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Resource recovery delays in harvested predator-prey models Griesmer, Stephen Joseph 1981

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RESOURCE RECOVERY DELAYS IN HARVESTED PREDATOR-PREY MODELS by STEPHEN J . GRIESMER B.S.E. P r i n c e t o n Univ. 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of I n t e r d i s c i p l i n a r y S t u d i e s ) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA 15 October 1981 (c) Stephen J . Griesmer, 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f 7JV)-f-P-rcLie.aplinear IT The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V ancouver, Canada V6T 1W5 Date Qc-Ub^- ? 3 f 11*1 i i ABSTRACT T h i s t h e s i s i n v e s t i g a t e s the e f f e c t of resource recovery d e l a y s , both continuous and d i s c r e t e , on the behaviour and p o p u l a t i o n dynamics of harvested predator-prey systems. Three models were t e s t e d with re s p e c t to l o c a l s t a b i l i t y , g l o b a l s t a b i l i t y , and state-space t r a j e c t o r i e s . The method of D-p a r t i t i o n s was used to determine l o c a l s t a b i l i t y p r o p e r t i e s while s i m u l a t i o n s were performed to d i s c o v e r g l o b a l p r o p e r t i e s . G e n e r a l l y , delays were found to induce m u l t i p l e t r a n s i t i o n s between l o c a l l y s t a b l e and unstable e q u i l i b r i a as harvest r a t e s are changed. T h i s phenomenon occurred under the management s t r a t e g i e s of c o n s t a n t - e f f o r t h a r v e s t i n g and constant-quota s t o c k i n g and h a r v e s t i n g . In a d d i t i o n , i n the predator-prey models with d e l a y , a high c o n v e r s i o n e f f i c i e n c y and a high s a t i a t i o n l i m i t f o r the predators can e f f e c t i n s t a b i l i t y ; t h i s i s not p o s s i b l e without d e l a y s . G l o b a l i n v e s t i g a t i o n s confirmed the l o c a l r e s u l t s and extended them to the simultaneous h a r v e s t i n g of the predator and prey f o r constant-quota h a r v e s t i n g and s t o c k i n g . The s i m u l a t i o n s a l s o showed b i f u r c a t i o n s i n l i m i t c y c l e s as the c a r r y i n g c a p a c i t y was a l t e r e d and as the s t o c k i n g r a t e s were v a r i e d . N u t r i e n t - l i m i t e d phytoplankton c u l t u r e s i n chemostats were analyzed with the mathematical techniques. I t i s shown that m u l t i p l e s t a b i l i t y t r a n s i t i o n s with changes in d i l u t i o n r a t e s are t h e o r e t i c a l l y p o s s i b l e but observed d e l a y s are not l a r g e enough to produce t h i s behaviour. i i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS x i i I n t r o d u c t i o n 1 L i t e r a t u r e Review 8 I. Methods and models 18 I n t r o d u c t i o n to l o c a l s t a b i l i t y 18 General model and assumptions 21 The D - p a r t i t i o n method 26 A s i n g l e - s p e c i e s example 30 Q u a s i - c h a r a c t e r i s t i c equations 38 Models i n v e s t i g a t e d 40 I I . Prey h a r v e s t i n g and delay 45 Constant-quota prey h a r v e s t i n g 45 C o n s t a n t - e f f o r t prey h a r v e s t i n g 65 Prey s t o c k i n g 79 Conc l u s i o n s 82 I I I . Predator h a r v e s t i n g and s t o c k i n g and delay 90 C o n s t a n t - e f f o r t predator h a r v e s t i n g 90 Constant-quota predator h a r v e s t i n g 110 Predator s t o c k i n g 136 Con c l u s i o n s 160 IV. Coexistence regions and g l o b a l s t a b i l i t y 163 i v I n t r o d u c t i o n 163 Methods 163 R e s u l t s 164 Conclusi o n s 217 V. Smoothed delays 219 I n t r o d u c t i o n 219 Method 223 R e s u l t s 224 Conclusi o n s 253 VI. Time delays i n chemostats 255 I n t r o d u c t i o n 255 Non-delayed model 258 Delayed model 259 D i s c u s s i o n 275 Con c l u s i o n 286 Con c l u s i o n 288 References c i t e d 295 Appendix A 310 The D - p a r t i t i o n method 310 Appendix B 318 Numerical i n t e g r a t i o n a l g o r i t h m 318 V LIST OF TABLES Table ( 2 . 1 ) — Equations f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and constant-quota prey h a r v e s t i n g 47 Table ( 2 . 2 ) — Equations f o r H o l l i n g ' s model with a type-II f u n c t i o n a l response and constant-quota prey h a r v e s t i n g 49 Table (2.3)-- Equations f o r I v l e v ' s model with c o n s t a n t -quota prey h a r v e s t i n g 51 Table (2.4)-- Equations f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and c o n s t a n t - e f f o r t prey h a r v e s t i n g 66 Table (2.5)-- Equations f o r H o l l i n g ' s model with a type-II f u n c t i o n a l response and c o n s t a n t - e f f o r t prey h a r v e s t i n g 68 Table ( 2 . 6 ) — Equations f o r I v l e v ' s model with c o n s t a n t -e f f o r t prey h a r v e s t i n g 70 Table ( 3 . 1 ) — Equations f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and c o n s t a n t - e f f o r t predator h a r v e s t i n g 92 Table ( 3 . 2 ) — Equations f o r H o l l i n g ' s model with a type-II f u n c t i o n a l response and c o n s t a n t - e f f o r t predator h a r v e s t i n g 94 Table ( 3 . 3 ) — Equations f o r I v l e v ' s model with c o n s t a n t -e f f o r t predator h a r v e s t i n g .' 96 Table (3.4)-- Equations f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and constant-quota predator v i h a r v e s t i n g 113 Table ( 3 . 5 ) — Equations f o r H o l l i n g ' s model with a type-II f u n c t i o n a l response and constant-quota predator h a r v e s t i n g 115 LIST OF FIGURES Fi g u r e ( 1 . 1 ) — S i n g l e - s p e c i e s D - p a r t i t i o n with d i s c r e t e delay 33 Fi g u r e (.1.2)-- S i n g l e - s p e c i e s D - p a r t i t i o n with smoothed delay 36 Fig u r e ( 2 . 1 ) — S t a b i l i t y region f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with delay and constant-quota prey h a r v e s t i n g 53 Fi g u r e ( 2 . 2 ) — S t a b i l i t y region f o r H o l l i n g ' s model with delay and constant-quota prey h a r v e s t i n g 56 Fi g u r e ( 2 . 3 ) — Type-II f u n c t i o n a l response 58 Fi g u r e ( 2 . 4 ) — Non-delayed and delayed p a r t i t i o n s with high and low s a t i a t i o n l i m i t s 62 Fig u r e ( 2 . 5 ) — S t a b i l i t y r e g ion f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , delay, and con s t a n t -e f f o r t prey h a r v e s t i n g 74 Fi g u r e (2.6)-- S t a b i l i t y region f o r H o l l i n g ' s model with delay and c o n s t a n t - e f f o r t h a r v e s t i n g 77 Fi g u r e ( 2 . 7 ) — S t a b i l i t y region f o r I v l e v ' s model with delay and c o n s t a n t - e f f o r t h a r v e s t i n g 80 Fi g u r e ( 2 . 8 ) — S t a b i l i t y r e g i o ns with prey s t o c k i n g i n I v l e v ' s model with delay 83 Fi g u r e ( 2 . 9 ) — Delayed and non-delayed p a r t i t i o n s with c o n s t a n t - e f f o r t prey h a r v e s t i n g ( I v l e v ' s model) 86 Fi g u r e (3.1)-- S t a b i l i t y r e g ion f o r the L o t k a - V o l t e r r a v i i i model with resource l i m i t a t i o n and delay with constant-e f f o r t predator h a r v e s t i n g 99 F i g u r e ( 3 . 2 ) — S t a b i l i t y r e g i o n f o r H o l l i n g ' s model with delay and c o n s t a n t - e f f o r t predator h a r v e s t i n g 102 F i g u r e ( 3 . 3 ) — Maximum a l l o w a b l e delay f o r H o l l i n g ' s model with c o n s t a n t - e f f o r t predator h a r v e s t i n g 104 F i g u r e ( 3 . 4 )— S t a b i l i t y region f o r I v l e v ' s model with delay and c o n s t a n t - e f f o r t predator h a r v e s t i n g 106 Fi g u r e ( 3 . 5 ) — Maximum a l l o w a b l e delay f o r I v l e v ' s model with c o n s t a n t - e f f o r t predator h a r v e s t i n g 108 F i g u r e ( 3 . 6 ) — Constant-quota h a r v e s t i n g and s t o c k i n g t r a j e c t o r y f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n 117 Fi g u r e ( 3 . 7 ) — P a r t i t i o n s l i c e s f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , d e l a y , and constant-quota predator h a r v e s t i n g 120 F i g u r e (3.8)-- Constant-quota h a r v e s t i n g and s t o c k i n g t r a j e c t o r y f o r H o l l i n g ' s model 126 F i g u r e ( 3 . 9 ) — P a r t i t i o n s l i c e s f o r H o l l i n g ' s model with delay and constant-quota predator h a r v e s t i n g 130 F i g u r e (3.10)— C o n s t a n t - e f f o r t h a r v e s t i n g and s t o c k i n g t r a j e c t o r y f o r I v l e v ' s model 137 F i g u r e ( 3 . 1 1 )— P a r t i t i o n s l i c e s f o r I v l e v ' s model with delay and constant-quota predator h a r v e s t i n g 139 F i g u r e (3.12)-- P a r t i t i o n s l i c e s f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , d e l a y , and predator s t o c k i n g ..146 F i g u r e ( 3 . 1 3 ) — P a r t i t i o n s l i c e s f o r H o l l i n g ' s model with and without delay and with predator s t o c k i n g 150 Fi g u r e (3.14)-- P a r t i t i o n s l i c e s f o r I v l e v ' s model with delay and predator s t o c k i n g 155 Fi g u r e ( 4 . 1 ) — H-G plane f o r H o l l i n g ' s model with r=2, K=40, J=20, L=10, M=1.r s=1, and T=0.75 165 Fi g u r e (4.2)-- H-G plane f o r H o l l i n g ' s model with r=1, K=60, J=20, L=10, M=1., s=1, and T=1.5 167 Fi g u r e (4.3)-- H-G plane f o r , H o l l i n g ' s model with r=1, K=40, J=20, L=10, M=1., s = 0.5, and T=1.5 169 Fi g u r e ( 4 . 4 ) — H-G plane f o r H o l l i n g ' s model with r=1, K=40, J=20, L=15, M=1., s=1, and T=1.5 171 Fi g u r e (4.5)-- H-G plane f o r H o l l i n g ' s model with r=1, K=40, J=20, L=10, M=3., s = 0.33, and T=1.5 173 Fi g u r e ( 4 . 6 ) — H-G plane f o r I v l e v ' s model with r=2, K=35, J = 20, b=1, c = 0.1, s=1, and T=0.9 175 Fi g u r e ( 4 . 7 ) — H-G plane f o r I v l e v ' s model with r=1, K=40, J = 30, b=1, c = 0.1, s=1, and T=1.5 177 Fi g u r e ( 4 . 8 ) — H-G plane f o r I v l e v ' s model with r=1, K=40, J=20, b=3, c = 0.1, s=0.33, and T=1.5 179 Fi g u r e ( 4 . 9 ) — H-G plane f o r I v l e v ' s model with r=1, K=40, J = 20, b=1, c = 0.05, s=1, and T=1.5 181 Fi g u r e (4.10)-- H-G plane f o r I v l e v ' s model with r=2, K=40, J=20, b=1, c = 0.1, s=1, and T=0.9 183 Fi g u r e ( 4 . 1 1 ) — H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r=2, K=2000, U=0.015, V=0.001, b=0.5, and T=1.1 ....185 F i g u r e ( 4 . 