a l l o c a t e d t o each of these c a t e g o r i e s changes over t i m e . By e x c l u d i n g p l a n t growth and m e t e o r o l o g i c a l i n f l u e n c e s I s i m p l i f y the mathematics and focu s a t t e n t i o n on the b a s i c b i o l o g i c a l p r o c e s s e s of epidemic development: i n f e c t i o n , l a t e n c y , s p o r u l a t o n , d i s p e r s a l , and 16 removal. The l o g i s t i c is a d i f f e r e n t i a l equation which describes changes through time in the proportion of vulnerable host tissue which is v i s i b l y diseased, x. Consequently, time periods of infectiousness and latency in the infect i o n cycle are represented by decay processes. Oort (1968) indicates that t h i s is not a serious drawback to modeling cereal rusts, at leas t . The fraction of vulnerable tissue which i s v i s i b l y diseased, x, accumulates d i r e c t l y through two processes (Ogle e_t a l . 1973). The f i r s t i s the appearance of uredia as they complete their latent periods. This occurs at rate f . The L second is the expansion of uredia over vulnerable tissue. I c a l l this rate f . Then the rate of accumulation of x is E dx/dt = f + f . (2.1) E L For future reference, Table 2.1 alphabetically l i s t s algebraic symbols used throughout the development of the model. If I is the mean length of the latent period and the age d i s t r i b u t i o n of infections is r e l a t i v e l y stable, then the rate at which latent infections become v i s i b l e can be taken as i t s average, f = L/i , L (2.2) 17 t o a f i r s t a p p r o x i m a t i o n . In f a c t , I i s p r o b a b l y somewhat g r e a t e r than the mean l e n g t h of the l a t e n t p e r i o d : a d i s p r o p o r t i o n a t e l y l a r g e number of l a t e n t i n f e c t i o n s are l i k e l y t o be i n the younger age c l a s s e s u n t i l d i s e a s e i n c r e a s e becomes l i m i t e d by a l a c k of h e a l t h y v u l n e r a b l e t i s s u e . C o m p e t i t i v e and/or s y n e r g i s t i c i n t e r a c t i o n s between u r e d i a have l i t t l e e f f e c t on t h i s r a t e (Metha and Zadoks, 1970; Teng and C l o s e , 1978). U r e d i a l E x p ansion C o m p e t i t i v e i n t e r a c t i o n s , however, must be c o n s i d e r e d i n d e r i v i n g f , the f u n c t i o n d e s c r i b i n g the r a t e of u r e d i a l E e x p a n s i o n . For i n s t a n c e , D a l y et. a_l. (1961) found t h a t wheat stem r u s t u r e d i a were s m a l l e r a t h i g h e r d e n s i t i e s of i n f e c t i o n . S i m i l a r r e s u l t s a re r e p o r t e d by Metha and Zadoks ( 1 9 7 0 ) f o r wheat l e a f r u s t and by Yarwood (1961) f o r bean r u s t . Under f i e l d c o n d i t i o n s K i n g s o l v e r e_t a_l. ( 1 9 5 9 ) o b s e r v e d a d e c l i n e i n the s e v e r i t y of wheat stem r u s t per u n i t amount of a p p l i e d inoculum when t h i s amount was i n c r e a s e d . P a r l e v l i e t (1979) c l a i m s t h a t c o m p e t i t i o n among u r e d i a over ho s t n u t r i e n t s i s e v i d e n t a t both h i g h and moderate u r e d i a d e n s i t i e s . I assume o n l y s p o r u l a t i n g u r e d i a can s i g n i f i c a n t l y a f f e c t the s p r e a d of d i s e a s e t h r o u g h u r e d i a l e x p a n s i o n , and, f o r the sake of argument, I c o n s i d e r l e a f r u s t of wheat and assume t h a t o n l y the l e a f b l a d e s a r e s u s c e p t i b l e t o i n f e c t i o n . T h e r e f o r e , the mean f r a c t i o n of v i s i b l y d i s e a s e d l e a f b l a d e t i s s u e i s e q u i v a l e n t t o the d i s e a s e s e v e r i t y , x. 18 A d a p t i n g Smith's (1963) r e a s o n i n g , I assume the average r a t e of e x p a n s i o n of i n f e c t i o u s t i s s u e , f , depends on the E a v a i l a b i l i t y of host n u t r i e n t s . S i n c e i n f e c t i o u s u r e d i a use n u t r i e n t s f o r expansion as w e l l as f o r maintenance and s p o r u l a t i o n , I approximate the r a t e of consumption of a v a i l a b l e n u t r i e n t s i n l e a f b l a d e s s u f f e r i n g a mean d i s e a s e s e v e r i t y x a s : V ( x ) = s v x - + s w f . (2.3) E Here s i s the f r a c t i o n of v i s i b l y d i s e a s e d t i s s u e which i s s p o r u l a t i n g , v i s the average r a t e a t which s p o r u l a t i n g u r e d i a consume n u t r i e n t s f o r m a i n t e n a n c e - s p o r u l a t i o n per f r a c t i o n of i n f e c t i o u s l e a f b l a d e t i s s u e , and w i s the r a t i o of the r a t e of consumption of a v a i l a b l e l e a f b l a d e n u t r i e n t s t o the r a t e of exp a n s i o n i n the f r a c t i o n of i n f e c t i o u s l e a f b l a d e t i s s u e . I seek a m a t h e m a t i c a l e x p r e s s i o n f o r f , the r a t e of E e x p a n s i o n of d i s e a s e d t i s s u e on l e a f b l a d e s s u p p o r t i n g u r e d i a on a mean f r a c t i o n x of t h e i r s u r f a c e . I assume t h a t f > 0 and E ~ t h a t 3f / 3x < 0 f o r 0 < x < 1. Perhaps the s i m p l e s t f u n c t i o n E _ ~ ~ which meets t h e s e r e q u i r e m e n t s i s of the form f = s x Z[V(1) - V ( x ) ] / V ( 1 ) , E (2.4) 19 where Z i s the mean r a t e of e x p a n s i o n per f r a c t i o n of i n f e c t i o u s l e a f b l a d e t i s s u e i n the absence of c o m p e t i t i v e i n h i b i t i o n . S i n c e f = 0 when x = 1, i t f o l l o w s from e q u a t i o n 2.3 t h a t E V(1) = sv. S u b s t i t u t i n g f o r V(x) and V(1) i n e q u a t i o n 2.4, the mean r a t e of ex p a n s i o n of i n f e c t i o u s t i s s u e i s f = x [ b - b x - c f ] , (2.5) E E where b = sZ i s the mean r a t e of u r e d i a l e x p a n sion per f r a c t i o n of v i s i b l y d i s e a s e d l e a f b l a d e t i s s u e i n the absence of c o m p e t i t i v e i n h i b i t i o n . Here c = sZw/v i s the r e l a t i v e c o m p e t i t i v e i n h i b i t i o n of u r e d i a l e x p a n s i o n due t o the demand on l e a f b l a d e n u t r i e n t s by t h i s e x p a n s i o n . R e a r r a n g i n g e q u a t i o n 2.5, the e x p e c t e d r a t e of i n c r e a s e i n the v i s i b l y d i s e a s e d p r o p o r t i o n of v u l n e r a b l e t i s s u e due t o u r e d i a l e x p a n s i o n i s f = bx(1 - x ) / ( 1 + c x ) . E L a t e n t l y I n f e c t e d T i s s u e Dynamics S u b s t i t u t i n g e q u a t i o n 2.2 and t h i s e x p r e s s i o n f o r f i n t o E e q u a t i o n 2.1, the r a t e of i n c r e a s e i n the p r o p o r t i o n of v u l n e r a b l e t i s s u e which i s v i s i b l y d i s e a s e d i s 20 dx/dt = bx (1 - X ) / ( 1 + cx) + L/£. ( 2 . 6 ) Because x i s the o n l y s t a t e v a r i a b l e c o n s i d e r e d i n the l o g i s t i c , L must be e x p r e s s e d i n terms of x i n e q u a t i o n 2 . 6 t o a l l o w d i r e c t c o m p a r i s o n s . T h i s r e q u i r e s a d e t a i l e d s tudy of the dynamics of the l a t e n t l y i n f e c t e d f r a c t i o n of v u l n e r a b l e t i s s u e , L. Changes i n L occur t h r o u g h two p r o c e s s e s : the e s t a b l i s h m e n t of new i n f e c t i o n s which o c c u r s at r a t e f , and the onset of T v i s i b l e symptoms which o c c u r s a t r a t e f . Hence, u s i n g e q u a t i o n 2 . 2 , the r a t e of change i n L can be w r i t t e n , dL/dt = f - L/£. ( 2 . 7 ) I Uredospore Dynamics The i n f e c t i o n r a t e , f , depends on the amount of h e a l t h y I and v u l n e r a b l e host t i s s u e (H) and the number . of v i a b l e u r e d o s p o r e s i n the d i s p e r s a l c l o u d ( U ) . The r a t e a t which U changes w i t h time can be w r i t t e n dU/dt = f - f + ( 2 . 8 ) 21 where f i s the r a t e of uredospore r e l e a s e and f i s the r a t e of + removal of ur e d o s p o r e s from the d i s p e r s a l c l o u d . Kochman and Brown (1975) found a l i n e a r r e l a t i o n s h i p between i n f e c t i o u s a r e a and uredospore p r o d u c t i o n f o r oat stem r u s t and oat crown r u s t . Assumptions t o t h i s e f f e c t a r e a l s o e v i d e n t i n t h e s i m u l a t i o n models of Shrum (1975) and Teng e t a l . (1980). A c c o r d i n g l y , f = Bsx, where B i s t h e r a t e of uredospore r e l e a s e i n t o the d i s p e r s a l c l o u d per u n i t amount of i n f e c t i o u s t i s s u e (1/B i s p r o p o r t i o n a l t o the average time i n t e r v a l between s u c c e s s i v e uredospore r e l e a s e s . ) I assume t h a t the mean r a t e of removal of ured o s p o r e s from the d i s p e r s a l c l o u d through d e a t h and/or d e p o s i t i o n i s f = DU, where D i s the r a t e parameter. L o s s e s t h r o u g h s u c c e s s f u l host c o n t a c t a re n e g l i g i b l e r e l a t i v e t o the s i z e of the uredospore c l o u d (Stakman and C h r i s t e n s e n , 1946). S t u d i e s on the d u r a t i o n of uredospore v i a b i l i t y (Shaner and Powelson, 1971; Yarwood and S y l v e s t e r , 1959) l e n d some q u a l i t a t i v e support t o t h i s r e l a t i o n s h i p . 22 S u b s t i t u t i n g these e x p r e s s i o n s i n t o e q u a t i o n 2.8, the mean r a t e of change i n the number of v i a b l e a i r b o r n e u r e d o s p o r e s becomes dU/dt = Esx - DU. The c h a r a c t e r i s t i c time s c a l e of t h i s e x p r e s s i o n i s 1/D, the mean l e n g t h of time between uredospore r e l e a s e and d e p o s i t i o n or l o s s of v i a b i l i t y . T h i s i s p r o b a b l y a matter of minutes or l e s s and c e r t a i n l y much l e s s than the time s c a l e at which n o t a b l e changes i n i n f e c t i o u s t i s s u e o c c u r . Hence the number of v i a b l e a i r b o r n e u r e d o s p o r e s , U, a d j u s t s q u i c k l y t o changes i n i n f e c t i o u s t i s s u e , sx, so t h a t , f o r any v a l u e of s x , U remains c l o s e t o i t s steady s t a t e v a l u e (the v a l u e of U a t which dU/dt = 0),: U z Bsx/D. (2.9) Here U i s e s t i m a t e d from i t s mean d a i l y v a l u e because the d i u r n a l p a t t e r n of uredospore r e l e a s e has been i g n o r e d i n d e r i v i n g e q u a t i o n 2.9. 23 T r a n s m i s s i o n To d e r i v e an e x p r e s s i o n f o r the i n f e c t i o n r a t e , f , I f i r s t I examine the d e t a i l s of the p r o c e s s of c e r e a l r u s t t r a n s m i s s i o n . C o n s i d e r the change i n the f r a c t i o n of v u l n e r a b l e h o s t t i s s u e e x p e r i e n c i n g i n i t i a l c o n t a c t w i t h a u r e d o s p o r e , I , o c c u r r i n g i n a s m a l l time i n t e r v a l , A t . I f At i s s m a l l enough t h a t r e l a t i v e changes i n the h e a l t h y p r o p o r t i o n of v u l n e r a b l e host t i s s u e , H, are n e g l i g i b l e , the change i n I can be approximated by: AI = ( J H U - I / G ) A t . Here J i s the r a t e of s u c c e s s f u l c o n t a c t per u n i t amount of h e a l t h y and v u l n e r a b l e t i s s u e per v i a b l e a i r b o r n e u r e d o s p o r e , U r e p r e s e n t s the number of v i a b l e a i r b o r n e u r e d o s p o r e s , and G i s the mean l e n g t h of time between i n i t i a l c o n t a c t and i n f e c t i o n e s t a b l i shment. The dynamics of A l i n t h i s e x p r e s s i o n o p e r a t e on a time s c a l e ' on the o r d e r of G which i s t y p i c a l l y a few hours (e.g., B u r l e i g h , 1965; Zadoks, 1961). In c o n t r a s t , i t has been assumed t h a t r e l a t i v e changes i n H are s m a l l i n At, and a c c o r d i n g t o e q u a t i o n 2.9, changes i n U oc c u r c o n c o m m i t e n t l y w i t h changes i n v i s i b l y d i s e a s e d t i s s u e . S i n c e t h i s time s c a l e i s of the order of a day or two a t l e a s t , I approaches i t s s t e a d y s t a t e v a l u e w i t h r e s p e c t t o h e a l t h y v u l n e r a b l e t i s s u e (H) and v i a b l e a i r b o r n e u r e d o s p o r e s (U). A p p r o x i m a t i n g I by i t s steady s t a t e v a l u e , I * , the mean r a t e of e s t a b l i s h m e n t of l a t e n t i n f e c t i o n s 24 becomes f ~I*/G = JHU . I S p a t i a l Aqqreqat i o n Uredospores are i m p l i c i t l y assumed to be randomly d i s p e r s e d over v u l n e r a b l e h o s t t i s s u e i n t h i s e q u a t i o n . However, the d i s t r i b u t i o n of p a r a s i t e s among h o s t s i s g e n e r a l l y aggregated or clumped w i t h i n h o s t p o p u l a t i o n s (Anderson and May, 1978). The n e g a t i v e b i n o m i a l model, which p r o v i d e s a s i n g l e parameter d e s c r i p t i o n of a g g r e g a t i o n , i s o f t e n used t o r e p r e s e n t such d i s t r i b u t i o n s ( e . g . G r i f f i t h s and H o l l i n g , 1969). F u r t h e r m o r e , Rouse e_t a l . (1980), Waggoner and R i c h (1981 ) , and Waggoner (1981) i n d i c a t e t h a t the n e g a t i v e b i n o m i a l can p r o v i d e a r e a s o n a b l e d e s c r i p t i o n of the a g g r e g a t i o n of d i s e a s e d t i s s u e i n c u l t i v a t e d p l a n t pathosystems. S i n c e most spores a re d e p o s i t e d c l o s e t o t h e i r source (Gregory, 1973), when a d i s p r o p o r t i o n a t e l y l a r g e amount of t i s s u e i s d i s e a s e d near t h i s , s o u r c e , a d i s p r o p o r t i o n a t e l y s m a l l number of c l o s e s i t e s a re s t i l l a v a i l a b l e f o r i n f e c t i o n . Hence, the a c t u a l i n f e c t i o n r a t e i s s m a l l e r than suggested i n the e q u a t i o n f o r f , above. U s i n g r e a s o n i n g s i m i l a r t o t h a t of I G r i f f i t h s and H o l l i n g (1969), Waggoner (1981) ' c o r r e c t s ' f o r t h i s e f f e c t by m o d i f y i n g the i n f e c t i o n r a t e e x p r e s s i o n t o 25 f = JHU(1 - x) 1 / k (2.10) I when clumping of d i s e a s e d t i s s u e can be d e s c r i b e d by the n e g a t i v e b i n o m i a l model. Here k i s a parameter v a r y i n g i n v e r s e l y w i t h the degree of a g g r e g a t i o n i n the d i s t r i b u t i o n of d i s e a s e d t i s s u e . Waggoner (1981) e s t i m a t e s k from the d i s t r i b u t i o n of u r e d i a per p l a n t . When k ->- oo , u r e d i a are i n d e p e n d e n t l y randomly ( P o i s s o n ) d i s t r i b u t e d among p l a n t s (the mean of the d i s t r i b u t i o n e q u a l s i t s v a r i a n c e ) . The s m a l l e r k, the g r e a t e r the clumping of u r e d i a among p l a n t s , the g r e a t e r the v a r i a n c e r e l a t i v e to the mean, and the more the mean u r e d i a l l o a d i s r e a l i z e d by h a v i n g few p l a n t s h e a v i l y i n f e c t e d and most p l a n t s u n i n f e c t e d . Of the many b i o l o g i c a l and p h y s i c a l p r o c e s s e s which can g e n e r a t e n e g a t i v e b i n o m i a l d i s t r i b u t i o n s ( B o s w e l l and P a t l , 1970; C r o f t o n , 1971) the major cause w i t h c e r e a l r u s t s may be d i f f e r e n c e s between p l a n t s i n exposure t o u r e d o s p o r e s . One m a n i f e s t a t i o n of t h i s i s the tendency of some ep i d e m i c s t o d e v e l o p f o c i (Kampmeijer and Zadoks, 1977). However, f o r our purposes h e r e , i t i s not of g r e a t importance t o know p r e c i s e l y what mechanisms generate the a g g r e g a t i o n of d i s e a s e d t i s s u e . Reduct i o n t o a S i n g l e Equat i o n For d i r e c t comparison w i t h the l o g i s t i c model, the system of e q u a t i o n s 2.6, 2.7, 2.9, and 2.10 must be reduced to a s i n g l e d i f f e r e n t i a l e q u a t i o n i n x, the p r o p o r t i o n of v i s i b l y d i s e a s e d v u l n e r a b l e t i s s u e . F i r s t , I w r i t e H i n terms of x by d e f i n i n g Q as the h e a l t h y f r a c t i o n of v u l n e r a b l e t i s s u e which i s not 26 v i s i b l y d i s e a s e d . Then the f r a c t i o n s of v u l n e r a b l e t i s s u e which are h e a l t h y and l a t e n t l y i n f e c t e d , r e s p e c t i v e l y , a r e H = Q(1 - x ) , and L = (1 - Q)(1 - x ) , (2.11) where Q i s a f u n c t i o n of x. Combining t h i s e x p r e s s i o n f o r K w i t h e q u a t i o n s 2.9 and 2.10, the i n f e c t i o n r a t e becomes f = a Q x( 1 - x ) ( l + 1 / k ) (2.12) I where a = B s J/D. A b i o l o g i c a l meaning f o r a emerges when e q u a t i o n 2.9 i s used i n t h i s e x p r e s s i o n g i v i n g a = JU/x. Thus, a i s the mean r a t e of s u c c e s s s f u l host c o n t a c t per u n i t d i s e a s e s e v e r i t y per u n i t amount of h e a l t h y and v u l n e r a b l e host t i s s u e . S u b s t i t u t i n g e q u a t i o n 2.12 i n t o e q u a t i o n 2.7 the r a t e of change i n the l a t e n t l y i n f e c t e d f r a c t i o n of v u l n e r a b l e t i s s u e becomes (1 + 1/k). dL/dt = a Q x(1 - x) - L/fc. (2.13) 27 The model has been reduced t o the system of two s i m u l t a n e o u s d i f f e r e n t i a l e q u a t i o n s 2.6, 2.13. F u r t h e r r e d u c t i o n t o a s i n g l e d i f f e r e n t i a l e q u a t i o n i n x r e q u i r e s an assumption which, a l t h o u g h not c o m p l e t e l y r e a l i s t i c , i s n e c e s s a r y to more c l e a r l y d e f i n e the b i o l o g i c a l l i m i t a t i o n s of l o g i s t i c d i s e a s e p r o g r e s s models. Once a g a i n I a p p e a l t o a d i f f e r e n c e i n dy n a m i c a l time s c a l e s i n making a s t e a d y s t a t e a p p r o x i m a t i o n and thus r e d u c i n g the system 2.6, 2.13, t o a s i n g l e d i f f e r e n t i a l e q u a t i o n . I assume t h a t l a t e n t l y i n f e c t e d t i s s u e , whose dynamics g e n e r a l l y o p e r a t e on a s l i g h t l y f a s t e r time s c a l e than those of i n f e c t i o u s t i s s u e (because the l a t e n t p e r i o d i s g e n e r a l l y somewhat s h o r t e r than the i n f e c t i o u s p e r i o d ) , remains near i t s steady s t a t e v a l u e w i t h r e s p e c t t o x i n e q u a t i o n 2.13: (1 + l / k ) L = a £ Q x ( 1 - x) S o l v i n g e q u a t i o n 2.11 f o r Q and u s i n g the r e s u l t i n the e x p r e s s i o n f o r L above, (1+l/k) L = a I x(1 - x ) /[1 + a I x(1 - x ) 1 / k ]. (2.14) S u b s t i t u t i n g t h i s e x p r e s s i o n i n t o e q u a t i o n 2.6, 28 (1+1/k) dx/dt = a x ( l - x ) / C l + a x f l - x ) 7 : + b x ( 1 - x ) / ( 1 - c x ) . (2.15) B i o l o g i c a l i n t e r p r e t a t i o n s f o r the parameters a r e g i v e n i n Table 2.1. The model has been reduced t o a s i n g l e d i f f e r e n t i a l e q u a t i o n i n x and i s now d i r e c t l y comparable t o the l o g i s t i c and o t h e r d i s e a s e p r o g r e s s e q u a t i o n s . D i s c u s s i o n Assumpt i o n s In d e v e l o p i n g e q u a t i o n 2.15 I have r e l a x e d a number of assumptions i m p l i c i t i n the l o g i s t i c model. In c o n t r a s t to the l o g i s t i c , e q u a t i o n 2.15 a l l o w s nonrandom uredospore d i s t r i b u t i o n over h o s t t i s s u e when k > 0 and i t a l l o w s u r e d i a l e x p a n sion co impose a d r a i n on the s u p p l y of h o s t n u t r i e n t s when c > 0. While t h i s f l e x i b i l i t y may r e p r e s e n t an improvement, some u n s a t i s f a c t o r y assumptions remain i n e q u a t i o n 2.15 b e s i d e s the e x c l u s i o n of host growth and m e t e o r o l o g i c a l i n f l u e n c e s . To reduce the model t o a s i n g l e e q u a t i o n I assumed t h a t the p r o p o r t i o n of l a t e n t l y i n f e c t e d v u l n e r a b l e host t i s s u e , L, remains a t i t s steady s t a t e v a l u e w i t h r e s p e c t to x. The e f f e c t of t h i s assumption i s examined i n F i g u r e 2.1. Curve c i l l u s t r a t e s the change of L over t i m e , t , i n the ' r e a l i t y * of the complete model, e q u a t i o n s 2.6, 2.13. Curve a i l l u s t r a t e s the ' r e a l ' s t e a d y s t a t e a p p r o x i m a t i o n : ' r e a l ' x v a l u e s c a l c u l a t e d by n u m e r c i a l i n t e g r a t i o n of the complete model ( e q u a t i o n s 2.6, 2.13) a r e s u b s t i t u t e d i n t o e q u a t i o n s 2.14 t o e s t i m a t e L. Curve b 29 i l l u s t r a t e s the ' e s t i m a t e d ' s t e a d y s t a t e a p p r o x i m a t i o n w i t h r e s p e c t t o x: x v a l u e s c a l c u l a t e d from the reduced model, e q u a t i o n 2.15, are s u b s t i t u t e d i n t o e q u a t i o n 2.14 t o e s t i m a t e L. S i n c e the steady s t a t e . a p p r o x i m a t i o n s are i n c l o s e r agreement w i t h each o t h e r than w i t h L's ' r e a l ' v a l u e , i t i s the steady s t a t e assumption i t s e l f , and not e r r o r i n c a l c u l a t i n g x, which i s m a i n l y r e s p o n s i b l e f o r d i f f e r e n c e s between L's ' r e a l ' v a l u e , and i t s a p p r o x i m a t i o n from the reduced model. In f a c t , the ' r e a l ' s teady s t a t e a p p r o x i m a t i o n i s a c t u a l l y l e s s a c c u r a t e than the ' e s t i m a t e d ' a p p r o x i m a t i o n d u r i n g almost 70% of the d i s e a s e season. T h i s s u g g e s t s t h a t e r r o r s i n the x v a l u e s c a l c u l a t e d from the reduced model compensate f o r p a r t of the e r r o r i n the ste a d y s t a t e a p p r o x i m a t i o n i n F i g u r e 2.1. F i g u r e 2.2 compares n u m e r i c a l i n t e g r a t i o n s of the reduced model ( c u r v e c) w i t h t h o s e of the c o r r e s p o n d i n g complete model, e q u a t i o n s 2.6, 2.13, f o r d i f f e r e n t i n i t i a l v a l u e s of L. The complete model i s more s e n s i t i v e t o the i n i t i a l p r o p o r t i o n of l a t e n t l y i n f e c t e d v u l n e r a b l e t i s s u e , L, than i t i s to the steady s t a t e assumption which a l l o w e d the r e d u c t i o n . Moreover, the reduced model i s a b e t t e r a p p r o x i m a t i o n when the complete model b e g i n s a t the more r e a s o n a b l e r a t i o of L/x = 10, r a t h e r than L/x = 0. Hence, i n a c c u r a c i e s caused by the assumption t h a t L remains at i t s ste a d y s t a t e v a l u e a r e p r o b a b l y s m a l l i n comparison to the p o t e n t i a l e f f e c t caused by i n a c c u r a c i e s i n e s t i m a t i n g L's i n i t i a l v a l u e . F i g u r e 2.2 a l s o h i g h l i g h t s a problem i n h e r e n t i n any model which does not e x p l i c i t l y account f o r the i n i t i a l amount of l a t e n t l y i n f e c t e d t i s s u e . Such a model must a s c r i b e the 30 d i f f e r e n c e between the h y p o t h e t i c a l d i s e a s e developments i l l u s t r a t e d by c u r v e s a anb b of F i g u r e 2.2 e i t h e r t o a d i f f e r e n c e i n some parameter v a l u e , or t o a d i f f e r e n c e i n the i n i t i a l d i s e a s e s e v e r i t i e s , or p a r t l y to both. In f a c t , t h e r e are no d i f f e r e n c e s between c o r r e s p o n d i n g parameter v a l u e s or i n the i n i t i a l d i s e a s e s e v e r i t i e s . In t h i s sense, a l l such models i n c l u d e parameters•which i m p l i c i t l y a g g r e g a t e a number of p r o c e s s e s . Of these models, the more complex, m e c h a n i s t i c ones have the advantage t h a t some imp o r t a n t p r o c e s s e s a r e e x p l i c i t l y a c counted f o r i n s e p a r a t e parameters. When these parameters are measured i n independent e x p e r i m e n t s , the number of p r o c e s s e s aggregated i n t o o t h e r parameters i s reduced and the n a t u r e of the p r o c e s s e s a g g r e g a t e d by these o t h e r parameters i s more c l e a r l y d e f i n e d . For i n s t a n c e , when the l o g i s t i c model i s f i t t e d t o d i s e a s e p r o g r e s s d a t a , i t s r a t e parameter n e c e s s a r i l y a g g r e g a t e s the p r o c e s s e s d e s c r i b e d by the f i v e parameters of e q u a t i o n 2.15. T h i s p r e c l u d e s a unique b i o l o g i c a l l y m e a n i n g f u l d e f i n i t i o n of the apparent i n f e c t i o n r a t e i n terms of the p r o c e s s e s u n d e r l y i n g each r e c u r r e n t i n f e c t i o n c y c l e ( e . g . , l a t e n c y , spore d e p o s i t i o n ) . I have a l s o assumed t h a t the a g g r e g a t i o n of d i s e a s e d t i s s u e i n space i n f l u e n c e s spore d i s p e r s a l but not u r e d i a l e x p a n s i o n . I f , as i s commonly suggested ( e . g . Kampmeijer and Zadoks, 1977; Stakman and C h r i s t e n s e n , 1946; Waggoner, 1981), spore d i s p e r s a l i s the major c o n t r i b u t o r t o c e r e a l r u s t s p r e a d , t h i s o m i s s i o n may not s e r i o u s l y l i m i t the u s e f u l n e s s of e q u a t i o n 2.15. A r e l a t e d a s s u m p t i o n , which i s a l s o i m p l i c i t i n Waggoner's (1981) model, i s t h a t the degree of u r e d i a a g g r e g a t i o n , as 31 measured by the n e g a t i v e b i n o m i a l parameter k, i s c o n s t a n t . T h i s i s u n l i k e l y but i n the absence of i n f o r m a t i o n d e s c r i b i n g how k changes, i t p r o v i d e s a ' n u l l h y p o t h e s i s ' as a s t a r t i n g p o i n t f o r f u r t h e r i n v e s t i g a t i o n . Throughout I have assumed t h a t a c o n s t a n t f r a c t i o n , s, of v i s i b l y d i s e a s e d t i s s u e i s i n f e c t i o u s . In r e a l i t y , s can be expected t o d e c l i n e as r u s t p r o g r e s s e s through i t s d i s e a s e season. Jack Bruhn ( p e r s o n a l c o m m u n i c a t i o n ) , u s i n g h i s p o t a t o l a t e b l i g h t s i m u l a t i o n model (Bruhn e_t a l . , 1980), has shown t h a t s i s l i k e l y t o be a c o m p l i c a t e d f u n c t i o n of x and t . A l t h o u g h the d e r i v a t i o n of t h i s f u n c t i o n i s beyond the scope of t h i s paper, i t i s enough to r e c o g n i z e t h a t the magnitudes of the parameters a and b de c r e a s e d u r i n g the d i s e a s e season. T h i s i n t r o d u c e s n e g a t i v e c u r v a t i v e t o p l o t s of l o g i t (x) a g a i n s t t and tends t o s t r a i g h t e n c u r v e s of - l n ( l n l / x ) a g a i n s t t . These are c h a r a c t e r i s t i c s of the Gompertz model which g i v e s the per u n i t r a t e of i n c r e a s e i n d i s e a s e s e v e r i t y as ( 1 / x ) ( d x / d t ) = g l n d / x ) where g i s a r a t e parameter. Comparison of D i s e a s e P r o g r e s s E q u a t i o n s In examining over 100 time s e r i e s of d i s e a s e i n c r e a s e d a t a from v a r i o u s c r o p s and e n v i r o n m e n t a l c o n d i t i o n s , Berger (1981) has found the Gompertz model t o be a much b e t t e r d e s c r i p t o r of d i s e a s e p r o g r e s s than the l o g i s t i c when broad ranges of x a r e 32 e n c o u n t e r e d . H o w e v e r , i t i s d i f f i c u l t t o p r o v i d e a b i o l o g i c a l r a t i o n a l e f o r t h e f o r m o f t h e f a c t o r f o r d e n s i t y d e p e n d e n t l i m i t a t i o n , 1n 1/x, i n t h e G o m p e r t z m o d e l . H e n c e , a l t h o u g h t h e G o m p e r t z m o d e l may p r o v i d e a n e x c e l l e n t d e s c r i p t i o n o f d i s e a s e p r o g r e s s , , i t d o e s n o t w a r r a n t s e r i o u s c o n s i d e r a t i o n a s a n e x p l a n a t o r y m o d e l . N o n e t h e l e s s , B e r g e r ' s ( 1 9 8 1 ) w o r k s u g g e s t s t h a t t h e G o m p e r t z t r a n s f o r m a t i o n p r o v i d e s a n e x c e l l e n t s t a n d a r d b y w h i c h t o c o m p a r e m o d e l s o f d i s e a s e p r o g r e s s . F i g u r e 2.3 i l l u s t r a t e s f o u r d i s e a s e p r o g r e s s e q u a t i o n s : c u r v e c r e p r e s e n t s t h e l o g i s t i c ; c u r v e d i l l u s t r a t e s W a g g o n e r ' s ( 1 9 8 1 ) m o d i f i c a t i o n o f t h e l o g i s t i c ; c u r v e s a a n d b d e s c r i b e t h e t w o t e r m s f a n d f , r e s p e c t i v e l y , w h i c h c o m p r i s e t h e d i s e a s e . E L p r o g r e s s m o d e l d e v e l o p e d h e r e f r o m e q u a t i o n 2 . 1 . T h e t e r m s f E a n d f r e s p e c t i v e l y r e p r e s e n t t h e c o n t r i b u t i o n s t o d i s e a s e L p r o g r e s s i n e q u a t i o n 2.15 o f u r e d i a l e x p a n s i o n o v e r h o s t t i s s u e a n d o f t h e e s t a b l i s h m e n t o f new i n f e c t i o u s t h r o u g h u r e d o s p o r e d i s p e r s a l . P a r a m e t e r v a l u e s i n F i g u r e 2.3 w e r e c h o s e n t o h i g h l i g h t t h e d i s t i n c t b e h a v i o r o f t h e s e e q u a t i o n s o v e r a w i d e r a n g e o f x v a l u e s w h i l e r e m a i n i n g w i t h i n b i o l o g i c a l l y r e a l i s t i c r a n g e s ( w h e r e s u c h r a n g e s a r e k n o w n ) . T h u s k i s s e t a t i t s l o w e r o b s e r v e d l i m i t o f 1 ( W a g g o n e r , 1 9 8 1 ) t o d i s t i n g u i s h W a g g o n e r ' s m o d i f i c a t i o n o f t h e l o g i s t i c f r o m t h e l o g i s t i c i t s e l f ; a n d l i s s e t a t i t s u p p e r c o m m o n l y o b s e r v e d l i m i t o f 14 d a y s ( e . g . , L e o n a r d , 1 9 6 9 ) t o d e m o n s t r a t e i t s p o t e n t i a l i n f l u e n c e m o s t c l e a r l y . F i g u r e 2.4 i l l u s t r a t e s t h e l o g i t t r a n s f o r m a t i o n s 33 c o r r e s p o n d i n g t o t h e p l o t s i n F i g u r e 2 . 3 . T h e a p p a r e n t i n f e c t i o n r a t e ( V a n d e r P l a n k , 1 9 6 3 ; F l e m i n g , 1 9 8 1 ) c o r r e s p o n d s t o t h e s l o p e o f t h e a p p r o p r i a t e c u r v e . T h u s , o f t h e f o u r e q u a t i o n s p l o t t e d , o n l y t h e l o g i s t i c ( c u r v e c ) p r e c l u d e s t h e p o s s i b i l i t y o f a d e c l i n e i n t h e a p p a r e n t i n f e c t i o n r a t e a s x i n c r e a s e s . S u c h a d e c l i n e i s f r e q u e n t l y o b s e r v e d ( e . g . , B e r g e r , 1 9 7 5 ; S i n c l a i r a n d C a m p a n a , 1 9 7 8 ) . B e r g e r ( 1 9 8 1 ) c l a i m s t h a t t h e l o g i s t i c g i v e s a c r e d i b l e s t a t i s t i c a l f i t o n l y i n t h e r a n g e o f .05 < x < 0.6. T h e G o m p e r t z t r a n s f o r m a t i o n s c o r r e s p o n d i n g t o t h e p l o t s o f F i g u r e 2.3 a r e s h o w n i n F i g u r e 2 . 5 . B e r g e r ( 1 9 8 1 ) i m p l i e s t h a t t h e l i n e a r i t y o f s u c h p l o t s p r o v i d e s a n e s t i m a t e o f t h e i r s u i t a b i l i t y a s d i s e a s e p r o g r e s s m o d e l s . H e n c e a c c o r d i n g t o t h i s m e a s u r e , t h e l o g i s t i c ( c u r v e c ) . i s t h e l e a s t a p p r o p r i a t e s i n c e i t h a s t h e w i d e s t v a r i a t i o n i n s l o p e . C o n v e r s e l y , t h e e x p r e s s i o n s f o r f ( c u r v e a ) a n d f ( c u r v e b) a r e t h e m o s t E L a p p r o p r i a t e s i n c e t h e y h a v e t h e t w o s m a l l e s t r a n g e s o f s l o p e . U n d e r t h i s c r i t e r i o n , e q u a t i o n 2.15 w h i c h c o m p r i s e s t h e s e t w o ' m o s t a p p r o p r i a t e ' e x p r e s s i o n s , i s a n i m p r o v e m e n t o v e r t h e l o g i s t i c a n d W a g g o n e r ' s ( 1 9 8 1 ) m o d i f i c a t i o n o f i t . H o w e v e r , b o t h W a g g o n e r ' s m o d e l a n d e q u a t i o n 2.15 h a v e a m a j o r d r a w b a c k c o m p a r e d t o t h e l o g i s t i c ; n e i t h e r c a n be r e a d i l y i n t e g r a t e d f o r g e n e r a l k. I f r a n d o m u r e d o s p o r e d i s t r i b u t i o n i s a s s u m e d ( i . e . k + oo) , W a g g o n e r ' s ( 1 9 8 1 ) m o d e l r e v e r t s t o t h e l o g i s t i c ; a n d e q u a t i o n 2 . 