1 2 ) — H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r=1, K=3000, U=0.015, V=0.001, b=0.5, and T=1.5 187 Fi g u r e ( 4 . 1 3 ) — H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r = 4, K=2000, U=0.015, V=0.001, b=0.5, and T=1.5 189 Fi g u r e ( 4 . 1 4 ) — H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r = 1 , K=2000, U=0.015, V=0.001, b=0.3, and T=1.5 191 Fig u r e ( 4 . 1 5 ) — H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r=1, K=2000, u=0.06, V=0.001, b=0.5, and T=1.5 193 Fi g u r e ( 4 . 1 6 ) — Allowable time delay f o r H o l l i n g ' s model .196 Fi g u r e ( 4 . 1 7 ) — L o c a l s t a b i l i t y curves i n the H-G plane f o r H o l l i n g ' s model with T=0, T=0.75, T=0.85 198 Fi g u r e ( 4 . 1 8 ) - - H-G plane f o r H o l l i n g ' s model without delay ( F i g u r e 4.2) 201 Fi g u r e (4.19)-- L i m i t c y c l e f o r H o l l i n g ' s model with delay and no h a r v e s t i n g 204 Fi g u r e (4.20).— L i m i t c y c l e f o r H o l l i n g ' s model without delay and no h a r v e s t i n g 206 Fi g u r e ( 4 . 2 1 ) — L i m i t c y c l e f o r H o l l i n g ' s model with delay and predator s t o c k i n g , G=-2 208 Fi g u r e ( 4 . 2 2 ) — L i m i t c y c l e f o r H o l l i n g ' s model with delay and predator s t o c k i n g , G=-1 210 Fi g u r e ( 4 . 2 3 ) — H-G plane f o r H o l l i n g ' s model with r = 1 , K=40, J=20, L=10, S=1, M=1, T=1.5 214 F i g u r e (5.1)-- Four smoothed d e l a y s .....221 F i g u r e . ( 5 . 2 ) — D - p a r t i t i o n f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , r e c t a n g u l a r delay, and c o n s t a n t - e f f o r t h a r v e s t i n g 226 F i g u r e ( 5 . 3 ) — Comparison of D - p a r t i t i o n s f o r d i s c r e t e and r e c t a n g u l a r delay 228 F i g u r e ( 5 . 4 ) — B i f u r c a t i o n i n the D - p a r t i t i o n f o r the t r i a n g u l a r delay of v a r i o u s widths .• 233 F i g u r e ( 5 . 5 ) — Comparison of D - p a r t i t i o n s f o r d i s c r e t e and 'strong' d e l a y s 238 F i g u r e (5.6)-- Comparison of D - p a r t i t i o n s f o r d i s c r e t e and 'weak' delays 241 F i g u r e (5.7)-- P a r t i t i o n s f o r H o l l i n g ' s model with predator h a r v e s t i n g 245 F i g u r e ( 5 . 8 ) — The CD i n t e r c e p t as a f u n c t i o n of A f o r the smoothed delay k e r n a l s 248 F i g u r e ( 5 . 9 ) — L o c a l s t a b i l i t y boundaries i n the H-G plane f o r H o l l i n g ' s model with the smoothed delays 251 F i g u r e (6.1)-- H a r v e s t i n g t r a j e c t o r y and D - p a r t i t i o n f o r Caperon's model 261 F i g u r e ( 6 . 2 ) — T r a n s i t i o n a l d e l a y s i n n i t r a t e - l i m i t e d chemostats 264 F i g u r e ( 6 . 3 ) — T r a n s i t i o n a l d e l a y s i n s i l i c o n - l i m i t e d chemostats 266 F i g u r e ( 6 . 4 ) — T r a n s i t i o n a l d e l a y s i n p h o s p h o r u s - l i m i t e d chemostats 268 F i g u r e ( 6 . 5 ) — P e r t u r b a t i o n boundary f o r Caperon's model with d i l u t i o n r a t e , w=0.009 per hour 270 F i g u r e ( 6 . 6 ) — P e r t u r b a t i o n boundary f o r Caperon's model with d i l u t i o n r a t e , w=0.02 per hour ....273 F i g u r e ( 6 . 7 ) — Chemostat s i m u l a t i o n runs with d i s c r e t e d e l a y , r e c t a n g u l a r smoothed delay, and without d e l a y ..276 F i g u r e ( A . 1 ) — Mapping between the s plane and the Q(s) plane: example 1 311 F i g u r e ( A . 2 ) — Mapping between the s plane and the Q(s) plane: example 2 316 x i i i ACKNOWLEDGMENTS An a c t i v e f i e l d of science i s l i k e an immense i n t e l l e c t u a l a n t h i l l ; the i n d i v i d u a l almost vanishes i n t o the mass of minds tumbling over each other, c a r r y i n g i n f o r m a t i o n from p l a c e to p l a c e , p a s s i n g i t around at the speed of l i g h t . — L e w i s Thomas The p r e p a r a t i o n of t h i s t h e s i s was supported by many who lended t h e i r ideas and obs e r v a t i o n s to a i d my i n t e l l e c t u a l development. I owe a great debt to my s u p e r v i s o r , Dr. A.C. Soudack, who guided me through t h i s arduous process and read t h i s t h e s i s three times. Through d i s c u s s i o n s and l e c t u r e s , Dr. C a r l Walters and Dr. C.S. H o l l i n g provided me with a f i r m foundation f o r my subsequent r e s e a r c h . In a d d i t i o n , Dr. Walter's i n c i s i v e comments on my f i r s t d r a f t are g r a t e f u l l y acknowledged. I a l s o wish to thank f e l l o w s t u d e n t s : E d i t h Krause, Ken Lertzman, K e i t h Lindsay, John Parslow, Tim Webb, and C h r i s Wood, f o r t h e i r enthusiasm, encouragement, and s t i m u l a t i n g d i s c u s s i o n s , but most of a l l f o r t h e i r f r i e n d s h i p . Tim Webb was of p a r t i c u l a r help i n the c r y s t a l l i z a t i o n of many ideas through c r i t i c a l d i s c u s s i o n s . T h i s t h e s i s was supported by a UBC f e l l o w s h i p . Many people pr o v i d e d u n s e l f i s h emotional and s o c i a l stays f o r the p e r i o d of my study. I a p o l o g i z e that the a t t e n t i o n was of t e n l e s s than r e c i p r o c a l a l t r u i s m on my p a r t . My f a m i l y , roommates, and f r i e n d s l i f t e d my burden more o f t e n than perhaps they r e a l i z e . Without these people, t h i s work would never have been completed. d e d i c a t e t h i s t h e s i s to my f a t h e r . 1 INTRODUCTION Time d e l a y s a r i s e i n the dynamics of n a t u r a l p o p u l a t i o n s through s e v e r a l b i o l o g i c a l mechanisms. Development time and other age-dependent e f f e c t s cause delays i n a p o p u l a t i o n ' s response to environmental c o n d i t i o n s or past p o p u l a t i o n d e n s i t i e s . Species with a g e - s p e c i f i c demographic c h a r a c t e r i s t i c s can t h e o r e t i c a l l y e x h i b i t s u s t a i n e d o s c i l l a t i o n s (Beddington 1974). Consequently, development time and age of ma t u r i t y rank as p o p u l a t i o n parameters that must be measured to enable p r e d i c t i o n of p o p u l a t i o n dynamics. In a d d i t i o n to a g e - r e l a t e d l a g s , a delay known as the resource recovery time occurs when resources f o r the p o p u l a t i o n l i m i t i t s growth because the consumer's demand o u t s t r i p s the r a t e of supply. T h i s d e l a y depends both on the consumer and resource dynamics and cannot be measured d i r e c t l y as a p o p u l a t i o n parameter but only i n d i r e c t l y through observed dynamics of both the consumer s p e c i e s and i t s resource. T h i s i s because the s i z e of the delay i s a complex f u n c t i o n of resource c o n v e r s i o n and resource u t i l i z a t i o n by the consumer as w e l l as the growth r a t e s of both the consumer and the r e s o u r c e . Because of i t s complexity, the delay i s measured by time s e r i e s a n a l y s i s of the past dynamics of the s p e c i e s r a t h e r than through independent e s t i m a t i o n of the l a g . The hypothesis of a resource recovery delay assumes that the food supply must be l i m i t i n g to the growth of i t s p r e d a t o r . If i t i s not, the recovery time w i l l be zero by d e f i n i t i o n . The resource recovery time must be at l e a s t of the same order 2 of magnitude as the growth-rate time scale of the consumer to affect any q u a l i t a t i v e difference in the dynamics of the consumer (May 1974). As a consequence of the assumption of food l i m i t a t i o n , i t follows that the predator cannot be regulated by other mechanisms such as self-regulation (Chitty 1960; Krebs 1978). In t h i s mechanism, populations stop growing because of a decay in the quality of individuals in that population. We define qu a l i t y as any i n d i v i d u a l , p h ysiological, or behavioural c h a r a c t e r i s t i c that a f f e c t s f i t n e s s . Of course, both forms of regulation could exist simultaneously; however, the inclusion of both in a model would confuse their individual e f f e c t s . The question of whether resource l i m i t a t i o n i s important in c o n t r o l l i n g the growth of populations has been a topic of great controversy. An argument against the importance of resource l i m i t a t i o n was presented by Hairston e t . a l . (i960) who reasoned that, because large-scale d e f o l i a t i o n i s not observed, food resource l i m i t a t i o n i s rare among herbivores. The transfer e f f i c i e n c y , defined as the r a t i o of secondary production to primary production, i s about 2 percent in t e r r e s t r i a l ecosystems (Steele 1974, pp. 98). Consequently, Hairston et.a_l. concluded that carnivores control herbivore biomass. A closer inspection of the mechanisms of resource l i m i t a t i o n reveals several weaknesses in t h i s argument. The b i o l o g i c a l control of weeds shows that insects are capable of destroying plants and thereby c o n t r o l l i n g their abundance (Huffaker 1957). The insect abundance w i l l then be adjusted to 3 the abundance of the p l a n t . According to Huffaker (1957), simple o b s e r v a t i o n s of d e f o l i a t i o n or damage of the b i o l o g i c a l l y c o n t r o l l e d Klamath weed by an e c o l o g i s t without knowledge of the h i s t o r y of the i n t e r a c t i o n would l e a d him to conclude that the phytophagous n a t u r a l enemies of the weed w i l l have l i t t l e e f f e c t on i t s d i s t r i b u t i o n or abundance. Only m a n i p u l a t i o n of the resource or the consumer s p e c i e s w i l l u l t i m a t e l y determine whether or not the resource i s l i m i t i n g to the growth of the i n s e c t or whether the i n s e c t l i m i t s the growth of the reso u r c e . One example of a system with resource l i m i t a t i o n was presented by Webb and Moran (1978). These experimenters pruned the thorn t r e e A c a c i a karroo to determine the e f f e c t of h a b i t a t abundance on the louse A c i z z i a r u s s e l l a e , which i s normally present at low abundance l e v e l s . Both s p e c i e s are indigenous to South A f r i c a . The pruning of the t r e e s to ground l e v e l i n c r e a s e d the amount of young f l u s h f o l i a g e , the h a b i t a t f o