1 5 , a l t h o u g h i t c a n now b e i n t e g r a t e d , p r o d u c e s a c u m b e r s o m e i n t e g r a l w h i c h p e r m i t s n o e a s y t r a n s f o r m a t i o n f o r p a r a m e t e r e s t i m a t i o n . B u t i f u r e d i a l e x p a n s i o n h a s l i t t l e e f f e c t o n c e r e a l r u s t p r o g r e s s ( i . e . b -> 0) a n d i f t h e P o i s s o n d i s t r i b u t i o n ( k -> « ) 34 a d e q u a t e l y d e s c r i b e s uredospore d i s p e r s a l , as James and S h i h (1973 ) i n d i c a t e f o r l e a f r u s t , then e q u a t i o n 2.15 s i m p l i f i e s t o dx/dt = ax(1 - x ) / ( 1 + a £ x ) . (2.16) T h i s e q u a t i o n i s i l l u s t r a t e d by c u r v e a i n F i g u r e s 2.3, 2.4, and 2.5 when the l a t e n t p e r i o d i s £ = 14 days and the r a t e parameter i s a = 0.14. By B e r g e r ' s (1981) c r i t e r i o n of l i n e a r i t y of the Gompertz t r a n s f o r m a t i o n , t h i s e q u a t i o n appears t o p r o v i d e a b e t t e r g e n e r a l d e s c r i p t i o n of d i s e a s e p r o g r e s s than e i t h e r the l o g i s t i c or Waggoner's (1981) m o d i f i e d l o g i s t i c ( F i g u r e 2.5). R e a r r a n g i n g i t s i n t e g r a l , a = l n [ x . / ( 1 - x . ) ] - l n [ x /(1 -x )] • t t o o t - £ l n [ ( 1 - x )/(1 - x )] where x and x are the d i s e a s e s e v e r i t i e s a t times 0 and t , o t r e s p e c t i v e l y . T h u s the r a t e parameter can be e s t i m a t e d from the s l o p e of p l o t s of l o g i t (x ) a g a i n s t t - £ l n [ ( l - x ) / ( l fc ° x ) ] . t Kiyosawa (1972) proposed a model t o d e s c r i b e the r a t e of i n c r e a s e i n the number of l e s i o n s or c u m u l a t i v e s p o r e s , y, over time f o r e p i d e m i c s of P y r i c u l a r i a o r y z a e : dy/dt = r y(1 - y/Y) 35 where Y i s the maximum v a l u e y a t t a i n s i n the y e a r . S i n c e the d i s e a s e s e v e r i t y t h a t year can be w r i t t e n as x y/Y, t h i s e q u a t i o n r e v e r t s t o the l o g i s t i c when both s i d e s are d i v i d e d by Y. F u r t h e r m o r e , s i n c e Riyosawa's r a t e parameter r i s not a f f e c t e d by t h i s r e s c a l i n g , r i s e q u i v a l e n t t o Van der P l a n k ' s (1963) a p p a r e n t i n f e c t i o n r a t e . Hence, p r e v i o u s remarks c o n c e r n i n g the l o g i s t i c a p p l y e q u a l l y w e l l t o Riyosawa's m o d i f i c a t i o n of i t . N e v e r t h e l e s s , Riyosawa's e q u a t i o n d i f f e r s from Van der P l a n k ' s l o g i s t i c e q u a t i o n i n one major r e s p e c t . In the e p i d e m i c s he s t u d i e d , Riyosawa found t h a t the r a t e of d i s e a s e p r o g r e s s d e c l i n e d near the end of the growing season r e g a r d l e s s of the l e v e l of d i s e a s e reached. Hence, he reasoned t h a t the d e c l i n e i n r a t e was not caused by l i m i t a t i o n i n the r e m a i n i n g amount of s u s c e p t i b l e h o s t t i s s u e , but by e x t e r n a l f a c t o r s . R esearch Recommendations There a r e a number of avenues open f o r f u r t h e r r e s e a r c h . On the t h e o r e t i c a l s i d e , S k y l a k a k i s (1980) has r e c e n t l y extended the concept of r e l a t i v e p a r a s i t i c f i t n e s s t o s i t u a t i o n s where c o m p e t i t i v e i n h i b i t i o n among d i s e a s e organisms i s s i g n i f i c a n t . A c c o r d i n g t o h i s model, i t i s i n e v i t a b l e t h a t the r a c e w i t h the g r e a t e r a p p arent i n f e c t i o n r a t e i n m i x t u r e s of two r a c e s growing on the same c r o p w i l l e v e n t u a l l y dominate the pathogen p o p u l a t i o n . H i s model i s b a s i c a l l y an e x t e n s i o n of the l o g i s t i c . P r e l i m i n a r y i n v e s t i g a t i o n s suggest t h a t u s i n g the more g e n e r a l e q u a t i o n 2.15 as a f o u n d a t i o n f o r e x p l o r i n g the c o m p e t i t i o n between two r a c e s may p r e s e n t a r i c h e r v a r i e t y of p o s s i b l e 36 outcomes ( i n c l u d i n g mutual c o e x i s t e n c e f o r both r a c e s ) . There i s g r e a t e r scope f o r i m m e d i a t e l y u s e f u l e x p e r i m e n t a l work. F i r s t , e q u a t i o n 2.16 needs t o be t e s t e d on a v a r i e t y of d i s e a s e p r o g r e s s d a t a . Second, a thorough study of the changes i n the age d i s t r i b u t i o n of the r u s t p o p u l a t i o n i n the f i e l d d u r i n g a d i s e a s e season would a l l o w e s t i m a t e s of the f u n c t i o n s ( x , t ) t o r e p l a c e s i n e q u a t i o n 2.15. I t might a l s o p e r m i t an e v a l u a t i o n of the assumption t h a t l a t e n t l y i n f e c t e d t i s s u e remains near i t s stea d y s t a t e v a l u e w i t h r e s p e c t t o d i s e a s e s e v e r i t y . Measurements of u r e d i a l l o a d s . and the p r o p o r t i o n s p o r u l a t i n g c o u l d be taken s i m u l t a n e o u s l y t o determine how s p o r u l a t i n g and n o n s p o r u l a t i n c u r e d i a are d i s t r i b u t e d among p l a n t s and l e a v e s . The assumption t h a t the degree of u r e d i a l . a g g r e g a t i o n i s c o n s t a n t d u r i n g the d i s e a s e season, and t h a t u r e d i a l l o a d s a re not c o r r e l a t e d w i t h the f r a c t i o n s p o r u l a t i n g , c o u l d both be checked w i t h t h i s d a t a . F i n a l l y , when e s t i m a t e s of the parameter v a l u e s , of the complete model e q u a t i o n s 2.6, 2.13, have been made from l a b and/or f i e l d work, they c o u l d be checked a g a i n s t parameter v a l u e s c a l c u l a t e d from f i t t i n g the model to independent d i s e a s e p r o g r e s s d a t a . T h i s would p r o v i d e a thorough t e s t of the model's e x p l a n a t o r y c a p a b i l i t i e s . Such a t e s t i s n e c e s s a r y because, a l t h o u g h the p r e d i c t i o n s of the model may be c o n s i s t e n t w i t h o b s e r v a t i o n , the mechanisms assumed i n the c o n s t r u c t i o n of the model may not r e f l e c t r e a l i t y ( e . g . , Solomon, 1979). 37 Summary A c o n t i n u o u s m a t h e m a t i c a l model of the p r o g r e s s of c e r e a l r u s t d i s e a s e was developed by i n c l u d i n g many of the n a t u r a l p r o c e s s e s u n d e r l y i n g each r e c u r r e n t i n f e c t i o n c y c l e . T h i s model i s s u b s e q u e n t l y reduced t o a s i n g l e e q u a t i o n and compared w i t h f o u r o t h e r models p r e v a l e n t i n the l i t e r a t u r e : the l o g i s t i c , Riyosawa's (1972) m o d i f i e d l o g i s t i c , Waggoner's (1981) m o d i f i e d l o g i s t i c , and the Gompertz. The e q u a t i o n d e v e l o p e d here p r o v i d e s a b a s i s f o r e x p l o r i n g c a u s a l l i n k a g e s between these f o u r models and i l l u m i n a t i n g some of t h e i r i m p l i c i t a s s u m p t i o n s . I t i s argued t h a t s i n g l e f i r s t o r d e r e q u a t i o n models must c o n t a i n one or more parameters which croup a number of e f f e c t s i n order t o be f i t t e d t o d i s e a s e p r o g r e s s d a t a . T h i s p r e c l u d e s unique b i o l o g i c a l l y m e a n i n g f u l d e f i n i t i o n s f o r parameters of such s i n g l e parameter models as the l o g i s t i c w i t h r e s p e c t to the p r o c e s s e s u n d e r l y i n g d i s e a s e p r o g r e s s (e.g. i n f e c t i o n , s p o r u l a t i o n ) . The d i f f e r e n t i a l e q u a t i o n dx/dt = ax (1 - X ) / ( 1 + a I x ) , where a i s a r a t e parameter, I i s the l e n g t h of the l a t e n t p e r i o d , t i s t i m e , and x i s d i s e a s e s e v e r i t y , i s o f f e r e d as a p o t e n t i a l l y u s e f u l a l t e r n a t i v e t o p r e v i o u s models of d i s e a s e p r o g r e s s . TABLE 2.1 DEFINITIONS OF ALGEBRAIC SYMBOLS* r a t e a t w h i c h v u l n e r a b l e h o s t t i s s u e i s s u c c e s s f u l l y c o n t a c t e d p e r u n i t d i s e a s e s e v e r i t y p e r u n i t a m o u n t o f h e a l t h y a n d v u l n e r a b l e h o s t t i s s u e ( t - 1 ) • v i a b l e u r e d o s p o r e r e l e a s e r a t e p e r u n i t i n f e c t i o u s t i s s u e ( U t - 1 ) u r e d i a l e x p a n s i o n r a t e p e r u n i t d i s e a s e s e v e r i t y i n t h e a b s e n c e o f c o m p e t i t i v e i n h i b i t i o n ( t - 1 ) r e l a t i v e c o m p e t i t i v e i n h i b i t i o n o n u r e d i a l e x p a n s i o n d u e t o t h e d e m a n d on h o s t n u t r i e n t s b y t h i s e x p a n s i o n (-) p e r v i a b l e u r e d o s p o r e r a t e o f r e m o v a l f r o m t h e d i s p e r s a l c l o u d , ( t _ 1 ) mean t i m e b e t w e e n i n i t i a l c o n t a c t a n d i n f e c t i o n e s t a b l i s h m e n t ( t ) h e a l t h y f r a c t i o n o f v u l n e r a b l e h o s t t i s s u e f r a c t i o n of vulnerable host tissue experiencing i n i t i a l contact with a viable uredospore (-) rate of successful contact per unit amount of healthy and vulnerable host tissue per viable airborne uredospore when uredospores disperse randomly over vulnerable host tissue ( U - 1 t _ 1 ) negative binomial parameter varying inversely with the degree of aggregation in the d i s t r i b u t i o n of uredia among plants (-) la t e n t l y infected proportion of vulnerable host tissue (-) mean length of the latent period (t) fraction of apparently healthy vulnerable host tissue which has not been l a t e n t l y infected (-) sporulating proportion of v i s i b l y diseased tissue (-) time (e.g., days) U : number of v i a b l e u r e d o s p o r e s i n the d i s p e r s a l c l o u d v : r a t e of consumption of a v a i l a b l e n u t r i e n t s f o r u r e d i a l maintenance and s p o r u l a t i o n per f r a c t i o n of i n f e c t i o u s v u l n e r a b l e t i s s u e ( e . g . , gms t ~ 1 ) w : r a t i o of the r a t e of consumption of a v a i l a b l e n u t r i e n t s t o the r a t e of e x p a n s i o n i n the f r a c t i o n of i n f e c t i o u s v u l n e r a b l e t i s s u e ( e . g . , gms) x : v i s i b l y d i s e a s e d f r a c t i o n of v u l n e r a b l e host t i s s u e (-) Z : u r e d i a l e x p a n s i o n r a t e per f r a c t i o n of i n f e c t i o u s v u l n e r a b l e t i s s u e i n the absence of c o m p e t i t i v e i n h i b i t i o n ( t _ 1 ) Footnote: *Brackets following d e f i n i t i o n s enclose the dimension-a l i t y , e.g. ( U t - 1 ) indicates B i s measured i n uredo-spores per u n i t time (e.g., day); (-) indicates c i s dimensionless. Rates r e f e r r e d to are mean instantaneous rates. FIGURES FIGURE 2.1 The proportion of la t e n t l y infected vulnerable host tissue, L , i s plotted against time, t. In curve c, L begins at .01 and is calculated numerically from the simultaneous equations 2 . 6 , 2 . 1 5 with 1 = 1 . Curve a gives the steady state approximation to L corresponding to curve c and calculated from equation 2 . 1 7 . Curve b i l l u s t r a t e s the steady state approximation to L , equation 2 . 1 7 , where x i s estimated from equation 2 . 1 8 . Parameter values were a . 1 3 4 , b = . 0 6 6 , c = 2, k = 1 , and i n i t i a l l y x = .001 . FIGURE 2.2 Disease severity, x, is plotted against time, t, for numerical integrations of the system of simultaneous d i f f e r e n t i a l equations 2 . 6 , 2 . 15 where L has i n i t i a l values of 0.01 and 0.0 for curve a and curve b, respectively; and for numerical integrations of equation 2 . 1 8 , curve c. For a l l plots a = . 1 3 4 , b = . 0 6 6 , c = 2, k = 1, 1 = 1 , and i n i t i a l l y x = . 0 0 1 . FIGURE 2.3 Disease severity, x, is plotted against time, t, for numerical integrations of four d i f f e r e n t equations: curve a, dx/dt = .14 x ( 1 - x ) / ( 1 + 2 x ) ; curve b, dx/dt .24 x (1 - X ) 2 / [ 1 + 3 . 3 6 x ( 1 - x ) ] ; c u r v e c , d x / d t = .09 x ( 1 - x ) ; a n d c u r v e d , d x / d t = .16 x ( 1 - x ) 2 . F I G U R E 2.4 L o g i t t r a n s f o r m a t i o n o f t h e . d i s e a s e p r o g r e s s c u r v e s o f F i g u r e 2 . 3 . T h e v a l u e s o f l o g i t ( x ) a t w h i c h x = .05 a n d . 6 a r e s h o w n . F I G U R E 2.5 G o m p e r t z t r a n s f o r m a t i o n o f t h e d i s e a s e p r o g r e s s c u r v e s o f F i g u r e 2 . 3 . T h e v a l u e s o f - 1 n ( m 1 / x ) a t w h i c h x .05 a n d . 6 a r e i n d i c a t e d . F I G U R E 2.1 F I G U R E 2 . 3 46 F I G U R E 2 . 4 47 FIGURE 2.5 PART I I I GENE-FOR-GENE RELATIONSHIPS 49 INTRODUCTION The t r i g g e r t o the p o s i t i v e feedback l o o p f o r d i s e a s e s p r e a d (Chapter 1) i s the s e l e c t i o n of v i r u l e n c e i n the r u s t p o p u l a t i o n . T h i s p r o c e s s of s e l e c t i o n i s w i d e l y b e l i e v e d t o be the r e s u l t of the . dynamics of p o p u l a t i o n g e n e t i c s o p e r a t i n g w i t h i n the s t r u c t u r e of gene-for-gene r e l a t i o n s h i p s . Gene-For-Gene R e l a t i o n s h i p s The gene-for-gene c o n c e p t was developed by F l o r (1956, 1971) w h i l e e x p e r i m e n t i n g w i t h f l a x (Linum u s i t a t i s s i m u m ) and the r u s t p a r a s i t i z i n g i t (Melampsora l i n i ) . He r e c o r d e d two q u a l i t a t i v e l y d i f f e r e n t t y p e s of d i s e a s e r e a c t i o n . S u s c e p t i b l e h o s t s s u f f e r e d n o t i c e a b l y g r e a t e r p a r a s i t i c development than h o s t s w i t h r e s i s t a n t d i s e a s e r e a c t i o n s . The observed p a t t e r n s of s e g r e g a t i o n i n d i c a t e d t h a t r e s i s t a n c e i n f l a x and v i r u l e n c e i n the r u s t a r e o f t e n d e t e r m i n e d by s i n g l e genes (Day, 1976). He n o t i c e d t h a t f o r each r e s i s t a n c e gene (R-qene) of the host a c o r r e s p o n d i n g v i r u l e n c e gene (V-gene) must be p r e s e n t i n the p a r a s i t e genotype b e f o r e t h e d i s e a s e can d e v e l o p (Table I I I ) . The response of the hos t t o the presence of the p a r a s i t e i s det e r m i n e d by t h i s i n t i m a t e i n t e r a c t i o n of s p e c i f i c l o c i . A t h e o r e t i c a l c o n s i d e r a t i o n of the o r i g i n of gene-for-gene r e l a t i o n s h i p s l e d Person (1968) t o the g e n e r a l c o n c l u s i o n t h a t " f o r o b l i g a t e p a r a s i t e s , whose environment i s l a r g e l y h o s t d e t e r m i n e d s p e c i f i c i n t e r a c t i o n s based on gene-for-gene r e l a t i o n s h i p s a re to be e x p e c t e d . " The f o l l o w i n g t h e o r e t i c a l s c e n a r i o i s a s l i g h t e x t e n s i o n of Person's i d e a s . C o n s i d e r a s p e c i e s o c c u p y i n g a c o n s i s t e n t environment. A 50 p a r t i c u l a r phenotype, say S, p r o b a b l y w i l l have been n a t u r a l l y s e l e c t e d . S i n c e phenotype i s the p r o d u c t of the i n t e r a c t i o n of genotype and environment, genotypes s e l e c t e d i n a p a r t i c u l a r environment are those whose D h e n o t y p e s a c a u i r e i t h e n e c e s s a r y r e s o u r c e s (energy, n u t r i e n t s , e t c . ) and then d i s t r i b u t e s these r e s o u r c e s among v a r i o u s b i o l o g i c a l p r o c e s s e s so as t o maximize r e l a t i v e r e p r o d u c t i v e s u c c e s s or f i t n e s s (Leon and Tumpson, 1975; Rapport, 1971). Suppose a p o p u l a t i o n of o b l i g a t e p a r a s i t e s , which consume the l i m i t e d r e s o u r c e s of members of the f i r s t ( h o s t ) s p e c i e s t o f u r t h e r t h e i r own r e p r o d u c t i v e s u c c e s s , becomes a s i g n i f i c a n t f a c t o r i n the environment. E v e n t u a l l y , t h a t p a r a s i t e phenotype, c a l l i t A, which i s most e f f e c t i v e a t d i v e r t i n g host r e s o u r c e s t o i t s own r e p r o d u c t i v e e f f o r t s , w i l l dominate the p a r a s i t e p o p u l a t i o n ( G i l l , 1974). I assume t h a t any d i f f e r e n t i a l s e l e c t i o n a c t i n g on the p a r a s i t e phenotypes when they are not i n f e c t i n g members of the h o s t s p e c i e s i s n e g l i g i b l e . Subsequent s e l e c t i o n on the host s p e c i e s i n t h i s environment of A - p a r a s i t e s w i l l f a v o r host phenotypes w i t h g r e a t e r - f i t n e s s than the S-phenotype. S i n c e S-hosts were s e l e c t e d i n the p a r a s i t e - f r e e environment, o t h e r host phenotypes are l e s s e f f i c i e n t a t r e s o u r c e a c q u i s i t i o n and a l l o c a t i o n . However, i n t h i s new environment, h o s t phenotypes which can i n t e r r u p t ( E l l i n g b o e , 1972, 1976; Person and Mao, 1974) the d r a i n of r e s o u r c e s by A - p a r a s i t e s may overcome any d e f i c i e n c y a t o b t a i n i n g and d i s t r i b u t i n g r e s o u r c e s . F u r t h e r m o r e , i n t e r f e r e n c e w i t h h o s t m e t a b o l i c p r o c e s s e s by A - p a r a s i t e s may markedly reduce the e f f i c i e n c y w i t h which i n f e c t e d S-hosts p r o c u r e r e s o u r c e s . 51 U l t i m a t e l y , s e l e c t i o n f a v o r s t h a t host phenotype ( c a l l i t R ) , which i s most e f f e c t i v e a t combining r e s o u r c e a c q u i s i t i o n and a l l o c a t i o n w i t h the p r e v e n t i o n of s u c c e s s f u l a t t a c k by A-p a r a s i t e s . In e f f e c t , the A - p a r a s i t e s are a v i r u l e n t on t h e R - h o s t s s e l e c t e d f o r r e s i s t a n c e t o them. E v e n t u a l l y the R-gene d i s p l a c e s the S-gene i n the h o s t p o p u l a t i o n . T h i s has two e f f e c t s . F i r s t , s e l e c t i o n f o r r e s i s t a n c e a t o t h e r host l o c i c e a s e s . Second, s e l e c t i o n f o r p a r a s i t e s which can s u c c e s s f u l l y d e v e l o p on R-hosts b e g i n s . By d e f i n i t i o n , t h e s e are v i r u l e n t p a r a s i t e phenotypes. U l t i m a t e l y , t h a t V-gene which produces the most e f f e c t i v e p a r a s i t e phenotype at d i v e r t i n g the r e s o u r c e s of R-hosts to i t s own p r o p a g a t i o n w i l l be s e l e c t e d . Subsequent i n c r e a s e i n the f r e q u e n c y of the V-gene i n the p a r a s i t e p o p u l a t i o n r e s u l t s s p e c i f i c a l l y from the p r e v a l e n c e of r e s i s t a n c e gene R i n the host p o p u l a t i o n . A gene-for-gene r e l a t i o n s h i p then e x i s t s (Table I I I ) . In f a c t , a gene-for-gene i n t e r a c t i o n p a t t e r n i s e x p e c t e d even i f a s t r i c t e q u i v a l e n c e between the numbers of l o c i i n v o l v e d i n the two p o p u l a t i o n s does not e x i s t ( P e r s o n , 1959). The i n c r e a s e i n frequency of gene V i n the p a r a s i t e p o p u l a t i o n i n f l u e n c e s the s e l e c t i v e f o r c e s on both p o p u l a t i o n s . S e l e c t i o n f o r v i r u l e n c e a t o t h e r p a r a s i t e l o c i and f o r R-hosts ceases w h i l e s e l e c t i o n f o r another h o s t r e s i s t a n c e gene, c a l l i t R , b e g i n s . Under the s e c i r c u m s t a n c e s a second d i r e c t i o n of s e l e c t i o n becomes i m p o r t a n t . I t can be e x p l a i n e d by c o n s i d e r i n g the 52 e f f e c t s of g e n o t y p i c change on p h e n o t y p i c t r a i t s . A s e l e c t e d genotype can be thought of as an arrangement of genes which i n t e r a c t and modify each o t h e r ' s e f f e c t s so as t o produce a r e g u l a t e d i n t e g r a t e d g e n e t i c system o p t i m a l l y s u i t e d f o r i t s environment (Mayr, 1976). Thus, when g e n e t i c changes a r e r e q u i r e d i n the genotype t o adapt the phenotype t o another environment, t h e s e g e n e t i c changes p r o b a b l y w i l l d i s r u p t p a r t s of the o r g a n i z a t i o n of g e n e t i c i n t e r a c t i o n s s e l e c t e d i n the o r i g i n a l environment ( S t e b b i n s , 1974). Hence, a l t e r a t i o n s of the genotype which adapt the phenotype t o a second environment a re l i k e l y t o reduce | i t s s u i t a b i l i t y f o r the o r i g i n a l j environment. E v i d e n c e of such i n t e r a c t i o n among v i r u l e n c e (V) genes i s a c c u m u l a t i n g i n the l i t e r a t u r e (Wolfe and Schwarzbach, 1978; Van der P l a n k , 1978). In a d d i t i o n , i f an "unneeded" gene i s r e s p o n s i b l e f o r the use of v a l u a b l e r e s o u r c e s i n the c o n s t r u c t i o n of gene p r o d u c t s which a r e themselves "unneeded" or i n f e r i o r , p h e n o t y p i c r e p r o d u c t i v i t y i s l i k e l y t o s u f f e r d i r e c t l y from t h i s i n e f f i c i e n c y ( H a r l a n , 1976). I t f o l l o w s t h a t "unneeded" R or V genes can be e x p e c t e d t o have d e t r i m e n t a l e f f e c t s on p h e n o t y p i c performance. Van der Plank (1963) has p o p u l a r i z e d t h i s concept as " s t a b i l i z i n g s e l e c t i o n " i n the p l a n t d i s e a s e l i t e r a t u r e . Note t h a t he uses the term " s t a b i l i z i n g s e l e c t i o n " t o d e s c r i b e the slow d i r e c t i o n a l s e l e c t i o n which r e s u l t s i n the d i s p l a c e m e n t of "unneeded" R or V genes by t h e i r S or A a l l e l e s , r e s p e c t i v e l y . A l t h o u g h Van der P l a n k ' s d e f i n i t i o n i s i n c o n s i s t e n t w i t h t h a t of p o p u l a t i o n g e n e t i c s (cf_. S p e i s s , 1977) i t has become common i n the p l a n t d i s e a s e l i t e r a t u r e and w i l l be 53 used h e r e . Why r e s i s t a n c e and v i r u l e n c e s h o u l d e v o l v e as monogenic r a t h e r than as p o l y g e n i c t r a i t s i s u n c e r t a i n . One p o s s i b l e e x p l a n a t i o n i s t h a t the number of gene s u b s t i t u t i o n s needed t o change a t r a i t may be i n v e r s e l y r e l a t e d t o the speed w i t h which the t r a i t can change i n these h o s t : p a r a s i t e systems. In t h i s c a s e , response t o new d i r e c t i o n s of s e l e c t i o n would be s w i f t e r i n genotypes w i t h monogenic d e t e r m i n a t i o n . Another p o s s i b l e e x p l a n a t i o n i s t h a t the d i s r u p t i o n of o r g a n i z a t i o n s u f f e r e d through e p i s t a t i c e f f e c t s by the background i n changing a t r a i t may be d i r e c t l y r e l a t e d t o the number of gene s u b s t i t u t i o n s r e q u i r e d t o make the change. Thus, s e l e c t i o n f o r m i n i m i z i n g the d i s r u p t i n g of g e n e t i c i n t e r a c t i o n i n the background genotype and/or s e l e c t i o n f o r a f a s t response to new s e l e c t i v e p r e s s u r e s may be r e s p o n s i b l e f o r the monogenic r a t h e r than p o l y g e n i c d e t e r m i n a t i o n of r e s i s t a n c e and v i r u l e n c e i n gene-for-gene r e l a t i o n s h i p s . R e t u r n i n g t o the gene-for-gene r e l a t i o n s h i p , host genotype RR r e q u i r e s o n l y gene R t o p r o v i d e r e s i s t a n c e a g a i n s t V-2 " 2 p a r a s i t e s so gene R i s unneeded and w i l l d ecrease i n f r e q u e n c y due t o s t a b i l i z i n g s e l e c t i o n . S i m i l a r y , when R becomes common .2 another gene, V , w i l l be s e l e c t e d i n the p a r a s i t e p o p u l a t i o n 2 and s t a b i l i z i n g s e l e c t i o n w i l l o p e r a t e a g a i n s t gene V. Thus, a dynamic p a t t e r n of gene-for-gene i n t e r a c t i o n s i s e xpected as a common f e a t u r e of h o s t : p a r a s i t e systems ( P e r s o n , 1966). Indeed, gene-for-gene r e l a t i o n s h i p s r e p r e s e n t the g e n e t i c a s p e c t s of what a r e , perhaps, the most t h o r o u g h l y documented 54 g e n e t i c feedback h o s t r p a r a s i t e systems. H o s t : p a r a s i t e systems w i t h gene-for-gene r e l a t i o n s h i p s i n c l u d e many of the major economic c r o p p l a n t s and t h e i r most d e s t r u c t i v e p a r a s i t e s , i n c l u d i n g the c e r e a l i c e r e a l r u s t systems ( S i d h u , 1975). These systems a re c h a r a c t e r i z e d by r e c u r r e n t changes i n the g e n o t y p i c f r e q u e n c i e s of the p a r a s i t e p o p u l a t i o n , making the c r o p ' s r e s i s t a n c e t o the p a r a s i t e d i f f i c u l t t o m a i n t a i n . I t has o f t e n been suggested t h a t damage to the c r o p i n such h o s t : p a r a s i t e systems might be s u b s t a n t i a l l y reduced i f the g e n o t y p i c f r e q u e n c i e s i n the p a r a s i t e p o p u l a t i o n c o u l d be s t a b i l i z e d . In t h i s r e g a r d the g r e a t e r ' b a l a n c e ' of the n a t u r a l c e r e a l r c e r e a l r u s t systems of the K i d d l e East (c_f. A n i k s t e r and Wahl, 1979; Browning, 1974) i s of s p e c i a l i n t e r e s t . The q u e s t i o n a r i s e s : I f removed from human i n t e r v e n t i o n and i s o l a t e d from o t h e r a s p e c t s of i t s n a t u r a l system, does a gene-for-gene r e l a t i o n s h i p w i t h s t a b i l i z i n g s e l e c t i o n i n h e r e n t l y impart s t a b i l i t y t o the' dynamics of c e r e a l : c e r e a l r u s t p o p u l a t i o n g e n e t i c s ? Chapter 3 r e p r e s e n t s a c o n t r i b u t i o n toward answering t h i s q u e s t i o n : a c o n t r o v e r s y i n the l i t e r a t u r e i s r e s o l v e d and i t i s shown t h a t s t a b i l i z i n g s e l e c t i o n a l o n e does not ensure the s t a b i l i t y of a gene-for-gene r e l a t i o n s h i p . 55 TABLE I I I A d i a l l e l i c gene-for-gene r e l a t i o n s h i p i l l u s t r a t i n g the complementary g e n e t i c s of a h a p l o i d host and i t s h a p l o i d p a r a s i t e . Host Genotype P a r a s i t e Genotype R e s i s t a n t (R) S u s c e p t i b l e (S) A v i r u l e n t (A) V i r u l e n t (V) Minus (-): P l u s (+): R e s i s t a n t h o s t d i s e a s e r e a c t i o n . S u s c e p t i b l e host d i s e a s e r e a c t i o n . 56 SELECTION. PRESSURES AND PLANT PATHOGENS: ROBUSTNESS OF THE MODEL Chapter 3 57 Leonard (1977) d e v e l o p e d a m a t h e m a t i c a l model t o examine the dynamics of a gene-for-gene r e l a t i o n s h i p . He d e r i v e d a l g e b r a i c e x p r e s s i o n s f o r the f r e q u e n c y of the s u s c e p t i b i l i t y gene i n the host p o p u l a t i o n and the f r e q u e n c y of the v i r u l e n c e gene i n the p a r a s i t e p o p u l a t i o n a t the s i n g l e n o n - t r i v i a l e q u i l i b r i u m p o i n t . At t h i s p o i n t , the two a l l e l e s of each p o p u l a t i o n a r e s i m u l t a n e o u s l y i n e q u i l i b r i u m and at non-zero f r e q u e n c i e s . By computer s i m u l a t i o n , Leonard (1977) found t h a t as time p r o g r e s s e s i n h i s model the p a i r of gene f r e q u e n c i e s d e p a r t e d from each s e t of non-zero i n i t i a l v a l u e s he used and s p i r a l e d i n towards the n o n - t r i v i a l e q u i l i b r i u m p o i n t when s t a b i l i z i n g s e l e c t i o n o c c u r r e d . T h i s s u g gests t h a t h i s model may be g l o b a l l y a s y m p t o t i c a l l y s t a b l e ( F i g . 3.1A). In common w i t h c u r r e n t use i n much of p o p u l a t i o n mathematics ( p a r t i c u l a r l y p o p u l a t i o n g e n e t i c s ) , the a d j e c t i v e " g l o b a l " a p p l i e s o n l y w i t h i n the a d m i s s i b l e phase space. S u b s e q u e n t l y S e d c o l e (1978) o f f e r e d computer s i m u l a t i o n and a n a l y t i c arguments c l a i m i n g t h a t Leonard's model i s i n h e r e n t l y u n s t a b l e . He found t h a t the p a i r of gene f r e q u e n c i e s , on l e a v i n g any s e t of non-zero i n i t i a l v a l u e s , would s p i r a l away from the n o n - t r i v i a l e q u i l i b r i u m p o i n t ( F i g . 3.1B). T h e r e f o r e , S e d c o l e d i s a g r e e d w i t h the s u g g e s t i o n t h a t Leonard's model e x p l a i n s the ' s t a b i l i t y ' of host-pathogen systems of the ' f e r t i l e c r e s c e n t ' a r e a of the M i d d l e E a s t . Leonard and Czochor (1978) l a t e r conceded t o Sedcole t h a t the n o n - t r i v i a l e q u i l i b r i u m p o i n t of Leonard's model i s a n a l y t i c a l l y u n s t a b l e . However i n c o n t r a s t to S e d c o l e , t h e i r computer s i m u l a t i o n s i n d i c a t e d t h a t Leonard's model i s s t a b l e . R e a l i z i n g t h a t S e d c o l e ' s (1978) a n a l y t i c t r e a t m e n t i s s t r i c t l y v a l i d o n l y i n the immediate v i c i n i t y of an e q u i l i b r i u m p o i n t , they r e s o l v e d t h i s apparent c o n f l i c t between the a n a l y t i c and t h e i r computer s i m u l a t i o n r e s u l t s by s u g g e s t i n g t h a t many c o n c e n t r i c l i m i t c y c l e s s u r r o u n d the n o n - t r i v i a l e q u i l i b r i u m p o i n t ( F i g . 3.1C). A l i m i t c y c l e i s a c l o s e d t r a j e c t o r y such t h a t no t r a j e c t o r y s u f f i c i e n t l y near i t i s a l s o c l o s e d . Thus Leonard and Czochor (1978) imply t h a t v e r y c l o s e t o the non-t r i v i a l e q u i l i b r i u m p o i n t S e d c o l e ' s a n a l y s i s a p p l i e s and the system s p i r a l s outward t o the innermost l i m i t c y c l e . F a r t h e r from the e q u i l i b r i u m p o i n t t h e i r computer s i m u l a t i o n s show t h a t the system can s p i r a l inward (Leonard and Czochor, 1978), presumably u n t i l i t reaches the f i r s t s t a b l e l i m i t c y c l e i n i t s p a t h . A l i m i t c y c l e i s s t a b l e or u n s t a b l e , r e s p e c t i v e l y , i f any i n f i n i t e s i m a l l y s m a l l p e r t u r b a t i o n t o the c l o s e d o r b i t decays or grows w i t h t i m e . For example, the i n n e r and o u t e r l i m i t c y c l e s of F i g . 3.1C are s t a b l e and u n s t a b l e , r e s p e c t i v e l y . Leonard and Czochor suggest t h a t the b e h a v i o r of host-pathogen systems i n the M i d d l e E a s t i s c o n s i s t e n t w i t h systems t h a t have u n s t a b l e e q u i l i b r i u m p o i n t s but s t a b l e l i m i t c y c l e s . T h i s c h a p t e r has two p u r p o s e s . F i r s t , i n c o n t r a s t - t o the c l a i m s of b o t h Sedcole (1978) and Leonard and Czochor (1978), i t shows t h a t Leonard's (1977) model i s not n e c e s s a r i l y l o c a l l y u n s t a b l e . Second, and more i m p o r t a n t , i t demonstrates t h a t Leonard's model i s not r o b u s t ; i . e . , s l i g h t changes i n the a ssumptions can d r a s t i c a l l y a f f e c t i t s b e h a v i o r . Three a l t e r n a t i v e s e t s of assumptions a r e examined. The g e n e t i c c o m p o s i t i o n s of host and pathogen p o p u l a t i o n s a r e assumed t o 59 change: i n a sequence of s i m u l t a n e o u s and d i s c r e t e s t e p s , i n a sequence of a l t e r n a t e and d i s c r e t e s t e p s , and c o n t i n u o u s l y . Each set of assumptions produces a q u a l i t a t i v e l y d i f f e r e n t type of s t a b i l i t y b e h a v i o r ( F i g . 3.1). In t h i s sense Leonard's model i s inadequate as an e x p l a n a t i o n of the ' s t a b i l i t y ' of gene-for-gene r e l a t i o n s h i p s i n the M i d d l e E a s t . Leonard's (1977) model i s based on T a b l e s 3. 1 and 3.2 which a r e a d a p t a t i o n s of T a b l e s 3. 2 and 3.3 r e s p e c t i v e l y i n h i s a r t i c l e . From T a b l e 3.1 Leonard e x p r e s s e s the f r e q u e n c y of the v i r u l e n c e gene i n the pathogen p o p u l a t i o n i n the ( i + 1 ) s t g e n e r a t i o n as n [1 - k + (1 - q 2 ) a ] (3.1) i n = i+1 1 - ( 1 - q 2 ) t + n ± [ ( 1 - q 2 ) ( a + t ) - k] S i m i l a r l y , from T a b l e 3.2 he w r i t e s the change i n f r e q u e n c y of the s u s c e p t i b i l i t y gene i n the host p o p u l a t i o n as (1 - q ) q 2 [ n s ( a + t ) + c - t s ] (3.2) Aq = ; 1 - s + nks - ( 1 - q 2 ) [ n s ( a + t ) + c - t s ] The system i s a t e q u i l i b r i u m when n = n and Aq = 0, i + 1 i s i m u l t a n e o u s l y . By e q u a t i o n s 3.1 and 3.2 60 n = n when n = 0, 1 and when q = q* = */~] - k/(a+t) i+1 i and (3.3) Aq = 0 when q = 0, 1 and when n = n* = ( t s - c ) / ( s a + s t ) G i v e n ' s t a b i l i z i n g s e l e c t i o n ' (sensu Van der P l a n k , 1968), Leonard (1977) has shown t h a t a s i n g l e n o n - t r i v i a l e q u i l i b r i u m e x i s t s a t ( n * , q * ) . In a d d i t i o n t h e r e are four t r i v i a l e q u i l i b r i a where each p o p u l a t i o n has l o s t an a l l e l e . These are (n,q) = ( 0 , 0 ) , (0 , 1 ) , . (1 , 0 ) , and ( 1 , 1 ) . Leonard and Czochor (1978) c l a i m t h a t (n,q) = ( 0 , q * ) , d , q * ) , ( n * , 0 ) , and ( n * , l ) a re a l s o non-t r i v i a l e q u i l i b r i a but e i t h e r n ? n or Aa ? 0 a t each of i + 1 i these p o i n t s (Eq. 3.3). I t i s agreed t h a t Leonard' model c y c l e s about the phase p l a n e ( t h e n x q p l a n e where 0 < n < 1, 0 < q < 1). The c o n t r o v e r s y ( F i g . 3.1) c o n c e r n s whether the system s p i r a l s i n t o the n o n - t r i v i a l e q u i l i b r i u m (n*,q*) as suggested by Leonard's (1977) computer s i m u l a t i o n s , or s p i r a l s away from i t as Sedcole (.1978) s t a t e s , or does b o t h , depending on i n i t i a l c o n d i t i o n s , w i t h the t r a j e c t o r y c o n v e r g i n g on a s t a b l e l i m i t c y c l e as Leonard and Czochor (1978) s u s p e c t . C o n s i d e r S e d c o l e ' s (1978) c l a i m s f i r s t , he i s c o r r e c t i n showing m a t h e m a t i c a l l y t h a t the system 61 n = g(n , q ), q = q + A q ( n ) = f ( n , q ) (3.4) i+1 i i i+1 i i i i i s l o c a l l y u n s t a b l e . F u r t h e r m o r e , t h e r e i s no e r r o r i n h i s computer s i m u l a t i o n which, s u g g e s t s g l o b a l i n s t a b i l i t y ( F i g . 3.1B). However, as Leonard and Czochor (1978) n o t e , S e d c o l e ' s system (Eq. 3.4), i n which Aq = Aq(n ), i s a m i s i n t e r p r e t a t i o n i of Leonard's model which i s of the form: n = g ( n , q ) , q =.q + A q ( n ) = f ( n , q ) (3.5) i+1 i i i+1 i i + l i+1 i A c c o r d i n g t o e q u a t i o n s 3.1, 3.2, and 3.5: 3g 3n. eq 39 3<3. = - 2q*n*(1-n*)(a+t) = ~ x 1 2 , say, eq 1 - ( 1 - q *2 ) t (3.6) 3f 3". ( 1 - q * ) q * 2 s ( a + t - k t ) = x , say, 2 1 eq ( l - s - n * k s ) [ 1 - ( 1 - q *2 ) t ] 62 1 - x 1 2 x 2 1 where x > 0. U s i n g t h e s e v a l u e s of the p a r t i a l d e r i v a t i v e s a t 12 the n o n - t r i v i a l e q u i l i b r i u m i t can be shown (c_f. S e d c o l e , 1978) t h a t the c h a r a c t e r i s t i c r o o t s a r e X = (2 - x x + ) / ( 2 - x x ) 2 - 4 ) / 2 . 12 21 12 21 When x x < 0 or x x > 4, |x| > 1 so t h e n o n - t r i v i a l 12 21 12 21 e q u i l i b r i u m p o i n t i s l o c a l l y u n s t a b l e . When 0 < x x < 4, | X. | 12 21 = 1 so the l i n e a r a n a l y s i s i n c o n c l u s i v e l y d e s c r i b e s the b e h a v i o r of the n o n - l i n e a r e q u a t i o n s (Eq. 3.5) near t h e n o n - t r i v i a l e q u i l i b r i u m p o i n t . Thus, S e d c o l e (1978), and Leonard and Czochor (1978) e r r i n u n e q u i v o c a l l y s t a t i n g t h a t the n o n - t r i v i a l e q u i l i b r i u m p o i n t of Leonard's model i s l o c a l l y u n s t a b l e . In f a c t , s u b s t i t u t i o n from the s u g g ested (Leonard, 1977) range of parameter v a l u e s i n t o Eq. 3.6 i n d i c a t e s t h a t g e n e r a l l y .0075 < x x < .02. Hence, the 12 21 63 l o c a l s t a b i l i t y of the n o n - t r i v i a l e q u i l i b r i u m p o i n t i s u n c e r t a i n f o r a l l r e p o r t e d computer s i m u l a t i o n s (Leonard, 1977; Leonard and C z o chor, 1978; S e d c o l e , 1978). Thus, n e i t h e r the l o c a l s t a b i l i t y a n a l y s i s , nor the r e l e v a n t computer s i m u l a t i o n s (Leonard, 1977; Leonard and C z o chor, 1978), r e s u l t i n any i n c o n s i s t e n c i e s w i t h the b e h a v i o r s p r e d i c t e d by e i t h e r a s y m p t o t i c s t a b i l i t y ( F i g . 