r A . r u s s e l l a e • The p o p u l a t i o n d e n s i t y of A . r u s s e l l a e i n c r e a s e d t e n - f o l d as a r e s u l t of the man i p u l a t i o n . N a t u r a l enemies, c o m p e t i t i o n , and c l i m a t e were a l s o t e s t e d and found to be i n s i g n i f i c a n t determinants of p o p u l a t i o n dynamics (Hoffman e_t.al. 1975; Webb 1977). The normally low abundance of the i n s e c t i s maintained by the p l a n t on which f l u s h p r o d u c t i o n i s u n p r e d i c t a b l e and i s o l a t e d . In response to t h i s ephemeral, patchy enviroment, A . r u s s e l l a e has developed r - s e l e c t i v e e x p l o i t a t i o n c h a r a c t e r i s t i c s . Webb and Moran's study u n d e r l i n e s s e v e r a l important 4 f e a t u r e s i n p l a n t - h e r b i v o r e i n t e r a c t i o n s i n p a r t i c u l a r and resource l i m i t a t i o n i n g e n e r a l . F i r s t , food may be l i m i t i n g although i t may not seem so because of the c o e v o l u t i o n of the pl a n t and i n s e c t . Some authors b e l i e v e that the s u r v i v a l of pl a n t s p e c i e s from h e r b i v o r e a t t a c k i s mainly due to t h e i r own d e f e n s i v e s t r a t e g i e s and ad a p t a t i o n s (Pimental and Soans 1970; Feeney 1976). Second, the q u a l i t y and type of the food i s s i g n i f i c a n t ; experimental d e f o l i a t i o n of the thorn t r e e s had no s i g n i f i c a n t e f f e c t on A . r u s s e l l a e d e n s i t i e s (Webb and Moran 1978) as i t d i d not in c r e a s e f l u s h abundance. In a d d i t i o n to p h y s i o l o g i c a l q u a l i t y , q u a l i t y i n terms of p l a n t chemistry both with r e q u i r e d n u t r i e n t s and h e r b i v o r e i n h i b i t o r s , o f t e n l i m i t s the e x p l o i t a t i o n of the resource by the h e r b i v o r e (van Emden and Way 1973; Feeney 1976; Lawton and M c N e i l l 1979). Given the e x i s t e n c e of resource l i m i t a t i o n , the hypothesis of resource recovery delay can be t e s t e d . Commonly c i t e d s t u d i e s of delay in density-dependent responses are the l a b o r a t o r y experiments by P r a t t (1943) and Slobodkin (1954) of Daphnia spp. P r a t t ' s experiments with Daphnia magna showed o o s c i l l a t i o n s i n d e n s i t y at 25 C because of delayed d e n s i t y -o dependent m o r t a l i t y . At 18 C, no such o s c i l l a t i o n s were observed. Slobodkin found s i m i l a r o s c i l l a t i o n s i n an experimental p o p u l a t i o n of Daphnia obtusa at 14°C; the o o s c i l l a t i o n s disappeared at 18-20 C. Hutchinson (1967, p. 591) e x p l a i n s that the lower temperature slowed growth and lowered the maintenance metabolism, induc i n g an i n c r e a s e i n the e f f i c i e n c y of food u t i l i z a t i o n and r e p r o d u c t i o n , e n a b l i n g an 5 overshoot i n r e p r o d u c t i o n . Although t h i s accounts f o r Slobodkin's r e s u l t s , i t does not e x p l a i n P r a t t ' s experiment. P r a t t (1943) showed that o s c i l l a t i o n s at 25° C in h i s experiment were due to a density-dependent death r a t e , which d i d not occur at 1 8 C. Frank (1960) more thoroughly i n v e s t i g a t e d l a g s i n Daphnia by f i t t i n g a m o d i f i e d l o g i s t i c model to h i s data. He found that the model with measured volume d e n s i t y , a delayed response of f e c u n d i t y to d e n s i t y , and a g e - s p e c i f i c f e c u n d i t y , growth, and m o r t a l i t y r a t e s e x h i b i t e d the best f i t to h i s data. In the Daphnia experiments, delay was hypothesized to account f o r o s c i l l a t i o n s . However, one should recognize that a system at e q u i l i b r i u m can a l s o have a resource recovery d e l a y . A p e r t u r b a t i o n of e i t h e r p o p u l a t i o n from e q u i l i b r i u m i s necessary to demonstrate the delay. The type of food, q u a l i t y of food, and the degree of damage a l l i n f l u e n c e the magnitude of the d e l a y . The delay i n the spruce budworm ( H o l l i n g e t . a l 1977) i s set by the g e n e r a t i o n time of spruce when budworm d e f o l i a t i o n i s f a t a l to the t r e e s . By c o n t r a s t , the d e l a y f o r the l a r c h bud moth i s set by a f o l i a g e regrowth time s c a l e s i n c e i t causes l i t t l e t r e e m o r t a l i t y ( B a l t e n s w e i l e r e t . a l . 1977). Density-dependent p l a n t defenses (Ryan and Green 1974) can a l s o i n t r o d u c e delays i n the resource a v a i l a b l e to consumers. Resource recovery delay seems to c h a r a c t e r i z e many s i t u a t i o n s i n resource management. In a g r i c u l t u r a l and f o r e s t s i t u a t i o n s , i n s e c t pests a t t a c k crops and t r e e s which have a slower recovery r a t e than the g e n e r a t i o n time of the i n s e c t . 6 Consequently, the a v a i l a b l e resource f o r one g e n e r a t i o n of the i n s e c t i s determined by the consumption or u t i l i z a t i o n of that resource by past g e n e r a t i o n s . The extent to which the growth of the i n s e c t p o p u l a t i o n i s i n h i b i t e d as a r e s u l t of the dim i n i s h e d resource depends on f o r a g i n g , migratory, and o v i p o s i t i o n h a b i t s . A p e r i o d i c i t y of pest outbreaks may r e s u l t from t h i s resource l i m i t a t i o n and d e l a y . The spruce budworm i n eas t e r n Canada serves as one example of the phenomenon of slow resource recovery. The budworm d e f o l i a t e s balsam and spruce t r e e s i n outbreaks o c c u r r i n g every 30 to 44 years ( H o l l i n g e t . a l . 1977). At high budworm d e n s i t i e s , d e f o l i a t i o n causes the t r e e s to d i e , e l i m i n a t i n g o v i p o s i t i o n s i t e s and food f o r f u t u r e g e n e r a t i o n s . If enough t r e e s p e r i s h , the budworm d e n s i t y w i l l plummet because of the slow recovery time of the f o r e s t r e s o u r c e . Indeed, i t has been shown by H o l l i n g that the f o l i a g e and branch d e n s i t y are necessary v a r i a b l e s to enable p r e d i c t i o n of the o s c i l l a t i o n s i n budworm p o p u l a t i o n s ( H o l l i n g e t . a l . 1977). Other examples of i n s e c t - r e s o u r c e systems with a p o t e n t i a l resource recovery delay that have outbreak c y c l e s are the mountain b i r c h B etula  t o r t u o s a - O p o r i n i a autumnata system i n northern F i n l a n d (Whittaker 1979) and the cinnabar moth - ragwort system i n B r i t a i n (Dempster 1971). An i n t e r e s t i n g q u e s t i o n a r i s i n g from the budworm study, which we w i l l be i n t e r e s t e d i n t h i s study, i s what r o l e ' h a r v e s t i n g ' ( i n s e c t spraying) a c t i v i t i e s p l a y in the r e g u l a t i o n of the budworm p o p u l a t i o n and the damping of the 7 outbreak c y c l e . By extending resource recovery delay to models of e x p l o i t e d p o p u l a t i o n s , we w i l l i n v e s t i g a t e the r e g u l a t o r y p o t e n t i a l of h a r v e s t i n g and the extent to which the c o n c l u s i o n s from u n e x p l o i t e d and non-delayed resource recovery models c a r r y over i n t o t h i s new s i t u a t i o n . T h i s study focuses on the e f f e c t of resource recovery time on predator-prey models which i n c l u d e h a r v e s t i n g . We w i l l review the l i t e r a t u r e behind t h i s delay, along with a general summary of other types of delay, and then present our r e s e a r c h and r e s u l t s . 8 LITERATURE REVIEW The t h e o r e t i c a l c o n c l u s i o n s d e r i v e d from non-delayed e c o l o g i c a l models o f t e n change when time d e l a y i s i n c o r p o r a t e d i n t o the model. The b a s i c p r o c e s s e s of d e n s i t y - d e p e n d e n t growth, p r e d a t i o n , c o m p e t i t i o n , m u t u a lism and h a r v e s t i n g i n the presence of d e l a y have a l l been s t u d i e d t o some e x t e n t by pa s t r e s e a r c h e r s . The g e n e r a l i t y of the c o n c l u s i o n s reached depends g r e a t l y on the m e t h o d o l o g i e s used. In some c a s e s , a v a r i e t y of model s t r u c t u r e s have been a n a l y z e d , w h i l e o t h e r s t u d i e s r e l y on r e s u l t s o b t a i n e d from a p a r t i c u l a r model. We w i l l summarize these i n v e s t i g a t i o n s , f o c u s i n g on t h e i r r o b u s t n e s s . In one of the e a r l i e s t papers on time d e l a y s i n e c o l o g y , H u t c h i n s o n (1948) i n t r o d u c e s the l i n k between d e n s i t y - d e p e n d e n t growth and the l a g i n h e r e n t i n t h i s feedback p r o c e s s . As an example, H u t c h i n s o n c i t e s Daphnia p o p u l a t i o n s i n which o s c i l l a t i o n s o c c u r which a r e a t l e a s t i n p a r t due t o the dependence of the f e c u n d i t y of p a r t h e n o g e n i c females on pa s t p o p u l a t i o n d e n s i t i e s t o which they have been exposed. The d e l a y i n t h i s system g i v e s r i s e t o overcompensation by the p o p u l a t i o n , c a u s i n g the d e n s i t y t o o v e r s h o o t i t s e q u i l i b r i u m v a l u e , e n a b l i n g the subsequent o s c i l l a t i o n s . Whether or not the s t a b i l i z i n g d e nsity-dependence can overcome the d e s t a b i l i z i n g r e s o u r c e l a g depends on the i n t r i n s i c r a t e of growth of the consumer s p e c i e s . Wangersky and Cunningham (1957a) e x p l i c i t l y show the i n f l u e n c e of the r e c o v e r y time and the r a t e of growth on the q u a l i t a t i v e b e h a v i o u r of t h e s o l u t i o n of a s i n g l e - s p e c i e s , l o g i s t i c model. 