3.1A), or c o n c e n t r i c l i m i t c y c l e s ( F i g . 3.1C), i n the p r e s c r i b e d (Leonard, 1977) range of parameter v a l u e s . There i s a n o t h e r a r e a of c r i t i c a l u n c e r t a i n t y i n Leonard's model. He ( L e o n a r d , 1977; Leonard and Czochor, 1978) d e f i n e s the parameter s (Table 3.2) as the r a t e of l o s s i n host f i t n e s s per u n i t amount of pathogen f i t n e s s . However, he p r o v i d e s no e v i d e n c e t o s u p p o r t h i s c h o i c e of v a l u e s : 0.05 < s < 0.2. In f a c t , x i s so s e n s i t i v e t o s (Eq. 3.6) t h a t the r e s u l t s of the 21 l o c a l s t a b i l i t y a n a l y s i s f o r the n o n - t r i v i a l e q u i l i b r i u m p o i n t are i n c o n c l u s i v e when s i s s m a l l but show i n s t a b i l i t y when s i s l a r g e . Computer s i m u l a t i o n s s t a r t i n g at n = 0.7 and q = 0.8366 produce inward s p i r a l s towards (n*,q*) f o r s = 0.8. T h i s d emonstrates t h a t Leonard's model must have at l e a s t one s t a b l e l i m i t c y c l e f o r these p a r t i c u l a r parameter v a l u e s (a = 0.0, c = 0.1, k = 0.3, t = 1.0 and s = 0.8). However, o t h e r s e t s of parameter v a l u e s may induce d i f f e r e n t b e h a v i o r s when s i s l a r g e . Hence, b o t h g l o b a l i n s t a b i l i t y ( F i g . 3.1B), and c o n c e n t r i c l i m i t c y c l e s ( F i g . 3.1C), r e p r e s e n t p o s s i b l e forms of phase p l a n e t r a j e c t o r i e s f o r Eq. 3.5 f o r l a r g e s. . At t h i s s t a g e i n the a n a l y s i s of Leonard's (1977) model, t h r e e b e h a v i o r s : g l o b a l a s y m p t o t i c s t a b i l i t y , ( c o n c e n t r i c ) l i m i t 64 c y c l e ( s ) , and g l o b a l i n s t a b i l i t y ( F i g . 3.1) must be e n t e r t a i n e d as p o s s i b i l i t i e s . A s i n g l e s t a b l e l i m i t c y c l e seems q u i t e l i k e l y and i s i n c l u d e d under ' ( c o n c e n t r i c ) l i m i t c y c l e ( s ) ' . U l t i m a t e l y , the n u m e r i c a l v a l u e s of the parameters may determine which b e h a v i o r the model (Eq. 3.5) e n a c t s . S e d c o l e ' s model (Eq. 3.4) assumes t h a t the g e n e t i c c o m p o s i t i o n s of host and pathogen p o p u l a t i o n s change i n a sequence of s i m u l t a n e o u s s t e p s . D u r i n g the growing season, the pathogen adapts t o the i n i t i a l c o m p o s i t i o n of the host and the host adapts t o the i n i t i a l c o m p o s i t i o n of the pathogen. Thus, the i n f l u e n c e e x e r t e d by one s p e c i e s on the o t h e r i s determined s o l e l y by i t s g e n e t i c c o m p o s i t i o n at the v e r y b e g i n n i n g of the growing season. S e d c o l e ' s model i m p l i c i t l y assumes t h a t the g e n e t i c feedback between the two p o p u l a t i o n s o c c u r s a t d i s c r e t e i n t e r v a l s and t h a t i t i s r e c i p r o c a l when i t does o c c u r . In c o n t r a s t , Leonard's model (Eq. 3.5) assumes t h a t the g e n e t i c c o m p o s i t i o n s of h o s t and pathogen p o p u l a t i o n s change i n a sequence of a l t e r n a t e s t e p s . D u r i n g the growing season the pathogen goes through a number of g e n e r a t i o n s i n which i t adapts to the unchanging g e n e t i c c o m p o s i t i o n of the h o s t . Thus, d u r i n g the growing season, the h o s t a f f e c t s the pathogen but the pathogen has no e f f e c t on the host w i t h r e s p e c t t o g e n e t i c c o m p o s i t i o n . At the end of the growing season, seeds a r e produced by the host t o i n i t i a t e a new host g e n e r a t i o n f o r the next season. The p r o p o r t i o n of seeds produced by s u s c e p t i b l e p l a n t s depends upon the g e n e t i c c o m p o s i t i o n of the h o s t p o p u l a t i o n , which has been c o n s t a n t s i n c e the b e g i n n i n g of the growing season, and the g e n e t i c c o m p o s i t i o n of the pathogen 65 p o p u l a t i o n a t the v e r y end of the growing season (Eq. 3 . 4 ) . Thus Leonard assumes t h a t the g e n e t i c feedback between the two p o p u l a t i o n s o c c u r s a t d i s c r e t e i n t e r v a l s and t h a t i t i s non-r e c i p r o c a l when i t does o c c u r . A c t u a l l y , the r e l a t i v e r e p r o d u c t i v i t y of a host genotype depends upon the r e l a t i v e amount of d i s e a s e i t has s u f f e r e d d u r i n g the growing season (Leonard and Czochor, 1978). T h i s , i n t u r n , i s r e l a t e d to the g e n e t i c c o m p o s i t i o n of the pathogen p o p u l a t i o n throughout the growing season (Van der P l a n k , 1968). However, the models p r e s e n t e d so f a r i m p l i c i t l y assume t h a t feedback o c c u r s o n l y at d i s c r e t e i n t e r v a l s . Hence, the r e l a t i v e r e p r o d u c t i v i t y of host genotypes i n these models i s r e l a t e d , not t o the r e l a t i v e amount of d i s e a s e s u f f e r e d and the g e n e t i c c o m p o s i t i o n of the pathogen throughout the growing season; but r a t h e r , t o the g e n e t i c c o m p o s i t i o n of the pathogen at a s i n g l e  i n s t a n t of the growing season. An a l t e r n a t i v e set of assumptions l e a d s to a t h i r d model. Suppose t h a t any change i n the g e n e t i c c o m p o s i t i o n of one s p e c i e s i m m e d i a t e l y i n f l u e n c e s the g e n e t i c c o m p o s i t i o n of the o t h e r ; Then, i f the p o p u l a t i o n s a r e l a r g e , the d i s c r e t e n e s s i n t r o d u c e d by i n d i v i d u a l " b i r t h s " and " d e a t h s " i s l o s t i n the l a r g e t o t a l d u r i n g the growing season and both s e l e c t i o n and feedback become a p p r o x i m a t e l y c o n t i n u o u s p r o c e s s e s . A model w i t h c o n t i n u o u s and r e c i p r o c a l g e n e t i c feedback r e s u l t s . In t h i s case the r e l a t i v e r e p r o d u c t i v i t y of host genotypes depends on the g e n e t i c c o m p o s i t i o n of the pathogen throughout the growing season. However t h i s model has a weakness t h a t n e i t h e r of the d i s c r e t e models has: i t i g n o r e s the d i s c o n t i n u i t i e s i n t r o d u c e d 66 by s e a s o n a l and l i f e - h i s t o r y phenomena i n the l i f e c y c l e s of the two s p e c i e s , and i t r e q u i r e s t h a t s e l e c t i o n be weak. The c o n t i n u o u s v a l u e c o r r e s p o n d i n g t o the d i s c r e t e f i t n e s s , W, i s m = l o g W = InW. e C o n v e r t i n g the f i t n e s s e s of T a b l e s 3. 1 and 3.2 w i t h t h i s e x p r e s s i o n , a c o n t i n u o u s v e r s i o n of the d i s c r e t e models can be adapted from Crow and Kimura (1970): dn/dt '= n(1 - n) [A(1 - q 2 ) + B q 2 ] where A = 1 n [ ( 1 - k + a ) / 1 - t ] > 0, and B = 1 n ( l - k ) < 0; and (3.7) dq/dt = q 2 (1-q)[Dn + H ( l - n ) ] where D = 1n 1-S(1-k) 1-c-s(1-k+a) > 0, and H = 1n 1-s 1 - c - s ( 1 - t ) 67 In common w i t h the models of Leonard and S e d c o l e , e q u a t i o n 3.7 i m p l i c i t l y assumes t h a t each p o p u l a t i o n has e i t h e r non-o v e r l a p p i n g g e n e r a t i o n s or a s t a b l e age d i s t r i b u t i o n . Age d i s t r i b u t i o n s a r e a p p r o x i m a t e l y s t a b l e when the r a t e of change i n t o t a l p o p u l a t i o n s i z e i s slow r e l a t i v e t o the l i f e s p a n . The n o n - t r i v i a l e q u i l i b r i u m of Eq. 3.7 i s ( n * , q*) = (H/(H-D), A/(A-B)) (3.8) The v a l u e s of the p a r t i a l d e r i v a t i v e s a t (n*,q*) are 3 dn 3n dt = 3 dg eq 3q dt = 0, eq dq 3n dt = (D-H)q* 2(1-q*) > 0, eq and 3 dn 3q dt = ( B - A ) n * ( 1 - n * ) 2 q * < 0 eq Kaplan (1958) shows t h a t because 3 dn 3n dt + 3 dq e q 3q dt = 0 eq 68 and 3 dn 3n dt eq x 3 dq 3q dt - 3 dn. eq 3 q dt x 3 dq eq 3n dt > 0 eq the s t a b i l i t y b e h a v i o r of the n o n - t r i v i a l e q u i l i b r i u m i s u n c e r t a i n . The d i r e c t method of Liapunov p r o v i d e s a means of e x t e n d i n g the s t a b i l i t y a n a l y s i s beyond the immediate v i c i n i t y of ( n * , q * ) . A g e n e r a l e x p r e s s i o n f o r Liapunov f u n c t i o n s of c o n t i n u o u s gene-for-gene r e l a t i o n s h i p s can be adapted from Goh (1978): n r f (u) n V(n,q) = J n* g (u) du r f (u) q g (u) d u (3.9) Here f (u) and g (u) a r e c o n t i n u o u s f u n c t i o n s of gene f r e q u e n c y , x x u e ( 0 , 1 ) , such t h a t f (u) > 0 when u > x* x f (u) = 0 when u = x* x f (u) < 0 when u < x* x 69 and g (u) > 0 . x In a d d i t i o n , f (u) and g (u) a l l o w V(n,q).«> as n or q •+ 0+ or x x 1-. The l o c a l s t a b i l i t y a n a l y s i s s u g g ested n e u t r a l s t a b i l i t y . N e u t r a l s t a b i l i t y e x i s t s g l o b a l l y ( F i g . 3.1D) i f V(n,q) = [f ( n ) / g ( n ) ] ( d n / d t ) +[f ( q ) / g ( q ) } ( d u / d t ) n n q q v a n i s h e s throughout the a d m i s s i b l e phase p l a n e . Hence, assume V(n,q) = 0 and l e t g (u) = U(1-U) and g (u) = U 2 ( 1 - U ) so t h a t , n q a f t e r s u b s t i t u t i n g e q u a t i o n 3.7 i n t o t h i s e x p r e s s i o n f o r V ( n , q ) , f (u) = (D-H)u + H and f (u) = ( A - B ) u 2 - A. S u b s t i t u t i n g these n q r e l a t i o n s h i p s i n t o e q u a t i o n 3.9 b e f o r e i n t e g r a t i n g , V(n,q) = D l n [ ( 1 - n * ) / ( 1 - n ) ] - B l n [ ( i - q * ) ] + A { l n ( q * / q ) + 1/q - 1/q*} - H1n(n*/n) T h i s p r o v e s m a t h e m a t i c a l l y t h a t t h i s c o n t i n u o u s v e r s i o n of Leonard's model has a b e h a v i o r d i f f e r e n t from any p r e v i o u s l y s u g g e s t e d : the maintenance of a r b i t r a r i l y l a r g e e l l i p t i c a l o r b i t s i n a c c e p t a b l e phase space. The s i z e of the o r b i t i s d e t e r m i n e d s o l e l y by the i n i t i a l c o n d i t i o n s ( F i g . 3.1D). In summary, i t has been shown t h a t t h r e e d i f f e r e n t models of the same gene-for-gene r e l a t i o n s h i p each produce q u a l i t a t i v e l y d i f f e r e n t t y p e s of s t a b i l i t y b e h a v i o r . T h i s v a r i a t i o n i n b e h a v i o r i s not the r e s u l t of m a t h e m a t i c a l e r r o r ; r a t h e r , i t i s due t o d i f f e r e n c e s i n assumptions made i n c o n s t r u c t i n g the models. In p a r t i c u l a r , the assumptions c o n c e r n i n g the r e c i p r o c i t y and c o n t i n u i t y of feedback between the host and p a r a s i t e g e n e t i c c o m p o s i t i o n s are r e s p o n s i b l e . In o t h e r words, the c o n c l u s i o n s of the models are not r o b u s t ; they are e x t r e m e l y s e n s i t i v e t o assumptions about the g e n e t i c feedback between the p o p u l a t i o n s . S i n c e the assumptions s p e c i f i c t o each model appear t o have both f a v o r a b l e and u n f a v o r a b l e aspects, the q u a l i t a t i v e s t a b i l i t y behavior of the gene-f o r -gene r e l a t i o n s h i p i s i n doubt. Hence none of t h e s e models p r o v i d e s a s a t i s f a c t o r y e x p l a n a t i o n of the s t a b i l i t y of gene f r e q u e n c i e s i n c e r e a l r u s t systems i n the M i d d l e E a s t . S t a b i l i z i n g s e l e c t i o n a l o n e i s not enough t o ensure the s t a b i l i t y of s i m p l e gene-for-gene r e l a t i o n s h i p s ; o t h e r f a c t o r s o p e r a t i n g i n c o n j u n c t i o n w i t h s t a b i l i z i n g s e l e c t i o n a re i n v o l v e d . 71 TABLES 3.1 R e l a t i v e pathogen f i t n e s s e s on d i f f e r e n t h o s t s r e l a t i v e pathogen f i t n e s s on r r ( s u s c e p t i b l e ) R ( r e s i s t a n t ) pathogen genotype frequency q 2 1 - q 2 v ( a v i r u l e n t ) 1 - n 1 1 - t V ( V i r u l e n t ) n 1 - k 1 - k + a k = c o s t of v i r u l e n c e , t = e f f e c t i v e n e s s of r e s i s t a n c e , and a = advantage of v i r u l e n t race on h o s t s w i t h c o r r e s p o n d i n g gene f o r r e s i s t a n c e . 72 TABLE 3.2 R e l a t i v e h o s t f i t n e s s e s i n t e r a c t i n g w i t h d i f f e r e n t pathogens r e l a t i v e h ost f i t n e s s e s when i n f e c t e d bv V ( a v i r u l e n t ) V ( v i r u l e n t ) h ost genotype f requency 1 - n n r r ( s u s c e p t i b l e ) q 2 1 - s 1 - s d - k) R- ( r e s i s t a n t ) 1-q 2 1 - c - s ( 1 - t ) 1 -c-s(1-k+a) c = c o s t of r e s i s t a n c e , s m u l t i p l i e d by pathogen f i t n e s s = d i s e a s s e v e r i t y . FIGURE 3. 1 Types of q u a l i t a t i v e b e h a v i o r which have been proposed f o r s i m p l e gene-for-gene r e l a t i o n s h i p s : A, g l o b a l a s y m p t o t i c s t a b i l i t y , B, g l o b a l i n s t a b i l i t y , C, (two) c o n c e n t r i c l i m i t c y c l e s , D, n e u t r a l l y s t a b l e c y c l e s . The p o s i t i o n of the n o n - t r i v i a l e q u i l i b r i u m p o i n t i s denoted by 'x' i n each phase p l a n e . FIGURE 3.IA - 3.ID n, frequency of virulence genes PART IV ALTERNATIVE MANAGEMENT STRATEGIES 76 INTRODUCTION "The problems of r u s t c o n t r o l a r e p s y c h o l o g i c a l as w e l l as p a t h o l o g i c a l . We must be open-minded. We must e l i m i n a t e myopia, p a r t i s a n s h i p , and dogmatism. We must combine depth of i n s i g h t w i t h b r e a d t h of v i s i o n . We must u t i l i z e a l l known methods of r u s t c o n t r o l and c o n t i n u a l l y s e a r c h f o r new ones." (Stakman, 1968) The need f o r a l t e r n a t i v e approaches t o c e r e a l r u s t c o n t r o l was o u t l i n e d i n Chapter 1. In P a r t IV, f o u r p o s s i b l e a l t e r n a t i v e s t r a t e g i e s , each of which appears t o have p o t e n t i a l u s e f u l n e s s , are examined. Ch a p t e r s 4 and 5 d e a l s p e c i f i c a l l y w i t h Wolfe and B a r r e t t ' s (1976) c o n c l u s i o n t h a t , a l t h o u g h monoculture has not always l e d t o e p i d e m i c s , " f o r many d i s e a s e s ... n o t a b l y those caused by l e a f pathogens of c e r e a l s , i n c r e a s e d u n i f o r m i t y of c r o p p r o d u c t i o n does p l a c e us on the k n i f e edge w i t h r e g a r d t o d i s e a s e and p e s t v u l n e r a b i l i t y . " Chapter 4 d e a l s w i t h the h y p o t h e s i s t h a t by u s i n g a m u l t i l i n e , a m i x t u r e of a g r o n o m i c a l l y i d e n t i c a l l i n e s but w i t h d i f f e r e n t r e s i s t a n c e genes, c e r e a l r u s t s pread from p l a n t - t o - p l a n t may be i m p a i r e d . Leonard and Czochor (1980) p r o v i d e a l i t e r a t u r e r e v i e w of t h i s a r ea (see a l s o Chapter 3 ) . In Chapter 4 i t i s s u g g ested that., c o n t r a r y to the commonly h e l d v i e w , m u l t i l i n e s a r e u n l i k e l y t o p r e v e n t a s i n g l e race from d o m i n a t i n g the r u s t p o p u l a t i o n u n l e s s the f requency of the m u l t i l i n e components ar e o f t e n a d j u s t e d . However a c c o r d i n g t o the assumptions of the model, even when such a d j u s t m e n t s occur o n l y r a r e l y , a m u l t i l i n e can always be s y n t h e s i z e d which s u f f e r s l e s s r u s t than the best m o n o c u l t u r e . So f a r I have d i s c u s s e d o n l y one a s p e c t of c e r e a l : c e r e a l r u s t p o p u l a t o n g e n e t i c s , namely t h a t r e l a t e d t o the s p e c i f i c 77 i n t e r a c t i o n s of major genes i n the gene-for-gene r e l a t i o n s h i p . In Chapter 5 I c o n s i d e r n o n s p e c i f i c p o l y g e n i c a l l y i n h e r i t e d t r a i t s and t h e i r l o n g - t e r m c o e v o l u t i o n i n n a t u r a l c e r e a l : c e r e a l r u s t pathosystems. I t i s proposed t h a t the r e l a t i o n s h i p between c e r e a l r u s t a g g r e s s i v e n e s s and c e r e a l r u s t f i t n e s s depends on the r e l a t i v e r u s t d i s p e r s a l s u c c e s s . The c o n c l u s i o n s support the h y p o t h e s i s t h a t t h i s second k i n d of r e s i s t a n c e , n o n s p e c i f i c p o l y g e n i c a l l y d e t e r m i n e d r e s i s t a n c e , w i l l be more d u r a b l e i n a g r i c u l t u r e than r e s i s t a n c e which i s s p e c i f i c and c o n t r o l l e d by major genes. C e r e a l r u s t d i s p e r s a l has been i m p l i c a t e d i n one way or a n o t her i n v i r t u a l l y a l l of the p r e v i o u s c h a p t e r s . I t i s d e a l t w i t h e x p l i c i t l y i n Chapter 6. Here r e a c t i o n - d i f f u s i o n models a r e used t o study the spread of p l a n t d i s e a s e i n a g r i c u l t u r a l r e g i o n s . I t i s proposed t h a t the s i z e and shape of f i e l d s i n an a g r i c u l t u r a l r e g i o n c o u l d be a d v a n t a g e o u s l y d e s i g n e d t o reduce c e r e a l r u s t s p r e a d . Methods f o r f i e l d t e s t i n g are s uggested. In Chapter 7 the l o n g range d i s p e r s a l of c e r e a l r u s t s i s d e s c r i b e d and i t i s proposed t h a t , under a p p r o p r i a t e c i r c u m s t a n c e s , n a t u r a l enemies c o u l d d e l a y epidemic development and thus h e l p i n r e d u c i n g y i e l d l o s s e s . Methods of f i e l d t e s t i n g a r e s u g g e s t e d . DISEASE CONTROL THROUGH USE OF MULTILINES: A THEORETICAL CONTRIBUTION Chapter 4 79 When a m u l t i l i n e i s grown w i d e l y , pathogens w i t h m u l t i p l e v i r u l e n c e ( i . e . , "complex r a c e s " c a p a b l e of i n c i t i n g d i s e a s e on two or more components of the m u l t i l i n e ) may be f a v o r e d by n a t u r a l s e l e c t i o n ( P e r s o n , et a l . , 1976). However, w i t h each component of the m u l t i l i n e c a r r y i n g o n l y a s i n g l e r e s i s t a n c e gene (R-gene), o n l y a s i n g l e v i r u l e n c e gene (v-gene) i s a c t u a l l y r e q u i r e d i n any p a r t i c u l a r i n t e r a c t i o n t h a t r e s u l t s i n d i s e a s e ; the one or more a d d i t i o n a l v-genes t h a t a re c a r r i e d by complex r a c e s a r e , i n f a c t , unneeded i n p a r t i c u l a r i n t e r a c t i o n s w i t h i n d i v i d u a l p l a n t s of the m u l t i l i n e . Assuming t h a t s t a b i l i z i n g s e l e c t i o n (Van der P l a n k , 1968) o p e r a t e s a g a i n s t unneeded v-genes, and t h a t every a d d i t i o n of an unneeded v-gene i s accompanied by a f i t n e s s l o s s , an en d p o i n t s h o u l d be reached a t which the a c c u m u l a t i o n by the complex race of y e t another v-gene i s no l o n g e r advantageous ( P e r s o n , et_ al., 1976). I t has been shown m a t h e m a t i c a l l y ( G r o t h , 1976) t h a t t h i s e n d p o i n t w i l l be det e r m i n e d i n d e p e n d e n t l y of the number of R-gene components i n the m u l t i l i n e . F u r t h e r m o r e , by e n l a r g i n g , the m u l t i l i n e t o i n c l u d e more components, the development of complex r a c e s w i l l p r oceed a t a sl o w e r r a t e and the p r o p o r t i o n of t h e m u l t i l i n e t h a t can be d i s e a s e d by any complex rac e a b l e t o m a i n t a i n i t s e l f a t a s i g n i f i c a n t q u a n t i t y w i l l be made s m a l l e r ( G r o t h , 1976). In t h i s c h a p t e r I hope t o c a r r y the ma t h e m a t i c a l development ' of m u l t i l i n e t h e o r y a s t e p f u r t h e r by c o n s i d e r i n g the r e l a t i o n s h i p s between the c h o i c e of c r o p c o m p o s i t i o n , the r e a d i n e s s of the p l a n t breeder t o change c r o p c o m p o s i t i o n , the y i e l d l o s s e s , and the c o m p o s i t i o n of the pathogen p o p u l a t i o n . I assume: ( i ) t h a t a breeder i s a b l e t o produce n d i f f e r e n t 80 genotypes of h o s t r e s i s t a n c e by u s i n g the a v a i l a b l e R-genes s i n g l y and i n c o m b i n a t i o n i n a s e r i e s of a g r o n o m i c a l l y i d e n t i c a l v a r i e t i e s ; ( i i ) t h a t a l l h o s t : p a r a s i t e i n t e r a c t i o n s a r e based on gene-for-gene r e l a t i o n s h i p s ( F l o r , 1956); ( i i i ) t h a t each of the n pathogen genotypes t h a t c o u l d be d i f f e r e n t i a t e d by phenotype oh the n h o s t r e s i s t a n c e genotypes a r e o r i g i n a l l y p r e s e n t i n the pathogen p o p u l a t i o n ; ( i v ) t h a t s t a b i l i z i n g s e l e c t i o n i s o p e r a t i n g as e x pected by experiment (Leonard, 1969; 1977; Van der P l a n k , 1968) and t h e o r y ( P e r s o n , 1959; P e r s o n , e t a l . , 1976; Van der P l a n k , 1968); (v) t h a t c o m p e t i t i v e or s y n e r g i s t i c i n t e r a c t i o n s between d i f f e r e n t u n i t s of inoculum on a common host p l a n t are n e g l i g i b l e ; ( v i ) t h a t removals are n e g l i g i b l e [as they would be d u r i n g the e a r l y s t a g e s of an epidemic (Van der P l a n k , 1968)] and ( v i i ) t h a t pathogens a r e randomly d i s t r i b u t e d i n terms of genotype among components of the m u l t i l i n e . M a t h e m a t i c a l Model of a M u l t i l i n e Now l e t m be the p r o p o r t i o n of the m u l t i l i n e t h a t i s of j component j , l e t Q (t). be the amount of pathogen genotype i i p r e s e n t a t time t , and l e t R be the r a t e of i n c r e a s e of i / j pathogen genotype i on h o s t genotype j . Then, assuming the m 's and R 's are c o n s t a n t j i» j r Q ( t ) = Q ( t - 1 ) e i i (4.1) 81 where n r j=1 m R 3 i/D (4.2) The f r e q u e n c y of pathogen genotype i a t time t i s f ( t ) = Q ( t - 1 ) e r ± / E Q ( t - 1 ) e r i i i i=1 i r n r. = f (t-1 )e V T, f (t-1 )e 1 i i = 1 i Assuming t h a t any d i f f e r e n t i a l s e l e c t i o n a c t i n g on the p a r a s i t e o u t s i d e the d i s e a s e season i s n e g l i g i b l e the g e n o t y p i c f r e q u e n c i e s would not change between d i s e a s e seasons a l t h o u g h the p o p u l a t i o n s i z e may be d r a s t i c a l l y reduced. Thus, i f time i s measured o n l y d u r i n g the d i s e a s e season, r . t n r . t f. (t) = f. (0)e 1 / z f (0)e 1 i=1 1 and £.(t)/f.(t) = [ f . ( 0 ) / f . ( 0 ) ] e x p [ r . - r . ) t ] (4.3) 82 A p p l i c a t i o n of the Model In the r e a l w o r l d , pathogen r e p r o d u c t i v i t y i s l i k e l y t o be g r e a t l y i n f l u e n c e d by weather and o t h e r e n v i r o n m e n t a l f a c t o r s . While t h e s e i n f l u e n c e s would i n t r o d u c e a c e r t a i n amount of " n o i s e " ( i . e . , u n p r e d i c t a b l e f l u c t u a t i o n s i n both the s i z e and g e n o t y p i c c o n t e n t of the p a r a s i t e p o p u l a t i o n ) , the p r o c e s s e s r e p r e s e n t e d by 4.3 would be e x p e c t e d t o p e r s i s t over the l o n g term. E q u a t i o n 4 . 3 makes i t c l e a r t h a t i f pathogen genotype i i s r e p r o d u c i n g mere q u i c k l y than genotype j ( i . e . , r > r ), then i j the r a t i o of t h e i r f r e q u e n c i e s increases exponentially w i t h t i m e . T h e r e f o r e , the frequency of t h a t genotype which reproduces most r a p i d l y w i l l approach 1.0 as t becomes l a r g e . Hence, i n analogy w i t h 4.1, the s i z e of the pathogen p o p u l a t i o n at time t , x ( t ) , when t i s l a r g e , can be w r i t t e n x ( t ) x ( t - 1 ) e (4.4) Where p a r a s i t e genotype k i s t h a t one which reproduces more q u i c k l y than any o t h e r on the p a r t i c u l a r host p o p u l a t i o n d e s c r i b e d by the s e t of m ' s . The r e l a t i v e r a t e of i n c r e a s e i n j s i z e of the pathogen p o p u l a t i o n , 1 dx, i s p l o t t e d a g a i n s t time x d t i n F i g u r e 4.1. Van der P l a n k , (1968) has i n d i c a t e d t h a t y i e l d l o s s e s i n a season are r o u g h l y p r o p o r t i o n a l t o the t o t a l amount of d i s e a s e , Z, p r e s e n t t h r o u g h o u t the season. 83 Z = / x ( t ) d t , o where t denotes the l e n g t h of the d i s e a s e season, f U s i n g 4.4 an upper bound can be p l a c e d on Z: Z < / x ( 0 ) e d t , max ~ 0 and t h e r e f o r e Z / x ( 0 ) < [ e x p ( r t ) - 1 ] / r . max k f k T h i s e q u a t i o n i s p l o t t e d i n F i g u r e 4.2. An o b j e c t i v e of the p l a n t b r e e d e r i s t o m i n i m i z e Z and thus maximize y i e l d and u l t i m a t e l y revenue w h i l e keeping c o s t s low. He would work toward t h i s o b j e c t i v e by d e t e r m i n i n g t h a t s e t of m 's f o r which Z i s m i n i m a l , j The response t i m e , T, the p r a c t i c a l minimum t o the l e n g t h of t i m e , i n pathogen g e n e r a t i o n s , between adjus t m e n t s of c r o p c o m p o s i t i o n , p r o v i d e s a measure of the r e a d i n e s s of the p l a n t breeder t o a l t e r the m ' s . C l e a r l y T depends upon a number of j f a c t o r s such as the c o s t / b e n e f i t r a t i o , the a c c e s s i b i l i t y of s u f f i c i e n t q u a n t i t i e s of the - v a r i o u s host R-genotypes i n a g r o n o m i c a l l y s i m i l a r v a r i e t i e s , and the ease w i t h which 84 a d j u s t m e n t s can be-made t o the m ' s . These f a c t o r s , i n t u r n , a re j i n f l u e n c e d by c r o p acreage and v a l u e , by a v a i l a b i l i t y of r e s i s t a n c e s o u r c e s and g e n e t i c i n f o r m a t i o n , by the natu r e of the c r o p ( e . g . , whether i n b r e e d i n g or o u t c r o s s i n g , annual or p e r e n n i a l ) , and by o t h e r f a c t o r s t h a t may determine how q u i c k l y the r e s u l t s of p l a n t b r e e d i n g a r e r e a l i z e d . Imagine t h a t at one extreme T i s v e r y s h o r t ; a t the o t h e r i t i s enormous. Where T i s s h o r t the p l a n t breeder w i l l have the o p t i o n of r e a c t i n g t o even s l i g h t s h i f t s i n the pathogen p o p u l a t i o n ; he w i l l attempt t o m i n i m i z e a of F i g u r e 4.1 by r e a d j u s t i n g the m ' s as o f t e n as i s p r a c t i c a b l e . T h i s would j amount to the r e p e a t e d i m p o s i t i o n of i n t e n s e l y d i s r u p t i v e s e l e c t i o n on the pathogen p o p u l a t i o n . I t i s o b v i o u s t h a t a s t r a t e g y such as t h i s would depend on a c c u r a t e g e n e t i c d a t a of the k i n d p r o v i d e d by a s e n s i t i v e m o n i t o r i n g program. With c o n t i n u a l a d j u stment of m 's i t i s u n l i k e l y t h a t any p a r t i c u l a r j genotype w i l l c o n s i s t e n t l y dominate the pathogen p o p u l a t i o n . Thus, a t t h i s extreme [where I assume response times a r e n e g l i g i b l e , and where a depends c r i t i c a l l y on the Q ( 0 ) ' s ] , no i g e n e r a l statement can be made c o n c e r n i n g the r e l a t i v e m e r i t s of mo n o c u l t u r e s and m u l t i l i n e s . E i t h e r c o u l d be p r e f e r r e d a c c o r d i n g t o the c i r c u m s t a n c e s . Where s t a b i l i z i n g s e l e c t i o n does not o p e r a t e , the most complex pathogen genotype i would q u i c k l y dominate and the p l a n t b r e e d i n g s t r a t e g y would be one of d e t e r m i n i n g the host genotype j which m i n i m i z e s R . Host genotype j would then be grown i n 85 m o n o c u l t u r e , pending f u r t h e r changes i n the system. At the o t h e r extreme, where the response time i s v e r y l o n g , an i n t r o d u c e d c r o p w i t h a p a r t i c u l a r set of m 's must be l e f t j unchanged f o r a l a r g e number of pathogen g e n e r a t i o n s . The pathogen p o p u l a t i o n w i l l have ample time t o adapt t o c r o p c o m p o s i t i o n and w i l l become dominated by a s i n g l e genotype. The p l a n t b r e e d e r t h e r e f o r e s h o u l d be a t t e m p t i n g t o m i n i m i z e r of F i g u r e 4.1 when he a l t e r s the m ' s . j Without s t a b i l i z i n g s e l e c t i o n , o n l y a s t a t i c monoculture i s needed (as a b o v e ) , and the response time i s no l o n g e r a r e l e v a n t f a c t o r . However i f s t a b i l i z i n g s e l e c t i o n i s o p e r a t i n g , then f o r each h o s t genotype j t h e r e i s a p a r t i c u l a r pathogen genotype, c a l l i t j f o r c o n v e n i e n c e , which r e p r o d u c e s more q u i c k l y than any o t h e r pathogen genotype on t h a t p a r t i c u l a r host genotype. In o t h e r words R > R , i * j . (4.5) . j r j i , j Pathogen genotype j i s t h a t genotype which has a v-gene c o r r e s p o n d i n g t o each R-gene c a r r i e d by host genotype j and no unneeded v-genes which reduce f i t n e s s (Van der P l a n k , 1968). Thus, i f pathogen genotype i c a r r i e s a l l the v-genes i n genotype j p l u s one e x t r a , then a measure of the s t r e n g t h (Van der P l a n k , 1968) of the R-gene c o r r e s p o n d i n g t o t h i s e x t r a v-gene i n terms of s t a b i l i z i n g s e l e c t i o n i s R R The l a r g e r t h i s i, D i f j 86 q u a n t i t y , the s t r o n g e r the s t a b i l i z i n g s e l e c t i o n o p e r a t i n g a g a i n s t the v-gene i n the absence of i t s c o r r e s p o n d i n g R-gene, and the s t r o n g e r the R-gene i_n t h i s p a r t i c u l a r g e n e t i c  background. S i m i l a r l y , pathogen genotype i reproduces more q u i c k l y on host genotype i than on any o t h e r host genotype; i . e . , R > R , i * j . (4.6) i r i i i j In t h i s case the p l a n t breeder can determine which pathogen genotype w i l l e v e n t u a l l y p r e v a i l but he i s s t i l l u nable t o p r e v e n t d o m i n a t i o n of the pathogen p o p u l a t i o n by one genotype. H i s o b j e c t i v e i s e s s e n t i a l l y t o f i n d t h a t s e t of m 's and t h a t j pathogen genotype k, f o r which r i s m i n i m a l ( F i g . 4.1). k Based on 4.2 and 4.5, the best c h o i c e of h o s t f o r a monoculture i s t h a t genotype h f o r which R i s m i n i m i z e d over a l l h ost genotypes j ; i . e . , R = min (R . ) . h,h j J , J Now c o n s i d e r a m u l t i l i n e which has m = 1-S and m = S h i where S i s s m a l l enough t h a t genotype h c o n t i n u e s t o dominate the pathogen p o p u l a t i o n . Then, by 4.2 and 4.6 87 r = ( 1 - S ) R + SR < R h h, h h, i h, h Hence, g i v e n s t a b i l i z i n g s e l e c t i o n , a m u l t i l i n e can always be formed f o r which the a s y m p t o t i c v a l u e of J_ dx i s l e s s than t h a t x d t of the best m o n o c u l t u r e . P r e d i c t i o n s of the Model The f o l l o w i n g c o n c l u s i o n s can be drawn from the above: ( i ) In the presence of s t a b i l i z i n g s e l e c t i o n the r e a d i n e s s w i t h which the c o m p o s i t i o n of the host c r o p can be a l t e r e d i s a key f a c t o r i n d e t e r m i n i n g the best s t r a t e g y of employing r e s i s t a n c e genes. ( i i ) I f the response time i s r e l a t i v e l y s h o r t the c r o p c o m p o s i t i o n (whether monoculture or m u l t i l i n e ) can be f r e q u e n t l y a d j u s t e d i n response t o the c h a n g i n g c o n s t i t u t i o n of the pathogen p o p u l a t i o n . Both the s i z e and make-up of the pathogen p o p u l a t i o n are under the c o n t r o l of the p l a n t b r e e d e r . M u l t i l i n e and m o n o c u l t u r e t e c h n i q u e s may prove u s e f u l under d i f f e r e n t c i r c u m s t a n c e s . ( i i i ) I f the response time i s s u f f i c i e n t l y l o n g 88 and a d j u s t m e n t s t o c r o p c o m p o s i t i o n s u f f i c i e n t l y i n f r e q u e n t to enable the pathogen t o become h i g h l y adapted, the m u l t i l i n e cannot p r e v e n t one genotype from d o m i n a t i n g the pathogen p o p u l a t i o n . However, i n these c i r c u m s t a n c e s , use of m u l t i l i n e s can m i n i m i z e the s i z e of the pathogen p o p u l a t i o n and, a c c o r d i n g l y , w i l l be p r e f e r a b l e t o mo n o c u l t u r e . ( i v ) The g r e a t e r the freq u e n c y of rearrangement (made t o the c r o p i n response t o changes i n the pathogen p o p u l a t i o n ) the lower the y i e l d l o s s e s w i l l be. I t s h o u l d be emphasized, however, t h a t i n r e a l i t y the c o s t s of m o n i t o r i n g , and the time and expense r e q u i r e d i n making a d j u s t m e n t s , w i l l g e n e r a l l y l i m i t the freq u e n c y w i t h which a d j u s t m e n t s can be made. Thus, even under the best of c o n d i t i o n s i t i s u n l i k e l y t h a t t h e p l a n t breeder w i l l be a b l e t o o p e r a t e i n the v i c i n i t y of a i n F i g u r e 4.1. M u l t i l i n e Implementation In n a t u r e , i t w i l l be i m p o s s i b l e to a c c u r a t e l y p r e d i c t the e f f e c t of a m u l t i l i n e on the g e n o t y p i c c o n t e n t of a pathogen p o p u l a t i o n u n t i l the m u l t i l i n e has been i n use f o r some time (Frey et a l . , 1973). 89 I n i t i a l d e c i s i o n s c o n c e r n i n g the c h o i c e of R-gene components and p r o p o r t i o n s of each i n the m u l t i l i n e must of n e c e s s i t y be based on the b e h a v i o r of those p a r a s i t e s t h a t happen t o be p r e s e n t and a v a i l a b l e f o r study when the m u l t i l i n e i s b e i n g assembled When a m u l t i l i n e i s used, and a new s e l e c t i v e regime i s imposed on the p a r a s i t e , i t w i l l be i m p o r t a n t t o c o n t i n u a l l y m o nitor the changes t h a t take p l a c e i n the p a r a s i t e p o p u l a t i o n . Data g a t h e r e d i n t h i s way d i c t a t e a d j u s t m e n t s needed i n the m u l t i l i n e f o r i m p r o v i n g i t s e f f e c t i v e n e s s . Because of the t h e o r e t i c a l e x p e c t a t i o n t h a t p a r a s i t e r e p r o d u c t i v i t y becomes maximal when a l l p a r a s i t e s are of one genotype ( F i g . 