9 If the product of the recovery time and the rate of growth i s increased beyond a certain l e v e l , sustained o s c i l l a t i o n s develop in the dynamics of the population. The importance of these two factors in the analysis of dynamics has been reiterated by studies of difference equations with a one-generation lag, which show that a large i n t r i n s i c rate of growth can lead to periodic behaviour and beyond— to chaotic p e r i o d i c i t i e s where the difference equation exhibits a solution which i s indistinguishable from random fluctuations in population density (Li and Yorke 1975; May 1976a). Chaotic dynamics have not been observed in nature. These two demographic parameters can also be related in a reciprocal fashion; the s t a b i l i t y of a single-species model depends on the r e l a t i v e magnitude of two time scales: the resource recovery time (the delay) and the growth rate time (1/r, where r i s the i n t r i n s i c rate of growth) (May 1973). I n s t a b i l i t y results when the resource recovery time exceeds the growth rate time by a factor of tr/2. One could also interpret these two time scales as those of a host and parasite. The conclusion would be that, i f the recovery time of the host from infection i s much slower than the reproductive rate of the parasite, the interaction would be unstable. While other possible s t a b i l i z i n g influences such as aggregation and f a c i l i t a t i o n have been ignored here, t h i s result does c l e a r l y indicate the d e s t a b i l i z i n g effect of a high host death rate in a slowly growing host population. (Barclay and van den Driessche 1975). 10 Some experimental and d a t a - f i t t i n g a c t i v i t i e s have d e a l t with delayed dynamics. Caperon (1969) f i t s a lagged r e s o u r c e -consumer model to d e n s i t y data from a chemostat p o p u l a t i o n of the algae I s o c h r y s i s qalbana i n a n i t r o g e n - l i m i t e d environment. With a simple smoothed (as opposed to d i s c r e t e ) delay, he was able to tra c k the dynamics of the p o p u l a t i o n when subjected to p e r t u r b a t i o n s ,in n i t r a t e c o n c e n t r a t i o n . T h i s was accomplished without i n t r o d u c i n g unobserved i n c r e a s i n g o s c i l l a t i o n s which were produced by the same model with a d i s c r e t e l a g . Apparently, smoothed la g s are more r e a l i s t i c and s t a b l e than d i s c r e t e ones as i n d i c a t e d by MacDonald (1977). We s h a l l look at t h i s q u e s t i o n i n t h i s t h e s i s . S e v e r a l r e s e a r c h e r s have attempted to e x p l a i n the o s c i l l a t i o n s produced i n Nich o l s o n ' s b l o w f l y experiments (Nicholson 1954). The c y c l e s are thought to be dependent on the delay i n resource r e c o v e r y . May (1974) f i t s N i c h o l s o n ' s data to a delayed l o g i s t i c equation and f i n d s the experimental and t h e o r e t i c a l r e s u l t s match i n the p e r i o d of o s c i l l a t i o n . More r e c e n t l y , Gurney e t . a l . (1980) repeat t h i s e x e r c i s e ; however, they separate the c o n t r i b u t i o n s of the lagged a d u l t recruitment and the non-lagged death r a t e . T h e i r model i s ab l e to mimic the experimental o b s e r v a t i o n s of two d i s c r e t e g e n e r a t i o n s per c y c l e found i n the l a r v a l - l i m i t e d environment and the merging of c y c l e s i n t o a continuous p e r i o d of r e p r o d u c t i v e a c t i v i t y when the a d u l t s are f o o d - l i m i t e d . T h i s study a l s o shows, through s p e c t r a l a n a l y s i s of N i c h o l s o n ' s data, that the c y c l e s are s e l f - s u s t a i n i n g and not simply 11 products of s t o c h a s t i c i t y or measurement u n c e r t a i n t y . O s c i l l a t i o n s can a r i s e through d e l a y s i n density-dependent feedback, although other mechanisms can e f f e c t c y c l i n g as w e l l . Two a l t e r n a t i v e hypotheses were t e s t e d i n the next study. In m o d e l l i n g the dynamics of l a b o r a t o r y p o p u l a t i o n s of the h o u s e f l y , Musca domestica T a y l o r and Sokal (1976) i n c l u d e a development time l a g between o v i p o s i t i o n and e c l o s i o n of a d u l t s i n a s i n g l e - s p e c i e s model. Run with independent parameter est i m a t e s , the model f o l l o w s the observed c y c l e s i n the p o p u l a t i o n , both in p e r i o d and i n amplitude for d i f f e r e n t female f e c u n d i t i e s . The genetic composition of the f l i e s was a l s o monitored to e l i m i n a t e the h y p o t h e s i s that s e l f - r e g u l a t i o n through g e n e t i c m o d i f i c a t i o n i s d r i v i n g the c y c l e s . The drawback to t h i s study i s that only the parameter space with d i v e r g e n t o s c i l l a t i o n s was t e s t e d ; no e q u i l i b r i u m d e n s i t y was obtained to v a l i d a t e the p o s i t i o n of the s t a b i l i t y boundary. Experimental i n v e s t i g a t i o n s of time l a g i n two-species systems have not been conducted i n s p i t e of the f a c t that t h e o r e t i c a l r e s e a r c h of time delays i n predator-prey models a c t u a l l y preceded s i n g l e - s p e c i e s i n v e s t i g a t i o n s . V o l t e r r a (1931) r e a l i z e d that the symmetry found in h i s model of predator-prey dynamics was u n r e a l i s t i c because of the instantaneous changes i n v o l v e d i n the predator's numerical response. He i n t r o d u c e d smoothed l a g s i n t o the i n t e r a c t i o n terms to c o r r e c t t h i s d e f i c i e n c y . U n f o r t u n a t e l y , t h i s change d i d not r e l i e v e the model of i t s s t r u c t u r a l i n s t a b i l i t y . V o l t e r r a 1 s only b i o l o g i c a l l y r e l e v a n t c o n c l u s i o n i n the study 12 was that the p e r i o d i c i t y of the o s c i l l a t i o n s had disappeared with the l a g and that the e q u i l i b r i u m became u n s t a b l e . In a somewhat analogous, although more r e a l i s t i c study, B r e l o t (1931) delayed the i n t e r a c t i o n terms of a L o t k a - V o l t e r r a model that i n c l u d e d density-dependent growth. V o l t e r r a concluded that the s o l u t i o n of the model w i l l be bounded i f c a r r y i n g c a p a c i t y i n the prey equation, which e x e r t s a s t a b i l i z i n g feedback i n f l u e n c e , i s i n c l u d e d i n the e q u a t i o n s . Boundedness im p l i e s some degree of b i o l o g i c a l r e a l i s m i n the s o l u t i o n . As understanding of p r e d a t i o n grew so d i d l a g i n v e s t i g a t i o n s i n predator-prey models. Using resource delay in the prey equation, May (1973) extended s t a b i l i t y r e s u l t s i n v o l v i n g d e s t a b i l i z i n g f u n c t i o n a l responses. F i r s t , he found that the a d d i t i o n of a predator to an unstable prey-resource system can l e a d to s t a b i l i t y because of the lengthening of the n a t u r a l time s c a l e of the system from 1/r to ( r b ) " 1 / * where b i s the i n t r i n s i c death r a t e of the predator and r i s the i n t r i n s i c growth r a t e of the prey. Second, May found t h a t , where the e q u i l i b r i u m prey p o p u l a t i o n i s s u b s t a n t i a l l y l e s s than the c a r r y i n g c a p a c i t y of the environment, the resource-l i m i t a t i o n e f f e c t must be stronger than the d e s t a b i l i z i n g f u n c t i o n a l response, as i n the non-delayed model. In a d d i t i o n , the time delay must be l e s s than the n a t u r a l time s c a l e of the predator-prey system, as i n the s i n g l e - s p e c i e s case. In a complementary study, H e l l e r (1978) compared the e f f e c t s of d i f f e r e n t f u n c t i o n a l responses on simulated p o p u l a t i o n t r a j e c t o r i e s with delayed models. He found that the most 13 s t a b l e f u n c t i o n a l response was a t y p e - I l l ('sigmoid response'), f o l l o w e d by a type-I ( l i n e a r response with a th r e s h o l d ) and type-II ('hyperbolic response'). T h i s s t a b i l i t y o r d e r i n g holds f o r non-delayed models as w e l l (Murdoch and Oaten 1975). His model a l s o showed that a type-II capture r a t e w i l l l e a d to a s t a b l e e q u i l i b r i u m only i f both s p e c i e s are l i m i t e d by other r e s o u r c e s . T h i s c o n c l u s i o n depends on the s p e c i f i c model used; there i s no reason to b e l i e v e that t h i s r e s u l t holds g e n e r a l l y . In non-delayed models, only the prey p o p u l a t i o n need be l i m i t e d f o r s t a b i l i t y (Rosenzweig and MacArthur 1963). Our r e s u l t s c o n t r a d i c t H e l l e r ' s c o n c l u s i o n about the type-II response and c o n f i r m the non-delayed f i n d i n g s . Delays i n the predator r a t e of i n c r e a s e ( V o l t e r r a 1931; B r e l o t 1931; Wangersky and Cunningham 1957a; Cushing 1977), development time, and i n t e r a c t i o n terms have i d e n t i c a l e f f e c t s as d elays i n resource l i m i t a t i o n . In the s i n g l e - s p e c i e s case, May e t . a l . (1974) proved that delay i n development time has the same consequences as the resource l a g . Other models of two-species processes i n which delay has been i n v e s t i g a t e d are competition and mutualism. For competing s p e c i e s , Caswell (1972) showed that delay can reverse the outcome of co m p e t i t i o n (See a l s o Gotatam and MacDonald 1975; Cushing 1977). More r e c e n t l y , Shibata and S a i t o (1980) i n v e s t i g a t e d ' c h a o t i c ' dynamics i n delayed c o m p e t i t i v e systems. They found t h a t , as the time delay i n the s e l f - l i m i t a t i o n terms i s i n c r e a s e d , the dynamics of the system can move from an o s c i l l a t i n g approach to e q u i l i b r i u m , to a p e r i o d i c s o l u t i o n , to 14 a p i t c h f o r k b i f u r c a t i o n , and f i n a l l y to a c h a o t i c s o l u t i o n . Gopalsamy (1980) e s t a b l i s h e s that g l o b a l l y s t a b l e c o e x i s t e n c e i s p o s s i b l e i n the L o t k a - V o l t e r r a c o m p e t i t i o n model with delays i n the i n t e r a c t i o n terms and when density-dependence i s stronger than the c o m p e t i t i v e i n t e r a c t i o n , as measured by the r a t i o of the i n t e r a c t i o n c o e f f i c i e n t s . F o l l o w i n g May (1976b, p. 65-69) and G i l b e r t and Raven (1976), MacDonald (1978) d i s c u s s e s l a g s i n the i n t e r a c t i o n terms of a model of o b l i g a t e mutualism. T h i s model has two e q u i l i b r i u m p o i n t s : an unstable lower-density e q u i l i b r i u m and a s t a b l e h i g h e r - d e n s i t y one. Time delay cannot change