4.1), the changes made t o the m u l t i l i n e w i l l be aimed at those components t h a t c a r r y the g r e a t e s t l o a d of the most common pathogen. These m u l t i l i n e components need not be removed a l t o g e t h e r ; i t may be s u f f i c i e n t t o reduce the p r o p o r t i o n s o c c u p i e d by them i n the m u l t i l i n e (a p r o c e d u r e t h a t would p r o b a b l y be a s s i s t e d , t o some e x t e n t , by n a t u r a l s e l e c t i o n ) . I t w i l l be e v i d e n t , of c o u r s e , t h a t the i n c o r p o r a t i o n of new R-genes (as they become a v a i l a b l e ) w i l l a l s o improve the performance of the m u l t i l i n e . Summary A m a t h e m a t i c a l model of a m u l t i l i n e i s used t o i l l u s t r a t e the r e l a t i o n s h i p s between the c o m p o s i t i o n of the h o s t c r o p , the r e a d i n e s s of the p l a n t breeder t o change c r o p c o m p o s i t i o n , the y i e l d l o s s e s , and the c o m p o s i t i o n of the pathogen p o p u l a t i o n . Under the assumptions of the model, i t i s shown t h a t an 90 i n f r e q u e n t l y a d j u s t e d m u l t i l i n e i s i n c a p a b l e of p r e v e n t i n g one genotype from d o m i n a t i n g the pathogen p o p u l a t i o n . I t i s a l s o shown t h a t the r e l a t i v e m e r i t s of m u l t i l i n e and monoculture s t r a t e g i e s of employing r e s i s t a n c e genes depend on the c i r c u m s t a n c e s . Important c o n s i d e r a t i o n s a r e s t a b i l i z i n g s e l e c t i o n and the r e a d i n e s s of the p l a n t b r eeder t o change c r o p c o m p o s i t i o n . P r a c t i c a l a s p e c t s of m u l t i l i n e i m p l e m e n t a t i o n a r e d i s c u s s e d . FIGURES FIGURE 4.1 P l o t s a g a i n s t time of the r e l a t i v e r a t e of i n c r e a s e i n s i z e of a pathogen p o p u l a t i o n , J_ dx ( s o l i d l i n e ) ; and x dt the p r o p o r t i o n of the pathogen p o p u l a t i o n composed of t h a t genotype, c a l l i t k, which reproduces more a u i c k l y than any o t h e r , f ( t ) (broken l i n e ) ; on the k p a r t i c u l a r h o s t p o p u l a t i o n d e s c r i b e d by the s e t of m ' s . The degree t o which the pathogen p o p u l a t i o n has k adapted t o the hos t p o p u l a t i o n i s i n d i c a t e d by f ( t ) . k I n i t i a l l y the r a t e of i n c r e a s e o f f ( t ) i s l i m i t e d by k the s c a r c i t y of aenotype k. When f ( t ) approaches 1.0 k i t s r a t e of i n c r e a s e i s a g a i n l i m i t e d but now because the e x c e s s i n f i t n e s s of pathogen genotype k over the p o p u l a t i o n average i s v e r y s m a l l . The r e l a t i v e r a t e of i n c r e a s e i n s i z e of the pathogen p o p u l a t i o n , j _ dx, x dt i n c r e a s e s a t a d e c e l e r a t i n g r a t e as the pathogen p o p u l a t i o n becomes adapted to the hos t p o p u l a t i o n . FIGURE 4.2 P l o t s of the r a t i o of the upper bound on d i s e a s e q u a n t i t y p r e s e n t throughout the season, Z , t o the max i n i t i a l amount, x ( 0 ) , a g a i n s t the d u r a t i o n of the d i s e a s e season i n pathogen g e n e r a t i o n s , t , when the f . ' maximum r a t e of i n c r e a s e i n d i s e a s e q u a n t i t y per pathogen g e n e r a t i o n , r , i s 1.0 ( s o l i d l i n e ) ; and r , iC K. f o r t = 1.0 (broken l i n e ) . Z / x ( 0 ) i n c r e a s e s more f max q u i c k l y w i t h t than w i t h r k . FIGURE 4.1 Z max /X (o ) FIGURE 4.2 20-1 t f o r r k THE CONSEQUENCES OF POLYGENIC DETERMINATION OF RESISTANCE AND AGGRESSIVENESS IN NONSPECIFIC HOST:PARASITE RELATIONSHIPS Chapter 5 96 I n t r o d u c t i o n G e n e t i c s t u d y of h o s t r p a r a s i t e i n t e r a c t i o n has f o c u s s e d m a i n l y on systems i n which M e n d e l i a n s e g r e g a t i o n of q u a l i t a t i v e l y d i f f e r e n t phenotypes can be o b t a i n e d . For t h i s k i n d of system the r e s i s t a n c e of the host i s e f f e c t i v e a g a i n s t c e r t a i n c u l t u r e s (= genotypes) of the pathogen and i n e f f e c t i v e a g a i n s t o t h e r s . Such r e s i s t a n c e , o f t e n r e f e r r e d t o as s p e c i f i c r e s i s t a n c e , was g i v e n the d e s c r i p t i v e l a b e l " v e r t i c a l " r e s i s t a n c e , by Van der Pl a n k ( 1 9 6 3 ) . Van der Plank d i s t i n g u i s h e d another k i n d of r e s i s t a n c e which he l a b e l l e d " h o r i z o n t a l " . H o r i z o n t a l r e s i s t a n c e i s e x p r e s s e d n o n s p e c i f i c a l l y , and to v a r i n g degrees, depending on the pathogen c u l t u r e w i t h which the host i s made t o i n t e r a c t . As w e l l , when a group of h o s t s has been i n o c u l a t e d w i t h a s i n g l e pathogen c u l t u r e and a r r a n g e d i n sequence a c c o r d i n g t o the l e v e l s of r e s i s t a n c e t h a t are e x p r e s s e d , the sequence (but not n e c e s s a r i l y the l e v e l s of r e s i s t a n c e ) remains unchanged when another pathogen c u l t u r e i s used. T h i s phenomenon i s r e f e r r e d t o as " c o n s t a n t r a n k i n g " . The d i s t i n g u i s h i n g f e a t u r e s of v e r t i c a l and h o r i z o n t a l r e s i s t a n c e a r e i l l u s t r a t e d i n F i g u r e s 5.1a and 5.1b r e s p e c t i v e l y . These f i g u r e s a l s o i l l u s t r a t e the d i f f e r e n c e s between h o s t : p a r a s i t e i n t e r a c t i o n s t h a t are s p e c i f i c ( F i g u r e 5.1a) and n o n s p e c i f i c ( F i g u r e 5.1b); the c o n s t a n t r a n k i n g of n o n s p e c i f i c i n t e r a c t i o n i s a l s o shown ( F i g . 5.1b). The g e n e t i c b a s i s of n o n s p e c i f i c i n t e r a c t i o n i s not w e l l u n d e r s t o o d . D r i v e r (1962a and 1962b), N e l s o n (1978), Robinson (1973), and Van der Plank (1963) have s t a t e d t h a t h o r i z o n t a l r e s i s t a n c e i s u s u a l l y under p o l y g e n i c c o n t r o l . E v i d e n c e t h a t 97 s u p p o r t s t h i s p o i n t of view i s now a c c u m u l a t i n g (cf_. Robinson, 1976). However, v e r y few s t u d i e s have been c a r r i e d out on pathogens t h a t show v a r i a b l e p a t h o g e n c i t y i n n o n s p e c i f i c i n t e r a c t i o n w i t h the h o s t . In these i n t e r a c t i o n s , p a t h o g e n i c i t y , l i k e h o r i z o n t a l r e s i s t a n c e , i s q u a n t i t a t i v e l y e x p r e s s e d and shows c o n s t a n t r a n k i n g . The h y p o t h e s i s t h a t p a t h o g e n i c i t y i s a l s o p o l y g e n i c a l l y d e t e r m i n e d seems t o be r e a s o n a b l e and i s not e n t i r e l y w i t h o u t s u p p o r t (Emara and S i d h u , 1974; Person e_t a l . , 1981; Watson, 1970). To a v o i d c o n f u s i o n I use the terms a g g r e s s i v e n e s s and v i r u l e n c e t o d i s t i n g u i s h between p a t h o g e n i c i t y as a p o l y g e n i c t r a i t and p a t h o g e n i c i t y as an o l i g o g e n i c (gene-for-gene) t r a i t , r e s p e c t i v e l y , where p a t h o g e n i c i t y i s the a b i l i t y t c i n c i t e d i s e a s e . Van der P l a n k (1975) and Robinson (1976) a l s o m a i n t a i n t h a t , compared w i t h s p e c i f i c r e s i s t a n c e (which has a h i s t o r y of b r e a k i n g down f o l l o w i n g a d a p t a t i o n i n the p a r a s i t e p o p u l a t i o n ) , r e s i s t a n c e which i s h o r i z o n t a l and n o n s p e c i f i c i s r e l a t i v e l y more s t a b l e . The e v i d e n c e they p r o v i d e f o r t h i s p o i n t of view i s l a r g e l y e m p i r i c a l . I assume t h a t f o r n o n s p e c i f i c systems the r e s i s t a n c e of the h o s t and the a g g r e s s i v e n e s s of the p a r a s i t e are b o t h p o l y g e n i c a l l y d e t e r m i n e d . My o b j e c t i v e i s , t o examine whether t h i s assumption w i l l l e a d i n d u c t i v e l y t o the e x p e c t a t i o n t h a t c o n s t a n t r a n k i n g w i l l be e x p r e s s e d , and t h a t the r e s u l t i n g system w i l l be r e l a t i v e l y more s t a b l e than a system of s p e c i f i c i n t e r a c t i o n s i n v o l v i n g major genes. 9 8 The Model S i n c e I assume t h a t host r e s i s t a n c e and p a r a s i t e a g g r e s s i v e n e s s a r e both p o l y g e n i c a l l y d e t e r m i n e d , t h e i r e x p r e s s i o n can be r e a s o n a b l y r e p r e s e n t e d by normal d i s t r i b u t i o n c u r v e s ( A l l a r d e t a l . , 1968). The f r e q u e n c y w i t h which i n d i v i d u a l s w i t h a t r a i t of magnitude x o c c u r i n a n o r m a l l y d i s t r i b u t e d p o p u l a t i o n i s f ( x ) = [ 1/U vT2T7 ]exp[ (x - y )2/{-2gz)] (5.1) X X where y and c , r e s p e c t i v e l y , are the mean and s t a n d a r d x x d e v i a t i o n of the d i s t r i b u t i o n . Examples of normal fre q u e n c y d i s t r i b u t i o n c u r v e s f o r host s u s c e p t i b i l i t y and p a r a s i t e a g g r e s s i v e n e s s are shown at the top of F i g u r e 5.2. For i l l u s t r a t i v e p u r p o s e s , each c u r v e i s d i v i d e d i n t o s i x c l a s s e s , d e s i g n a t e d by numbers zer o t o f i v e , i n c l u s i v e ; the d i v i s i o n s r e p r e s e n t one, two, and t h r e e s t a n d a r d d e v i a t i o n s from the mean. S i n c e I have chosen to d i s c u s s the i n t e r a c t i o n s of the model system i n terms of the e x t e n t t o which d i s e a s e i s d e v e l o p e d and e x p r e s s e d , c l a s s e s z e r o t o f i v e of the p a r a s i t e p o p u l a t i o n r e p r e s e n t i n c r e a s i n g l e v e l s of a g r e s s i v e n e s s (denoted by A) and, I assume, i n c r e a s i n g l e v e l s of p a r a s i t e r e p r o d u c t i v i t y . As f o r the host p o p u l a t i o n , the z e r o c l a s s ( r e p r e s e n t i n g maximal r e s i s t a n c e ) d e s i g n a t e s the host c l a s s w i t h minimal s u s c e p t i b i l i t y (denoted by S) and, I assume, w i t h 99 g r e a t e s t r e p r o d u c t i v i t y . Because a g g r e s s i v e n e s s and s u s c e p t i b i l i t y a r e e x p r e s s e d o n l y when host and p a r a s i t e i n t e r a c t , the a r b i t r a r i l y a s s i g n e d numbers must be regarded as r e p r e s e n t i n g the p o t e n t i a l f o r g e n e r a t i n g d i f f e r i n g l e v e l s of d i s e a s e e x p r e s s i o n . However, i t i s not known how n a t u r e t r a n s l a t e s t h i s p o t e n t i a l i n t o the r e a l i z e d l e v e l s of d i s e a s e e x p r e s s i o n and how t h i s t r a n s l a t i o n can be d e s c r i b e d m a t h e m a t i c a l l y . In the f o l l o w i n g I c o n s i d e r one p o s s i b i l i t y by which the i n t e r a c t i o n of p a r a s i t i c a g g r e s s i v e n e s s , A, and h o s t s u s c e p t i b i l i t y , S, c o u l d produce d i s e a s e : an a d d i t i v e model ( F i g u r e 5.2) i n which D = A + S. (5.2) (An extreme a l t e r n a t i v e would be the m u l t i p l i c a t i v e h y p o t h e s i s D = A X S). In e q u a t i o n 5.2, D r e p r e s e n t s the s e v e r i t y of d i s e a s e r e a c t i o n . With h o s t and p a r a s i t e p o p u l a t i o n s each s u b d i v i d e d i n t o s i x c l a s s e s i n F i g u r e 5.2 a m a t r i x of t h i r t y - s i x i n t e r a c t i o n s i s o b t a i n e d . In accordance w i t h e q u a t i o n 5.2, the t r a n s l a t i o n from p o t e n t i a l t c r e a l i z e d l e v e l s of d i s e a s e e x p r e s s i o n was made, a r b i t r a r i l y , by a d d i n g the numbers which had been a s s i g n e d t o the i n t e r s e c t i n g h o s t and p a r a s i t e c l a s s e s . The l e v e l s of d i s e a s e i n the i n t e r a c t i o n m a t r i x t h e r e f o r e range from z e r o ( l e a s t s u s c e p t i b l e h o s t : l e a s t a g g r e s s i v e p a r a s i t e ) t o ten (most s u s c e p t i b l e host:most a g g r e s s i v e p a r a s i t e ) . As shown by t h i s i n t e r a c t i o n m a t r i x , the assumption of 100' a d d i t i v i t y l e a d s t o t h e e x p e c t a t i o n t h a t " c o n s t a n t r a n k i n g " o f h o s t s a n d p a r a s i t e s w i l l b e a d i s t i n g u i s h i n g f e a t u r e o f p o l y g e n i c a l l y b a s e d p a r a s i t i c s y s t e m s f o r w h i c h t h e i n t e r a c t i o n s a r e n o n s p e c i f i c . N e x t we a s c e r t a i n t h e r e l a t i v e s e n s i t i v i t y o f d i s e a s e e x p r e s s i o n t o c h a n g e s i n a g g r e s s i v e n e s s a n d s u s c e p t i b i l i t y . T h i s i s a n e c e s s a r y c o n s i d e r a t i o n when d i s c u s s i n g t h e d u r a b i l i t y o f r e s i s t a n c e . P a r z e n ( i 9 6 0 ) s h o w s t h a t , r e g a r d l e s s o f t h e f o r m o f t h e p r o b a b i l i t y d e n s i t y f u n c t i o n s , f ( A ) a n d f ( S ) , E [ A + S ] = E [ A ] + E [ S ] , (5.3) a n d V a r [ A +S] = V a r [ A ] + V a r [ S ] + 2 C o v [ A , S ] . (5.4) C o m b i n i n g e q u a t i o n s 5.2 a n d 5.3, y = u + u (5.5) D A S T h u s , i n t h e a d d i t i v e m o d e l , t h e m e a n d i s e a s e r e a c t i o n , P , i s D t h e sum o f t h e mean a g g r e s s i v e n e s s , u , a n d mean A s u s c e p t i b i l i t y , u . S 101 The j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n f o r A and S, f ( A , S ) , g i v e s the p r o p o r t i o n of h o s t r p a r a s i t e i n t e r a c t i o n s i n v o l v i n g h o s t s of s u s c e p t i b i l i t y S and p a r a s i t e s of a g g r e s s i v e n e s s A. Given random a s s o c i a t i o n between h o s t s and p a r a s i t e s . f ( A , S) = f ( A ) + I f ( S ) ( 5 . 6 ) A c c o r d i n g t o t h i s r e l a t i o n s h i p , A and S are independent and hence u n c o r r e l a t e d random v a r i a b l e s . T h e r e f o r e Cov[A, S ] = 0 , ( 5 . 7 ) and by e q u a t i o n s b.2, 5.4. and 5 . 7 , c2 = <r2 + cr 2 . ( 5 . 8 ) D A S Hence, i n the a d d i t i v e model, the v a r i a n c e of d i s e a s e r e a c t i o n , c2 , i s the sum of the v a r i a n c e i n a a g r e s s i veness , <s2 , and the D A v a r i a n c e i n s u s c e p t i b i l i t y , <r2. F u r t h e r m o r e , because A and S a r e independent n o r m a l l y d i s t r i b u t e d random v a r i a b l e s , i t can be shown ( P a r z e n , 1 9 6 0 , pp. 3 1 7 - 3 1 9 ) t h a t the c o n v o l u t i o n of t h e i r p r o b a b i l i t y d e n s i t y f u n c t i o n s , f(D) i n the a d d i t i v e model, i s a l s o normal. In s h o r t , g i v e n random h o s t : p a r a s i t e a s s o c i a t i o n w i t h d i s e a s e r e a c t i o n (D) 102 b e i n g the sum of a g g r e s s i v e n e s s (A) and s u s c e p t i b i l i t y ( S ) , d i s e a s e r e a c t i o n i s n o r m a l l y d i s t r i b u t e d , i . e . f(D) s a t i s f i e s e q u a t i o n 5.1 w i t h mean u = p + p and v a r i a n c e <s2 = e2 + D A S D A c ~ . S The d i f f e r e n t k i n d s of h o s t : p a r a s i t e i n t e r a c t i o n i d e n t i f i e d i n the m a t r i x of F i g u r e 5.2 a r e summarized by the normal frequency d i s t r i b u t i o n c u r v e of d i s e a s e r e a c t i o n below the m a t r i x . T h i s c u r v e i l l u s t r a t e s the f a c t t h a t a p h e n o t y p i c change, t a k i n g p l a c e i n e i t h e r of the two i n t e r a c t i n g p o p u l a t i o n s , t r a n s l a t e s t o a r e l a t i v e l y s m a l l e r change when i t s e f f e c t i s e x p r e s s e d i n terms of d i s e a s e development. For example, suppose f o r some reason t h a t the mean s u s c e p t i b i l i t y of the c r o p changes by an amount A . Then, assuming no change i n the d i s t r i b u t i o n P S of a g g r e s s i v e n e s s i n the p a r a s i t e p o p u l a t i o n , i t f o l l o w s from e q u a t i o n 5.5 t h a t the mean l e v e l of d i s e a s e r e a c t i o n a l s o changes by A . However, bv e q u a t i o n 5.8, I A /c\ < IA / a I .V S ' yD D ' ' \ S because e > 0. Hence, a g i v e n change i n the mean host A s u s c e p t i b i l i t y (or mean p a t h o g e n i c a g g r e s s i v e n e s s ) produces a s m a l l e r r e l a t i v e change ( a l t h o u g h an e q u a l a b s o l u t e change) i n the mean s e v e r i t y of d i s e a s e r e a c t i o n . Thus, changes i n a g g r e s s i v e n e s s or s u s c e p t i b i l i t y appear t o be "damped" i n terms of t h e i r p h e n o t y p i c e x p r e s s i o n i n d i s e a s e r e a c t i o n . We w i l l r e f e r t o t h i s phenomenon as " p h e n o t y p i c damping". I t i m p l i e s t h a t once a h o r i z o n t a l l y r e s i s t a n t h o s t c u l t i v a r has been d e v e l o p e d , the o c c u r r e n c e of n o t i c e a b l y 103 i n c r e a s e d amounts of d i s e a s e would r e q u i r e a s u b s t a n t i a l i n c r e a s e i n a g g r e s s i v e n e s s of the p a r a s i t e p o p u l a t i o n . But c o n v e r s e l y , i t a l s o i m p l i e s t h a t a s u b s t a n t i a l i n c r e a s e i n h o r i z o n t a l r e s i s t a n c e i s r e q u i r e d t o b r i n g about a n o t i c e a b l e r e d u c t i o n i n d i s e a s e . G e n e r a l l y , p h e n o t y p i c damping of the mean d i s e a s e r e a c t i o n , y , w i t h r e s p e c t t o the mean of a t r a i t x o c c u r s when D A /<r |< | A / c r | , >: = A or S, (5.9) y D y .' x M D Mx i where A i s the change i n y a s s o c i a t e d w i t h a change of A , , D yx i n t he mean of t r a i t x, y . Here x r e p r e s e n t s e i t h e r p a r a s i t i c x a g g r e s s i v e n e s s (A) or host s u s c e p t i b i l i t y ( S ) . In the a d d i t i v e model, p h e n o t y p i c damping of the v a r i a b i l i t y i n d i s e a s e r e a c t i o n a l s o o c c u r s . To s i m p l i f y a l g e b r a i c m a n i p u l a t i o n s , I measure t h i s e f f e c t i n terms of v a r i a n c e r a t h e r than s t a n d a r d d e v i a t i o n . | A U 2 ) A 2 | < | A U 2 ) A 2 I, x = A or S. (5.10). 1 D D 1 X X 1 When t h i s r e l a t i o n s h i p i s s a t i s f i e d , p h e n o t y p i c damping of the v a r i a n c e of the d i s e a s e r e a c t i o n , c*, w i t h r e s p e c t t o the v a r i a n c e i n a t r a i t x o c c u r s . Here A ( * 2 ) i s the change i n c 2 D D a s s o c i a t e d w i t h a change of A(<*2) i n the v a r i a n c e of t r a i t x, 104 e2^. The d i f f e r e n c e i n the power of the d i v i s o r s i n e q u a t i o n s 5.9 and 5.10 i s d i c t a t e d by the d e s i r e t o m a i n t a i n comparisons between d i m e n s i o n l e s s q u a n t i t i e s . Duncan (1953) o u t l i n e s the advantages of d o i n g so. A c c o r d i n g t o e q u a t i o n 5.8, U 2 ) = A(<r2 ) , x = A or S. D x Together, t h i s e q u a l i t y and e q u a t i o n 5.8 s a t i s f y i n e q u a l i t y 5.10. T h e r e f o r e the v a r i a n c e of d i s e a s e r e a c t i o n i s a l s o p h e n o t y p i c a l l y damped i n the a d d i t i v e model. Di s c u s s i o n I t i s w i d e l y a c c e p t e d t h a t h o r i z o n t a l r e s i s t a n c e i s u s u a l l y p o l y g e n i c . In t h i s paper I ask whether the p r i n c i p a l c h a r a c t e r i s t i c s c o n v e n t i o n a l l y a s s o c i a t e d w i t h h o r i z o n t a l r e s i s t a n c e , namely " c o n s t a n t r a n k i n g " and d u r a b i l i t y , f o l l o w d i r e c t l y from p o l y g e n i c d e t e r m i n a t i o n . "Constant r a n k i n g " . I have shown t h a t " c o n s t a n t r a n k i n g " i c a n r e s u l t from a d d i t i v e i n t e r a c t i o n of a g g r e s s i v e n e s s and s u s c e p t i b i l i t y when they a re p o l y g e n i c . In f a c t , i f the i n t e n s i t y of d i s e a s e as a f u n c t i o n of a g g r e s s i v e n e s s , A, and s u s c e p t i b i l i t y , S, i s D = D(A, S ) , 105 t h e n , g e n e r a l l y , " c o n s t a n t r a n k i n g " ensues i f and o n l y i f D( A , js) > D (A - a, S - 6 ) (5. 11) f o r a l l A > a > 0 , S > 6 > 0 . Hence, p o l y g e n i c d e t e r m i n a t i o n produces " c o n s t a n t r a n k i n g " not o n l y f o r a d d i t i v e i n t e r a c t i o n of a g g r e s s i v e n e s s and s u s c e p t i b i l i t y , but f o r a l l m a t h e m a t i c a l s t r u c t u r e s of i n t e r a c t i o n which s a t i s f y e q u a t i o n 5 . 1 1 . " P h e n o t y p i c damping" i s the p r o p e r t y whereby a change i n the mean or v a r i a n c e of the p h e n o t y p i c d i s t r i b u t i o n of e i t h e r i n t e r a c t i n g p o p u l a t i o n t r a n s m i t s as a r e l a t i v e l y s m a l l e r change i n the c o r r e s p o n d i n g parameters of the d i s t r i b u t i o n of d i s e a s e r e a c t i o n . P o t e n t i a l l y i t can c o n t r i b u t e t o the d u r a b i l i t y of h o r i z o n t a l r e s i s t a n c e but we are u n c e r t a i n as t o the g e n e r a l i t y of t h i s p r o p e r t y . There a re numerous f u n c t i o n s D(A, s) s a t i s f y i n g e q u a t i o n 5.11 i n which the d i s e a s e r e a c t i o n i s n e i t h e r the sum nor the product of a g g r e s s i v e n e s s and s u s c e p t i b i l i t y . P h e n o t y p i c damping has been demonstrated f o r none of t h e s e . In t h i s c o n t e x t i t would be u s e f u l t o m a t h e m a t i c a l l y d e f i n e the e f f e c t s of a g g r e s s i v e n e s s and s u s c e p t i b i l i t y on p a r a s i t i c development. D i f f e r e n t modes of i n t e r a c t i o n can sometimes be d i s t i n g u i s h e d by the shape of the f r e q u e n c y d i s t r i b u t i o n c u r v e f o r d i s e a s e r e a c t i o n . For i n s t a n c e , when the magnitudes of a g g r e s s i v e n e s s and s u s c e p t i b i l i t y a r e measured i n u n i t s of 106 s t a n d a r d d e v i a t i o n s about the mean, m u l t i p l i c a t i v e i n t e r a c t i o n c o u l d be d i s c e r n e d from a d d i t i v e i n t e r a c t i o n by the presence of skewness. However, o t h e r more complex modes of i n t e r a c t i o n may r e s u l t i n d i s e a s e r e a c t i o n d i s t r i b u t i o n s which a r e e x p e r i m e n t a l l y i n d i s t i n g u i s h a b l e from each o t h e r and/or from the d i s t r i b u t i o n s produced by s i m p l e a d d i t i v e or m u l t i p l i c a t i v e i n t e r a c t i o n . In a r e l a t e d c o n t e x t , P a r l e v l i e t and Zadoks (1977) suggest t h a t gene-for-gene i n t e r a c t i o n among minor genes a t many l o c i can be e x p e r i m e n t a l l y i n d i s t i n g u i s h a b l e from a d d i t i v e gene i n t e r a c t i o n among minor genes a t many l o c i . Slow response t o s e l e c t i o n . I t i s l i k e y t h a t the slow response t o s e l e c t i o n which i s f r e q u e n t l y a s s o c i a t e d w i t h p o l y g e n i c t r a i t s i s a much more im p o r t a n t c o n t r i b u t o r t o the d u r a b i l i t y of h o r i z o n t a l r e s i s t a n c e than " p h e n o t y p i c damping". Because s u b s t i t u t i o n s f o r many genes are n e c e s s a r y to produce a s u b s t a n t i a l change i n a p o l y g e n i c a l l y based p h e n o t y p i c t r a i t , such a change t a k e s a l o n g time t o occur i n a new s e l e c t i v e regime. In c o n t r a s t , a gene s u b s t i t u t i o n a t a s i n g l e pathogen l o c u s i s o f t e n enough to p r o v i d e v i r u l e n c e t o an o t h e r w i s e v e r t i c a l l y r e s i s t a n t c u l t i v a r ( F l o r , 1971; P e r s o n , 1959). At l e a s t two d i f f e r e n t p r o c e s s e s of a d a p t a t i o n can account f o r the r e l a t i v e i n e r t i a of p o l y g e n i c t r a i t s i n t h e i r response t o changes i n the d i r e c t i o n of s e l e c t i o n . G a l l e g l y (1968) s u g g e s t s a s t e p w i s e p r o g r e s s i o n whereby a s p e c i f i c s u b s t i t u t i o n a t a p a r t i c u l a r l o c u s i s n e c e s s a r y b e f o r e a subsequent s u b s t i t u t i o n at an o t h e r l o c u s i s a d a p t i v e . A change of phenotype becomes e v i d e n t o n l y a f t e r a number of such s t e p s i n v o l v i n g p a r t i c u l a r gene s u b s t i t u t i o n s . 107 On the o t h e r hand, H a r l a n (1976) suggests t h a t each polygene may be enmeshed i n an i n t e g r a t e d m a t r i x of p l e i o t r o p i c a l l y i n t e r a c t i n g p o l y g e n e s . He proposes t h a t such complexes of c c - a d a p t e d (Dobzhansky, 1970) polygenes as a whole d e f i n e a g g r e s s i v e n e s s i n the pathogen and s u s c e p t i b i l i t y i n the h o s t . Under such c i r c u m s t a n c e s , S t e b b i n s (1974) suggests t h a t a d a p t a t i o n i s l i k e l y t o p r o c e e d q u i c k l y t h rough a few g e n e t i c changes which o n l y a d j u s t the e x i s t i n g gene complex even when the r e s u l t i n g phenotype i s not o p t i m a l f o r the new c o n d i t i o n s . E x t e n s i v e r e o r g a n i z a t i o n of a gene complex r e q u i r e s s i m u l t a n e o u s g e n e t i c changes at many i n t e r a c t i n g l o c i and, t h e r e f o r e , i s l i k e l y t o be an e x c e e d i n g l y r a t e event, perhaps even on e v o l u t i o n a r y time s c a l e s These c o n s i d e r a t i o n s a l l l e a d i n t u i t i v e l y to the e x p e c t a t i o n t h a t n o n s p e c i f i c , p o l y g e n i c a l l y d etermined host r e s i s t a n c e w i l l be more d u r a b l e than r e s i s t a n c e which i s c o n t r o l l e d by one, or by a few genes. E v o l u t i o n a r y dynamics. The model of p o l y g e n i c i n t e r a c t i o n i l l u s t r a t e d i n F i g u r e 5.2 a l s o has i m p l i c a t i o n s f o r e v o l u t i o n a r y dynamics of u n c u l t i v a t e d h o s t : p a r a s i t e r e l a t i o n s h i p s . I t seems r e a s o n a b l e t o assume t h a t a g g r e s s i v e n e s s and r e s i s t a n c e w i l l b o t h c o r r e l a t e d i r e c t l y w i t h r e p r o d u c t i v i t y (of p a r a s i t e s and of i n f e c t e d h o s t s , r e s p e c t i v e l y ) ; t h u s , when the upper c u r v e s of these f i g u r e s a r e taken t o r e p r e s e n t r e p r o d u c t i v i t y , s e l e c t i o n f o r t h i s parameter i s n e c e s s a r i l y d i r e c t i o n a l . However, F i g u r e 5.2 shows t h a t t h e s e s e l e c t i v e f o r c e s , when they are t r a n s l a t e d through an i n t e r a c t i o n m a t r i x and t o the c u r v e of d i s e a s e r e a c t i o n s below, are d i a m e t r i c a l l y opposed. A l t h o u g h s t a b i l i t y 108 c o u l d r e s u l t f r o m a b a l a n c i n g o f t h e s e . t w o o p p o s i n g f o r c e s o f d i r e c t i o n a l s e l e c t i o n , t h e a c h i e v e d s t a b i l i t y w o u l d h a v e i t s o r i g i n i n d i r e c t i o n a l , r a t h e r t h a n s t a b i l i z i n g , s e l e c t i o n . T h u s , i n c o n t r a s t t o V a n d e r P l a n k ' s ( 1 9 7 5 ) s u g g e s t i o n , t h e e x p e c t e d s t a b i l i t y may n o t d e r i v e f r o m s t a b i l i z i n g s e l e c t i o n o p e r a t i n g i n d e p e n d e n t l y i n e a c h o f t h e t w o i n t e r a c t i n g p o p u l a t i o n s . F o r e x a m p l e , i m a g i n e t h a t r e c e n t a d a p t a t i o n s t o g r e a t e r r e s i s t a n c e i n t h e h o s t p o p u l a t i o n i n h i b i t d i s e a s e d e v e l o p m e n t i n a n u n c u l t i v a t e d h o s t : p a r a s i t e s y s t e m . W i t h a s u b s t a n t i a l r e d u c t i o n i n d i s e a s e t h e i n t e n s i t y o f d i r e c t i o n a l s e l e c t i o n f a v o r i n g h o s t r e s i s t a n c e i s l i k e l y t o be r e d u c e d , e i t h e r t h r o u g h a r e d u c t i o n i n t h e n u m b e r s o f p u s t u l e s o r l e s i o n s t h a t a r e i n i t i a t e d , o r t h r o u g h t h e e x t e n t t o w h i c h t h e s e d e v e l o p on i n d i v i d u a l h o s t p l a n t s , o r t h r o u g h a n i n c r e a s e i n t h e f r a c t i o n o f t h e t o t a l h o s t p o p u l a t i o n w h i c h e s c a p e s d i s e a s e , o r by o t h e r m e a n s . H o w e v e r t h e r e w i l l be no r e l a x a t i o n o f d i r e c t i o n a l s e l e c t i o n f a v o r i n g a g g r e s s i v e n e s s . A s P i m e n t e l ( 1 9 6 1 ) h a s p o i n t e d o u t , t h e r e d u c t i o n i n d i s e a s e w o u l d a l s o i n t r o d u c e . a d i f f e r e n t i a l i n r e l a t i v e r e p r o d u c t i v i t i e s o f h o s t a n d p a r a s i t e w h i c h , w i t h t h e p r o g r e s s o f t i m e , w o u l d f u r t h e r r e d u c e t h e i n t e n s i t y o f s e l e c t i o n f a v o r i n g h o s t r e s i s t a n c e . G e n o t y p e s w o u l d c h a n g e , a g g r e s s i v e n e s s w o u l d i n c r e a s e f a s t e r t h a n r e s i s t a n c e , a n d a b a l a n c e w o u l d be r e s t o r e d . S t a b i l i t y a l s o r e q u i r e s t h a t h o s t r e s i s t a n c e i n c r e a s e m o r e q u i c k l y t h a n p a t h o g e n i c a g g r e s s i v e n e s s when d i s e a s e i s a b u n d a n t . H o w e v e r , s t a b i l i z i n g s e l e c t i o n o f some s o r t s e e m s t o be n e c e s s a r y t o l i m i t t h e i n t e n s i t y o f s e l e c t i o n f o r a g g r e s s i v e n e s s i n o b l i g a t e p a r a s i t e s . T h e a l t e r n a t i v e i s t o a s s u m e t h a t 109 r e s i s t a n c e can i n h e r e n t l y e v o l v e more q u i c k l y than a g g r e s s i v e n e s s . At l a s t two k i n d s of c i r c u m s t a n c e s c o u l d g i v e r i s e t o s t a b i l i z i n g s e l e c t i o n i n these h o s t : p a r a s i t e systems. F i r s t , v e r y heavy d i s e a s e may k i l l most h o s t s l e a v i n g j u s t a few s c a t t e r e d p a t c h e s of p l a n t s s u i t a b l e f o r the pathogen. The next g e n e r a t i o n of p a r a s i t e s would then be c o n f i n e d t o these s m a l l i s o l a t e d groups of s u r v i v i n g h o s t s . The more a g g r e s s i v e p a r a s i t e s would soon decimate the h o s t s they were i n f e c t i n g ; and w i t h s u c c e s s f u l d i s p e r s a l from one p a t c h t o another b e i n g an e x c e p t i o n a l l y r a t e e vent, f u t u r e g e n e r a t i o n s would be doomed t o a l a c k of a v a i l a b l e h o s t s and l i k e l y d e a t h . In c o n t r a s t , by the time the l e s s a g g r e s s i v e pathogens were t h r e a t e n i n g the v i a b i l i t y of t h e i r . h o s t s , the h o s t p o p u l a t i o n c o u l d have r e c o v e r e d t o the p o i n t where enough new p a t c h e s had formed t h a t s u c c e s s f u l d i s p e r s a l between groups of host p l a n t s was a g a i n f e a s i b l e . Thus, w i t h s m a l l i s o l a t e d groups of host p l a n t s , s t a b i l i z i n g s e l e c t i o n f o r reduced a g g r e s s i v e n e s s i s to be e x p e c t e d . When i t i s combined . w i t h the s t r o n g d i r e c t i o n a l s e l e c t i o n f o r g r e a t e r host r e s i s t a n c e , the b a l a n c e between host and p a r a s i t e c o u l d be r e s t o r e d . B l a c k (.1975) p r o v i d e s e v i d e n c e t h a t a s i m i l a r e x p l a n a t i o n can account f o r the s c a r c i t y o f the more " a g g r e s s i v e " s t r a i n s of many i n f e c t i o u s d i s e a s e s i n p r i m i t i v e human s o c i e t i e s R i g o r o u s m a t h e m a t i c a l support f o r the s c e n a r i o d e v e l o p e d above can be found i n Anderson (1979), G i l p i n (1975), and L e v i n and p i m e n t e l ( 1 9 8 i ) . A l e x a n d e r and B o r g i a (1978) d i s c u s s a number of r e l e v a n t c o n c e p t s . 1 1 0 Another type of s i t u a t i o n a l s o promotes s e l e c t i o n a g a i n s t e x c e s s i v e a g g r e s s i v e n e s s . For example, S h a t t o c k (1976) has j observed t h a t p o t a t o l a t e b l i g h t can j o n l y s u r v i v e the w i n t e r i n t u b e r s and v e r y a g g r e s s i v e i s o l a t e s may not s u r v i v e the w i n t e r because e x t e n s i v e t u b e r decay w i l l e i t h e r k i l l the fungus or p r e v e n t i n f e c t e d shoots from b e i n g produced i n the f o l l o w i n g s p r i n g . S i m i l a r l y , C h e s t e r (1946) and Zadoks (1965) have noted t h a t the s t r e s s of r u s t i n f e c t i o n added t o t h a t of extreme temperatures i s o f t e n enough t o k i l l wheat l e a v e s d u r i n g the non-growing season. In both c a s e s l e s s a g g r e s s i v e pathogens would seem to have a s u r v i v o r s h i p advantage d u r i n g the non-growing season when d i s p e r s a l s u c c e s s i s r e l a t i v e l y r a t e . The e x t e n t of t h i s p o t e n t i a l l y i m p o r t a n t phenonmenon among h o s t i p a r a s i t e systems i s g e n e r a l l y unknown. A complex r e l a t i o n s h i p between the a g g r e s s i v e n e s s of a p a r a s i t e genotype and i t s f i t n e s s or c a p a c i t y t o g a i n r e p r e s e n t a t i o n i n t h e ' n e x t g e n e r a t i o n i s i m p l i c i t i n the two s c e n a r i o s above. When the p a r a s i t e ' s o p p o r t u n i t i e s f o r d i s p e r s a l s u c c e s s a r e . good, I suggest t h a t s e l e c t i o n w i l l f a v o r p a r a s i t e r e p r o d u c t i v i t y . Under these c i r c u m s t a n c e s a p o s i t i v e c o r r e l a t i o n between a g g r e s s i v e n e s s and f i t n e s s i s e x p e c t e d . But when s u c c e s s f u l d i s p e r s a l i s r a r e , l e s s a g g r e s s i v e p a r a s i t e s , which a l l o w l o n g e r h o s t s u r v i v a l , w i l l p r o b a b l y be s e l e c t e d . In t h i s s i t u a t i o n a g g r e s s i v e n e s s and f i t n e s s are n e g a t i v e l y c o r r e l a t e d . Thus i t seems t h a t r e l a t i v e d i s p e r s a l s u c c e s s l a r g e l y d e t e r m i n e s the n a t u r e of the c o r r e l a t i o n between p a r a s i t i c f i t n e s s and a g g r e s s i v e n e s s . Moreover, when d i s p e r s a l i s p a s s i v e and g e n o t y p i c d i f f e r e n c e s i n i n t r i n s i c d i s p e r s a l a b i l i t y a r e I l l n e g l i g i b l e , I suggest t h a t , f o r g i v e n a b i o t i c c o n d i t i o n s , r e l a t i v e d i s p e r s a l s u c c e s s i s f i x e d by the a v a i l a b i l i t y and c o n t i n u i t y of s u s c e p t i b l e h o s t s i n space. For c o e v o l u t i o n over the l o n g - t e r m p a r a s i t i s m may be viewed as a t w o - d i m e n s i o n a l system i n which the d i s t r i b u t i o n c u r v e s f o r a g g r e s s i v e n e s s and r e s i s t a n c e a r e l o c a t e d on, and move a l o n g , two d i f f e r e n t axes ( F i g u r e 5.4). At any p o i n t i n t i m e , t h e s e c u r v e s , r e g a r d l e s s of t h e i r p o s i t i o n , would i n t e r s e c t t o form an i n t e r a c t i o n m a t r i x . A c c u m u l a t i o n or l o s s of polygenes i n one p o p u l a t i o n would be matched by a c c u m u l a t i o n or l o s s of polygenes i n the o t h e r . P r o t r a c t e d p e r i o d s of imbalance c o u l d r e s u l t i n e x t i n c t i o n f o r the system. Viewed i n t h i s way, t h e r e seems to be no means by which, through an a c c u m u l a t i o n of polygenes f o r r e s i s t a n c e based on s e l e c t i o n , the host c o u l d " o u t r u n " the p a r a s i t e . But s i n c e i t cannot be a s s u r e d e i t h e r t h a t the numbers of polygenes c a p a b l e of b e i n g a c c u m u l a t e d , or t h a t the r a t e s of a c c u m u l a t i o n of pol y g e n e s i n host and pathogen are i d e n t i c a l , i t seems n e c e s s a r y t o i nvoke " g e n e t i c feedback" ( P i m e n t e l , 1961) or some such mechanism f o r ho m e o s t a s i s i n o r d e r t o ensure the c o n t i n u i n g b a l a n c e on which l o n g - t e r m c o e v o l u t i o n would depend. 