the c h a r a c t e r of these e q u i l i b r i u m p o i n t s . With d i a g o n a l term l a g s , the s t a b i l i t y c r i t e r i o n turned out to be too complex to reach any d e f i n i t e c o n c l u s i o n . As d i s c u s s e d e a r l i e r , one other process i n v o l v i n g two s p e c i e s which would seem to be m o d i f i e d i n the presence of delay i s p a r a s i t i s m , p a r t i c u l a r l y because the generation time of the host o f t e n exceeds the g e n e r a t i o n time of the p a r a s i t e . The time delay which a r i s e s i n p a r a s i t i s m i s development time. As d i s c u s s e d by May and Anderson (1978), t h i s l a g may be represented e i t h e r e x t r i n s i c a l l y or i n t r i n s i c a l l y . The i n t r i n s i c form i s of course a delay-dependent rate of p a r a s i t e i n c r e a s e ; the e x p l i c i t form i s given by s p e c i f y i n g the p o p u l a t i o n dynamics of f r e e - l i v i n g i n f e c t i v e stages. For both cases, the s t a b i l i t y c r i t e r i o n depends, as i n other models, on the magnitude of the time s c a l e f o r changes i n the r a t e of l o s s of i n f e c t i v e stages r e l a t i v e to the development time d e l a y . In 15 the same paper, May and Anderson showed that i f the time d e l a y i s s m a l l , the system can be s t a b i l i z e d by o v e r d i s p e r s i o n or aggregation of p a r a s i t e s . As the l a g i n c r e a s e s , t h i s s t a b i l i z a t i o n becomes im p o s s i b l e . The r o l e of aggregation i n s u c c e s s f u l b i o l o g i c a l c o n t r o l programs has been d i s c u s s e d by H a s s e l l (1978). Because of the enhanced r e a l i s m of delay models and t h e i r consequences i n the above e c o l o g i c a l phenomena, one might be i n t e r e s t e d i n e x p l o r i n g the e f f e c t s of h a r v e s t i n g i n time delay models. The management concepts of s t a b i l i t y , t h r e s h o l d harvest r a t e s , maximum s u s t a i n a b l e y i e l d , and optimal approach paths are used to a s c e r t a i n the q u a n t i t a t i v e and q u a l i t a t i v e d i f f e r e n c e s among h a r v e s t i n g s t r a t e g i e s . Whether or not the a d d i t i o n of delay changes these s t r a t e g i e s c o u l d b r i n g to q u e s t i o n the g e n e r a l i t y of the a p p l i c a t i o n of these fundamental concepts. In t h i s v e i n , C l a r k (1976a) s t u d i e d a s i n g l e - s p e c i e s delay model of baleen whales and compares i t s behaviour with the standard l o g i s t i c model with r e a l i s t i c parameter es t i m a t e s . He found that the delay d i d not a f f e c t the optimal e q u i l i b r i u m escapement l e v e l but m o d i f i e d the optimal approach to t h i s e q u i l i b r i u m . In the example, however, numerical i n t e g r a t i o n of the t r a j e c t o r y i n d i c a t e s that the most-rapid approach c l o s e l y approximates the non-delayed optimal approach. In c o n t r a s t to C l a r k ' s r e s u l t s , Kilmer and Probert (1979) present a resource d e p l e t i o n model with a primary resource with recovery delay and a g r a z i n g consumer. They f i n d t h a t a c o n s t r a i n e d approach path to the optimal stock l e v e l i s 16 o s c i l l a t o r y , not asymptotic as the non-delayed most r a p i d approach would r e q u i r e . Kilmer and Probert give an example i l l u s t r a t i n g that delayed dynamics can i n c r e a s e the p r e d i c t e d harvest by n e a r l y 50 percent over that p r e d i c t e d by the non-delayed model. P h y s i c a l l y , the h a r v e s t e r s are t a k i n g advantage of the overshoots i n the stock d e n s i t y caused by the d e l a y . C l e a r l y , more i n v e s t i g a t i o n i s necessary before any d e f i n i t i v e c o n c l u s i o n s can be reached i n the area of optimal c o n t r o l . Two s t u d i e s concerned with the s t a b i l i t y of s i n g l e - s p e c i e s delayed models with h a r v e s t i n g come to c o n t r a d i c t o r y c o n c l u s i o n s . Brauer (1977) i n s e r t s a d i s c r e t e delay i n t o the standard l o g i s t i c model. His a n a l y t i c a l r e s u l t s show that an i n c r e a s e i n harvest rate a l l o w s a g r e a t e r range of d e l a y s f o r which the system i s s t a b l e . A more e c o l o g i c a l l y r e l e v a n t c o n c l u s i o n i s that delayed dynamics can be s t a b i l i z e d through h a r v e s t i n g . Cushing (1977) repeats t h i s e x e r c i s e with a continuous delay and d i s c o v e r s that h a r v e s t i n g cannot s t a b i l i z e an unstable e q u i l i b r i u m . Since both r e s u l t s appear to be c o r r e c t , one must conclude that a q u a l i t a t i v e d i f f e r e n c e e x i s t s between these two types of d e l a y . We w i l l address t h i s q u e s t i o n l a t e r i n t h i s study. Few s t u d i e s combining predator-prey i n t e r a c t i o n s with time delays and h a r v e s t i n g have been presented because of the a n a l y t i c complexity of the problem. Brauer (1977) forwards two numerical examples from p r e d a t o r - p r e y models with a N i c h o l s o n ( l i n e a r ) f u n c t i o n a l response and with a H o l l i n g type-II f u n c t i o n a l response. Brauer s t a t e s that the e f f e c t of prey 17 h a r v e s t i n g on s t a b i l i t y i s completely d i f f e r e n t i n the two models: i n the former, constant h a r v e s t i n g can s t a b i l i z e a delayed system; i n the l a t t e r , i t cannot. T h i s study i s intended to r e s o l v e some of these c o n f u s i n g c o n c l u s i o n s r e g a r d i n g the r e l a t i o n s h i p between h a r v e s t i n g and time delay i n predator-prey systems. F i r s t , we w i l l explore the l o c a l s t a b i l i t y of three d i f f e r e n t i a l equation models of pre d a t o r - p r e y systems: the L o t k a - V o l t e r r a model with resource l i m i t a t i o n i n the prey equation, H o l l i n g ' s model with a type-II f u n c t i o n a l response, and I v l e v ' s model. The mathematical method of D - p a r t i t i o n s was used f o r l o c a l s t a b i l i t y a n a l y s i s . H a r v e s t i n g of both the predator and prey s p e c i e s has been s t u d i e d . The types of h a r v e s t i n g i n c l u d e d are c o n s t a n t - e f f o r t h a r v e s t i n g , constant-quota h a r v e s t i n g , and constant-quota s t o c k i n g . For each of these cases, both a d i s c r e t e and a more r e a l i s t i c , smooth delay w i l l be a p p l i e d . Second, we extend some of these r e s u l t s to a g l o b a l l e v e l u s ing numerical i n t e g r a t i o n of the equa t i o n s . The e f f e c t s of d i f f e r e n t parameters on s t a b i l i t y are i n v e s t i g a t e d . 18 I. METHODS AND MODELS I n t r o d u c t i o n to l o c a l s t a b i l i t y The s t a b i l i t y of an e c o l o g i c a l model answers s e v e r a l c r u c i a l q u e s t i o n s about the behaviour of the model. An unstable e q u i l i b i u m w i l l r e p e l system t r a j e c t o r i e s causing e x t i n c t i o n of one of the s p e c i e s , a l i m i t c y c l e around the e q u i l i b r i u m , or a convergence to another s t a b l e e q u i l i b r i u m of the model. The type of t r a j e c t o r y which a r i s e s determines the po p u l a t i o n dynamics of the i n t e r a c t i n g species under d i f f e r e n t management p o l i c i e s , which i n turn serve as c r i t e r i a f o r judging the s u i t a b i l i t y of the p o l i c i e s . The l o c a l s t a b i l i t y of an e q u i l i b r i u m can supply i n f o r m a t i o n about the recovery time of a p o p u l a t i o n from a small p e r t u r b a t i o n around the e q u i l i b r i u m , from some e x t r i n s i c mechanism such as weather or immigration. In mathematical terms, t h i s recovery time i s the c h a r a c t e r i s t i c r e t u r n time (May e t . a l . 1974; Beddington e t . a l . 1976a; Brauer 1979a). Rel a t e d to the magnitude of the l a r g e s t eigenvalue of the l i n e a r i z e d system, r e t u r n time i n d i c a t e s how q u i c k l y the t r a j e c t o r y w i l l r e t u r n to i t s e q u i l i b r i u m v a l u e . More r i g o r o u s l y , the r e t u r n time measures the time needed f o r the e f f e c t of a d i s t u r b a n c e to be reduced by a f a c t o r of 1/e i n a system with an a s y m p t o t i c a l l y s t a b l e e q u i l i b r i u m . The r e t u r n time a l s o measures the time h o r i z o n over which e x p l o i t a t i o n w i l l a f f e c t the dynamics of the system. The a l g e b r a i c enumeration of s t a b i l i t y c o n d i t i o n s 19 s p e c i f i e s which parameters w i l l enhance s t a b i l i t y and which w i l l endanger i t . Although demographic parameters are assumed to be constant, i n r e a l i t y , they change due to the pressure of n a t u r a l s e l e c t i o n , g e n e t i c d r i f t , or any other mechanism which may modify the genetic composition of the p o p u l a t i o n . These parameters can a l s o be i n f l u e n c e d by the c o n d i t i o n of the environment, a dynamic f a c t o r which i s u s u a l l y hidden i n these c o n s t a n t s . I f a parameter i s absent from the s t a b i l i t y c o n d i t i o n s , we know that i t p l a y s no r o l e i n the q u a l i t a t i v e dynamics of the model and that p e r t u r b a t i o n s i n t h i s parameter are r e l a t i v e l y benign. Moreover, the s t a b i l i t y c o n d i t i o n s o f t e n p rovide a guide f o r condensing s e v e r a l model parameters to reduce complexity (Ludwig e t . a l . . 1 978), perhaps l e a d i n g to a new key parameter by t h i s r e d e f i n i t i o n (See May and Anderson 1978 f o r an example). F i n a l l y , the s t a b i l i t y of a system can guide c l a s s i f i c a t i o n of d i f f e r e n t system components. For i n s t a n c e , May (1974) d i s t i n g u i s h e s between the s t a b i l i z i n g i n f l u e n c e of density-dependence and .the d e s t a b i l i z i n g f u n c t i o n a l response i n predator-prey systems. The u t i l i t y of s t a b i l i t y a n a l y s i s conceals i t s p i t f a l l s . Because e c o l o g i c a l processes and consequently models are predominantly n o n l i n e a r , l o c a l s t a b i l i t y a n a l y s i s dominates the l i t e r a t u r e s i n c e g l o b a l c o n c l u s i o n s are o f t e n not p o s s i b l e without a l a r g e number of s i m u l a t i o n s . As H o l l i n g (1973) and May (1977b) mention, l o c a l a n a l y s i s obscures the o v e r a l l landscape of s t a b i l i t y regions f o r a model i n favor of marking the peaks of the unstable e q u i l i b r i a and the v a l l e y s of the 20 s t a b l e e q u i l i b r i a . L o c a l a n a l y s i s r e v e a l s the nature of the e q u i l i b r i a but not the r e s i l i e n c e of the system as a whole, that i s , the a b i l i t y of the system to absorb d i s t u r b a n c e s from o u t s i d e i n f l u e n c e s ( H o l l i n g 1973). G l o b a l s t a b i l i t y analyses and s i m u l a t i o n models capture t h i s c h a r a c t e r i s t i c b e t t e r . G e n e r a l l y , s t a b i l i t y a n a l y s i s p r o v i d e s few c l u e s about the t r a j e c t o r i e s of s t a t e v a r i a b l e s , although recovery time does f u n c t i o n as a t r a j e c t o r y measure. Kolmogorov's theorem (May 1974) says that predator-prey systems sub j e c t to some c o n s t r a i n t s have e i t h e r a s t a b l e l i m i t c y c l e or a s t a b l e e q u i l i b r i u m p o i n t f o r a l l i n i t i a l c o n d i t i o n s : l o c a l s t a b i l i t y i m p l i e s g l o b a l s t a b i l i t y . However, time-delayed models and h a r v e s t i n g models g e n e r a l l y v i o l a t e the c o n d i t i o n s of t h i s theorem. The e x i s t e n c e of a saddle p o i n t i n a d d i t i o n s to a s t a b l e e q u i l i b r i u m v i o l a t e s the c o n s t r a i n t s of Kolmogorov's theorem. Furthermore, even with the theorem, a r e g i o n of l i m i t c y c l e s and one of s t a b l e nodes cannot be d i s t i n g u i s h e d without a c t u a l numerical c a l c u l a t i o n of the t r a j e c t o r i e s . Mathematical methods f o r determining the e x i s t e n c e of p e r i o d i c s o l u t i o n s such as the Hopf b i f u r c a t i o n technique ( K n o l l e 1976; MacDonald 1977; Cushing 1977) and the Kry l o v -B o g o l i u b o v - M i t r o p o l s k i method (Bojadziev and Chan 1979) have the same l i m i t a t i o n s as l o c a l s t a b i l i t y a n a l y s i s : they only apply around the e q u i l i b r i u m or at boundaries of s t a b l e parameter space. Consequently, the d i s t i n c t i o n between l i m i t c y c l e s and s t a b l e p o i n t s i s best determined v i a s i m u l a t i o n . S t a b i l i t y a n a l y s i s i s only a l i m i t e d a i d i n t h i s r e s p e c t . 21 General model and assumptions Having reviewed the s t r e n g t h s and weaknesses of l o c a l s t a b i l i t y a n a l y s i s , we w i l l proceed to analyze both the l o c a l and g l o b a l s t a b i l i t y c h a r a c t e r i s t i c s of s e v e r a l harvested and delayed predator-prey models, emphasizing how delay m o d i f i e s the p r e d i c t i o n s of non-delayed analyses (Brauer e t . a l . 1976; Brauer and Soudack 1978, 1979). We w i l l begin with an o u t l i n e of the method and then present our r e s u l t s with g r a p h i c a l examples. In t h i s t h e s i s we c o n s i d e r f o r c e d predator-prey equations of the form: dx/dt = r x ( 1 - x ( t - T ) / K ) - f l ( x ) y - H(x,y) dy/dt = f 2 ( x ) y - f3(y) - G(x,y) where x = the d e n s i t y of the prey p o p u l a t i o n , y = the d e n s i t y of the predator p o p u l a t i o n , r = the i n t r i n s i c r a t e of i n c r e a s e of the prey, K = the c a r r y i n g c a p a c i t y of the environment f o r the prey, f l ( x ) = the f u n c t i o n a l response of the prey s u b j e c t to the predator, f2(x) = the numerical response of the predator to prey d e n s i t y , f3( y ) = the death r a t e of the predator, H(x,y) = the harvest r a t e of the prey, G(x,y) = the harvest r a t e of the p r e d a t o r . 22 Many e c o l o g i c a l assumptions are contained w i t h i n these equations. The f o l l o w i n g i s a l i s t of these assumptions and a short e x p l a n a t i o n of t h e i r i n v e s t i g a t e d death r a t e . (1) . Delay i n resource recovery rate We assume that a delay e x i s t s between the rate of inc r e a s e of the prey and the prey density-dependent e f f e c t . (2) . Prey r a t e of increase 1imited by the c a r r y i n g c a p a c i t y of  the environment T h i s i s a common s u b s t i t u t i o n f o r a constant per c a p i t a growth r a t e . T h i s l o g i c a l assumption guarantees t h a t , i n the absence of p r e d a t i o n , the prey w i l l be l i m i t e d by some s o r t of environmental c o n s t r a i n t such as food (See May 1974; P i e l o u 1969, pp.19-21, f o r review). (3) . Random se a r c h i n g by predator The c h o i c e s of f l i n t h i s study r e s t r i c t represented d i s t r i b u t i o n s of prey. As d e s c r i b e d e a r l i e r i n the host-p a r a s i t e d i s c u s s i o n , aggregation of the prey can s t a b i l i z e an unstable predator-prey i n t e r a c t i o n . May and Anderson (1978) d i s c u s s the r o l e of d i s t r i b u t i o n of p a r a s i t e s on h o s t - p a r a s i t e dynamics. G r i f f i t h s and H o l l i n g (1969) and H a s s e l l (1968) give examples of such aggregation by two p a r a s i t e s . Beddington e t . a l . (1978) and H a s s e l l (1978, p. 174) suggest that the patchy d i s t r i b u t i o n of a prey and the d i f f e r e n t i a l e x p l o i t a t i o n of these patches by the predator i s the key to s u c c e s s f u l 23 b i o l o g i c a l c o n t r o l . (4) . No predator i n t e r f e r e n c e Predator i n t e r f e r e n c e i s e s p e c i a l l y p r e v a l e n t when predator aggregation o c c u r s . I t reduces the a v a i l a b l e search time i n p r o p o r t i o n to the frequency of encounters with the prey ( H a s s e l l 1978, p. 81; Beddington 1975). (5) . Numerical response i s a 1inear f u n c t i o n of the number of  prey eaten Although t h i s assumption i s not i m p l i c i t i n the above predator-prey model, a l l of the models analyzed here w i l l c o n t a i n t h i s assumption. H a s s e l l (1978) argues that t h i s assumption i s most a p p r o p r i a t e to h o s t - p a r a s i t e systems as opposed to predator-prey systems. More r e a l i s t i c a l l y , the predator r a t e of i n c r e a s e should be dependent on the development r a t e of the p r e d a t o r , the s u r v i v a l r a t e of the l a r v a e , and the f e c u n d i t y of the a d u l t female (Beddington 1975; Lawton e t . a l . 1975; Beddington e t . a l . 1976b; H a s s e l l 1978). Two of our models, however, c o n t a i n a parameter J which d e f i n e s the minimum number of prey needed f o r a p o s i t i v e r a t e of predator i n c r e a s e which i s s l i g h t l y more r e a l i s t i c than other models. (6) . There are no prey refuges Adding prey refuges to the system s t a b i l i z e s the system. Huffaker's (1958) experiment with two s p e c i e s of mites i s a 24 good example of the i n f l u e n c e of refuges on p o p u l a t i o n dynamics. T h e o r e t i c a l a n a l yses of refuge i n c l u d e Maynard Smith and S l a k i n (1972) and Maynard Smith (1974). A t y p e - I l l f u n c t i o n a l response ( H o l l i n g 1959) e f f e c t i v e l y p r o v i d e s a refuge with a s t a b i l i z i n g rate of a t t a c k at low prey d e n s i t i e s ( H a s s e l l and May 1973). (7) . Only one s p e c i e s of predator and one s p e c i e s of prey are  i n c l u d e d . T h i s assumption l i m i t s the number of i n t e r a c t i n g s p e c i e s to two while e l i m i n a t i n g c e r t a i n e c o l o g i c a l mechanisms from c o n s i d e r a t i o n . One such mechanism i s switching by the predator between prey items which c o n t r i b u t e s to the s t a b i l i z a t i o n of the system (Paine 1966; Royama 1971; May 1977b). Competition for r esources i s a l s o prevented by t h i s assumption. (8) . H a r v e s t i n g types assume constant h a r v e s t i n g m o r t a l i t y . The h a r v e s t i n g types in t h i s model are c o n s t a n t - e f f o r t h a r v e s t i n g and constant-quota h a r v e s t i n g . Constant-quota h a r v e s t i n g removes a f i x e d number i n d i v i d u a l s or i n d i v i d u a l s per area per u n i t time; c o n s t a n t - e f f o r t h a r v e s t i n g removes a f i x e d percentage. I f the h a r v e s t i n g m o r t a l i t y i s known, we have an i n v a r i a n t measure of e f f o r t or y i e l d from these d e f i n i t i o n s ( R o t h s c h i l d 1977). In our models, we assume that the h a r v e s t i n g m o r t a l i t y i s h e l d constant over a long management p e r i o d . Feedback p o l i c i e s are not i n v e s t i g a t e d i n t h i s study. In a c t u a l management p r a c t i c e , measuring the 25 m o r t a l i t y due to h a r v e s t i n g i s d i f f i c u l t (Jones 1977); keeping i t constant i s next to i m p o s s i b l e . The terms constant-quota and c o n s t a n t - e f f o r t suggest that each h a r v e s t i n g type a r i s e s from d i f f e r e n t management p r a c t i c e s . The r e l a t i o n s h i p between h a r v e s t i n g m o r t a l i t y and e f f o r t , measured i n boat-days (S c h a e f f e r 1957) or p e s t i c i d e dosage (Shoemaker 1973), i s o f t e n assumed to be l i n e a r . I f t h i s r e l a t i o n s h i p holds, then c o n s t a n t - e f f o r t h a r v e s t i n g , as d e f i n e d i n the p r e v i o u s sentence, w i l l remove a f i x e d percentage of i n d i v i d u a l s per u n i t time from the stock. In a g r i c u l t u r e and f i s h e r i e s , however, the m o r t a l i t y i s o f t e n not l i n e a r l y r e l a t e d to e f f o r t . In a pest management study, Conway (1976) models r e s i d u a l p e s t i c i d e e f f e c t s which cause n o n l i n e a r i t i e s i n the m o r t a l i t y - e f f o r t curve. In f i s h e r i e s , gear c o m p e t i t i o n , timing of h a r v e s t , v e s s e l power, and stock abundance can skew the l i n e a r r e l a t i o n s h i p ( R o t h s c h i l d 1977). The other management