112 Summary Model systems of p a r a s i t i s m , based on the assumption that host r e s i s t a n c e and p a r a s i t e a g g r e s s i v e n e s s are both n o n s p e c i f i c and p o l y g e n i c a l l y d e t e r m i n e d , l e a d t o the e x p e c t a t i o n t h a t such systems w i l l e x h i b i t c o n s t a n t r a n k i n g and t h a t the r e s i s t a n c e w i l l be r e l a t i v e l y more s t a b l e than r e s i s t a n c e which i s s p e c i f i c and c o n t r o l l e d by major genes. I d i s c u s s the l o n g - t e r m c o e v o l u t i o n of such systems and suggest t h a t the r e l a t i o n s h i p between a g g r e s s i v e n e s s and p a r a s i t i c f i t n e s s depends on the r e l a t i v e d i s p e r s a l s u c c e s s of the p a r a s i t e . The concept of p h e n o t y p i c damping i s i n t r o d u c e d whereby changes i n the mean or v a r i a n c e o f the host or p a r a s i t e p h e n o t y p i c d i s t r i b u t i o n s r e s u l t i n r e l a t i v e l y s m a l l e r changes i n the c o r r e s p o n d i n g parameters of the d i s e a s e r e a c t i o n d i s t r i b u t i o n . FIGURES FIGURE 5.1 (a) V e r t i c a l R e s i s t a n c e . " P l u s " (+) d i s e a s e r e a c t i o n s s i g n i f y t h a t the pathogen c u l t u r e has been s u c c e s s f u l i n e s t a b l i s h i n g i t s e l f and c a u s i n g d i s e a s e on the host c u l t i v a r of c o n c e r n . "Minus" (-) r e a c t i o n s i n d i c a t e t h a t the pathogen c u l t u r e has f a i l e d t o e s t a b l i s h i t s e l f l e a v i n g the hos t c u l t u r e f r e e of d i s e a s e . Each of the t h r e e pathogen c u l t u r e s can s u c c e s s f u l l y a t t a c k one, and o n l y one, of the t h r e e host c u l t i v a r s . C u l t u r e s , a , 0 , and j are s p e c i f i c t o , and i n c i t e d i s e a s e on, hos t c u l t i v a r s A, B, and C, r e s p e c t i v e l y . D i f f e r e n t i a l s e l e c t i o n i s e v i d e n t ; the r e l a t i v e r a n k i n g of host c u l t i v a r s i n terms of r e s i s t a n c e depends s p e c i f i c a l l y on the pathogen c u l t u r e used f o r i n o c u l a t i o n . (b) H o r i z o n t a l R e s i s t a n c e . The d i s e a s e r e a c t i o n s here d i f f e r i n degree r a t h e r than i n k i n d so the e x t e n t of d i s e a s e development must be s c a l e d t o i l l u m i n a t e d i f f e r e n c e s . In t h i s f i g u r e the s c a l i n g i s n u m e r i c a l from a minimum of 0 t o a maximum of 4 . The non-s p e c i f i c i t y of d i s e a s e r e a c t i o n i s i l l u s t r a t e d ; each of the t h r e e pathogen c u l t u r e s i n c i t e s some d i s e a s e on each ho s t c u l t i v a r . Constant r a n k i n g i s a l s o e v i d e n t ; r e g a r d l e s s of the i n f e c t i n g pathogen c u l t u r e , host c u l t i v a r C i s more r e s i s t a n t than host c u l t i v a r B, and host c u l t i v a r B i s more r e s i s t a n t than host c u l t i v a r A. S i m i l a r l y , on each h o s t c u l t i v a r t h e r e i s a c o n s t a n t r a n k i n g among pathogen c u l t u r e s : c u l t u r e x b e i n g mor a g g r e s s i v e than y which i n t u r n i s more a g g r e s s i v e than z (adapted from Person and Ebba, 1975). FIGURE 5.2 A d d i t i v e Model. Here f ( x ) i s the freq u e n c y w i t h which i n d i v i d u a l s w i t h amount x of the t r a i t under c o n s i d e r a t i o n occur i n the p o p u l a t i o n of c o n c e r n . Because of t h e i r p o l y g e n i c d e t e r m i n a t i o n we a s s i g n normal frequency d i s t r i b u t i o n s t o p a r a s i t i c a g g r e s s i v e n e s s , A, and hos t s u s c e p t i b i l i t y , S. Assuming the amount of d i s e a s e i s D = A + S, the m a t r i x r e c o r d s the s e v e r i t y of d i s e a s e f o r the v a r i o u s i n d i c a t e d c l a s s e s of hos t : p a r a s i t e i n t e r a c t i o n . The. r e s u l t i n g p r o b a b i l i t y d e n s i t y f u n c t i o n f o r D, f ( D ) , i s n o r m a l l y d i s t r i b u t e d . ( T h i s F i g u r e was used w i t h p e r m i s s i o n by Raoul Robinson, 1976.) FIGURE 5.3 C o e v o l u t i o n of host r e s i s t a n c e and p a r a s i t e a g g r e s s i v e n e s s from time t t o time t . Normal c u r v e s 1 2 on the axes r e p r e s e n t the f r e q u e n c y d i s t r i b u t i o n s of a g g r e s s i v e n e s s or r e s i s t a n c e f o r times t and t . 1 2 These c u r v e s i n t e r s e c t i n d i s e a s e i n t e r a c t i o n m a t r i c e s as i n F i g u r e s 5.2 and 5.3. There i s no p a r t i c u l a r reason why t h i s diagram s h o u l d be s y m m e t r i c a l nor why the normal c u r v e s s o l d a l l have e a u a l s p r e a d s . 116 FIGURE 5.1a D ,. \ H o s t PathogenX \ A B c OC 4 - — — fi' + — 4 — : — + FIGURE 5.1b Pathogen Host A B C CL 4 3 2 3 2 1 4 2 1 0 187 117 F I G U R E 5 . 2 10 8 6 4 2 0 DISEASE REACTION (D) 118 THE EFFECT OF FIELD GEOMETRY ON THE SPREAD OF CROP DISEASE Chapter 6 120 I n t r o d u c t i o n A e r i a l d i s p e r s a l has l o n g been r e c o g n i z e d as a s i g n i f i c a n t f a c t o r i n the s p r e a d of many of the most d e s t r u c t i v e p a r a s i t i c d i s e a s e s of .man's c r o p s . The importance of d e t e r m i n i n g how the d e s i g n of agroecosystems a f f e c t epidemic development stems d i r e c t l y from man's s t r u g g l e t o f e e d h i s e x p l o d i n g p o p u l a t i o n s i n the f a c e of i n c r e a s i n g l o s s e s i n a g r i c u l t u r a l p r o d u c t i o n t o p a r a s i t e s and pathogens ( P i m e n t e l , 1977). When a g i v e n t o t a l acreage w i t h i n a r e g i o n i s p l a n t e d w i t h a c e r t a i n c r o p or v a r i e t y , the chances of d i s e a s e s p r e a d i n g from one f i e l d t c another may be a d i r e c t consequence of the s i z e and shape of the f i e l d s . Van der. Plank (1948, 1949, 1960) and Waggoner (1962) ' s t u d i e d t h i s problem f o r pathogens which a r e p a s s i v e l y d i s p e r s e d by wind. They assumed t h a t a l l f i e l d s a r e the same s i z e and t h a t shape and o r i e n t a t i o n are c o n s t a n t . N o t i n g t h a t much inoculum f a l l s back i n t o the f i e l d from which i t o r i g i n a t e s , Van der P l a n k reasoned t h a t changes i n the amount of e s c a p i n g inoculum would be i n c o n s e q u e n t i a l t o d i s e a s e p r o g r e s s w i t h i n the f i e l d but c r u c i a l t o the spread of d i s e a s e between f i e l d s . A c c o r d i n g t o h i s m a t h e m a t i c a l model, the p r o b a b i l i t y of i n f e c t i o n per u n i t a rea i n one f i e l d by p r o p a g u l e s r e l e a s e d from a n o t h e r d e c r e a s e s as the d i s t a n c e between f i e l d s i n c r e a s e s . S i n c e the d i s t a n c e between f i e l d s i s p r o p o r t i o n a l t o the s i z e of i n d i v i d u a l f i e l d s when a c o n s t a n t acreage i s grown, he recommended l a r g e f i e l d s . S u b s e q u e n t l y , Waggoner (1962) suggested t h a t s i n c e d i s e a s e spreads r e l a t i v e l y e a s i l y once e s t a b l i s h e d w i t h i n a f i e l d , the p r o b a b i l i t y of i n f e c t i o n per f i e l d may be a more a p p r o p r i a t e 121 i n d i c a t o r of p o t e n t i a l epidemic development than Van der P l a n k ' s measure, the p r o b a b i l i t y per u n i t a r e a w i t h i n a f i e l d . Waggoner based h i s a n a l y s i s of the a e r i a l s pread of d i s e a s e between f i e l d s on Gregory's (1945) e q u a t i o n . Gregory adapted S u t t o n ' s (1932) t h e o r y of a t m o s p h e r i c eddy d i f f u s i o n t o d e s c r i b e t h e decrease i n d e n s i t y of inoculum d e p o s i t e d w i t h d i s t a n c e from the p o i n t of r e l e a s e . Waggoner (1962) i n d i c a t e d t h a t the p r o b a b i l i t y of i n f e c t i o n i n one f i e l d by inoculum from another d e c r e a s e s as the f i e l d a r e a d e c r e a s e s . In c o n t r a s t t o Van der Plank (1948, 1949, 1960, 1963), he c o n c l u d e d t h a t d e c r e a s i n g the s i z e , and t h e r e f o r e the i s o l a t i o n , of f i e l d s i n t o which a r e g i o n ' s c r o p l a n d i s s u b d i v i d e d c o u l d i n h i b i t epidemic development. Van der Plank and Waggoner have a p p a r e n t l y m a i n t a i n e d t h e i r o p p o s i n g v i e w p o i n t s ever s i n c e (Zadoks and Kampmeijer, 1977). However, b o t h Van der P l a n k and Waggoner i m p l i c i t l y assumed steady s t a t e c o n d i t i o n s f o r t h e i r models. Hence, t h e i r c o n c l u s i o n s a r e s u s p e c t i f e i t h e r the spread of d i s e a s e . over space, or the r a t e of inoculum p r o d u c t i o n at the s o u r c e , or b o t h , change w i t h t i m e . S i n c e both d i s p e r s a l and p r o d u c t i o n a r e time dependent d u r i n g epidemic development ( A y l o r , 1978; Berger and Luke, 1979; Cammack, 1958; Stakman and H a r r a r , 1957), t h i s problem d e s e r v e s f u r t h e r a n a l y s i s . In t h i s c h a p t e r a d i f f u s i o n model i s used t o study the s p r e a d of p l a n t d i s e a s e when both d i s p e r s a l and inoculum p r o d u c t i o n a r e time dependent. In S e c t i o n s 1-6 I examine the r e l a t i o n s h i p between i n o c u l u m d i s p e r s a l and p r o d u c t i o n i n s m a l l i s o l a t e d ' f i e l d s ' . A n a l y t i c and n u m e r i c a l t e c h n i q u e s are used i n 122 S e c t i o n 7 t o study the e f f e c t of c h a n g i n g the s i z e or shape of a s s o c i a t e d f i e l d s on d i s e a s e s p r e a d . I f i n i s h w i t h a d i s c u s s i o n of the g e n e r a l p r i n c i p l e s b e h i n d the r e s u l t s . J_. The Model The model i s d e v e l o p e d around a s t r u c t u r e s i m i l a r t o t h a t used by F l e m i n g (1980b) and c o n c e r n s the spread of c e r e a l r u s t when pathogen biomass i s s m a l l . L e t N(x, y, t ) be the pathogen d e n s i t y , the t o t a l pathogen biomass (mycelium, p u s t u l e s , and s p o r es) per u n i t a r e a , a t p o s i t i o n ( x , y) a t time t . Then the r a t e of change of pathogen d e n s i t y w i t h time can be w r i t t e n . 3N/9t = F (N) + F (N) (6.1.1) G D where F (N) and F (N) d e s c r i b e the r a t e s of change of N(x, y, t ) G D due t o p o p u l a t i o n growth and d i s p e r s a l r e s p e c t i v e l y . P o p u l a t i o n growth, F (N), can be s e p a r a t e d i n t o two G components: F (N) = G(N) - L(N) (6.1.2) G where G(N) and L ( N ) , r e s p e c t i v e l y , r e p r e s e n t the r a t e s a t which i n c r e a s e s and d e c r e a s e s i n p a r a s i t e d e n s i t y a t (x, y) occur as a r e s u l t of l o c a l p o p u l a t i o n dynamics. 123 L e t B be the per u n i t r a t e of pathogen biomass p r o d u c t i o n ( t h r o u g h s p o r u l a t i o n and l e s i o n e x p a n s i o n ), and l e t H be the f r a c t i o n of t h i s p r o d u c t i o n l o s t when spores l a n d i n g i n the f i e l d f a i l t o i n f e c t v u l n e r a b l e host t i s s u e ( t h r o u g h e.g., d e p o s i t i o n on the grou n d ) . Then G(N) = B . f ( N ) . (1 - H) (6.1.3) where B depends on the age s t r u c t u r e of the pathogen p o p u l a t i o n , B and H depend on the m i c r o c l i m a t i c c o n d i t i o n s , and f(N) i s the e f f e c t i v e p a r a s i t e d e n s i t y . The f u n c t i o n f(N) d e s c r i b e s the e f f e c t s of d e n s i t y dependent c o m p e t i t i o n among pathogens f o r the f i n i t e r e s o u r c e s such as space, energy, or n u t r i e n t s of the host p l a n t s . Gregory ( 1S 7 2) has shown t h a t e a r l y i n the growing season, when the p r o p o r t i o n of h e a l t h y v u l n e r a b l e h o s t t i s s u e i s h i g h , the p r o b a b i l i t y of m u l t i p l e i n f e c t i o n i s low. Under these c o n d i t i o n s c o m p e t i t i o n i s n e g l i g i b l e and f(N ) = N (6.1.4) In 6.1.2 L(N) r e p r e s e n t s the r a t e of l o s s of pathogen biomass per u n i t a r e a due t o : p a r a s i t e senescene on h e a l t h y host t i s s u e , d i s e a s e - c a u s e d host d e a t h s , p l u s n a t u r a l host d e a t h . S i n c e c o m p e t i t i v e or s y n e r g i s t i c i n t e r a c t i o n s among i n d i v i d u a l 124 as a s h i f t i n the c e n t e r of t h i s d i s t r i b u t i o n t o a p o i n t downwind of the mid-width l i n e , the magnitude of t h i s s h i f t b e i n g r o u g h l y p r o p o r t i o n a l t o the v e c t o r wind v e l o c i t y . A l t h o u g h l e s s inoculum would be l o s t a t the upwind edge of the f i e l d , the l o s s would be g r e a t e r a t the downwind edge, and o v e r a l l , the net l o s s from the f i e l d would be g r e a t e r than i f inoculum were randomly d i s s e m i n a t e d . Hence, our a n a l y s i s u n d e r e s t i m a t e s the p o t e n t i a l i n h i b i t i o n of epidemic development when d i s p e r s a l i s d i r e c t e d . Moreover, by the same r e a s o n i n g , t h i s u n d e r e s t i m a t i o n i s g r e a t e s t f o r f i e l d s o r i e n t e d p e r p e n d i c u l a r l y t o the p r e v a i l i n g wind d i r e c t i o n and l e a s t f o r f i e l d s w i t h an o r i e n t a t i o n p a r a l l e l t o wind d i r e c t i o n . Hence e l o n g a t i n g square f i e l d s i s l i k e l y t o be much more r e w a r d i n g i n terms of r u s t c o n t r o l than d e c r e a s i n g f i e l d s i z e when a s p e c i f i c f r a c t i o n of the r e g i o n i s devoted t o s u s c e p t i b l e c e r e a l s . Of c o u r s e , economic c o n s i d e r a t i o n s w i l l be i m p o r t a n t d e t e r m i n a n t s of the r e l a t i v e b i a s toward e i t h e r approach i n any mixed s t r a t e g y t o c h a n g i n g f i e l d geometry. An i m p o r t a n t q u e s t i o n i s whether the g a i n i n c e r e a l r u s t c o n t r o l i s g r e a t enough f o r r e a s o n a b l e f i e l d d imensions t o be of p r a c t i c a l use. The answer must come from f i e l d t e s t s , e s p e c i a l l y s i n c e the p r i n c i p l e p a r a m e t e r s : R , R , and D, a l l depend i o c r i t i c a l l y on l o c a l c o n d i t i o n s . However, the models c e r t a i n l y suggest a p o t e n t i a l f o r p r a c t i c a l use: F i g u r e 6.10 i n d i c a t e s t h a t f o r a p p a r e n t l y p l a u s i b l e parameter v a l u e s , e l o n g a t i n g a 200 m x 200 m f i e l d i n t o a 2000 m x 20 m f i e l d c o u l d reduce pathogen biomass a t h a r v e s t by 70%. The e f f e c t on y i e l d i s u n c l e a r but the r e s u l t i s e n c o u r a g i n g because 20 m seems wide enough t o 125 p a r a s i t e s a r e i n s i g n i f i c a n t d u r i n g the i n i t i a l s t a g e s of epidemic development (Van der P l a n k , 1963) L(N) = M N (6.1.5) where M i s the i n t r i n s i c r a t e of " n a t u r a l " pathogen m o r t a l i t y . " N a t u r a l " m o r t a l i t y i n c l u d e s a l l forms of m o r t a l i t y except l o s s e s due t o n a t u r a l enemies which a r e assumed t o be n e g l i g i b l e . The term F (N) i n 6.1.1 a c c o u n t s f o r the f u n d a m e n t a l l y D p a s s i v e and a i r b o r n e movement of d i s e a s e inoculum (e.g. s p o r e s ) . I assume t h a t the e f f e c t of inoculum d i s p e r s a l on N(x, y, t ) i s a d i f f u s i o n p r o c e s s . A f t e r a s h o r t time 6t I assume t h a t the o f f s p r i n g of a s i n g l e r e p r o d u c i n g p a r a s i t e w i l l be n o r m a l l y d i s t r i b u t e d about t h e i r p a r e n t w i t h v a r i a n c e * 2 6 t when d i s p e r s a l o c c u r s over a u n i f o r m s u r f a c e of u n l i m i t e d e x t e n t i n a l l d i r e c t i o n s . S k e l l a m (1951) g i v e s the maximum l i k e l i h o o d e s t i m a t e of tf2, the mean square d i s p e r s i o n r a t e d u r i n g the time i n t e r v a l (0, fit), as tf2 = I r 2 / ( n 6 t ) . i-1 Here r i s the d i s t a n c e of the i t h u n i t of pathogen biomass i 126 ( e . g . , s p o r e ) f r o m i t s p a r e n t ( e . g . , p u s t u l e ) a n d n i s t h e n u m b e r o f o b s e r v e d v a l u e s o f r . i T h e r a n d o m d i f f u s i o n h y p o t h e s i s i m p l i c i t l y a s s u m e s t h a t a c o n s t a n t p r o p o r t i o n o f t h e p a t h o g e n p o p u l a t i o n i s d i s p e r s i n g a t a l l t i m e s ; t h a t d i s p e r s a l r e s u l t s i n t r a n s p o r t w h i c h i s u n c o r r e l a t e d a n d r a n d o m w i t h r e s p e c t t o d i r e c t i o n a n d d i s t a n c e ; a n d ( i n d i r e c t l y ) t h a t t h e p r o b a b i l i t y o f new i n f e c t i o n d e c r e a s e s m o n o t o n i c a l l y w i t h d i s t a n c e f r o m t h e s o u r c e . T h e e f f e c t s o f w i n d o n p r o p a g u l e d i s s e m i n a t i o n , w h i c h c a n i n v a l i d a t e a l l o f t h e s e a s s u m p t i o n s , a r e c o n s i d e r e d i n t h e d i s c u s s i o n s e c t i o n . U n d e r t h e a s s u m p t i o n s o f r a n d o m d i f f u s i v e d i s p e r s a l , S k e l l a m (1951) s h o w s t h a t F ( N ) = D 2 [ ( 3 2 / 3 x 2 ) + ( 3 2 / 3 y 2 ) + ( 3 2 / 3 z 2 ) ] N (6.1.6) D w h e r e D 2 = «r 2/2q f o r d i f f u s i o n i n q d i m e n s i o n s , q = 1, 2, o r 3. I m p l i c i t i n 6.1.6 a r e t w o a s s u m p t i o n s r e q u i r i n g t h e s p a t i a l s c a l e ' o f o b s e r v a t i o n t o b e l a r g e : F i r s t , t h e a s s u m p t i o n t h a t D a n d R a r e c o n s t a n t i n t h e f i e l d r e q u i r e s t h a t t h e f i e l d b e v i e w e d a s a u n i f o r m s u b s t r a t e , a ' c a r p e t * r a t h e r t h a n a c o l l e c t i o n o f i n d i v i d u a l l y i s o l a t e d p l a n t s . S e c o n d , 6.1.6 i s d e r i v e d f r o m f i r s t p r i n c i p l e s ( c _ f . O k u b o , 1 9 8 0 , p p . 9-11) b y v i s u a l i z i n g p r o p a g u l e d i s s e m i n a t i o n a s a r a n d o m w a l k p r o c e s s i n w h i c h t h e l e n g t h o f e a c h r a n d o m s t e p i s v e r y s m a l l c o m p a r e d t o t h e s p a t i a l s c a l e o f o b s e r v a t i o n . S u b s t i t u t i n g 6.1.4 i n t o 6 . 1 . 3 , 127 G(N) = BN(1 - H); and then c o mbining t h i s e x p r e s s i o n and 6.1.5 i n 6.1.2, the l o c a l p o p u l a t i o n dynamics a r e d e s c r i b e d by F (N) = (B - BH - M)N = RN. (6.1.7) G U s i n g 6.1.6 and 6.1.7 i n 6.1.1 t o d e s c r i b e the growth of the d i s e a s e p o p u l a t i o n i n two d i m e n s i o n s , 3N/3t = RN + D 2[ (3 2 / 3 x 2 ) + (3 2 / 3 y 2 ) ] N (6.1.8) Here i t i s i m p l i c i t l y assumed t h a t the pathogen's age d i s t r i b u t i o n i s s t a b l e ; t h a t any d e c l i n e i n the d i f f u s i o n r a t e , D, w i t h c r o p growth i s n e g l i g i b l e ; and t h a t net changes i n the growth r a t e of the p a r a s i t e p o p u l a t i o n , R, due t o s h i f t i n g e n v i r o n m e n t a l c o n d i t i o n s or g e n e t i c a d a p t a t i o n t o the host a r e s m a l l . I have d e v e l o p e d e q u a t i o n 6.1.8 i n terms of pathogen biomass. However, f o l l o w i n g Okubo (1980, pp. 217-220), the same e q u a t i o n can be d e r i v e d when N r e p r e s e n t s the number of pathogens i n a d i s e a s e p o p u l a t i o n undergoing s i m p l e b i r t h and death p r o c e s s e s and age-dependent d i s p e r s a l . ( C e r e a l r u s t d i s p e r s a l i s age-dependent d u r i n g the d i s e a s e season: 128 u r e d o s p o r e s a r e d i s p e r s e d w h i l e t h e i r p a r e n t a l s p o r u l a t i n g u r e d i a remain f i x e d on h o s t t i s s u e ) . In t h i s case the " m a c r o s c o p i c " parameters R and D r e p r e s e n t c o r r e s p o n d i n g q u a n t i t i e s a veraged over the ( m i c r o s c o p i c ) age s t r u c t u r e . Thus, e q u a t i o n 6.1.8 can be thought of i n terms of pathogen numbers as w e l l as pathogen biomass d e n s i t y . I t i s i n s t r u c t i v e t o compare the r e p r e s e n t a t i o n of space i n 6.1.8 w i t h t h a t of s i m u l a t i o n models which have a l s o been used t o d e s c r i b e the growth and s p r e a d of p l a n t d i s e a s e i n time and space. Both Kampmeijer and Zadoks (1977) and Shrum (1975) d i v i d e d space i n t o a number of compartments or c e l l s , each of which c o u l d i n c l u d e a l a r g e number of host p l a n t s . They adopted f u n c t i o n s which d e c r e a s e d m o n o t o n i c a l l y w i t h d i s t a n c e t o d e s c r i b e the " l o n g range" movement of spores between c e l l s . But by s p r e a d i n g s p o r e s e v e n l y w i t h i n c e l l s " , they i m p l i c i t l y assumed t h a t f o r " s h o r t range" d i s p e r s a l the p r o b a b i l i t y of i n f e c t i o n i s not a f u n c t i o n of d i s t a n c e from the s o u r c e . I n c o n s i s t e n c i e s a r i s e on the b o u n d a r i e s between c e l l s . For example, c o n s i d e r two p l a n t s c o l i n e a r w i t h a p o i n t source of i n o c u l u m i n such a s i m u l a t i o n model. The p r o b a b i l i t y of i n f e c t i o n may be q u i t e d i f f e r e n t a t the two p l a n t s i f they a r e s e p a r a t e d by a c e l l boundary, w h i l e the p r o b a b i l i t i e s would be e x a c t l y the same a t the two p l a n t s even i f they were s e p a r a t e d by a much g r e a t e r d i s t a n c e , p r o v i d e d they l a y w i t h i n the same c e l l . Kiyosawa (1976) a v o i d e d t h e s e i n c o n s i s t e n c i e s by modeling d i s p e r s a l from p l a n t t o p l a n t but h i s r e s u l t i n g computer program was so cumbersome and time consuming t h a t he was f o r c e d t o 129 r e s t r i c t h i s s i m u l a t i o n s t o p l o t s of o n l y 97 x 97 p l a n t s . E q u a t i o n 6.1.8, a l l o w s c o n s i d e r a t i o n of r e a l i s t i c f i e l d s i z e s w i t h o u t h a v i n g t o " c o m p a r t m e n t a l i z e " space and t o l e r a t e the a t t e n d a n t i n c o n s i s t e n c i e s . 2. C r i t i c a l Dimensions f o r R e c t a n g u l a r F i e l d s : No S u r v i v a l  O u t s i d e I b e g i n by s t u d y i n g the e f f e c t of f i e l d s i z e on pathogen p o p u l a t i o n growth where d e a t h i s i n s t a n t a n e o u s o u t s i d e the f i e l d . C o n s i d e r a r e c t a n g u l a r f i e l d w i t h s i d e s of l e n g t h a and b, r e s p e c t i v e l y , i n the x and y d i r e c t i o n s . I t has one c o r n e r a t the o r i g i n , ( x , y) = (0, 0) and i t s d i a g o n a l l y o p p o s i t e c o r n e r a t ( a , b ) . O u t s i d e the f i e l d I assume t h a t the environment i s so p h y s i o l o g i c a l l y u n s u i t a b l e f o r the p a r a s i t e t h a t i t cannot s u r v i v e t h e r e . T h i s r e q u i r e s t h a t the d e n s i t y N(x, y, t ) v a n i s h on the f i e l d b o u n d a r i e s . Hence I l o o k f o r s o l u t i o n s of e q u a t i o n 6.1.8 which s a t i s f y the boundary c o n d i t i o n s N(x, y, t ) = 0 at x=0, x=a, y=0, y=b; t > D (6.2.1) and i n i t i a l c o n d i t i o n s N(x, y, 0) = N (x, y ) , 0 < x < a, 0 < y < b. (6.2.2) I The f u n c t i o n N ( x , y) d e s c r i b e s the pathogen d e n s i t y a t time I 130 z e r o and hence i s p o s i t i v e somewhere w i t h i n the f i e l d and n e g a t i v e nowhere. U s i n g s t a n d a r d methods (c_f. P i p e s , 1958, p. 501), i t can be v e r i f i e d d i r e c t l y t h a t s i n ( m l l x / a ) s i n ( n l l y / b ) exp[F(m, n ) t ] (6.2.3) where F(m, n) = R - n 2 D 2 ( m 2 / a 2 + n 2 / b 2 ) , i s a s o l u t i o n of 6.1.8 and 6.2.1 f o r any p o s i t i v e i n t e g e r s m, n. S i n c e any smooth i n i t i a l d e n s i t y N ( x , y) can be w r i t t e n as a I s u p e r p o s i t i o n of the s i n e f u n c t i o n s o c c u r r i n g i n 6.2.3, ( P i p e s , 1958, p. 501), the s o l u t i o n of 6.1.8 and 6.2.1 which s a t i s f i e s 6.2.2 can be w r i t t e n N(x, y, t ) = E Z A sin(mnx/a) s i n ( n n y / b ) e x p [ F ( m , n ) t ] mn m = l !r ^ *1 (6.2.4) To f i n d the c o n s t a n t s A , I s e t t = 0 i n e q u a t i o n 6.2.4 and mn m u l t i p l y b o t h s i d e s by s i n ( p n x / a ) s i n ( q n y / b ) 131 where p and q a r e i n t e g e r s , and i n t e g r a t e the r e s u l t from x = 0 t o a and from y = 0 t o b. On r e a r r a n g i n g the i n t e g r a l ( P i p e s , 1958, p. 502), a b A = (4/ab) / / N ( x , Y) s i n ( m x/a) s i n ( n y/b)dy dx. o o 1 As t i n c r e a s e s , the b e h a v i o r of e x p a n s i o n 6.2.4 becomes dominated by i t s f i r s t term (the c o e f f i c i e n t A i s n e c e s s a r i l y f o r l a r g e t . Thus, a t any p o i n t ( x , y) i n s i d e the f i e l d , the pathogen d e n s i t y N e v e n t u a l l y i n c r e a s e s e x p o n e n t i a l l y w i t h t i m e , remains t h e same, or d e c r e a s e s e x p o n e n t i a l l y w i t h t i m e , depending on whether 1 1 > 0 ) : N(x, y, t ) z A s i n ( n x / a ) sin(ny/b) e x p [ F ( l , l ) t ] , 1 1 F( 1 , 1 ) > 0, F( 1 , 1 ) = 0, or F ( 1 , 1 ) < 0. (6.2.5) Here F(1 , 1 ) = R - n 2D 2/d 2 and d 2 = (a z + b 2). 132 i s a measure of the s i z e of the f i e l d i n terms of d i f f u s i o n . Hence, a c c o r d i n g t o the model, any i n i t i a l pathogen p o p u l a t i o n e v e n t u a l l y e i t h e r d e c r e a s e s or i n c r e a s e s as d i s r e s p e c t i v e l y e i t h e r l e s s or g r e a t e r than d where c d = D/n R . c (6.2.6) The q u a n t i t y d , w i t h d i m e n s i o n s of l e n g t h , i s the c r i t i c a l c v a l u e of d (above) f o r the o c c u r r e n c e or o t h e r w i s e of pathogen biomass s p r e a d w i t h i n the f i e l d . S i n c e the l o s s of inoculum o c c u r s through the f i e l d b o u n d a r i e s , the r a t e of l o s s i s r o u g h l y p r o p o r t i o n a l t o the p e r i m e t e r . In c o n t r a s t , the p r o d u c t i o n r a t e i s p r o p o r t i o n a l t o the f i e l d ' s a r e a because p r o d u c t i o n o c c u r s a t e v e r y p o i n t w i t h i n the f i e l d . S i n c e a s m a l l e r f i e l d has a g r e a t e r p e r i m e t e r r e l a t i v e t o i t s a r e a than a l a r g e r f i e l d , d i s p e r s a l has a r e l a t i v e l y s t r o n g e r . e f f e c t i n s m a l l e r f i e l d s . E q u a t i o n 6.2.6 s p e c i f i e s the s i z e (d ) of f i e l d s f o r which p r o d u c t i o n e x a c t l y c compensates f o r d i s p e r s a l l o s s e s . In s m a l l e r f i e l d s (d < d ) the c pathogen p o p u l a t i o n d e c r e a s e s w i t h time because i t s growth r a t e i n the f i e l d cannot o f f s e t the l o s s e s due t o d i s p e r s a l out -of 133 the f i e l d . On the o t h e r hand, i n l a r g e r f i e l d s where d > d , c p o p u l a t i o n growth exceeds d i s p e r s a l l o s s e s so the number of pathogens i n c r e a s e s w i t h time ( F i g u r e 6.1). Note, however, t h a t t r a n s i e n t e f f e c t s can oc c u r b e f o r e the lowest term i n the s e r i e s (m = n = 1) has a c h i e v e d dominance i n 6.2.4 and e s t a b l i s h e d a s u b s e q u e n t l y c o n s i s t e n t t r e n d . A p p l i c a t i o n and i n t e r p r e t a t i o n of the c r i t i c a l f i e l d s i z e concept r e q u i r e s c a r e . Pathogen biomass d e n s i t y , N, i s d i s t i n c t from d i s e a s e s e v e r i t y , x, the v i s i b l y d i s e a s e d f r a c t i o n of s u s c e p t i b l e h o s t t i s s u e . In c o n t r a s t t o N, x n e c e s s a r i l y i n c r e a s e s m o n o t o n i c a l l y d u r i n g the d i s e a s e season as u r e d i a become v i s i b l e on e n t e r i n g t h e i r i n f e c t i o n s p e r i o d (Chapter 2 ) , except perhaps when the hos t grows u n u s u a l l y q u i c k l y . Hence, r e p o r t s t h a t x i n c r e a s e d on s m a l l p l o t s (e.g. P a r l e v l i e t and van Ommeren, 1975) s h o u l d not be used t o i n f e r t h a t N a l s o i n c r e a s e d , and t h e r e f o r e , t h a t the c r i t i c a l f i e l d s i z e concept d i d not a p p l y . On the o t h e r hand, because 6.1.6 h o l d s o n l y when the s p a t i a l s c a l e of o b s e r v a t i o n i s l a r g e , the c r i t i c a l f i e l d s i z e concept i s not s t r i c t l y a p p l i c a b l e when r e a l i s t i c parameter v a l u e s p r e d i c t i t t o be ve r y s m a l l ( e . g . , i n 6.2.6). 3_. E f f e c t s of F i e l d S i z e on P r o d u c t i o n and Ex i t Rates S i n c e the t o t a l pathogen biomass i n the r e c t a n g u l a r f i e l d of S e c t i o n 2 a t time t i s a b P ( t ) = / / N(x, y, t ) dy dx, o o (6.3.1) 1 3 4 where N(x, y, t ) i s g i v e n by 6.2.4, i t s r a t e of change i s a b dP/dt = 1 1 (3N/3t)dy dx. o o Hence, by 6.1.8, I l a b jdP/dt = / / {RN + D 2 [ ( 3 2 N / 3 x 2 ) + ( 3 2N/8y 2) ]}dy dx I I o o = RP - P , where b a '; P = D 2 { 9 / 3 x ) / Ndy + (3/3y) / N dx) , (6.3.2) E o o Here RP i s the net r a t e a t which pathogen biomass i s c r e a t e d w i t h i n the f i e l d , w h i l e P i s the r a t e a t which biomass i s l o s t E as inoculum l e a v e s the f i e l d a c r o s s the boundaries. T h e r e f o r e , dP/dt = (R - E)P, where E = P /P. (6.3.3) E Here E i s the per u n i t r a t e a t which pathogen biomass l e a v e s the f i e l d . I t i s the t w o - d i m e n s i o n a l analogue of K i e r s t e a d and Slobodan's (1953) " l e a k a g e " . G e n e r a l l y , the e x i t r a t e E i s time-dependent, but as t becomes l a r g e , P and P become dominated by the l o w e s t terms i n 1 3 5 t h e i r s e r i e s . Hence, f o r st e a d y s t a t e v a l u e s , m = n = 1 i n 6.2.4, 6.3.1, and 6.3.2, and s u b s t i t u t i n g f o r P and P i n 6.3.3, E K i e r s t e a d and S l o b o d k i n (1953) e s t i m a t e t h a t E * n 2 D 2/d 2 when t i s l a r g e . (6.3.4) T h i s r e l a t i o n s h i p shows c l e a r l y t h a t the per u n i t e x i t r a t e , E, depends on the f i e l d d i m e n s i o n s , d. U s i n g t h i s a p p r o x i m a t i o n i n 6.3.3, a t the s t e a d y s t a t e ( l / P ) ( d P / d t ) = R - I I 2 D 2/d 2. (6.3.5) Hence, the s m a l l e r the s i z e of the f i e l d i n terms of d i f f u s i o n , d 2 , the sl o w e r the 'per c a p i t a ' r a t e of pathogen biomass i n c r e a s e w i t h i n an i s o l a t e d f i e l d . T h i s r e l a t i o n s h i p runs c o u n t e r t o the c o n c l u s i o n s of Van der Pl a n k (1960). He c l a i m e d t h a t " p a r a s i t e d i s p e r s a l between f i e l d s would be reduced by p l a n t i n g a few l a r g e f i e l d s which were f a r a p a r t when a c e r t a i n p r o p o r t i o n of a r e g i o n i s a l l o c a t e d t o a c r o p . However, taken t o the extreme, 6.3.5 su g g e s t s t h a t i f f i e l d s a r e s m a l l enough ( i . e . d < d ), d i s e a s e may be unable t o p e r s i s t w i t h i n s m a l l c f i e l d s . 