s t r a t e g y c o n s i d e r e d , constant-quota h a r v e s t i n g , i m p l i e s that a f i x e d number of i n d i v i d u a l s are removed per season where the quota i s set by the resource management agency. For a d i s c u s s i o n of an a l t e r n a t i v e management s t r a t e g i e s based on the dynamics of the stock, see Aron (1979). Beddington (1979) i n v e s t i g a t e s the dynamics of a p o p u l a t i o n s u b j e c t to a h a r v e s t i n g f u n c t i o n which i n c o r p o r a t e s a f i n i t e h a n d l i n g time, as observed f o r n a t u r a l p r e d a t o r s . S t a b i l i z i n g mechanisms, besides resource l i m i t a t i o n , were excluded from t h i s study to s i m p l i f y a n a l y s i s of time l a g on 26 the r e s o u r c e - l i m i t e d predator-prey system. In f u t u r e s t u d i e s , other s t a b i l i z i n g i n f l u e n c e s and delay should be i n v e s t i g a t e d . In determining the l o c a l s t a b i l i t y c o n d i t i o n s of p r e d a t o r -prey systems with time delay and h a r v e s t i n g , Brauer (1977) found a l i m i t i n g c o n d i t i o n on the magnitude of the time delay as a f u n c t i o n of the demographic parameters f o r constant-quota prey h a r v e s t i n g . In g e n e r a l , i t i s not yet known how to d e r i v e the closed-form l i m i t i n g c o n d i t i o n s f o r a l l types of h a r v e s t i n g . T h i s should not deter us from i n v e s t i g a t i n g l o c a l s t a b i l i t y through computer s i m u l a t i o n s . Although t h i s approach l a c k s the r i g o r and elegance of f u n c t i o n a l a n a l y s i s , the s i m u l a t i o n r e s u l t s are j u s t as i n f o r m a t i v e from a b i o l o g i c a l s t a n d p o i n t . The D - p a r t i t i o n method The f i r s t step i n l o c a l s t a b i l i t y a n a l y s i s i s the d e r i v a t i o n of l i n e a r i z e d equations f o r our general model. The delayed equations with c o n s t a n t - e f f o r t prey h a r v e s t i n g and constant delay are • dx/dt = r x ( 1 - x ( t - T ) / K ) - f l ( x ) y - Hx (1.1) dy/dt = f 2 ( x ) y - f3(y) (1.2) L i n e a r i z i n g these equations with x(t)=x*+e(t) and y(t)=y*+n(t) where (x*,y*) i s the e q u i l i b r i u m , we o b t a i n de/dt = - ( r x * / K ) e ( t - T ) + r ( 1 - x * / K ) e ( t ) 27 -(f 1 ' (x*)y*+H)e(t) - f K x * ) n ( t ) (1.3) dn/dt = (f2'(x*)y*)e(t) + (f2(x*) - f3'(y*))n(t) (1.4) To simplify these equations, we make the following substitutions A = -rx*/K (1.5) B = r d - x * / K ) - f 1 ' ( x * ) y * - H (1.6) C = f1(x*) (1.7) D = f 2'(x*)y* (1.8) F = f2(x*) -. f3'(y*) (1.9) With these s u b s t i t u t i o n s , (1.3) and (1.4) become de/dt = Ae(t-T) + Be(t) - Cn(t) (1.10) dn/dt = De(t) + Fn(t) (1.11) Because of the delayed term e ( t - T ) , we cannot f i n d the eig e n v a l u e s i n the usual way. But by t r a n s f o r m i n g these equations with the Laplace transform, we can d e r i v e e q u i v a l e n t s t a b i l i t y c o n d i t i o n s . The s t a b i l i t y c o n d i t i o n that the eigenvalues have negative r e a l p a r t s i s transformed to the c o n d i t i o n that a l l roots of the equation Q(s) = det [sI-W(s)] = 0 (1.12) have negative r e a l p a r t s where W(s) i s the Laplace transform of 28 the l i n e a r i z e d e q u a t i o n . The term Q(s) i s c a l l e d the q u a s i -c h a r a c t e r i s t i c e q u a t i o n ( E l ' s g o l ' t s 1966). . The L a p l a c e t r a n s f o r m s of e q u a t i o n s (1.10) and (1.11) are s E ( s ) = A E ( s ) e x p ( - s T ) + BE(s) - CN(s) + e(0) (1.13) s N ( s ) = D E ( s ) + F N ( s ) + n ( 0 ) (1.14) where E ( s ) i s the t r a n s f o r m of e ( t ) and N(s) i s the t r a n s f o r m of n ( t ) . In m a t r i x form, l e t t i n g e ( 0 ) and n(0) be z e r o , (1.13) and (1.14) change t o s-Aexp(-sT)-B C -D s-F The d e t e r m i n a n t of the f i r s t m a t r i x i n (1.15) i s Q(s) where Q(s) = ( s 2 - (B+F)s + CD)exp(sT) - A ( s - F ) + B F (1.16) I f we l e t z = s T , we f i n d t h a t Q(z) = ( z 2 - T(B + F ) z + C D T 2 ) e x p ( z ) - AT(z - FT) + BFT 2 (1.17) An assumption i n the models we are s t u d y i n g i s t h a t t h e r e i s no p r e d a t o r i n t e r f e r e n c e . M a t h e m a t i c a l l y , t h i s says t h a t f 3 ( y ) E ( s ) N(s) = 0 (1.15) 29 v a r i e s l i n e a r l y with the predator d e n s i t y . From the d i f f e r e n t i a l equation (1.2) at e q u i l i b r i u m and the above assumption, F = f 2 ( x * ) - f 3 ' ( y * ) = 0 (1.18) and the q u a s i - c h a r a c t e r i s t i c equation (1.17) i s reduced to Q(z) = ( z 2 - TBz + CDT 2)exp(z) - ATz (1.19) From Brauer (1977), the c o n d i t i o n that (1.10) and. (1.11) be s t a b l e leads to the s t a b i l i t y c o n d i t i o n T < (p*/2CD)[(A 2-B 2+4CD) 1/ 2 - (A 2 - B 2 ) 1 / 2 ] (1.20) where p* = a r c c o s (-B/A). Brauer used t h i s formula f o r two d i f f e r e n t models and found that the s t a b l e l i m i t f o r T decreased with i n c r e a s e d h a r v e s t i n g i n the case of the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , and i n c r e a s e d with i n c r e a s e d h a r v e s t i n g i n the same model with a type-II f u n c t i o n a l response. One i s i n t e r e s t e d i n whether t h i s i s simply a r e s u l t of the models used or a more gen e r a l r e s u l t . In a d d i t i o n , one would l i k e to check which b i o l o g i c a l parameters augment the e f f e c t s of delay and which c o u n t e r a c t them. T h i s i n f o r m a t i o n can be obtained by f i n d i n g the l i m i t on T, analogous to (1.20), f o r d i f f e r e n t harvest r a t e s i n other models through s i m u l a t i o n runs. A l t e r n a t i v e l y , we can make a second t r a n s f o r m a t i o n of the 30 l i n e a r i z e d equation to map the s t a b i l i t y boundary where Q(z) develops roots with p o s i t i v e r e a l p a r t s i n t o the parameter space of the q u a s i - c h a r a c t e r i s t i c equation. T h i s approach allows a g r a p h i c a l comparison between the non-delayed and delayed models and among d i f f e r e n t models. T h i s t r a n s f o r m a t i o n method i s c a l l e d the D - p a r t i t i o n method, the D-composition method, or the 0-root p l a t e a u ( E l ' s g o l ' t s 1966; E l ' s g o l ' t s and Norkin 1973; Pinney 1958). The mathematical j u s t i f i c a t i o n f o r t h i s method i s provided with examples in Appendix A. For now, i t s u f f i c e s to give a simple example of the method. A s i n g l e - s p e c i e s example The delayed, s i n g l e - s p e c i e s l o g i s t i c model with constant-quota h a r v e s t i n g i s L i n e a r i z i n g (1.21) with e ( t ) = x ( t ) - x * , where x* i s the e q u i l i b r i u m p o p u l a t i o n s i z e , we o b t a i n dx/dt = rx ( 1 - x ( t - T ) / K ) - H (1.21) de/dt = - ( r x * / K ) e ( t - T ) + r ( 1 - x * / K ) e ( t ) (1 .22) L e t t i n g a -(rx*/K) (1 .23) b r(1-x*/K) ( 1 .24) so that 31 de/dt = ae(t-T) + be(t) (1 .25) The Laplace transform of (1.25) i s sE(s) = aE(s)exp(-sT) + bE(s) + e(0) (1 .26) where E(s) i s the transform of e ( t ) . Using the n o t a t i o n i n equation (1.12), The q u a s i - c h a r a c t e r i s t i c equation f o r the s i n g l e - s p e c i e s d i f f e r e n t i a l equation i s from (1.12). If we l e t s=x + i y , then the d i v i d i n g l i n e between s t a b l e and unstable regions i n the (x,y)-plane w i l l be the y a x i s with negative x s t a b l e and p o s i t i v e x u n s t a b l e . To f i n d t h i s d i v i d e r i n the (a,b)-plane, we can l e t s=iy/T and s o l v e f o r a and b i n terms of y — i n e f f e c t , p a r a m e t e r i z i n g the s t a b i l i t y boundary. From s=iy/T, equation (1.28) becomes W(s) = aexp(-sT) + b ( 1 .27) Q(s) = s - a exp(-sT) - b = 0 (1 .28) i y - aTexp(-iy) - bT = 0 (1 .29) Equating r e a l and imaginary p a r t s to zero, we o b t a i n 3 2 -aT cos y - b? = 0 (1.30) y + aT s i n y = 0 (1.31) S o l v i n g for a and b i n terms of y, (1.30) and (1.31) change t o b = (y/T) cot y (1.32) a = -(y/T) esc y (1.33) I f we l e t y vary from 0 to J T / 2 , the D - p a r t i t i o n equations map put l i n e (1) i n Fi g u r e (1.1) f o r T=1. The l i n e l a b e l l e d (2) i s the p a r t i t i o n l i n e f o r y=0, a + b = 0. The p a r t i t i o n l i n e does not change as H i s a l t e r e d . The usefulness of t h i s technique i s c l e a r l y seen when we superimpose the h a r v e s t i n g e q u i l i b r i u m l i n e , which shows the r e l a t i o n between b and a as h a r v e s t i n g i n c r e a s e s . Because b « r(1-x*/K) and a = -rx*/K, b = r + a (1.34) We w i l l c a l l t h i s h a r v e s t i n g e q u i l i b r i u m l i n e , the h a r v e s t i n g t r a j e c t o r y . This l i n e i s i n s e r t e d f o r r=1.8 on F i g u r e (2.1). Two f a c t s can be e x t r a c t e d d i r e c t l y from t h i s diagram. F i r s t , as H i n c r e a s e s , b and a i n c r e a s e ; t h e r e f o r e , h a r v e s t i n g can s t a b i l i z e an unstable system provided t h a t r < 2. When r=1.8, we see that s t a b i l i z a t i o n of the e q u i l i b r i u m i s p o s s i b l e . Second, when r < tr/2 (T=1), the system i s always s t a b l e unless x*>K/2. . 33 Figure ( 1 . 1 ) — D - p a r t i t i o n f o r the s i n g l e - s p e c i e s , l o g i s t i c with d i s c r e t e delay, T=1 . ( , D - p a r t i t i o n ; , h a r v e s t i n g t r a j e c t o r y ) . The parameter r i s 1 . 8 ; K does not enter the s t a b i l i t y c o n d i t i o n s . The l i n e l a b e l l e d ( 1 ) i s obtained from the D - p a r t i t i o n equations and l i n e ( 2 ) i s the p a r t i t i o n l i n e f o r s = 0 , a + b = 0 . The arrow i n a l l our i l l u s t r a t i o n s p o i n t s i n the d i r e c t i o n of i n c r e a s e d h a r v e s t i n g or s t o c k i n g . 