136 4. G e n e r a l C r i t i c a l D imensions: No S u r v i v a l O u t s i d e M c M u r t r i e (1978) has g e n e r a l i z e d e q u a t i o n 6.2.6 f o r the c r i t i c a l v a l u e of d, the s i z e of the f i e l d i n terms of d i f f u s i o n , t o d = k D//R", (6.4.1) c where k i s a c o n s t a n t of o r d e r u n i t y . The v a l u e of k, l i k e the e x p r e s s i o n f o r d, depends on the shape of the f i e l d . For i n s t a n c e , S k e l l a m (1951) and K i e r s t e a d and S l o b o d k i n (1953) have shown t h a t i f 6.1.8 d e s c r i b e s the growth of a p o p u l a t i o n i n a s t r i p of i n f i n i t e l e n g t h i n one d i m e n s i o n and of w i d t h W i n the o t h e r , then d = W and k = n . (6.4.2) T h i s r e s u l t f o l l o w s d i r e c t l y from 6.2.5 by l e t t i n g b -> °° and p u t t i n g a = W. A number of workers (e.g. L a n d a h l , 1959; S k e l l a m , 1951) have shown t h a t 6.4.1, 6.4.2 a l s o a p p l y t o p o p u l a t i o n s undergoing l o g i s t i c growth i n an i n f i n i t e s t r i p of w i d t h W i n an o t h e r w i s e c o m p l e t e l y u n f a v o r a b l e environment. For l o g i s t i c growth 6.1.7 t a k e s the form 137 F (N) = RN(1 - N/N ) (6.4.3) G max where N i s the maximum p o p u l a t i o n d e n s i t y the h a b i t a t can max s u s t a i n . As b e f o r e , the p o p u l a t i o n c o l l a p s e s when W < W , the _ c c r i t i c a l w i d t h , but when W > W p o p u l a t i o n growth i s now l i m i t e d t o an e q u i l i b r i u m d e n s i t y N* where 0 < N* < N . T h i s max e q u i l i b r i u m d e n s i t y , N*, i n c r e a s e s w i t h W, the w i d t h of the s t r i p . B r a d f o r d and P h i l i p (1970) f u r t h e r extended t h e s e r e s u l t s . They showed t h a t c r i t e r i a s i m i l a r t o 6.4.1, 6.4.2 a r i s e f o r any form of l o c a l p o p u l a t i o n growth, F (N), which i s a p o s i t i v e G f u n c t i o n f o r 0 < N < N w i t h F (0) = F (N ) = 0. max G G max S k e l l a m (1951) and K i e r s t e a d and S l o b o d k i n (1953) have a l s o shown t h a t i f 6.1.8 d e s c r i b e s the growth of a p o p u l a t i o n i n a c i r c u l a r a r e a of r a d i u s a i n an o t h e r w i s e t o t a l l y i n h o s p i t a b l e e n v i r o n m e n t , then k = 2.4048 and d = a. (6.4.4) 5. C r i t i c a l Dimensions f o r I s o l a t e d C i r c u l a r A r e a s : S u r v i v a l  O u t s i d e Having g e n e r a l i z e d 6.4.1 i n terms of f i e l d geometry and p o p u l a t i o n growth, I now add r e a l i s m by a l l o w i n g pathogen 138 s u r v i v a l o u t s i d e the f i e l d . E q u a t i o n 6.2.1 i m p l i e s t h a t inoculum s u f f e r s i n s t a n t a n e o u s death as soon as i t reaches a f i e l d boundary. More r e a l i s t i c a l l y , s u r v i v a l i s d i f f i c u l t , not i m p o s s i b l e , o u t s i d e the f i e l d . To . i n v e s t i g a t e how the h o s t i l i t y of the e x t e r i o r environment a f f e c t s the c r i t i c a l d i m e n s i o n s , c o n s i d e r d i s p e r s a l i n two d i m e n s i o n s w i t h r a d i a l symmetry and assume t h a t a p a t c h of s u s c e p t i b l e h o s t s i s i s o l a t e d i n an o t h e r w i s e i n h o s p i t a b l e environment. S i n c e R, as d e f i n e d i n 6.1.7, r e p r e s e n t s the s u r v i v a l and r e p r o d u c t i v e a b i l i t i e s of the p a r a s i t e , i t seems r e a s o n a b l e (Dimond and H o r s f a l l , 1965; Yarwood and S y l v e s t e r , 1959) t o s e t R = R i n s i d e the p a t c h : r < a, (6.5.1) i and R = - R o u t s i d e the p a t c h : r > a. o Both R and R are p o s i t i v e , i o T r a n s f o r m i n g 6.1.8 i n t o p o l a r c o o r d i n a t e s ( P i p e s , 1958, p. 504), the r a t e of change i n d i s e a s e s e v e r i t y a t time t at a d i s t a n c e r from the c e n t r e of the p a t c h i s 3N/3t = RN + D 2 [ ( 3 2 N / 3 r 2 ) + ( 3 N / 3 r ) / r ] . 139 The stea d y s t a t e s o l u t i o n s of t h i s e q u a t i o n determine how the s e v e r i t y of the e x t e r n a l environment, which i n c r e a s e s w i t h R , o a f f e c t s the c r i t i c a l r a d i u s of the p a t c h . P u t t i n g 3N/3t = 0 and r e a r r a n g i n g the r e s u l t i n g steady s t a t e e q u a t i o n s i n t o the forms of B e s s e l ' s e q u a t i o n of o r d e r z e r o (R > 0 ) and B e s s e l ' s m o d i f i e d e q u a t i o n of o r d e r z e r o (R < 0 ) , r e s p e c t i v e l y ( P i p e s , 1958, pp. 346, 355): d 2N/dQ 2 + (dN/dQ)/Q + NR / jR | = 0 , where (6.5.2) Q = r /[RT/D. I f pathogen d e n s i t i e s i n s i d e and o u t s i d e the p a t c h a re N ( r ) and N ( r ) , r e s p e c t i v e l y , then bounded s o l u t i o n s a t the i o c e n t r e of the p a t c h r e q u i r e t h a t dN / d r = 0 a t r = 0, (6.5.3) i At the edge of the p a t c h , where r = a, I make the b i o l o g i c a l l y r e a s o n a b l e assumptions ( S k e l l a m , 1951) of c o n t i n u i t y i n the d e n s i t y and f l u x , r e s p e c t i v e l y , of pathogen biomass: N (a) = N (a) i o 140 and (6.5.4) dN / d r = dN /dr at r = a i o Without c o n t i n u i t y i n d e n s i t y , an i n f i n i t e l y l a r g e f l u x must e x i s t a t the p a t c h p e r i m e t e r ; and w i t h o u t c o n t i n u i t y i n f l u x , an i n f i n i t e l y l a r g e change i n d e n s i t y must e x i s t i n an i n f i n i t e s i m a l l y s m a l l neighborhood of the p e r i m e t e r . For r > a, the s o l u t i o n N ( r ) of 6.5.2 which i s bounded and o n o n n e g a t i v e f o r l a r g e r (c_f. P i p e s , 1958, pp. 355-356) i s N ( r ) = BK ( r /| R |/D) o o o where B > 0 i s c o n s t a n t and K i s the m o d i f i e d B e s s e l f u n c t i o n o of the second k i n d of o r d e r z e r o . For r < a, the s o l u t i o n of N ( r ) s a t i s f y i n g 6.5.3 ( P i p e s , i 1958, p. 505) i s N ( r ) = AJ ( r / T T 7 D ) i o i where A > 0 i s c o n s t a n t and J i s the B e s s e l f u n c t i o n of the o f i r s t k i n d of o r d e r z e r o . M a t c h i n g the s o l u t i o n s N ( r ) and N ( r ) g i v e n above a t the o i 141 f i e l d boundary r = a w i t h c o n d i t i o n s 6.5.4, i t f o l l o w s from the B e s s e l f u n c t i o n i d e n t i t i e s ( e . g . W y l i e , 1975, p. 410) t h a t n o n - t r i v i a l s o l u t i o n s f o r N ( r ) e x i s t o n l y i f J (a/R /D)K (a/R~7D)- /R /R J (a/W/D)K (a/R~7D)=0. (6.5.5) 1 i o o o i o i l o i i Here J and K I a r e the f i r s t o r d e r c o u n t e r p a r t s t o J and K , 1 1 o o r e s p e c t i v e l y . F i g u r e 6.2 i s a t y p i c a l p l o t of t h i s r e l a t i o n s h i p . To d e t e r m i n e a r e p r e s e n t a t i v e c r i t i c a l a r e a f o r c e r e a l -1/2 r u s t s , I e s t i m a t e d D = 1 m day from Cammack's (1958) d a t a -1 f o r P u c c i n i a p o l y s o r a and I e s t i m a t e d R = 0.01 day f o r • o P u c c i n i a c o r o n a t a from Yarwood and S y l v e s t e r (1958). In F i g u r e 6.2, the c r i t i c a l r a d i u s c o r r e s p o n d i n g t o R = 0.01 i s o a p p r o x i m a t e l y a = 0 . 5 m so the p r e d i c t e d c r i t i c a l a r ea i s c a p p r o x i m a t e l y n a 2 = 0.8m2. However, a c c o r d i n g to the c o n c e r n s c about 6.1.6 d i s c u s s e d above, i t i s not c l e a r t h a t the c r i t i c a l a r e a concept a p p l i e s a t a l l a t such a s m a l l s p a t i a l s c a l e of o b s e r v a t i o n . N o n e t h e l e s s , c r i t i c a l a r e a e f f e c t s may be commonly r e a l i z e d but r a r e l y r e c o g n i z e d f e a t u r e s of c e r e a l r u s t systems: the p r e d i c t e d c r i t i c a l a r e a depends on R , R , and D, and hence o i i t w i l l change d u r i n g the growing season, i t w i l l depend on the host v a r i e t y and r u s t race i n v o l v e d , and i t w i l l depend on the 142 c h a r a c t e r i s t i c s of the a e r o b i o l o g i c a l zone of c o n c e r n . 6. C r i t i c a l Dimensions f o r Exponent i a l and L o q i s t i c Growth i n  I s o l a t e d I n f i n i t e S t r i p s : S u r v i v a l O u t s i d e The o n e - d i m e n s i o n a l analogue of the p r e v i o u s problem c o n c e r n s a s t r i p of i n f i n i t e l e n g t h i n the y - d i r e c t i o n and of w i d t h W i n the x - d i r e c t i o n . R = R i n s i d e the s t r i p : 1x1 < W/2, and l (6.6.1) o u t s i d e the s t r i p : I x j > W/2, o where R and R are p o s i t i v e , i o At Steady State i n o n e - d i m e n s i o n 6.1.8 becomes d 2N/dx 2 + RN/D2 = 0. (6.6.2) G i v e n the boundary c o n d i t i o n s dN = 0 a t x = 0, dx N (W/2) = N (W/2) (6.6.3) i o and dN /dx = dN /dx i o a t x = W/2 143 where N and N are r e s p e c t i v e l y the s o l u t i o n s f o r | x | < W/2 and i o |x| > W/2; and g i v e n the r e q u i r e m e n t t h a t N (x) be n o n - n e g a t i v e o and bounded f o r x > W/2, Ludwig e t a_l. (1979) showed t h a t the c r i t i c a l w i d t h of the s t r i p i s W = (2D/ /~TT) a r c t a n / R~7R . (6.6.4) c i o i F i g u r e 6.3 i s a t y p i c a l graph of the c r i t i c a l w i d t h , W , c a g a i n s t R , the h o s t i l i t y of the e x t e r i o r environment, o R e p r e s e n t a t i v e c r i t i c a l s t r i p w i d t h s f o r c e r e a l r u s t s (R = 0.01 o d a y - 1 ) a r e a p p r o x i m a t e l y o. 2 m. I Ludwig e_t a_l. (1979) have extended t h i s r e s u l t t o l o g i s t i c growth, 6.4.3, i n s i d e the s t r i p . However, as i n the s i t u a t i o n w i t h i n s t a n t a n e o u s d e a t h at the boundary, p a r a s i t e growth i s l i m i t e d t o a maximum d e n s i t y , 0 < N* < N , when the s t r i p max w i d t h exceeds the c r i t i c a l w i d t h . T h i s d e n s i t y i s p l o t t e d i n F i g u r e 6.4 a g a i n s t the s t r i p w i d t h f o r v a r i o u s v a l u e s of R when o D = N = R = 1. The p o i n t s of i n t e r s e c t i o n w i t h the W a x i s max i a r e the c r i t i c a l w i d t h s d e f i n e d i n 6.6.4. F i g u r e 6.4 adds more support t o Waggoner's (1962) c o n c l u s i o n s : i t s u g g e s t s t h a t even when the f i e l d d i m e n s i o n s exceed the c r i t i c a l d i m e n s i o n s , s m a l l e r f i e l d s w i l l have lower pathogen d e n s i t i e s . 144 1_. D i s p e r s a l Between F i e l d s The c oncept of c r i t i c a l d i m e n s i o n s has been d i s c u s s e d f o r a v a r i e t y of d i f f e r e n t g e o m e t r i e s ; f o r l o g i s t i c , e x p o n e n t i a l , and o t h e r forms of p o p u l a t i o n growth; and f o r a wide range of s u r v i v a l r a t e s o u t s i d e an i s o l a t e d f i e l d . The broad a p p l i c a b i l i t y of the c r i t i c a l d i m e n s i o n s concept s u g g e s t s t h a t g e n e r a l l y p a r a s i t e p o p u l a t i o n s may be slow t o i n c r e a s e on s m a l l f i e l d s , and i f t h e r e i s l i t t l e d i s e a s e i n the f i e l d s t h e r e can be l i t t l e d i s p e r s a l between them. I now a d d r e s s t h i s h y p o t h e s i s e x p l i c i t l y . I c o n s i d e r an i n f i n i t e number of i d e n t i c a l s t r i p s of w i d t h W r u n n i n g p a r a l l e l t o each o t h e r and each e x t e n d i n g f o r an i n f i n i t e . d i s t a n c e i n the. y - d i r e c t i o n . Due t o the i n h e r e n t symmetry of the problem o n l y the r e g i o n 0 < x < b, where H = 2b - W i s the d i s t a n c e between s t r i p s and a s t r i p i s c e n t e r e d at the o r i g i n , ( F i g . 6.5) need be c o n s i d e r e d . My o b j e c t i v e i s t o s o l v e the system d 2N/dx 2 + RN/D2 = 0, (6.7.1) where R = R when 0 < x < W/2 l and R = -R when W/2 < x < b, o 145 s u b j e c t t o the boundary c o n d i t i o n s 6.6.3 and the e x t r a c o n d i t i o n , a r i s i n g from c o n t i n u i t y and symmetry about x = b, dN / d t = 0 a t x = b. (6.7.2) o When 0 < x < W/2, the s o l u t i o n of 6.7.1 s a t i s f y i n g the symmetry c o n d i t i o n 6.6.3 a t x = 0 i s N (x) = A cos (x /R -7D) i i where A > 0 i s c o n s t a n t (c_f. P i p e s , 1958, p. 501). S i m i l a r l y when W/2 < x < b, the g e n e r a l s o l u t i o n i s N (x) = B co s h (x /TT-/D) + C s i n h ( X /R~/D) o o o where B and C a r e c o n s t a n t s ( W y l i e , 1975, p. 375). U s i n g 6.7.2 and the boundary c o n d i t i o n s 6.6.3 a t x = W/2, i t can be shown, a f t e r some c a l c u l a t i o n , t h a t the c r i t i c a l s t r i p w i d t h i s W = (2D/ / R ) a r c t a n [ / R /R tanh (H v/R~/2D) ] , (6.7.3) c c o i o where H = 2b - W. 146 In F i g u r e 6.6 the c r i t i c a l s t r i p w i d t h , W , i s p l o t t e d c a g a i n s t the square r o o t of the decay r a t e o u t s i d e , R , f o r o d i f f e r e n t v a l u e s of H, the d i s t a n c e between p a r a l l e l s t r i p s . For a p a r t i c u l a r s e p a r a t i o n d i s t a n c e (H), those c o m b i n a t i o n s of s t r i p w i d t h (W) and h o s t i l i t y of the e x t e r i o r environment ( /R ) o r e p r e s e n t e d by p o i n t s above the c r i t i c a l c u r v e a l l o w d i s e a s e development. In c o n t r a s t , pathogen p o p u l a t i o n s d e c l i n e f o r c o m b i n a t i o n s below the c u r v e . Hence, r e g a r d l e s s of the i n i t i a l d i s t r i b u t i o n of inoculum i n a r e g i o n w i t h a c e r t a i n c r o p a c r e a g e , i f the f i e l d s a r e narrow enough, the pathogen p o p u l a t i o n cannot s u s t a i n i t s e l f . R e p r e s e n t a t i v e c r i t i c a l s t r i p w i d t h s f o r t h e c e r e a l r u s t s a r e so s m a l l ( i . e . f o r R = 0.01, o w= o.05m) t h a t the a p p l i c a b i l i t y of the concept i s q u e s t i o n a b l e . F i g u r e 6.6 shows t h a t i n c r e a s i n g the s t r i p w i d t h , W, or r e d u c i n g the h o s t i l i t y of the e x t e r i o r environment, R , f o r a o g i v e n s t r i p s e p a r a t i o n , H, c o u l d t u r n v e r y s p a r s e p o p u l a t i o n s i n t o e p i d e m i c s . I n c r e a s e s i n W appear c o n c o m i t a n t l y w i t h r e c e n t t r e n d s towards l a r g e s c a l e mechanized a g r i c u l t u r e ( A p p l e , 1978). Moreover, m o d e r n i z a t i o n of c u l t u r a l p r a c t i c e s f r e q u e n t l y d e c r e a s e s R , the h o s t i l i t y of the environment o u t s i d e the o f i e l d . For i n s t a n c e , Rotem and P a l t i (1969) r e p o r t t h a t the i n t r o d u c t i o n of i r r i g a t i o n f o r non-host c r o p s nearby can enhance pathogen s u r v i v a l by i t s a f f e c t on the m i c r o c l i m a t e . F i g u r e 6.7 c l a r i f i e s the r e l a t i o n s h i p between the c r i t i c a l s t r i p w i d t h , W , and the f r a c t i o n of the a g r i c u l t u r a l r e g i o n c 147 sown t o s u s c e p t i b l e h o s t s , W /(H+W ), f o r v a r i o u s v a l u e s of R c c o i n 6.7.3. F i g u r e 6.7 s u p p o r t s the common b e l i e f (e.g.. Johnson, 1953) t h a t i n c r e a s i n g the r e l a t i v e a r e a p l a n t e d t o a p a r t i c u l a r c r o p , W/(H+W), can i n c r e a s e d i s e a s e s e v e r i t y . F i g u r e 6.7 a l s o shows t h a t i n c r e a s i n g f i e l d s i z e s from W < W t o W > W , w h i l e c c m a i n t a i n i n g the same f r a c t i o n of t o t a l a r e a a l l o c a t e d t o the c r o p , c o u l d change a s i t u a t i o n of p o p u l a t i o n d e c l i n e i n t o one of p o p u l a t i o n growth. T h e r e f o r e , i n c o n t r a s t t o Van der P l a n k (1948, 1949, 1960), i n c r e a s i n g the a r e a , and hence, i s o l a t i o n of the f i e l d s i n t o which a r e g i o n ' s c r o p acreage i s s u b d i v i d e d , can i n c r e a s e the danger of epidemic development. So f a r I have c o n s i d e r e d o n l y s t e a d y s t a t e s o l u t i o n s of 6.1.8 f o r boundary c o n d i t i o n s which r e p r e s e n t d i s p e r s a l between f i e l d s . However, t r a n s i e n t s o l u t i o n s a r e o f t e n more r e l e v a n t t o p l a n t d i s e a s e management. For i n s t a n c e , s u c c e s s i n c e r e a l r u s t c o n t r o l i s f r e q u e n t l y measured i n terms of how l o n g epidemic development can be d e l a y e d ( e . g . F l e m i n g and P e r s o n , 1978; K r a n z , 1978). Time-dependent s o l u t i o n s to the system 6.6.3, 6.7.1, and 6.7.2 were o b t a i n e d by n u m e r i c a l i n t e g r a t i o n . Moore Lee (1978) d e s c r i b e s the n u m e r i c a l i n t e g r a t i o n program i n d e t a i l . In F i g u r e s 6.8-6.11 the w i d t h of the s t r i p i s 200m, the l e n g t h of the s i d e of a square 4 ha f i e l d , a r e p r e s e n t a t i v e s i z e f o r European c o n d i t i o n s ( P o t t s and V ickerman, 1974). As -1 -1/2 e s t i m a t e d b e f o r e , R =0.01 day and D = 1 m day , and I s e t o -1 R =0.05 day . T h i s v a l u e of R appears t o be r e a s o n a b l e f o r a o o 148 v a r i e t y w i t h good q u a n t i t a t i v e r e s i s t a n c e ( P a r l e v l i e t and Ommeren, T975) . F i g u r e 6.8 i l l u s t r a t e s the change i n r e l a t i v e pathogen d e n s i t y over space a f t e r f i v e c o n s e c u t i v e twenty-day i n t e r v a l s . I n i t i a l l y , pathogen d e n s i t i e s are u n i f o r m i n the f i e l d and z e r o o u t s i d e . C o n f i d e n c e i n models i s o f t e n e s t a b l i s h e d by n o t i n g q u a l i t a t i v e s i m i l a r i t y between the o v e r a l l p a t t e r n s of model p r e d i c t i o n s and the a p p r o p r i a t e f i e l d d a t a ( H o l l i n g , ed., 1978). In t h i s c o n t e x t I note t h a t i n f i e l d t e s t s of c e r e a l r u s t d i s p e r s a l , R o e l f s (1972) o b t a i n e d f l a t - t o p p e d g r a d i e n t s s i m i l a r t o t hose of F i g u r e 6.8. I n i t i a l l y u n i f o r m and e q u a l pathogen d e n s i t i e s i n s i d e and o u t s i d e the f i e l d s were used t o d e s c r i b e the e f f e c t of s i m u l t a n e o u s spore showers o r i g i n a t i n g from o u t s i d e .the a g r i c u l t u r a l r e g i o n a t the b e g i n n i n g of the d i s e a s e seasons. The 100-day p r o f i l e s i n F i g u r e 6.8 were e s s e n t i a l l y unchanged when i n i t i a l pathogen d e n s i t i e s were z e r o o u t s i d e the f i e l d . These l a s t i n i t i a l c o n d i t i o n s r e p r e s e n t the o v e r w i n t e r i n g of p a r a s i t e s i n the f i e l d s . F i g u r e 6.7 s u g g e s t s t h a t i n c r e a s i n g s t r i p w i d t h s t h r o u g h the c r i t i c a l s i z e c o u l d change d e c l i n i n g pathogen p o p u l a t i o n s i n t o growing ones. F i g u r e 6.9 extends these c o n c l u s i o n s . I t shows t h a t e n l a r g i n g f i e l d s f u r t h e r beyond the c r i t i c a l s i z e l e a d s t o even g r e a t e r growth r a t e s f o r p o p u l a t i o n s of d i s e a s e o r g a n i s m s . F i g u r e 6.10 g i v e s the r e l a t i v e pathogen biomass a t h a r v e s t as a f u n c t i o n of f i e l d s i z e when a t e n t h of the r e g i o n has been p l a n t e d t o the c r o p . We assume t h a t the c e r e a l has a growing 149 season of 100 days as i s common f o r o a t s (Frey et a l . , 1977) and wheat ( C h e s t e r , 1943; P e t e r s o n , 1965) and note t h a t the c e r e a l i s s u s c e p t i b l e t o a t t a c k t h r o u g h o u t t h i s p e r i o d ( P e t e r s o n , 1965) . T o g e t h e r , F i g u r e 6.7, 6.9, and 6.10 suggest t h a t when a l l f i e l d s a r e the same s i z e , w i t h shape and o r i e n t a t i o n c o n s t a n t , y i e l d l o s s e s t o d i s e a s e d e c r e a s e as a r e g i o n ' s t o t a l c r o p acreage i s p r o g r e s s i v e l y s u b d i v i d e d i n t o an i n c r e a s i n g number of s m a l l e r f i e l d s . Waggoner (1962) and Kampmeijer and Zadoks (1977) a l s o i n d i c a t e d o v e r a l l advantages t o h a v i n g many s m a l l f i e l d s i n terms of epidemic c o n t r o l . In c o n t r a s t , Van der Plank (1948, 1949, 1960, 1963) and d u r i n g the 1971 e p i d e m i o l o g y c o n f e r e n c e i n Wageningen (Zadoks and Kampmeijer, 1977) has s t e a d f a s t l y recommended a few l a r g e f i e l d s . F i g u r e 6.11 shows t h a t r e d u c i n g r e l a t i v e s t r i p s i z e i s g e n e r a l l y more e f f e c t i v e a t i n h i b i t i n g pathogen i n c r e a s e f o r s m a l l e r s t r i p s than f o r l a r g e r s t r i p s . For example, d e c r e a s i n g the s t r i p w i d t h from 200 m t o 100 m d e c r e a s e s pathogen biomass a t h a r v e s t by o n l y 8% whereas a r e d u c t i o n from 20 m t o 10m l e a d s ' t o an 82% d e c r e a s e . T h e r e f o r e , i t i s i m p o r t a n t to d e t e r m i n e the w i d t h of an i n f i n i t e s t r i p , W, c o r r e s p o n d i n g t o a f i e l d . In F i g u r e 6.8 the w i d t h i s 200 m, the s i d e of a square 4 ha f i e l d . When an i n f i n i t e d e a t h r a t e e x i s t s o u t s i d e , 6.2.5. g i v e s a measure of the s i z e of such a 200 m x 200 m f i e l d i n terms of d i f f u s i o n : d = 141 m. The v a l u e of W which p r o v i d e s the same v a l u e of d i n 6.2.5 f o r an i n f i n i t e s t r i p i s 141 m. N o n e t h e l e s s , F i g u r e 6.11 shows t h a t 'even a t t h i s lower v a l u e of W, h a l v i n g the s t r i p 150 w i d t h reduces c e r e a l r u s t biomass a t h a r v e s t by j u s t 14%. Under the s e c i r c u m s t a n c e s , the economics of the s i t u a t i o n w i l l d i c t a t e the v a l u e of c h a n g i n g f i e l d s i z e s . However, F i g u r e 6.1 suggests t h a t f i e l d shape may a l s o be an i m p o r t a n t f a c t o r i n d i s e a s e p r o g r e s s . The e f f e c t of f i e l d shape can be measured i n 6.2.5 by d 2 , the s i z e of a r e c t a n g u l a r f i e l d i n terms of d i f f u s i o n when pathogens cannot s u r v i v e o u t s i d e . For example, i f a 4 ha f i e l d i s square then d 2 = 20000 m2, but i f i t s d i m e n s i o n s are 20 m x 2000 m then d 2 i s o n l y 400 m2. S i m i l a r l y , d 2 = W2 f o r an i n f i n i t e s t r i p of w i d t h W. Thus, i n f i n i t e s t r i p s of w i d t h s W = / 20000 = 141 m and W = /400m2 i 20 m, j r e s p e c t i v e l y , are of the same s i z e i n terms of d i f f u s i o n as 200 m x 200 m and 20 m x 2000 m f i e l d s w i t h no s u r v i v a l o u t s i d e . T h e r e f o r e , assuming 6.2.5 h o l d s a p p r o x i m a t e l y when R =0.01 o per day, F i g u r e 6.10 i n d i c a t e s t h a t the 20 m x 2000 m f i e l d (W = 20 m) w i l l have 70% l e s s c e r e a l r u s t biomass at h a r v e s t than a square f i e l d of the same 4 ha a r e a (W = 141 m) when R =o.05 per o day. ' G e n e r a l l y , i f 6.2.5 h o l d s r e a s o n a b l y w e l l f o r r e a l i s t i c decay r a t e s o u t s i d e the f i e l d , then the s i z e of the f i e l d i n terms of d i f f u s i o n , d 2 , i s more s e n s i t i v e t o f i e l d shape than t o f i e l d a r e a and e l o n g a t i n g f i e l d s of a g i v e n area may s u b s t a n t i a l l y reduce d i s e a s e ( F i g u r e 6.12). 151 D i s c u s s i o n Van der Plank (1948, 1949, 1960) and Waggoner (1962) s t u d i e d the s i g n i f i c a n c e of f i e l d geometry on the spread of p l a n t d i s e a s e w i t h models which i g n o r e the e f f e c t s of time dependence i n p o p u l a t i o n growth and d i s p e r s a l . G e n e r a l l y , however d i s p e r s a l and p o p u l a t i o n growth a re time-dependent p r o c e s s e s which i n t e r a c t s i m u l t a n e o u s l y t o produce changes i n p o p u l a t i o n d e n s i t i e s over space and t i m e . T h i s has been emphasized by much r e c e n t work i n p o p u l a t i o n dynamics on a range of i n s e c t ( e.g. C l a r k , 1979; Jo n e s , 1977; Myers and Campbell, 1976; W e l l i n g t o n , 1964), l a r g e mammal (e. g . S i n c l a i r and N o r t o n -G r i f f i t h s , eds., .1979), marine ( e . g . S t e e l e , ed., 1978), m i c r o t i n e ( e . g . Krebs and Myers, 1974), p l a n t pathogen (e.g. Kiyosawa, 1976; Shrum, 1975), and t h e o r e t i c a l (e.g. L e v i n , 1978) systems. D i f f u s i o n e q u a t i o n s have proved to be p a r t i c u l a r l y s u c c e s s f u l i n d e c r i b i n g the f o r m a t i o n of p a t c h e s of p h y t o p l a n k t c n . U s i n g e m p i r i c a l l y e s t i m a t e d parameter v a l u e s S t e e l e (1976) demonstrated t h a t the r e s u l t 6.4.1, 6.4.4 i s su p p o r t e d by the g e n e r a l o b s e r v a t i o n t h a t p l a n k t o n p a t c h e s i n the open sea appear a t s c a l e s of the o r d e r of 10-100 km. Moreover, P i a t t and Denman (1975) and Wroblewski e_t a l . ( 1 975) show t h a t m o d i f i c a t i o n s t o t h i s model t o account f o r z o o p l a n k t o n g r a z i n g produce r e l a t i v e l y s m a l l a d j u s t m e n t s t o the c r i t i c a l d i m e n s i o n s . Hence, the r e s u l t 6.4.1, 6.4.4 i s g e n e r a l l y regarded as a t h e o r e t i c a l l y and e m p i r i c a l l y r o b u s t d e s c r i p t i o n of p l a n k t o n p a t c h i n e s s ( L e v i n , 1978). The d i f f u s i o n models i n d i c a t e t h a t the s i z e and shape of 152 f i e l d s can be i m p o r t a n t d e t e r m i n a n t s of epidemic development. Because the net e f f l u x of d i f f u s i n g inoculum from a f i e l d o c c u r s a c r o s s the s i d e s of the f i e l d , the r a t e i s a p p r o x i m a t e l y p r o p o r t i o n a l to the p e r i m e t e r . However, s i n c e p a r a s i t e s are c r e a t e d a t a l l p o i n t s w i t h i n a f i e l d , the p r o d u c t i o n r a t e i s p r o p o r t i o n a l t o the a r e a . Hence the p e r i m e t e r t o a r e a r a t i o of a f i e l d i s r o u g h l y p r o p o r t i o n a l t o the r a t i o of d i f f u s i o n l o s s e s t o pathogen p r o d u c t i o n i n t h a t f i e l d . S i n c e the p e r i m e t e r t o a r e a r a t i o i n c r e a s e s as a r e a d e c r e a s e s f o r f i e l d s of a g i v e n shape, s m a l l e r f i e l d s may h i n d e r d i s e a s e development. C l a r k et_ a_l. (1978) have suggested t h a t the same mechanism o p e r a t e s d u r i n g o u t b r e a k s of s p r u c e budworm (Chor i s t o n e u r a  fumi f e r a n a Clem.) i n the b o r e a l f o r e s t s of e a s t e r n N o r t h A m e r i c a . Here, s m a l l i s o l a t e d s t a n d s of even the most s u s c e p t i b l e f o r e s t a r e u n l i k e l y t o e x p e r i e n c e o u t b r e a k s of these d e f o l i a t i n g i n s e c t s (Van R a a l t e , 1972). D i s p e r s a l l o s s e s a c r o s s the s t a n d p e r i m e t e r s have been i m p l i c a t e d ever s i n c e M o r r i s and Mott (1963) f i r s t made the s u g g e s t i o n . E x t e n d i n g t h i s l i n e of r e a s o n i n g t o an a g r i c u l t u r a l r e g i o n as a whole, the r a t e of d i s p e r s a l l o s s i s r o u g h l y p r o p o r t i o n a l t o the sum of f i e l d p e r i m e t e r s w h i l e the p r o d u c t i o n r a t e i s p r o p o r t i o n a l t o the f r a c t i o n of the r e g i o n growing s u i t a b l e h o s t s . Hence, f o r a g i v e n p r o d u c t i o n r a t e , d i s e a s e s p r e a d can be reduced by i n c r e a s i n g the sum of f i e l d p e r i m e t e r s . There a r e two extreme approaches by which i n c r e a s e s i n the sum of f i e l d p e r i m e t e r s can be a c h i e v e d : f i r s t , by changing the shape of f i e l d s such t h a t the l e n g t h t o ' w i d t h r a t i o i n c r e a s e s w h i l e t h e i r s i z e remains the same, and second, by i n c r e a s i n g the number of 153 f i e l d s t h r o u g h a r e d u c t i o n i n t h e i r s i z e w h i l e t h e i r shape i s f i x e d . T h i s second method s u p p o r t s the recommendations of Waggoner (1962) and Zadoks and Kampmeijer (1977) f o r s u b d i v i d i n g a r e g i o n ' s c r o p l a n d i n t o many s m a l l f i e l d s . I t argues a g a i n s t Van der P l a n k (1948, 1949, 1960, 1963) who c l a i m e d t h a t few l a r g e f i e l d s would p r o v i d e s u p e r i o r d i s e a s e c o n t r o l . B o r l a u g (1959) and Browning and F r e y (1969) were among the f i r s t t o implement m u l t i l i n e s , c u l t i v a r s composed of s e v e r a l c o n s t i t u e n t l i n e s each c a r r y i n g d i f f e r e n t r e s i s t a n c e genes, f o r d i s e a s e c o n t r o l . In some ways each p l a n t i n a f i e l d sown t o a m u l t i l i n e c u l t i v a r can be thought of as r e p r e s e n t i n g the l i m i t t o d e c r e a s i n g f i e l d s i z e s . A l t h o u g h our models are no l o n g e r s t r i c t l y a p p l i c a b l e when the t y p i c a l d i s t a n c e t r a v e l l e d by a pathogen i s l a r g e compared w i t h " f i e l d s i z e " , the same g e n e r a l p r i n c i p l e s may a p p l y . In f a c t , many workers (e.g. Burdon, 1978; Frey e_t a_l. , 1977; T r e n b a t h , 1977) have suggested t h a t d i s p e r s a l l o s s e s a r e s i g n i f i c a n t f o r s i m p l e p a t h o g e n i c r a c e s a t t a c k i n g m u l t i l i n e s . The computer s i m u l a t i o n s of Kiyosawa (1976) and Kampemeijer and Zadoks (1977) p r o v i d e s u p p o r t f o r t h i s i d e a . Waggoner (1962) has emphasized the importance of o r i e n t i n g f i e l d s w i t h r e s p e c t t o the p r e v a i l i n g wind d i r e c t i o n . For example, c o n s i d e r a f i e l d e l o n g a t e d p e r p e n d i c u l a r l y t o wind d i r e c t i o n . Assuming d i s p e r s a l i s random, our model p r e d i c t s t h a t inoculum i s s y m m e t r i c a l l y d i s t r i b u t e d a c r o s s the w i d t h of the f i e l d w i t h r e s p e c t t o the mi d - w i d t h l i n e ( d i s t a n c e = 0 m i n F i g u r e 6.8). In s i m p l e terms, the e f f e c t of wind can be imagined as a s h i f t i n the c e n t e r of t h i s d i s t r i b u t i o n t o a p o i n t downwind of the mid-width l i n e , t he magnitude of t h i s s h i f t 154 b e i n g r o u g h l y p r o p o r t i o n a l t o the v e c t o r wind v e l o c i t y . A l t h o u g h l e s s i n oculum would be l o s t a t the upwind edge of the f i e l d , the l o s s would be g r e a t e r a t the downwind edge, and o v e r a l l , the net l o s s from the f i e l d would be g r e a t e r than i f inoculum were randomly d i s s e m i n a t e d . Hence, our a n a l y s i s u n d e r e s t i m a t e s the p o t e n t i a l i n h i b i t i o n of e p idemic development when d i s p e r s a l i s d i r e c t e d . Moreover, by the same r e a s o n i n g , t h i s u n d e r e s t i m a t i o n i s g r e a t e s t f o r f i e l d s o r i e n t e d p e r p e n d i c u l a r l y t o the p r e v a i l i n g wind d i r e c t i o n and l e a s t f o r f i e l d s w i t h an o r i e n t a t i o n p a r a l l e l t o wind d i r e c t i o n . Hence e l o n g a t i n g square f i e l d s i s l i k e l y t o be much more r e w a r d i n g i n terms of r u s t c o n t r o l than d e c r e a s i n g f i e l d s i z e when a s p e c i f i c f r a c t i o n of the r e g i o n i s devoted t o s u s c e p t i b l e c e r e a l s . Of c o u r s e , economic c o n s i d e r a t i o n s w i l l be i m p o r t a n t d e t e r m i n a n t s of the r e l a t i v e b i a s toward e i t h e r approach i n any mixed s t r a t e g y to changing f i e l d geometry. An i m p o r t a n t q u e s t i o n i s whether the g a i n i n c e r e a l r u s t c o n t r o l i s g r e a t enough f o r r e a s o n a b l e f i e l d d i m ensions t o be of p r a c t i c a l use. The answer must come from f i e l d t e s t s , e s p e c i a l l y s i n c e the p r i n c i p l e p a r a m e t e r s : R , R , and D, a l l depend i o c r i t i c a l l y on l o c a l c o n d i t i o n s . However, the models c e r t a i n l y suggest a p o t e n t i a l f o r p r a c t i c a l • u s e : F i g u r e 6.10 i n d i c a t e s t h a t f o r a p p a r e n t l y p l a u s i b l e parameter v a l u e s , e l o n g a t i n g a 200 m x 200 m f i e l d i n t o a 2000 m x 20 m f i e l d c o u l d reduce pathogen biomass a t h a r v e s t by 70%. The e f f e c t on y i e l d i s u n c l e a r but the r e s u l t i s e n c o u r a g i n g because 20 m seems wide enough to a l l o w e f f i c i e n t use of a g r i c u l t u r a l machines. 155 Summary The s p r e a d of p l a n t d i s e a s e s which have p a s s i v e a i r b o r n e d i s p e r s a l s t a g e s i s s t u d i e d u s i n g m a t h e m a t i c a l models which i n c l u d e time-dependent f e a t u r e s of pathogen p o p u l a t i o n growth and d i s p e r s a l . These d i f f u s i o n models, which a r e based on c e r e a l r u s t b i o l o g y , demonstrate t h a t f i e l d geometry can have a major i n f l u e n c e on ep i d e m i c development. Assuming pathogen d i s p e r s a l a p p r o x i m a t e s a random d i f f u s i o n p r o c e s s , I show t h a t the g r e a t e r the p e r i m e t e r t o area r a t i o of the f i e l d s i n an a g r i c u l t u r a l r e g i o n , the s l o w e r the i n c r e a s e of d i s e a s e w i t h i n those f i e l d s . Hence, when a c o n s t a n t p r o p o r t i o n of a r e g i o n ' s acreage i s a l l o c a t e d t o a c r o p , the models i n d i c a t e t h a t d e c r e a s i n g f i e l d s i z e and e l o n g a t i n g f i e l d shape can r e t a r d d i s e a s e p r o g r e s s and thus reduce y i e l d l o s s e s . Moreover, knowledge of the p r e v a i l i n g d i r e c t i o n of pathogen d i s p e r s a l by wind can be used t o f u r t h e r augment the advantages of e l o n g a t i n g f i e l d . The r e s u l t s a r e d i s c u s s e d i n terms of p r e v i o u s work i n p l a n t e p i d e m i o l o g y i n p a r t i c u l a r and p o p u l a t i o n dynamics i n g e n e r a l . FIGURES FIGURE 6.1 The l o c i of d - i s o p l e t h s i n terms of the dimensions of a r e c t a n g u l a r f i e l d i s o l a t e d i n an o t h e r w i s e c o m p l e t e l y u n i n h a b i t a b l e r e g i o n ( e q u a t i o n 6.2.5). The parameter d measures the s i z e of the f i e l d i n terms of the d i f f u s i n g pathogen p o p u l a t i o n ; the l a r g e r d, the g r e a t e r the l i k e l i h o o d t h a t d i s e a s e w i l l i n c r e a s e i n the f i e l d . FIGURE 6.2 The l o c u s of c r i t i c a l r a d i i , a of an i s o l a t e d c c i r c u l a r p a t c h of s u s c e p t i b l e h o s t s a g a i n s t the square r o o t of the e x p o n e n t i a l decay r a t e i n the e x t e r i o r e n v i r o n m e n t , /R -. T h i s i l l u s t r a t e s e q u a t i o n 6.5.5 f o r o D = R = 1 . i FIGURE 6.3 The c r i t i c a l w i d t h , W of a s t r i p of i n f i n i t e l e n g t h c i s p l o t t e d a g a i n s t the h o s t i l i t y of the e x t e r i o r e n vironment, v'R a c c o r d i n g t o e q u a t i o n 6.6.4 f o r D = 157 FIGURE 6.4 P l o t of the stea d y s t a t e d e n s i t y , N*, a g a i n s t the w i d t h of the s t r i p , W, f o r v a r i o u s decay r a t e s o u t s i d e the s t r i p , R , when l o g i s t i c growth o c c u r s i n s i d e . The o parameters a r e : d i f f u s i o n r a t e , D = 1; c a r r y i n g c a p a c i t y , N = 1 ; and i n t r i n s i c r a t e of i n c r e a s e i n max the s t r i p , R = 1 . ( a f t e r Ludwig, et a l . , 1979) i FIGURE 6.5 Geometry used t o st u d y d i s p e r s a l between f i e l d s . The r a t e of e x p o n e n t i a l pathogen biomass growth, R, i s p l o t t e d a g a i n s t d i s t a n c e from the o r i g i n . A d i s t a n c e H s e p a r a t e s i n f i n i t e s t r i p s of w i d t h W. One s t r i p i s c e n t e r e d a t the o r i g i n and o t h e r s a r e c e n t e r e d e v e r y 2b d i s t a n c e u n i t s on e i t h e r s i d e of i t . FIGURE 6.6 P l o t of the c r i t i c a l w i d t h of s t r i p s i n a r e g i o n , W , c as a f u n c t i o n of the decay r a t e i n the e x t e r i o r e n v i r o n m e n t , R , f o r d i f f e r e n t d i s t a n c e s between o s t r i p s . H. The p a r a s i t e grows e x p o n e n t i a l l y at r a t e R i = 1 i n the s t r i p and d i f f u s e s a t r a t e D = 1. The graph i l l u s t r a t e s e q u a t i o n 6.7.3. FIGURE 6.7 The l o c u s of c r i t i c a l v a l u e s i n 6.7.3 f o r D = R = 1. i An i n f i n i t e number of p a r a l l e l s t r i p s of w i d t h , W , a c d i s t a n c e H a p a r t , a r e p l a n t e d t o s u s c e p t i b l e h o s t s f o r v a r i o u s v a l u e s of R , the h o s t i l i t y of the e x t e r i o r o environment. W /(H + W ) i s the p r o p o r t i o n of the c c r e g i o n p l a n t e d t o s u s c e p t i b l e h o s t s . FIGURE 6.8 R e g i o n a l dynamics. R e l a t i v e pathogen d e n s i t y a c r o s s an i n f i n i t e s t r i p of w i d t h W = 200 m f o r f i v e c o n s e c u t i v e 20-day i n t e r v a l s a f t e r an i n i t i a l l y u n i f o r m i n f e c t i o n w i t h i n the s t r i p , N ( x , o ) . The N(x, t ) r e p r e s e n t i n u m e r i c a l s o l u t i o n s of the system 6.1.8, 6.6.3, and 6.7.2 w i t h D = 1, R = .05, and R = .01. The i o d i f f e r e n c e between H = W and H = 5W was n e g l i g i b l e . FIGURE 6.9 The e f f e c t of f i e l d s i z e on r e g i o n a l d i s e a s e p r o g r e s s . The t o t a l p a r a s i t e biomass i n the s t r i p a t time t , P ( t ) , r e l a t i v e t o the u n i f o r m l y d i s t r i b u t e d i n i t i a l number, P(O), i s p l o t t e d a g a i n s t time f o r s t r i p s of i n f i n i t e l e n g t h and w i d t h W. Parameter v a l u e s and i n i t i a l c o n d i t i o n s a r e i d e n t i c a l t o tho s e used i n F i g u r e 6-8. 159 FIGURE 6.10 Pathogen spread over a r e g i o n . The e f f e c t of s t r i p s i z e , W, on r e l a t i v e pathogen biomass a t h a r v e s t , P ( t ! ) / P ( 0 ) . The graph i l l u s t r a t e s the r e s u l t s of H n u m e r i c a l s o l u t i o n s t o e q u a t i o n s 6.6.3, 6.1.8, and 6.7.2 f o r R = .05 and .025 when H = 5W and a growing i season of t = 100 days. I n i t i a l c o n d i t i o n s and the H v a l u e s of D and R a r e i d e n t i c a l t o those of F i g u r e s o 6.8 and 6.9. FIGURE 6.11 The e f f e c t of h a l v i n g f i e l d s i z e on r e g i o n a l d i s e a s e development. P ( t ) and P ( t ) a r e the pathogen W H w/2 H biomasses a t h a r v e s t i n i n f i n i t e s t r i p s of w i d t h W and W/2, r e s p e c t i v e l y . Parameter v a l u e s and i n i t i a l c o n d i t i o n s a r e i d e n t i c a l t o those of F i g u r e 6.10 except t h a t R = .05, o n l y , i FIGURE 6.12 The e f f e c t of f i e l d shape on the r e l a t i v e pathogen biomass i n a r e g i o n ' s f i e l d s a t h a r v e s t . The v e r t i c a l a x i s , P ( t )/P ( t ), g i v e s the pathogen biomass a t w H 41 1 H h a r v e s t , t = 100 days, i n an i n f i n i t e s t r i p of w i d t h H 0 < W < 141 m r e l a t i v e t o t h a t i n an i n f i n i t e s t r i p of w i d t h W = 141 m. The h o r i z o n t a l a x i s g i v e s the w i d t h of t he 4 ha f i e l d which has the same v a l u e of d 2 i n e q u a t i o n 6.2.5, the f i e l d s i z e i n terms of d i f f u s i o n when s u r v i v a l o u t s i d e i s i m p o s s i b l e , as the i n f i n i t e s t r i p of w i d t h o < W < 141 m. The maximum v a l u e of d 2 a t t a i n a b l e by a 4 ha r e c t a n g u l a r f i e l d o c c u r s when the f i e l d i s square; the c o r r e s p o n d i n g v a l u e of W i s 141 m. 162 F I G U R E 6 . 2 F I G U R E 6.3 164 F I G U R E 6 . 4 1 FIGURE 6.5 - j r Right half 1 i Left half • i of s t r ip 1 i of the | i c e n t e r e d 1 i next str ip at or ig in 1 < H > i -i i 1 i o w/2 b 2 b Distance from origin FIGURE 6.6 F I G U R E 6.7 F I G U R E 6.8 Distance f r o m c e n t e r of s t r i p (m) 169 T i n e ' ( J a y s ) F I G U R E 6 . 1 1 FIGURE 6.12 172 1S3AJEI] |B S S E l U O l q 3SC3S|p SAI|E|aa 173 THE POTENTIAL FOR CONTROL OF CEREAL RUST BY NATURAL ENEMIES Chapter 7 174 I n t r o d u c t i o n H o l l i n g (1959) and Takahashi (1964) proposed a mechanism t o e x p l a i n the sudden t r a n s i t i o n of some i n s e c t p o p u l a t i o n s from low or endemic d e n s i t i e s t o outbreak d e n s i t i e s . They imagined t h a t t h e s e i n s e c t p o p u l a t i o n s were s u b j e c t t o a r e l a t i v e l y c o n s t a n t degree of p r e d a t i o n . They assumed t h a t c i r c u m s t a n c e s such as the a v a i l a b i l i t y of a l t e r n a t e prey m a i n t a i n e d p r e d a t o r d e n s i t i e s when the i n s e c t p o p u l a t i o n was a t endemic d e n s i t i e s ; and t h a t o t h e r c i r c u m s t a n c e s such as much l o n g e r p r e d a t o r g e n e r a t i o n t i m e s p r e v e n t e d p r e d a t o r d e n s i t i e s from i n c r e a s i n g w i t h the d e n s i t i e s of o u t b r e a k i n g i n s e c t p o p u l a t i o n s . They p o s t u l a t e d t h a t t h i s background p r e d a t i o n g e n e r a l l y p r e v e n t s r a p i d i n c r e a s e s i n i n s e c t d e n s i t y a t endemic p o p u l a t i o n l e v e l s . However, s t o c h a s t i c phenomena, such as unusual weather p a t t e r n s , might o c c a s i o n a l l y a l l o w the i n s e c t t o escape the c o n t r o l of th e s e polyphagous n o n - s y n c h r o n i z e d p r e d a t o r s . They suggested t h a t under t h e s e c i r c u m s t a n c e s an outbreak would occur which would not c o l l a p s e u n t i l the i n s e c t had o v e r - e x p l o i t e d i t s r e s o u r c e s . Recent e v i d e n c e i n d i c a t e s t h a t background p r e d a t o r and p a r a s i t e c o n t r o l of endemic p o p u l a t i o n s may be a common element of i n s e c t outbreak systems (e.g. Campbell and S l o a n , 1977; Clar.k e t a l . , 1978; McLeod, 1979; M o r r i s , 1963; Southwood and Comins, 1976). In t h i s paper I c o n s i d e r the f e a s i b i l i t y of augmenting the e f f e c t s of n a t u r a l p r e d a t i o n and p a r a s i t i s m of c e r e a l r u s t s . A s i m p l e model of c e r e a l r u s t p o p u l a t i o n dynamics i s used t o i l l u s t r a t e the o f t e n u n r e a l i z e d p o t e n t i a l of background n a t u r a l 175 The Model The model w i l l be used t o d e s c r i b e the t r a n s i t i o n of a c e r e a l r u s t p o p u l a t i o n from endemic t o epidemic d e n s i t i e s . I t i s assumed t h a t under c o n d i t i o n s of low d e n s i t y the r u s t has a n e g l i g i b l e i n f l u e n c e on the d e n s i t y and g e n o t y p i c c o n t e n t of the c r o p . T h e r e f o r e , s i n c e the model i s concerned o n l y w i t h the i n i t i a t i o n of the t r a n s i t i o n t o o u t b r e a k r u s t d e n s i t i e s and not w i t h the dynamics of the o u t b r e a k , host dynamics w i l l be e x c l u d e d from the model. There i s e v i d e n c e ( D i n o o r , 1967; R o e l f s , 1974; Stewart et a l . , 1970; Watson and Cass Smith, 1962) t h a t the l a r g e c e r e a l r u s t r e g i o n s of the w o r l d can be d i v i d e d i n t o zones and sub-zones a c c o r d i n g t o t h e i r r e l a t i v e degree of independence w i t h r e s p e c t t o a e r o b i o l o g i c a l r e l a t i o n s h i p s . For example, Stakman and C h r i s t e n s e n (1946) c o n s i d e r the P a c i f i c c o a s t of N o r t h America t o be a d i s t i n c t zone because the Rocky Mountains h i n d e r spore exchange w i t h the n e i g h b o r i n g zones i n the Great P l a i n s . Hence the model w i l l c o n s i d e r low d e n s i t y r u s t dynamics i n an a e r o b i o l o g i c a l zone. I f N i s the p o p u l a t i o n d e n s i t y of the c e r e a l r u s t ( i n p u s t u l e s / a r e a ) then i t s r a t e of change i n a g i v e n a e r o b i o l o g i c a l sub-zone can be d e s c r i b e d as dN/dt = G(N) - L ( N ) , (7.1) where G(N) and L(N) a r e the r a t e s per u n i t a r e a a t which the-r u s t p o p u l a t i o n g a i n s and l o s e s p u s t u l e s , r e s p e c t i v e l y . Time i s 176 enemies to improve the c o n t r o l of t h e s e p a t h o g e n i c f u n g i . The arguments o u t l i n e d above were d e v e l o p e d by assuming a type I I I p r e d a t o r f u n c t i o n a l response ( H o l l i n g , 1959); I show t h a t s i m i l a r arguments a p p l y t o c e r e a l r u s t s when a s i m p l e r type I I response i s assumed. P r e s e n t l y , the c o n t r o l of c e r e a l r u s t s i s almost e n t i r e l y based on the c u l t i v a t i o n of r e s i s t a n t h o s t v a r i e t i e s . However i n many c e r e a l growing r e g i o n s , r e s i s t a n c e has f r e q u e n t l y p r o v e d inadequate due t o the e v o l u t i o n of new r a c e s (Johnson, 1961). There are a l a r g e number of organisms which prey upon the uredospore stage of c e r e a l r u s t f u n g i . P a r l u c a f i l u m appears t o be the most common and w i d e s p r e a d m y c o p a r a s i t e of c e r e a l r u s t s ( C a r l i n g et a l . , 1976; Cooke, 1977; De Vay, 1956). Other f u n g i a l s o a t t a c k the c e r e a l r u s t s ( B i a l i e t a l . , 1972; Keener, 1934). Pon et a l . (1954) have shown t h a t b a c t e r i a too can p a r a s i t i z e c e r e a l r u s t p u s t u l e s . In a d d i t i o n , a number of i n s e c t s , p a r t i c u l a r l y C o l e o p t e r a and D i p t e r a , o f t e n a t t a c k r u s t p u s t u l e s on c e r e a l s ( C h e s t e r , 1946; Cunningham, 1967; Darpoux, 1 960).. Moreover, the g e n e r a l l a c k of i n v e s t i g a t i o n c o n c e r n i n g t h i s complex of c e r e a l r u s t p r e d a t o r s and p a r a s i t e s i m p l i e s t h a t the agents mentioned above may r e p r e s e n t o n l y a s m a l l f r a c t i o n of a l l the c o n t r i b u t o r s t o the complex ( B a r n e t t and B i n d e r , 1973). 177 r e p r e s e n t e d by T. I f B i s the r a t e of spore p r o d u c t i o n per p u s t u l e G(N) = B . f ( N ) . T + I T (7.2) L I where T and T a r e , r e s p e c t i v e l y , the t r a n s m i s s i o n f a c t o r s f o r L I l o c a l and immigrant s p o r e s , the p r o p o r t i o n s of the s p o r e s which s u c c e s s f u l l y i n i t i a t e new p u s t u l e s i n t h i s a e r o b i o l o g i c a l zone ( c f . Anderson and May, 1978; MacDonald, 1961). S i n c e the ' p a r e n t s ' of l o c a l l y produced spore genotypes have been exposed to l o c a l f o r c e s of n a t u r a l s e l e c t i o n , l o c a l l y produced spore genotypes ought t o be b e t t e r adapted t o l o c a l c o n d i t i o n s than immigrant spore genotypes. Hence, g e n e r a l l y T > T . L I In 7.2 f(N) i s the e f f e c t i v e number of p u s t u l e s per u n i t a r e a . I t a l l o w s f o r the e f f e c t s of d e n s i t y dependent c o m p e t i t i o n over a c e r e a l p l a n t f o r f i n i t e r e s o u r c e s such as space, energy, or n u t r i e n t s . However, i n the e a r l y s t a g e s of an e p i d e m i c , c o m p e t i t i o n i s n e g l i g i b l e (Van der P l a n k , 1975) so f(N ) = N. (7.3) I i s the d e n s i t y independent r a t e of spore i m m i g r a t i o n per u n i t a r e a . I t i s the r e s u l t of c o n t i n e n t a l c e r e a l r u s t d i s s e m i n a t i o n (Zadoks, 1965). For i n s t a n c e , i n N o r t h America 178 stem r u s t of wheat ( P u c c i n i a g r a m i n i s f. sp. t r i t i c i ) o v e r w i n t e r s i n Mexic o , Texas, and L o u i s i a n a . I n the s p r i n g stem r u s t s pores a r e blown n o r t h , and s i n c e the c r o p s a r e l a t e r t h e r e , the d i s e a s e f o l l o w s the wheat phenology g r a d i e n t . Oversummering o c c u r s i n Canada, i n the n o r t h e r n ' ; U n i t e d S t a t e s , and i n c e r t a i n c o o l mountain a r e a s of Mexic o . Zadoks ( 1 9 6 8 ) has emphasized t h a t a r e a s of h i g h e l e v a t i o n are o f t e n e p i d e m i o l o g i c a l l y s i g n i f i c a n t , p a r t i c u l a r l y i n warm c l i m a t i c zones. Here the r u s t s can p e r s i s t a l l year because t h e r e a r e always f r e s h green c e r e a l c r o p s or g r a s s e s a t some a l t i t u d e . In the f a l l the w i n t e r wheat a r e a s of the so u t h e r n U n i t e d S t a t e s and Mexico r e c e i v e spores blown south from n o r t h e r n r e g i o n s and spores d i s p e r s i n g from oversummering s i t e s i n the Mexican mountains. The ev i d e n c e f o r a l l of t h i s comes from a v a r i e t y of s o u r c e s : f i e l d r e c o r d s of e p i d e m i c s , g e o g r a p h i c a l d i s t r i b u t i o n of d i f f e r e n t r a c e s , spore t r a p p i n g , and m e t e o r o l o g i c a l d a t a ( C r a i g i e , 1 9 4 5 ; Stakman and H a r r a r , 1 9 5 7 ) . In s h o r t , every a e r o b i o l o g i c a l zone i n N o r t h America i s s u b j e c t t o i n v a s i o n of v a r i a b l e i n t e n s i t y by m i g r a t i n g r u s t spores throughout i t s growing season. Most of thes e spores come from o t h e r a r e a s where host phenology i s more advanced. S i m i l a r p a t t e r n s of d i s s e m i n a t i o n a r e e v i d e n t f o r o t h e r c e r e a l r u s t s i n N o r t h America ( C h e s t e r , 1 9 4 6 ; F r e y e t a l . , 1 9 7 3 ) and f o r c e r e a l r u s t s i n o t h e r p a r t s of the w o r l d ( D i n o o r , 1 9 6 7 ; Hogg e t a l . , 1 9 6 9 ; Nagarajan e t a l . , 1 9 7 6 ) . The d e n s i t y dependent l o s s r a t e can be w r i t t e n 179 L(N) = f (N) + f (N) + f D E (N) , (7.4) where f (N) i s the sum of l o s s e s due t o n a t u r a l h o s t d e a t h s , D p l u s l o s s e s from r u s t i n d u c e d host d e a t h s , p l u s l o s s e s due t o p u s t u l e senescene on h e a l t h y c e r e a l t i s s u e (adapted from Anderson and May, 1978). In the absence of c o m p e t i t i v e or s y n g e r g i s t i c e f f e c t s a t low r u s t d e n s i t i e s (Van der P l a n k , 1975) f'(N) = DN (7.5) Here D i s the i n t r i n s i c r a t e of ' n a t u r a l ' r u s t m o r t a l i t y . ' N a t u r a l ' m o r t a l i t y e x c l u d e s l o s s e s due t o p r e d a t i o n or p a r a s i t i s m on the r u s t f u n g i but i n c l u d e s a l l o t h e r forms of m o r t a l i t y . L o s s es due t o the e s s e n t i a l l y p a s s i v e and wind-mediated p r o c e s s of e m i g r a t i o n (Stakman and C h r i s t e n s e n , 1946) a r e p r o p o r t i o n a l t o the r u s t p o p u l a t i o n d e n s i t y (Gregory, 1972; Waggoner, 1962), Hence where E i s the i n s t a n t a n e o u s r a t e of e m i g r a t i o n . L o s s e s of t h e r u s t f u n g i t o the complex of background p r e d a t o r s and p a r a s i t e s a r e r e p r e s e n t e d by f (N). R e a r r a n g i n g D f (N) = EN E (7.6) P 180 the H o l l i n g (1965) d i s c e q u a t i o n i n t o a M i c h a e l i s - M e n t o n form and assuming random d i s t r i b u t i o n of n a t u r a l enemies among the members of the r u s t p o p u l a t i o n I. f (N) = AN(Z + N ) ~ 1 \ (7.7) Here A i s the maximum a t t a c k r a t e ( p u s t u l e s / t i m e ) and Z i s the r u s t d e n s i t y a t which the r a t e of l o s s e s t o the complex of background p r e d a t o r s and p a r a s i t e s i s A/2. E q u a t i o n '7.7 s i m p l y i m p l i e s t h a t t h e r e must be some maximum r a t e of a t t a c k per p r e d a t o r or p a r a s i t e when prey are u n l i m i t e d i n number. S i m p l e r ( m a s s - a c t i o n ) assumptions a r e b i o l o g i c a l l y u n r e a l i s t i c . At low p u s t u l e d e n s i t y the i n f l u e n c e of n a t u r a l enemies i s l i m i t e d by the number of p u s t u l e s they can f i n d . T h i s ' s e a r c h i n g e f f i c i e n c y ' of the n a t u r a l enemy complex i s i n v e r s e l y r e l a t e d t o the parameter Z. As p u s t u l e d e n s i t i e s i n c r e a s e , n a t u r a l enemies become more s u c c e s s f u l a t c o n t a c t i n g r u s t p u s t u l e s so they b e g i n t o spend more time consuming and l e s s time s e a r c h i n g f o r t h e i r p r e y . E v e n t u a l l y a l l t h e i r time i s spent e a t i n g and d i g e s t i n g c e r e a l r u s t . At t h i s p o i n t f (N) = A and the n a t u r a l enemy p complex has become s a t i a t e d . I t i s no l o n g e r c a p a b l e of i n c r e a s i n g i t s consumption as the d e n s i t y of r u s t p u s t u l e s i n c r e a s e s . I t has been i m p l i c i t l y assumed i n 7.7 t h a t any change i n the numbers of n a t u r a l enemies r e s u l t i n g from changes i n p u s t u l e d e n s i t i e s i s n e g l i g i b l e r e l a t i v e to the magnitude of p u s t u l e 181 d e n s i t y changes. T h i s r e q u i r e s t h a t the p r e d a t o r s and p a r a s i t e s have a l t e r n a t e food s o u r c e s which can m a i n t a i n t h e i r p o p u l a t i o n s when the r u s t p o p u l a t i o n i s a t low d e n s i t y , and t h a t the r u s t have a much g r e a t e r i n t r i n s i c r a t e of i n c r e a s e d u r i n g outbreak than the n a t u r a l enemies. The a v a i l a b l e e v i d e n c e ( M a d e l i n , 1968; Pon e_t a_l. , 1954; Rambo and Bean, 1970; Stakman and C h r i s t e n s e n , 1946) i n d i c a t e s t h a t the system meets t h e s e r e q u i r e m e n t s . T h i s i s a c o n s e r v a t i v e a ssumption i n terms of the c o n t r o l e x e r t e d by n a t u r a l enemies on d i s e a s e p r o g r e s s . Any n u m e r i c a l response of the n a t u r a l enemies t o i n c r e a s e s i n p u s t u l e d e n s i t y can o n l y s t r e n g t h e n t h e i r e f f e c t i v e n e s s . In f a c t , i n c r e a s e s i n the d e n s i t y of n a t u r a l enemies r e s u l t i n p r o p o r t i o n a t e i n c r e a s e s i n the maximum a t t a c k r a t e , A, i n 7.7. T h i s , i n t u r n , produces p r o p o r t i o n a t e i n c r e a s e s i n the r a t e of l o s s e s t o the complex of background p r e d a t o r s and p a r a s i t e s , f (N), i n 7.4. P S u b s t i t u t i n g 7.3 i n t o 7.2, G(N) = BNT + IT ; L L then , combining 7.5, 7.6, and 7.7 i n 7.4, L(N) = (D + E)N + A N(Z + N ) ~ ' . Hence 7.1 can be w r i t t e n 182 i i dN/dT = ( B T - D - E ) N + I T - A N(Z + N ) - 1 (7.8) L I For f u t u r e r e f e r e n c e the v a r i a b l e s and parameters of t h i s model have been l i s t e d i n Ta b l e 7.1. D i m e n s i o n a l A n a l y s i s The P i theorem (Duncan, 1953) can be used t o f i n d a d i m e n s i o n l e s s form of 7.8. F i r s t , d e f i n e the d i m e n s i o n l e s s c o e f f i c i e n t s of r u s t p o p u l a t i o n d e n s i t y and t i m e , r e s p e c t i v e l y , as n = N/Z and t = AT/Z. Then 7.8 can be e x p r e s s e d i n coef f i c i e n t s dn/dt = g(n) -Jt(n) where the r a t e of l o s s e s i n d u c e d p a r a s i t e s i s (7.9) terms of these d i m e n s i o n l e s s (7.10) by background p r e d a t o r s and £(n) = n(1 + n ) - 1 , (7.11) 183. and the g r o s s p r o d u c t i o n r a t e i s g(n) rn + 1 In 7.12 (7.12) r = Z ( B T - D - E)/A (7.13) and i = IT /A (7.14) I a r e n o n - d i m e n s i o n a l parameters d e s c r i b i n g r e s p e c t i v e l y the r a t e s of e x p o n e n t i a l growth and net i m m i g r a t i o n of n. G r a p h i c a l . A n a l y s i s S i n c e the c l a s s i c a l paper of Rosenzweig and MacArthur (1963) g r a h i c a l t e c h n i q u e s have been f r e q u e n t l y used to a n a l y z e s i m p l e p r e d a t o r - p r e y systems (e.g. C a n a l e , 1970; Ludwig et, a l . , 1978; May, 1971; Rosenzweig, 1973). The a p p l i c a t i o n of these t e c h n i q u e s has o f t e n drawn a t t e n t i o n t o a s p e c t s of system dynamics whose importance had g e n e r a l l y been u n r e c o g n i z e d (e.g. Noy-Meir, 1975; Pet-erman et. a l . , 1979; Rosenzweig, 1971). C o n s i d e r F i g u r e 7.1. A c c o r d i n g to 7.10 the d i f f e r e n c e i n 184 magnitude of the c u r v e s i n F i g u r e 7.1 d e f i n e s the cur v e f o r dn/dt i n F i g u r e 7.2. The e q u i l i b r i u m d i s e a s e d e n s i t i e s n and n E R se p a r a t e the d e n s i t i e s a t which the r u s t p o p u l a t i o n i s i n c r e a s i n g (n < n , n > n ) from those a t which i t i s d e c r e a s i n g E R (n < n < n ) . E R Thus when n < n , t h e r e l e a s e p o i n t , the p o p u l a t i o n tends R toward endemic d e n s i t i e s , n . When n < n the e f f e c t of n a t u r a l E E enemies i s l i m i t e d by t h e i r s e a r c h i n g c a p a b i l i t i e s a t low r u s t d e n s i t i e s and the gross p r o d u c t i o n exceeds the l o s s e s t o n a t u r a l enemies ( F i g . 7 . 1 ) so the p o p u l a t i o n i n c r e a s e s t o n . When n < E E n < n the s i t u a t i o n i s r e v e r s e d ( F i g . 7.1) and the comDlex of R background p r e d a t o r s and p a r a s i t e s d r i v e s the p o p u l a t i o n down to n . However, when n > n , the complex of background p r e d a t o r s E R and p a r a s i t e s i s s a t i a t e d , g r o s s p r o d u c t i o n exceeds the l o s s e s and the d i s e a s e 'escapes' the c o n t r o l of i t s n a t u r a l enemy complex. From F i g u r e 7.1 i t i s c l e a r t h a t the l o c a t i o n s of the endemic r u s t d e n s i t y , n , and the r e l e a s e p o i n t , n , depend upon E R the r e l a t i v e p o s i t i o n of the g r o s s p r o d u c t i o n c u r v e , g ( n ) , w i t h r e s p e c t t o the cu r v e d e f i n i n g the l o s s e s t o n a t u r a l enemies lin). T h i s l a t t e r c u r v e i s f i x e d i n F i g u r e 7.1 a c c o r d i n g t o 7.11. Hence, any change i n the r e l a t i v e p o s i t i o n s of the c u r v e s of F i g u r e 7.1 depends e n t i r e l y upon the parameters, r and i , of the g r o s s p r o d u c t i o n l i n e , 7.12. 185 D u r i n g the non-disease season i t i s almost c e r t a i n t h a t r < 0 i n 7.13 because poor weather c o n d i t i o n s w i l l d e p r ess spore p r o d u c t i o n , B, and boost the r a t e of ' n a t u r a l ' m o r t a l i t y , D. F u r t h e r m o r e , the s c a r c i t y of a v a i l a b l e l i v i n g host t i s s u e r e s u l t s i n low t r a n s m i s s i o n f a c t o r s which d e c r e a s e both r and i . The n o n - d i m e n s i o n a l r a t e of r u s t i m m i g r a t i o n , i , may be low f o r o t h e r reasons as w e l l . The approximate synchrony i n d i s e a s e season between n e i g h b o r i n g a e r o b i o l o g i c a l zones, which s u p p l y the b u l k of l o c a l i m m i g r a t i o n (Stakman and C h r i s t e n s e n , 1946), r e s u l t s i n a c o r r e l a t i o n between i and the d i m e n s i o n l e s s r u s t e x p o n e n t i a l growth r a t e , r . Thus, w i t h i s m a l l and r < 0 d u r i n g the n o n - d i s e a s e season, i t i s c l e a r t h a t the r u s t w i l l b e g i n the d i s e a s e season a t low d e n s i t y . T h i s i s l i k e l y t o be t r u e even i f the e f f e c t s of n a t u r a l enemies decrease s u b s t a n t i a l l y o u t s i d e the d i s e a s e season. As the d i s e a s e season p r o g r e s s e s the weather improves f o r the r u s t and c r o p growth i n c r e a s e s the amount of l i v i n g h o s t t i s s u e . F u r t h e r m o r e , n a t u r a l s e l e c t i o n b e g i n s to g e n e t i c a l l y a d j u s t the l o c a l r u s t p o p u l a t i o n t o the r e s i s t a n c e e x p r e s s e d by the c e r e a l ( F l e m i n g and P e r s o n , 197 .8). Hence i and r i n c r e a s e t h r ough the e a r l y p a r t of the d i s e a s e season. The consequences of i n c r e a s e s i n i and r d u r i n g the r u s t season can be v i s u a l i z e d i n F i g u r e 7.1 by a l t e r i n g the o r i e n t a t i o n of the l i n e g ( n ) . As the r u s t season b e g i n s i i s s m a l l and r < 0 so the l i n e g(n) s l o p e s downward t o the r i g h t from i t s i n t e r c e p t a t i ( F i g u r e 7.1). At t h i s stage the s c a l e d e q u i l i b r i u m r u s t endemic d e n s i t y , n , i s v e r y s m a l l and no E 186 s c a l e d r e l e a s e d d e n s i t y , n , e x i s t s . E p i d e m i c s cannot o c c u r . R Assuming, f o r s i m p l i c i t y , t h a t i i s f i x e d , the i n c r e a s e s i n r as the d i s e a s e season p r o g r e s s e s a re m a n i f e s t e d i n an a n t i c l o c k w i s e r o t a t i o n of g(n) about i t s i n t e r c e p t , i , i n F i g u r e 7 . 1 . As r , the d i m e n s i o n l e s s e x p o n e n t i a l growth r a t e , i n c r e a s e s , the endemic p o p u l a t i o n l e v e l r i s e s s l i g h t l y . When r > 0 the s c a l e d r e l e a s e d e n s i t y , n , a p p e a r s , and as r i n c r e a s e s f u r t h e r n R R | -d e c r e a s e s v e r y r a p i d l y w h i l e n I c o n t i n u e s t o show s l i g h t E growth. E v e n t u a l l y r reaches a c r i t i c a l v a l u e , r , where n = n c E R and the n a t u r a l enemy complex i s no l o n g e r a b l e t o check r u s t i n c r e a s e . The e f f e c t of i n c r e a s e s i n r on the s c a l e d e q u i l i b r i u m r u s t endemic d e n s i t y , n , and on the s c a l e d r e l e a s e d e n s i t y , n , are E R diagrammed i n F i g u r e 7 . 3 . A s i m i l a r s c e n a r i o can be d e v e l o p e d f o r the i n f l u e n c e of the d i m e n s i o n l e s s r a t e of r u s t i m m i g r a t i o n , i , on n and n E R ( F i g u r e 7.4). In F i g u r e s 7.3 and 7.4, An r e p r e s e n t s the s i z e of i n s t a n t a n e o u s p o s i t i v e f l u c t u a t i o n i n n needed t o r e l e a s e a p o p u l a t i o n of s i z e n when r = .3 and i = . 1 . Parameter E i n c r e a s e s of Ar = r - r or of A i = i - i are both s u f f i c i e n t c c i n t h emselves t o r e l e a s e the r u s t f u n g i from endemic d e n s i t i e s . Appendix A d e s c r i b e s the methods of d e t e r m i n i n g An and the c r i t i c a l v a l u e s , r and i . c c 187 The l o c u s of c r i t i c a l v a l u e s f o r the system parameters i s graphed i n F i g u r e 7.5. A l l c o m b i n a t i o n s of i , the d i m e n s i o n l e s s i m m i g r a t i o n r a t e , and r , the d i m e n s i o n l e s s p o p u l a t i o n growth r a t e , which l i e under the c u r v e a l l o w the complex of background p r e d a t o r s and p a r a s i t e s t o c o n t r o l r u s t d e n s i t y . The p l o t s of F i g u r e s 7.3 and 7.4 c o r r e s p o n d t o s l i c e s taken a l o n g i = .1 and r = .3 which c u l m i n a t e i n c r i t i c a l p o i n t s a t 3 and 4 r e s p e c t i v e l y i n F i g u r e 7.5. F i g u r e 7.6 i l l u s t r a t e s the a s s o c i a t i o n of parameters i and r w i t h the s t a t e v a r i a b l e n a t c r i t i c a l p o i n t s . P o i n t s 3 and 4 on the n - a x i s c o r r e s p o n d w i t h F i g u r e 7.5 and i n d i c a t e the p o s i t i o n s of the c r i t i c a l v a l u e s , from F i g u r e s 7.3 and 7.4, r e s p e c t i v e l y . When i i s l a r g e and r i s s m a l l the n a t u r a l enemy c c complex i s c a p a b l e of c o n t r o l l i n g l a r g e r r u s t p o p u l a t i o n s than when i i s s m a l l and r i s l a r g e . F i g u r e s 7.5 and 7.6 demonstrate t h a t , w i t h i n l i m i t s , management can t r a d e o f f i , the s c a l e d r u s t i m m i g r a t i o n r a t e , a g a i n s t r , the d i m e n s i o n l e s s r a t e of e x p o n e n t i a l growth, i n a t t e m p t i n g t o m a i n t a i n n a t u r a l enemy c o n t r o l of the fungus. However, c o n c e n t r a t i o n on keeping r low seems t o be advantageous f o r a c o u p l e of re a s o n s . S i n c e r i s a f f e c t e d p r e d o m i n a n t l y by events w i t h i n the a e o r b i o l o g i c a l zone of c o n c e r n , l o c a l management presumably has a more d i r e c t i n f l u e n c e on i t . S e c o n d l y , when r i s s m a l l An becomes v e r y l a r g e ( F i g u r e 7.3, Appendix B) so s t o c h a s t i c v a r i a t i o n s i n the r e l a t i v e r u s t p o p u l a t i o n d e n s i t y , n, must a l s o be v e r y l a r g e b e f o r e they can r e l e a s e the fungus. F i n a l l y , a f t e r 188 o u t b r e a k , the t o t a l damage t o the c r o p i s much more s e n s i t i v e t o s l i g h t v a r i a t i o n s i n r than to s l i g h t v a r i a t i o n s i n i ( F i g u r e 7.1). D i s c u s s i o n I t has been shown above t h a t a n a t u r a l enemy complex may be c a p a b l e of r e s t r i c t i n g r u s t d e n s i t y t o endemic l e v e l s u n t i l r , the r e l a t i v e e x p o n e n t i a l growth r a t e , and i , the r e l a t i v e i m m i g r a t i o n r a t e , have exceeded c r i t i c a l v a l u e s . Without t h e s e l o s s e s t o background p r e d a t o r s and p a r a s i t e s , the r u s t p o p u l a t i o n c o u l d b u i l d up smoothly t o l a r g e l e v e l s even b e f o r e r and i have passed c r i t i c a l v a l u e s . Thus, when the d i s e a s e e n t e r s i t s phase of e x p o n e n t i a l growth a f t e r a c r i t i c a l p o i n t has been passe d , i t s d e n s i t y may be c o n s i d e r a b l y below what i t would have been i n the absence of l o s s e s t o n a t u r a l enemies. Due t o the e x p o n e n t i a l n a t u r e of subsequent p o p u l a t i o n growth i n the remainder of the d i s e a s e season, e a r l y l o s s e s t o n a t u r a l enemies w i l l be compounded many times b e f o r e h a r v e s t i n terms of c e r e a l c r o p y i e l d ( K r a n z , 1974; Van der P l a n k , 1975). F i g u r e 7.7 demonstrates how an e f f e c t i v e n a t u r a l enemy complex might d e l a y epidemic development even i f c r i t i c a l parameter v a l u e s have a l r e a d been passed. Parameter v a l u e s were chosen so t h a t the r a t e of change of r u s t p u s t u l e d e n s i t y i n 7.8 i s dN/dT =0.5N +0.1 - AN /O+N) 189 Thus when A, the maximum r a t e of a t t a c k by the complex on r u s t p u s t u l e s , i s 1.0, e q u a t i o n s 7.8, 7.13, and 7.14 g i v e the r e l a t i v e e x p o n e n t i a l growth and i m m i g r a t i o n r a t e s of the r u s t as r =0.5 and i =0.1, r e s p e c t i v e l y . Then F i g u r e 7.3 i n d i c a t e s t h a t r i s g r e a t e r than i t s c r i t i c a l v a l u e , r . V i s u a l i z i n g F i g u r e 7.1 c w i t h r s l i g h t l y g r e a t e r than r , i t can be seen t h a t a c ' b o t t l e n e c k ' i n net p r o d u c t i o n w i l l o c c u r near the c r i t i c a l r u s t p u s t u l e d e n s i t y , N . By 7.9, N = n because Z has been set a t c c c 1.0. At t h i s ' b o t t l e n e c k ' g(n) i s b a r e l y g r e a t e r than i ( n ) so the r a t e of d i s e a s e i n c r e a s e i s s h a r p l y c u r t a i l e d . The c u r v e w i t h A = 1 i n F i g u r e 7.4 demonstrates t h i s e f f e c t . The complex of background p r e d a t o r s and p a r a s i t e s d e l a y s u n r e s t r i c t e d e x p o n e n t i a l growth u n t i l T > 20. When the parameters are chosen so t h a t r < r , the r u s t p u s t u l e d e n s i t y asymptotes at N < N c c and the p o p u l a t i o n never e n t e r s a phase of u n l i m i t e d e x p o n e n t i a l growth when T becomes l a r g e . G e n e r a l l y , the p o t e n t i a l r o l e of n a t u r a l enemies i n r u s t c o n t r o l seems t o have been i g n o r e d i n the l i t e r a t u r e . T h i s may be due t o a p r e c o n c e p t i o n t h a t the e f f e c t of any agent which c o n t r o l s c e r e a l r u s t w i l l i n c r e a s e p r o p o r t i o n a l l y w i t h r u s t p o p u l a t i o n d e n s i t y . Indeed, an a b i l i t y t o e s c a l a t e the a t t a c k r a t e i n response t o the growth of the t a r g e t p o p u l a t i o n i s c o n v e n t i o n a l l y c o n s i d e r e d t o be an e s s e n t i a l i n g r e d i e n t f o r s u c c e s s f u l c o n t r o l . Commonly suggested means of a c h i e v i n g t h i s a r e u s i n g n a t u r a l enemies w i t h l a r g e r e p r o d u c t i v e c a p a c i t i e s ( e . g . H a s s e l l , 1978; Southwood, 1977) and h i g h l y a g g r e g a t i v e 190 movement t o l o c a t i o n s of h i g h prey d e n s i t y (e.g. Beddington e_t a l . , 1978; H a s s e l l , 1976). However, as shown above, the e f f e c t of polyphagous n o n - s y n c h r o n i z e d p r e d a t o r s and p a r a s i t e s i s l i k e l y t o be swamped d u r i n g r u s t e p i d e m i c s . The e f f e c t s h o u l d be most pronounced a t low d i s e a s e d e n s i t i e s . T h i s i n t r o d u c e s a n o t h e r problem. I t i s q u i t e c o n c e i v a b l e t h a t such a c o n t r o l l i n g e f f e c t may go u n n o t i c e d f o r y e a r s due t o inadequacy of c o n v e n t i o n a l methods of measurement a t low d e n s i t y (Wolfe and Schwarzbach, 1978). Indeed, the importance of system dynamics a t low p o p u l a t i o n d e n s i t i e s seems t o be a common f e a t u r e of many renewable r e s o u r c e management problems (e.g. Campbell and S l o a n , 1977; C l a r k et a l . , 1978; McLeod, 1978; Peterman, 1977; Southwood and Comins, 1976). T e s t i n g f o r the r e g u l a t o r y e f f e c t s of a n a t u r a l enemy complex i s c r i t i c a l l y dependent on the a p p r o p r i a t e c h o i c e of t e s t a r e a s . These a r e a s must be l a r g e enough t o p r e v e n t e x c e s s i v e m i g r a t i o n of the c e r e a l r u s t s or n a t u r a l enemies r e l a t i v e t o t h e i r l o c a l p o p u l a t i o n d e n s i t i e s . C u l t u r a l p r a c t i c e s i n the a r e a s s h o u l d a l l o w the n a t u r a l enemy t o r e a l i z e as much of i t s r e g u l a t o r y p o t e n t i a l as p o s s i b l e . Two methods of e v a l u a t i n g n a t u r a l enemy c o n t r o l a re commonly su g g e s t e d ( e . g . , DeBach and H u f f a k e r , 1971). One method compares an a r e a b e f o r e and a f t e r the m a n i p u l a t i o n of the n a t u r a l enemy complex. T h i s method i s e a s i l y performed. I t can be r e p e a t e d i n b i o l o g i c a l l y and c l i m a t i c a l l y d i s s i m i l a r a r e a s and can d e a l w i t h f a i r l y l a r g e m i g r a t i o n r a t e s . Another method compares s e t s of ' i d e n t i c a l ' p l o t s where the n a t u r a l enemy : complex has and has not been m a n i p u l a t e d . Care must be taken t o 191 ensure t h a t the n e c e s s a r y s e p a r a t i o n of the m a n i p u l a t e d and unmanipulated p l o t s t o negate d i s p e r s a l between them i s not gr e a t enough t o produce s i g n i f i c a n t l y d i f f e r e n t e n v i r o n m e n t a l c o n d i t i o n s . I n t e r p r e t a t i o n of the r e s u l t s i s r e l a t i v e l y . s t r a i g h t f o r w a r d . Two ways of m a n i p u l a t i n g n a t u r a l enemy complexes may be p r a c t i c a l f o r c e r e a l r u s t s . C h e m i c a l t e c h n i q u e s would use f u n g i c i d e s and i n s e c t i c i d e s t o s e l e c t i v e l y reduce the e f f e c t of n a t u r a l enemies w h i l e h a v i n g a m i n i m a l i n f l u e n c e on the r u s t s because of the dosage, t i m i n g , t o x i c i t y , f o r m u l a t i o n s , e t c . Hence, i f the n a t u r a l enemies a r e n o r m a l l y e f f e c t i v e , the r u s t p o p u l a t i o n s i n t r e a t e d a r e a s s h o u l d i n c r e a s e t o h i g h e r d e n s i t i e s when compared w i t h those i n u n t r e a t e d a r e a s . Augmentation t e c h n i q u e s .would i n v o l v e amending c u l t u r a l p r a c t i c e s or i n t r o d u c i n g f o r e i g n enemies t o i n c r e a s e the r e g u l a t o r y e f f e c t of the complex of n a t u r a l enemies. E f f e c t i v e i n t r o d u c t i o n s or changes i n c u l t u r a l p r a c t i c e s ought t o reduce y i e l d l o s s e s i n the m a n i p u l a t e d a r e a s . The magnitude and t i m i n g of the c o n t r o l e x e r t e d by n a t u r a l enemy'complexes i s l i a b l e t o show g r e a t v a r i a b i l i t y i n d i f f e r e n t a r e a s . The same s p e c i e s i n d i f f e r e n t environments o f t e n behave d i f f e r e n t l y (Cunningham, 1967; Keener, 1934) and the complex co m p r i s e s d i f f e r e n t s p e c i e s i n d i f f e r e n t environments (Keener, 1934; Pon e_t a l . , 1954). In some a r e a s the c o n t r o l may be i n s i g n i f i c a n t due t o a s c a r c i t y of n a t u r a l enemies. T h i s i s m a n i f e s t e d as a v e r y low maximum a t t a c k r a t e , A, i n e q u a t i o n 7.7. Then, t h r o u g h 7.14, i > 1 i n F i g u r e 7.1, and as soon as r > 1 the a p p r o x i m a t e l y e x p o n e n t i a l phase of . d i s e a s e i n c r e a s e 192 b e g i n s . In o t h e r a r e a s the n a t u r a l enemies may be i n e f f e c t i v e e a r l y i n the r u s t season f o r reasons such as l a t e emergence or u n f a v o r a b l e e n v i r o n m e n t a l c o n d i t i o n s . Under th e s e c i r c u m s t a n c e s the r u s t may be pa s t i t s r e l e a s e d e n s i t y b e f o r e the complex has a s s e r t e d i t s e l f . C h e s t e r (1946) has suggested t h a t i n s t i l l o t h e r a r e a s p r e d a t o r y i n s e c t s may s e r v e as v e c t o r s i n l o c a l r u s t d i s s e m i n a t i o n . He notes t h a t t h e r e i s no e v i d e n c e t h a t i n s e c t s are i n v o l v e d i n l o n g d i s t a n c e r u s t d i s p e r s a l . T h i s a i d i n l o c a l d i s p e r s a l t a k e s the form of l a r g e r l o c a l t r a n s m i s s i o n f a c t o r s , T . Through 7.13 these l a r g e r v a l u e s of T produce l a r g e r L L e x p o n e n t i a l growth r a t e s , r , which may more than compensate the r u s t p o p u l a t i o n f o r p r e d a t i o n l o s s e s . Hence, r e g u l a t i o n may be weak i n the s e a r e a s . N o n e t h e l e s s , the background p r e d a t o r s and p a r a s i t e s of c e r e a l r u s t s may be u n n o t i c e d but im p o r t a n t c o n t r o l l i n g agents i n a number of a e o b i o l o g i c a l zones. These zones i n p a r t i c u l a r o f f e r the p o s s i b i l i t y of i n c r e a s i n g the e f f e c t of the n a t u r a l enemy complex e i t h e r by a d j u s t m e n t s t o the complex i t s e l f or by changes t o the a g r i c u l t u r a l environment which b e n e f i t the e x i s t i n g n a t u r a l enemy complex or by b o t h . I n c r e a s i n g the e f f e c t of background p r e d a t o r s and p a r a s i t e s d i r e c t l y may i n v o l v e i n t r o d u c t i o n and e s t a b l i s h m e n t of non-n a t i v e s p e c i e s i n t o the l o c a l complex and/or a r t i f i c a l l y b o o s t i n g the e f f e c t i v e n e s s of the n a t i v e n a t u r a l enemy complex, p a r t i c u l a r l y a t the b e g i n n i n g of the d i s e a s e season. The i m p o r t a t i o n of new n a t u r a l enemies g e n e r a l l y o f f e r s the g r e a t e s t hope f o r i n c r e a s i n g the r e g u l a t o r y e f f e c t of n a t u r a l enemies, e s p e c i a l l y i n the r e l a t i v e l y s i m p l e environments of 193 c e r e a l a g r i c u l t u r e ( B e i r n e , 1974). C a n d i d a t e s f o r i n t r o d u c t i o n must be a b l e t o a t l e a s t p a r t i a l l y l i m i t the r u s t d u r i n g the d i s e a s e season w i t h o u t p r o d u c i n g s e r i o u s d e t r i m e n t a l s i d e -e f f e c t s on the agroecosystem. Many s u c c e s s f u l i m p o r t s f o r b i o l o g i c a l c o n t r o l have come from c l o s e l y r e l a t e d host s p e c i e s (Reeks and Cameron, 1971). P a r a s i t e s or p r e d a t o r s which o p e r a t e i n s i m i l a r m i c r o h a b i t a t s on a wide v a r i e t y of host or prey s p e c i e s are a l s o p o t e n t i a l l y u s e f u l (DeBach, 1974). Programs f o r a r t i f i c i a l l y b o o s t i n g the n a t i v e n a t u r a l enemy complex i n v o l v e e i t h e r mass p r o d u c t i o n and p e r i o d i c c o l o n i z a t i o n or p l a n n e d g e n e t i c improvement. The h i g h c o s t s of development and a p p l i c a t i o n of these t e c h n i q u e s a l o n g w i t h the v a s t n e s s and low v a l u e per a c r e of a n n u a l c e r e a l c r o p s make a r t i f i c i a l b o o s t i n g a l a s t r e s o r t . These methods are most s u c c e s s f u l w i t h n a t u r a l enemies which have been shown to be e f f e c t i v e by r e s e a r c h but a r e i n h i b i t e d because they are i n t o l e r a n t of weather extremes, or are not s y n c h r o n i z e d w i t h the n e c e s s a r y h o s t s t a g e s , or are rendered i n e f f e c t i v e by p e r i o d i c e n v i r o n m e n t a l u n f a v o r a b i 1 i t y (DeBach, 1974). P a r l u c a f i l u m appears t o be the p r i m a r y c a n d i d a t e f o r a r t i f i c i a l b o o s t s . A c c o r d i n g t o Rambo and Bean (1970): " I t i s w i d e s p r e a d i n d i s t r i b u t i o n , e a s i l y d i s s e m i n a t e d , can s u r v i v e f o r l o n g p e r i o d s , and does not i n f e c t h e a l t h y t i s s u e . " I t can a l s o be an e f f e c t i v e h y p e r p a r a s i t e : H a r d i s o n (1942) r e p o r t s t h a t sometimes s p o r u l a t i o n of the r u s t i s almost or e n t i r e l y p r e v e n t e d . F u r t h e r m o r e , i t i s e s i l y c u l t u r e d ( B a r n e t t and B i n d e r 1973) and the b i o l o g i c a l s p e c i a l i z a t i o n w i t h i n the s p e c i e s towards d i f f e r e n t r u s t s (Keener, 1934) suggests t h a t g e n e t i c 194 improvement may be p o s s i b l e . A l t h o u g h d i r e c t l y a d j u s t i n g the n a t u r a l enemy complex may p r o v i d e the most promise, p r a c t i c a l d i f f i c u l t i e s are i n h e r e n t i n such o p e r a t i o n s , e s p e c i a l l y those i n v o l v i n g a r t i f i c i a l b o o s t s (Baker and Cook, 1974). These c o n t r o l p o s s i b i l i t i e s are w e l l worth g r e a t e r i n v e s t i g a t i o n but p r a c t i c a l a p p l i c a t i o n seems t o be a few y e a r s away. On the o t h e r hand, m a n i p u l a t i o n of the a g r i c u l t u r a l environment to a m p l i f y the e f f e c t of background p r e d a t o r s and p a r a s i t e s , a l t h o u g h p r o b a b l y l i m i t e d i n scope, may be i m m e d i a t e l y a p p l i c a b l e . The p l a n t breeder would c o n c e n t r a t e on the f a c t o r s he can a d j u s t t o d e c r ease r and i i n e q u a t i o n s 7.13 and 7.14, p a r t i c u l a r l y a t the b e g i n n i n g of the d i s e a s e season. T h i s might i n v o l v e s y n c h r o n i z i n g the appearance of c e r e a l t i s s u e w i t h h i g h d e n s i t i e s of n a t u r a l enemies of r u s t to i n c r e a s e the r a t e of a t t a c k on r u s t p u s t u l e s , A; or c o n c e n t r a t i o n on h i g h l e v e l s of host r e s i s t a n c e e a r l y i n the d i s e a s e season t o d e c r e a s e the r a t e of spore p r o d u c t i o n , B, and i n c r e a s e the ' n a t u r a l ' , m o r t a l i t y r a t e , D; or u s i n g p a r t i c u l a r p a t t e r n s of r e s i s t a n c e gene deployment i n space ( F r e y , et a l . , 1973) t o d e c r e ase the t r a n s m i s s i o n f a c t o r s , T and T . M o d i f i c a t i o n of I L the a g r i c u l t u r a l m i c r o h a b i t a t t o b e n e f i t the background p r e d a t o r s and p a r a s i t e s might be f e a s i b l e t h r o u g h changes i n the m a s s - h a r v e s t i n g p r a c t i c e s or changes i n the p h e n o t y p i c p r o p e r t i e s of i n d i v i d u a l c e r e a l p l a n t s (DeBach, 1974). Hagen and H a l e (1974) have suggested t h a t the p r o v i s i o n of p a r t i c u l a r r e q u i r e m e n t s of the n a t u r a l enemies such as a l t e r n a t e h o s t s or p rey may prove u s e f u l i n c e r t a i n s i t u a t i o n s . 195 Whatever the means, m a n i p u l a t i o n of the a g r i c u l t u r a l environment must overcome the s h o r t c r o p p i n g p e r i o d and the r o t a t i o n p r a c t i c e s o f t e n a s s o c i a t e d w i t h c e r e a l c u l t u r e . These commonly d i s r u p t the b u i l d u p of n a t u r a l enemy p o p u l a t i o n s (Hagen, et a l . , 1976). I n t e g r a t e d c o m b i n a t i o n s of e n v i r o n m e n t a l m a n i p u l a t i o n s , a r t i f i c i a l b o o s t s , and i n t r o d u c t i o n s t o the complex of background p r e d a t o r s and p a r a s i t e s may p r o v i d e the b e s t c o n t r o l i n many a e r o b i o l o g i c a l zones. P e s t r e g u l a t i o n by n a t u r a l enemies i s s u i t e d t o the i n c r e a s i n g l y p r e v a l e n t l a r g e s c a l e c e r e a l a g r i c u l t u r e which can a f f o r d t o d e v e l o p l o n g - t e r m r e s e a r c h programs. Indeed Hagen e_t a l . , (1976) l i s t s e v e r a l examples where the m a n i p u l a t i o n of n a t u r a l enemies has been used t o c o n t r o l g r a i n p e s t s . Summary A m a t h e m a t i c a l model demonstrates t h a t a complex of polyphagous n o n - s y n c h r o n i z e d p r e d a t o r s and p a r a s i t e s i s l i k e l y t o c o n t r o l o n l y low d e n s i t y c e r e a l r u s t p o p u l a t i o n s . N o n e t h e l e s s , c o n t r o l a t low r u s t d e n s i t y can d e l a y epidemic development and thus reduce y i e l d l o s s e s . Methods of l e n g t h e n i n g t h i s d e l a y a r e d i s c u s s e d . These methods i n v o l v e a d j u s t i n g the complex of background n a t u r a l enemies d i r e c t l y or changing some c h a r a c t e r i s t i c s of the a g r i c u l t u r a l environment t o a l l o w the complex t o a s s e r t i t s e l f more f o r c e f u l l y . 196 TABLE 7.1 D e f i n i t i o n s of the v a r i a b l e s and parameters of the model (7.8) l i s t e d i n a l p h a b e t i c a l o r d e r . U n i t s of measurement f o l l o w i n p a r e n t h e s e s . A = maximum a t t a c k r a t e ( p u s t u l e s / a r e a / t i m e ) B = r a t e of spore p r o d u c t i o n per p u s t u l e ( s p o r e s / p u s t u l e / t i m e ) D = i n t r i n s i c r a t e of ' n a t u r a l ' r u s t m o r t a l i t y e x c l u d i n g l o s s e s t o p r e d a t i o n or p a r a s i t i s m ( t i m e " 1 ) E = i n s t a n t a n e o u s e m i g r a t i o n r a t e ( t i m e - 1 ) I = r a t e of spore i m m i g r a t i o n per u n i t a r e a ( s p o r e s / a r e a / t i m e ) N = p o p u l a t i o n d e n s i t y of the c e r e a l r u s t ( p u s t u l e s / a r e a ) T = time T , T = t r a n s m i s s i o n f a c t o r s f o r l o c a l and immigrant L i ' s p o r e s , r e s p e c t i v e l y . These g i v e the p r o p o r t i o n of s pores which s u c c e s s f u l l y i n i t i a t e new p u s t u l e s i n the a e r o b i o l o g i c a l zone of c o n c e r n , ( p u s t u l e / s p o r e s ) v a l u e of N a t which the r a t e of l o s s e s t o the n a t u r a l enemy complex, f (N), i s A/2 i n P ( 7 . 7 ) . ( p u s t u l e s / a r e a ) . 1 9 8 FIGURE FIGURE 7.1 P l o t s of the g r o s s p r o d u c t i o n r a t e , g ( n ) , and the r a t e of l o s s e s t o background n a t u r a l enemies, &(n), a g a i n s t the r e l a t i v e r u s t d e n s i t y , n. Here, as i n F i g u r e s 7.2, 7.3, and 7.4, n and n a r e the endemic and E R r e l e a s e d e n s i t i e s , r e s p e c t i v e l y , f o r i = .1 and r = .3. FIGURE 7.2 The r a t e of change of the r e l a t i v e d i s e a s e i n c i d e n c e a g a i n s t the r e l a t i v e d i s e a s e i n c i d e n c e , n, w i t h r e s p e c t t o the r e s c a l e d time t ( e q u a t i o n 7.9). R e l a t i v e d e n s i t i e s n and n are d e f i n e d as i n F i a u r e 7 . 1 . R FIGURE 7.3 The e f f e c t of the r e l a t i v e e x p o n e n t i a l growth r a t e , r , on the r u s t endemic and r e l e a s e d e n s i t i e s when i = .1. The s o l i d and dashed c u r v e s r e p r e s e n t the l o c u s of the endemic and r e l e a s e d e n s i t i e s r e s p e c t i v e l y . D e n s i t i e s n E and n R c o r r e s p o n d w i t h F i g u r e s 7.1, 7.2, and 7.4. When r = .3 the i n c r e a s e i n n needed t o r e l e a s e t h e r u s t p o p u l a t i o n from n a t u r a l enemy c o n t r o l i s An. The 199 c r i t i c a l p o i n t , where An = n - n = 0, i s R E c i r c u m s c r i b e d by the c i r c l e . FIGURE 7.4 The e f f e c t of the s c a l e d i m m i g r a t i o n r a t e , i , on the r u s t endemic and r e l e a s e d e n s i t i e s when r = .3. The s o l i d and dashed c u r v e s and An a r e a l l d e f i n e d as i n F i g u r e 7.3. D e n s i t i e s n and n c o r r e s p o n d w i t h F i g u r e s E R 7.1, 7.2, and 7.3. The c i r c l e c i r c u m s c r i b e s t h e c r i t i c a l p o i n t which i s d i f f e r e n t from the one i n F i g u r e 7.3. FIGURE 7.5 The l o c u s of c r i t i c a l v a l u e s . The complex of background p r e d a t o r s and p a r a s i t e s p r e v e n t s u n l i m i t e d r u s t i n c r e a s e f o r a l l combina t i o n s of i and r l y i n g u nderneath the c u r v e , F i g u r e s 7.3 and 7.4 c o r r e s p o n d t o s l i c e s t a k e n a l o n g i = .1 and r = .3 r e s p e c t i v e l y . The r e s p e c t i v e c r i t i c a l p o i n t s a re i n d i c a t e d by the c i r c l e d numbers 3 and 4. FIGURE 7.6 Magnitudes of i and r p l o t t e d a g a i n s t the r e l a t i v e r u s t d e n s i t y a t the c r i t i c a l p o i n t . The symbols, 3 and 4, on the n a x i s c o r r e s p o n d w i t h F i g u r e 7.5 i n i d e n t i f y i n g the p o s i t i o n s of the c r i t i c a l v a l u e s from F i g u r e s 7.3 and 7.4 r e s p e c t i v e l y . FIGURE 7.7 D i s e a s e p r o g r e s s c u r v e s . C e r e a l r u s t p o p u l a t i o n d e n s i t y , N, i s p l o t t e d a g a i n s t t i m e , T, a c c o r d i n g t o 7.8 when Z = 1, IT I .1, and BT - D - E = .5. Two v a l u e s of the L maximum a t t a c k r a t e a re compared: the s o l i d l i n e has A = 1 and the dashed l i n e has A= 0 i n d i c a t i n g a c o m p l e t e l y i n e f f e c t i v e complex of background p r e d a t o r s and p a r a s i t e s . When A = 1, e q u a t i o n 7.14 has i = .1 and 7.13 has r = .5 which i s s l i g h t l y g r e a t e r than the c r i t i c a l v a l u e ( F i g u r e 7.4). T h i s c r i t i c a l r v a l u e o c c u r s at N = .46 when Z = 1 and i = c .1. F i g u r e 7.7 shows t h a t the presence of an e f f e c t i v e complex of background n a t u r a l enemies can s u b s t a n t i a l l y r e t a r d d i s e a s e p r o g r e s s (when T = 32, N > 32 m i l l i o n f o r A = 0 ) . A l l parameter v a l u e s were chosen f o r ma t h e m a t i c a l c o n v e n i e n c e . 2 0 1 F I G U R E 7 . 1 magnitude (dimensionless) O I FIGURE 7.2 2 0 3 F I G U R E 7 .3 FIGURE 7.4 r = .3 locus of rig- \ locus of n| 0 critical point R An I n (dimensionless) FIGURE 7 . 5 FIGURE 7.6 (SS9|U0!SU9UJip) 9pn| ju6DLU FIGURE 7.7 0 1 10 2 0 30 T (time) PART V C O N C L U S I O N S CHAPTER 8 CONCLUSIONS: THE BROADER CONTEXT 210 B r i e f Summary of Purpose, Contribution, and Limitations of the Research A number of s p e c i f i c roles can be attributed to s c i e n t i s t s i n general. A common element to each of those roles i s the urge to understand how nature -works. Once a natural phenomenon has been observed and described, the s c i e n t i s t synthesizes, organizes, and summarizes the information i n various ways to provide new perspectives. New insight i s gained from these new perspectives, and from t h i s new insight new hypotheses are ventured as conjectural explanations. The s c i e n t i s t then deduces the l o g i c a l consequences of these new hypotheses and compares them with the natural phenomenon under study. This comparison provides a test of his understanding and the results of t h i s test provide new information which can be combined with previous information as input to the next cycle through the s c i e n t i f i c method. This thesis has aimed at r e v i s i n g and extending previous explanations of the dynamic relationships between cereals and t h e i r r u s t s . This research has been carried out on a broad front: In terms of population genetics I have considered both s p e c i f i c oligogenic interactions (chapter 3) and polygenic interactions (Chapter 5); i n terms of population dynamics, I have considered the d e t a i l s of the processes of disease progress (chapter 2 ) , the effects of a higher trophic l e v e l (Chapter 7), and s p a t i a l effects (Chapter 6 ) ; 1 have also considered the interaction of population genetics and population dynamics (Chapters 4 and 5). However, broad as t h i s front may be, nowhere has i t been pierced to any great depth; nowhere has the s c i e n t i f i c method been taken f u l l 211 cycle by t e s t i n g the revised and extended explanations i n the f i e l d . Rather, the s k i l l s of the author have l i m i t e d him to suggesting approaches to f i e l d t e s t i n g . Nonetheless, Browning (Chapter l ) , Zadoks and Schein (Chapter l ) , and Stakman (Part IV: Introduction), a group comprising some of history's most important contributors to plant epidemiology, have a l l been quoted as either e x p l i c i t l y or i m p l i c i t l y endorsing t h i s type of broad approach. In ecological terms, the accompanying lack of depth might be viewed as a resource a l l o c a t i o n problem. Leonard (1980) has observed, "In science, innovative thinking i s not always correct, but i t i s nearly always valuable." Hopefully the research described above w i l l prove as valuable in stimulating further investigation as has the work which was c r i t i c i z e d above. In conclusion, t h i s research i s considered i n a broader context i n two respects: F i r s t , as a contribution towards a comprehensive program f o r developing new approaches "to cereal rust management; and second, as an indicator of the p o t e n t i a l f o r ecological research i n plant epidemiology. Comprehensive Approach to Cereal Rust Management Besides monoculture and the four approaches studied i n Part IV, a few other strategies f o r cereal rust management have also been pro-posed. However, l i k e the m u l t i l i n e approach (chapter 4 ) , most of these other strategies r e l y on using s p e c i f i c resistance genes to impose disruptive selection on a temporal and/or s p a t i a l basis ( c f . Person et a l . , 1976). Fungicides are an exception but they have generally proven to 212 be uneconomic for cereal rust control (Wain and Carter, 1972). The positive feedback loop f o r disease spread, whereby l o c a l disease progress increases the contamination of other l o c a l i t i e s and thus creates more l o c a l disease progress, etc., was outlined i n Chapter 1. The success of the t r a d i t i o n a l monoculture strategy depends on having a poorly adapted rust population everywhere. A single combination of crop resistance genes i s r e l i e d upon to prevent entrance into the positive feedback loop of disease spread. But once a viru l e n t rust race arises anywhere, the positive feedback loop can quickly spread i t to other l o c a l i t i e s and thus cause devastating y i e l d losses. Local f a i l u r e of resistance can quickly grow into a regional, or even continental, problem because nothing backs up the monoculture's resistance. In Part IV, four p o t e n t i a l methods of providing such support were investigated. The use of natural enemies (Chapter 7), horizontal resistance (Chapter 5 ) , and s p e c i f i c f i e l d geometries (Chapter 6) could a l l be used to support either monoculture or multiline (Chapter 4) deployment of s p e c i f i c resistance genes. Furthermore, there i s no obvious reason why these potential back-up strategies, when combined, should be mutually i n h i b i t o r y (as the use of fungicides and natural enemies might be). For example, i f long narrow f i e l d s are used, as recommended i n Chapter 6, and i f the s t r i p s between f i e l d s provide habitat for mycophagus insects (and not cereal pests), then having a l l growing areas r e l a t i v e l y close to natural enemy habitat might enhance the effect of natural enemies. 2 1 3 In contrast to using the pure monoculture approach where one aims at preventing entry into the positive feedback, loop f o r disease spread, i n using the back-up strategies one regards entry into t h i s loop through the eventual breakdown of resistance as inevitable and aims at i n h i b i t i n g the positive feedback loop i t s e l f . This i n h i b i t i o n would be effected by delaying and/or reducing the rate at which disease develops l o c a l l y . In turn, these delays and rate reductions would be achieved by imposing v a r i a b i l i t y ( i n space through, e.g., f i e l d design; i n time through, e.g-j changing m u l t i l i n e composition; i n species d i v e r s i t y through, e.g_., natural enemies; and i n genetic d i v e r s i t y through, e.g., using multilines and polygenic resistance) and exp l o i t i n g whatever i n s t a b i l i t i e s are associated with these forms of v a r i a b i l i t y (e.g., Chapters 4 , 6, and 7 )• Furthermore, by replacing monoculture by mult i l i n e gene deploy-ment, one can t h e o r e t i c a l l y s h i f t the associated r i s k structure. In monoculture the probability of any s i g n i f i c a n t rust damage i n any year may be r e l a t i v e l y low, but when i t does occur i t can quickly become severe because of the positive feedback loop for disease spread. In contrast, the l i k e l i h o o d of some races being adapted to some m u l t i l i n e components i s t h e o r e t i c a l l y much higher, but because t h e i r a b i l i t y to spread i s supposedly much reduced, we expect that mu l t i l i n e s w i l l suffer some rust more often but that they w i l l suffer severe rust less often. This s h i f t i n r i s k structure should be welcomed " i n the r e l a t i v e l y common si t u a t i o n where the farmer i s r i s k averse"; that i s where the farmer w i l l sustain a s l i g h t decrease i n his long term 214 average y i e l d i n order to prevent occasional but catastrophic e p i -demics (Conway, 1977)• For the same reason, the mu l t i l i n e approach should be used with as much background genetic heterogeneity as i s agronomically acceptable. This should provide some "buffering" against non-target pests and pathogens and against environmental uncertainty (Wolfe, 1978). The research and development required to implement any of the cereal rust management strategies studied above w i l l have various expenses associated with i t . Hence, i t may seldom be worthwhile to investigate more than one of them i n d e t a i l f o r any pa r t i c u l a r time and place. Moreover, the epidemological environment may preclude the usefulness of certain strategies i n certain agro-ecosystems altogether. Nonetheless, conditions change over time and space so that even when a l l strategies can't be used or aren't needed simultaneously, a dif f e r e n t time and/or place may require a different combination of strategies. Under these circumstances i t would be useful f o r any grower to have easy access to guidelines and descriptions concerning the basic technology f o r each strategy. This would help i n i d e n t i -f y i n g and implementing the appropriate combination of strategies and thus could provide substantial savings i n time, expense, and human suffering. National or international i n s t i t u t i o n s may be the most appro-priate depositories f o r guidelines and descriptions of the basic technology associated with each cereal rust management strategy. To best provide up-to-date guidelines and descriptions these i n s t i t u t i o n s should themselves be leaders i n developing t h i s technology. One 21S approach to such leadership might be f o r an i n s t i t u t i o n to i d e n t i f y p a r t i c u l a r locations within i t s j u r i s d i c t i o n where immediate useful-ness and promise f o r each i n d i v i d u a l strategy exists and then develop i t s associated technology 'under f i r e ' . Eventually the i n s t i t u t i o n would be able to provide technological expertise on an array of strategies. Any grower could then i d e n t i f y the combination which appeared most suitable t o h i s conditions (economic, environmental, technological, s o c i a l , . . . ) and r e a d i l y acquire up-to-date knowledge of the technology involved i n implementing that combination. Ecological Research i n Plant Epidemiology Ecology i s the study of the habits and modes of l i f e of l i v i n g organisms and of t h e i r relationships with t h e i r environment (Fowler and Fowler, 1975). By t h i s d e f i n i t i o n plant epidemiology ( i n both natural and a g r i c u l t u r a l ecosystems) i s a branch of ecology. But f o r a l l i t s importance, ecologists have shown surprisingly l i t t l e interest i n plant epidemiology. To be sure there has been some e f f o r t , but i t has been small compared to the e f f o r t s of ecologists evident i n f i s h e r i e s management, or the control of insect outbreaks i n forestry and agriculture, or studies of eutrophication and a l g a l blooms, or f o r e s t - f i r e prevention, or range management. For instance, i n his review of e c o l o g i c a l work on widely f l u c t u a t i n g populations, H o l l i n g (1973) c i t e s work i n a l l of these areas except plant epidemiology. Agriculture offers ecologists the opportunity to investigate the effects of w e l l defined (e.g., species or race s p e c i f i c ) pertur-bations of many kinds on a variety of time and space scales i n 2 1 6 r e l a t i v e l y simple, but nonetheless s o c i a l l y very important, ecosystems. One reason for the s c i e n t i f i c appeal of agro-ecosystems i s t h e i r sim-p l i c i t y . This s i m p l i c i t y i s manifest i n the r e l a t i v e l y small trophic webs, i n the largely man-determined and therefore quite w e l l known population dynamics of a pr i n c i p l e species (e.g., crop, grazer), and i n the regular and r e l a t i v e l y homogeneous s p a t i a l patterns. In addi-t i o n , many plant pathosystems feature the extraordinarily simple and rather w e l l documented gene-for-gene genetic feedback rel a t i o n s h i p . This r e l a t i v e s i m p l i c i t y of agro-ecosystems should be par t i c u -l a r l y appealing to the mathematical modelers of th e o r e t i c a l ecology. Standard i n i t i a l assumptions which grossly oversimplify natural eco-systems may be appropriate f o r agro-ecosystems (e.g., the regular shape and constant size of patches in a patchy habitat i n Chapter 6 ). With agro-ecosystems, as opposed to natural ecosystems, the modeler i s not as limited to studying small pieces of the whole; h o l i s t i c understand-ing i s much more accessible. Besides the s i m p l i c i t y of agro-ecosystems i n general, many plant pathosystems have another property of perhaps even greater potential interest to ecologists. Many economically and/or s o c i a l l y important renewable resource systems have proven to be extremely d i f f i c u l t to manage because system behavior often evolves into patterns almost d i a -m etrically opposed to that intended by management. This has prompted intense interest i n the s t a b i l i t y properties of ecological systems and i n how these properties change under p a r t i c u l a r management regimes 217 (e.g., H o l l i n g , 1973)' In general, such studies have recommended that management redefine i t s objectives (e.g., f i s h i n g at rates below the maximum sustainable y i e l d rate) to allow for uncertainty i n the changes of system s t a b i l i t y properties (e.g., H o l l i n g , 1973). An alternative approach to t h i s problem has been undertaken here with regard t o cereal:cereal rust pathosystems. Rather than a change i n management objectives, alterations of system structure have been suggested i n the hope that these alterations w i l l allow the evolution of system s t a b i l i t y properties t o be directed and used to achieve the o r i g i n a l management objectives. Plant pathosystems in pa r t i c u l a r and perhaps agro-ecosystems i n general offer appropriate grounds f o r studying the e f f i c a c y of system redesign to meet such problems. Af t e r a l l , system redesign through changes i n host r e s i s -tance i s s t i l l the conventional approach to disease management i n many plant pathosystems. To a large extent the meeting of ecology and plant epidemi-ology i s mutually b e n e f i c i a l . Zadoks and Schein (198O) argue that ecology can provide plant epidemiology with a necessary orientation towards p r i n c i p l e s at the population l e v e l . Plant epidemiology, i n turn, can provide ecology with additional s o c i a l relevance and a variety of well-defined perturbation experiments i n r e l a t i v e l y s i m p l i s t i c systems unique to ecology. 218 LITERATURE CITED A l e x a n d e r , R.D. and G. B o r g i a . 1978. 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Kampmeijer. 1977. The r o l e of c r o p p o p u l a t i o n s and t h e i r deployment, i l l u s t r a t e d by means of a s i m u l a t o r , EPIMUL 76. Ann. N.Y. Acad. S c i . 287: 164-190. Zadoks, J . C , and R.D. S c h e i n . 1980. E p i d e m i o l o g y and p l a n t -d i s e a s e management, the known and the needed. Pages 1-17 i n J . P a l t i and J . Kranz (eds.) Comparative E p i d e m i o l o g y : T o o l f o r B e t t e r D i s e a s e Management. Pudoc, Wageningen, The N e t h e r l a n d s . 2 39 APPENDIX A: : DETERMINATION OF CRITICAL FLUCTUATIONS, A n, AND CRITICAL POINTS, r c AND i c . The intensity of instantaneous change i n n , the dimensionless c o e f f i c i e n t of rust population density, required to release an endemic population from natural enemy regulation has been defined as A n . The magnitude of An i s the difference between the two roots ( n^ > n^) of (10). Solving for n in $ = g(n) - £(n) = r n + i - n ( l + n)" 1 = 0, we find An = n R - n^ = >/(r + i - l ) 2 - 4ri '/r. At the c r i t i c a l parameter values, r c and i , the curves g(n) and 2.(n) of F'jig. 1 are equal to each other in slope and magnitude. Thus c r i t i c a l values can be deduced from the simultaneous solution of the equations g(n) = l(n) and d g(n) = d £(n) dn dn The relationship between n, r, and i at a c r i t i c a l point follows: r c M l + n ) " 2 and i c = n 2 (1 + n ) ~ 2 = n 2 r c . 

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