35 Comparing these results to Brauer (1977), the f i r s t statement corresponds to the argument that an increase in the harvest rate increases the range of T for which the system i s l o c a l l y asymptotically stable. His s t a b i l i t y conditions can be transformed to those of the D-partition. Notice also that the condition r < tr/2 for T=1 and H=0 follows from Hutchinson (1948) and May (1973). Furthermore, the size of the s t a b i l i t y region varies with T. Since a and b depend l i n e a r l y on T, when T i s decreased by a factor of 2, the (a,b)-coordinates of the s t a b i l i t y region are doubled. With discrete delay, the population size T time units ago determines the present rate of increase of the population. Smoothing the delay can al t e r the conclusions found above for discrete delay. Smoothing d i s t r i b u t e s the density dependence over many past populations instead of a single one. The weighting function used in the d i s t r i b u t i o n of delay i s c a l l e d the delay kernal. For the smoothed delay with a 'strong' kernal (see Figure 5.1(c)) from Cushing (1977) with r=1.8 and T=1, the equilibrium i s stable at H=0 and becomes unstable after a small increment in harvesting rate. The p a r t i t i o n and trajectory are shown in Figure (1.2). The l i m i t for rT with no harvesting has been increased by smoothing but the i n i t i a l slope of the p a r t i t i o n at H=0 i s smaller. When r i s greater than 2, d e s t a b i l i z i n g the equilibrium at H=0, no amount of harvesting can s t a b i l i z e the equilibrium. 36 Figure ( 1 . 2 ) — D-partition for the single-species, l o g i s t i c with smoothed delay, T=1. ( , D-partition; , harvesting t r a j e c t o r y ) . The parameter r is 1 . 8 ; K does not enrer the s t a b i l i t y conditions. 38 Q u a s i - c h a r a c t e r i s t i c equations Having demonstrated the D - p a r t i t i o n technique i n a s i n g l e -s p e c i e s context, we can now advance to predator-prey systems. We w i l l begin by c a l c u l a t i n g the q u a s i - c h a r a c t e r i s t i c equations under d i f f e r e n t h a r v e s t i n g regimes. Then we w i l l i n t r o d u c e models with s p e c i f i c forms f o r f 1 , f2, and f 3 , and use the D-p a r t i t i o n to f i n d the s t a b i l i t y regions and to d i s c o v e r how T a f f e c t s these r e g i o n s . 1. C o n s t a n t - e f f o r t prey harvest From the p r e v i o u s s e c t i o n , equation (1.19) i s 2. Constant-quota prey h a r v e s t The only change i n t h i s type of h a r v e s t i n g from that of c o n s t a n t - e f f o r t i s the B term. Here Q(z) = ( z 2 - TBz + CDT 2) exp(z) - ATz (1 .35) B = rx*(1-x*/K) - y * f 1 \ ( x * ) (1 .36) Q(z) i s the same as i n c o n s t a n t - e f f o r t prey h a r v e s t . 3. C o n s t a n t - e f f o r t predator harvest In t h i s case, dx/dt = r x ( 1 - x ( t - T ) / K ) - f l ( x ) y (1 .37) 39 dy/dt = f 2 ( x ) y - f 3 ( y ) - Hy (1.38) The development of the q u a s i - c h a r a c t e r i s t i c i s the same as i n the c o n s t a n t - e f f o r t prey h a r v e s t i n g case up t o e q u a t i o n ( 1 . 1 7 ) , w i t h B as g i v e n i n ( 1 . 3 6 ) , so t h a t Q(z) = ( z 2 - T(B+F)z +CDT 2) ex p ( z ) - AT(z - FT) + BFT 2 (1.39) But the e q u i l i b r i u m c o n d i t i o n s t a t e s t h a t d y / d t = y * F = f 2 ( x * ) -f 3 ' ( x * ) - H y * = 0 and F=0 so t h a t the q u a s i - c h a r a c t e r i s t i c e q u a t i o n becomes Q(z) = ( z 2 - TBz + C D T 2 ) e x p ( z ) - ATz (1.40) as above. 4. C o n s t a n t - q u o t a p r e d a t o r h a r v e s t i n g U n l i k e the c o n s t a n t - e f f o r t c a s e , F i s non-zero. We, t h e r e f o r e , add t h i s parameter e q u a t i o n ( 1 . 4 0 ) . The r e s u l t i s Q(z) = ( z 2 + T(B+F) + C D T 2 ) e x p ( z ) - AT(z-FT) + BFT 2 (1.41) where F = f 2 ( x * ) - f 3 ' ( y * ) (1 .42) 40 Models i n v e s t i g a t e d 1. L o t k a - V o l t e r r a model with r e s o u r c e - l i m i t a t i o n The s i m p l e s t model most f r e q u e n t l y used fo r predator-prey i n t e r a c t i o n s i s the L o t k a - V o l t e r r a model (May 1973). Because we are s t u d y i n g resource l i m i t a t i o n d e l a y , we a l s o i n c l u d e a prey density-dependent growth term i n the equations. The model i s dx/dt = rx ( 1 - x ( t - T ) / K ) - uxy (1.43) dy/dt = vxy - by (1.44) The i n t r i n s i c r a t e of growth i n t h i s model i s d e f i n e d by parameter r ; K g i v e s the c a r r y i n g c a p a c i t y o f the environment; u i s the predator r a t e of e f f e c t i v e search; v i s the numerical response constant; and b i s the predator death r a t e . May (1973) i n d i c a t e s that the L o t k a - V o l t e r r a model with resource l i m i t a t i o n c h a r a c t e r i z e s the s t a b i l i t y of a wider c l a s s of models as the f i r s t - t e r m T a y l o r s e r i e s approximation. S t e e l e (1976) o b j e c t s to t h i s s e r i e s approximation because i t e l i m i n a t e s the i n t r i n s i c n o n l i n e a r c h a r a c t e r of b i o l o g i c a l responses. Wangersky and Cunningham (1957a) d e a l with t h i s system with d i s c r e t e lags i n the i n t e r a c t i o n terms. Although May (1973) c r i t i z e s t h i s as u n r e a l i s t i c s i n c e the r e s u l t i n g p r edator-prey system i s l e s s s t a b l e than the s i n g l e - s p e c i e s system, i t should be recognized t h a t V o l t e r r a (1931) argues 41 c o n v i n c i n g l y that there should be a delay i n the numerical response of the p r e d a t o r . V o l t e r r a a l s o i n c l u d e s a delay i n the f u n c t i o n a l response f o r symmetry. A delay i n the f u n c t i o n a l response should be much smaller than the numerical response d e l a y . Brauer (1977) obtained some r e s u l t s f o r t h i s model with h a r v e s t i n g . He found that as h a r v e s t i n g i n c r e a s e d , the maximum al l o w a b l e delay shrunk. We w i l l t r y to d i s c o v e r i f t h i s r e s u l t i s g e n e r a l l y t r u e . From l i n e a r i z i n g (1.43) and (1.44), the l o c a l s t a b i l i t y c o n d i t i o n f o r t h i s model without h a r v e s t i n g or delay i s - r ( l - 2 x * / K ) - uy* < 0 (1.45) Since x*=b/v and y*=(r/u)(1-b/vK), (1.45) becomes -rb/(vK) < 0 (1.46) I t f o l l o w s that the e q u i l i b r i u m i s always l o c a l l y s t a b l e i n the absence of delay and h a r v e s t i n g . 2. H o l l i n g ' s model with a type-II f u n c t i o n a l response H o l l i n g (1966) intr o d u c e s the type-II response f l = Mx/(L+x) as a more r e a l i s t i c a l t e r n a t i v e to the l i n e a r N i c h o l s o n - B a i l e y response. T h i s n o n l i n e a r response d e f i n e s a s a t u r a t i o n d e n s i t y of prey above which an i n c r e a s e i n d e n s i t y w i l l have no e f f e c t on the number of prey eaten per p r e d a t o r . 42 The s e a r c h i n g time i s not constant i n t h i s response as i n the l i n e a r one but assumes that some time must be expended i n k i l l i n g and e a t i n g each prey before search resumes. H o l l i n g f i t t h i s response to data from the p r a y i n g mantis, H.crassa. The model i s dx/dt = rx( 1-x(t-T)/K) - M(xy/(x+D) (1.47) dy/dt = sM(xy/(x+L) - Jy/(J+D) (1.48) In t h i s model, r and K are d e f i n e d as the L o t k a - V o l t e r r a model with resource l i m i t a t i o n . The parameter M i s the maximum consumption r a t e per pre d a t o r ; L i s the prey d e n s i t y at which consumption i s L/2. Equation (1.48) a l s o i n c l u d e s the assumption that a t h r e s h o l d prey d e n s i t y J must be reached before r e p r o d u c t i o n can occur ( H a s s e l l 1978). Brauer (1977) i n v e s t i g a t e s the e f f e c t of resource delay on H o l l i n g ' s model as w e l l . He found, c o n t r a r y to the r e s u l t s with the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , that the a l l o w a b l e time delay i n c r e a s e s 'with h a r v e s t i n g as i n the s i n g l e - s p e c i e s l o g i s t i c model. The l o c a l s t a b i l i t y c o n d i t i o n s f o r t h i s model without h a r v e s t i n g or delay are rd-2x*/K) - y*ML/(x*+L) 2 < 0 (1.49) M 2sLx/(x+L) 3 > 0 ( 1 .50) from l i n e a r i z i n g (1.47) and (1.48). With x*=J, 43 y * = ( r / M ) ( J+ L ) ( 1 - J / K ) , f o r the system without h a r v e s t i n g or delay, (1.49) becomes K - J < J + L . (1.51) C o n d i t i o n (1.50) i s true f o r a l l x>0 since the parameters are p o s i t i v e . 3. I v l e v ' s model Gause (1934) f i r s t d e r i v e d t h i s model and I v l e v (1961) a p p l i e d i t to h i s f i s h e r i e s s t u d i e s . As i n the p r e v i o u s model, the r a t e of a t t a c k approaches an asymptote, given by the parameter b below. The model i s dx/dt = r x ( 1 - x(t-T)/K) - by(1 - exp(-cx)) (1.52) dy/dt = s(ex p ( - c J ) - e x p ( - c x ) ) . (1.53) Parameter c i s analogous to L i n the previous model with L=1/C; b i s analogous to M. S t r u c t u r a l l y , I v l e v ' s model i s i d e n t i c a l to H o l l i n g ' s model with a type-II f u n c t i o n a l response. The remainder of the parameters r e t a i n the meaning employed i n the p r e v i o u s models. From l i n e a r i z i n g (1.52) and (1.53), the s t a b i l i t y c o n d i t i o n s without delay or h a r v e s t i n g are r ( l - x * / K ) - y*bcexp(-cx*) < 0 s b 2 c e x p ( - c x * ) ( 1 - e x p ( - c x * ) ) > 0 (1 .54) (1 .55) 44 With no h a r v e s t i n g or delay, x*=J, and y*=rJ[1-J/K]/[b(1-exp(-c x * ) ) ] . The s t a b i l i t y c o n d i t i o n (1.54) becomes J/(K - J) > 1 - c J / ( e x p ( c J ) - 1). (1.56) C o n d i t i o n (1.55) i s true f o r a l l x*>0. In t h i s chapter we i n t r o d u c e d the method of D - p a r t i t i o n s to determine the l o c a l s t a b i l i t y i n our s i n g l e - s p e c i e s examples. We a l s o o u t l i n e d the predator-prey models we w i l l a nalyze i n subsequent c h a p t e r s . With t h i s i n t r o d u c t i o n concluded, we w i l l proceed with the i n v e s t i g a t i o n of resource recovery l a g s i n harvested predator-prey models. 45 I I . PREY HARVESTING AND DELAY Constant-quota prey harvest ing Equation (1.35) g i v e s the q u a s i - c h a r a c t e r i s t i c equation f o r constant-quota prey h a r v e s t i n g f o r our three models. The D - p a r t i t i o n method maps the q u a s i - c h a r a c t e r i s t i c equation i n t o a parameter space, d i v i d i n g the space i n t o s t a b l e and unstable r e g i o n s . The (B,CD) plane i s convenient space to compare the delayed and non-delayed r e s u l t s as the non-delayed p a r t i t i o n l i n e i s v e r t i c a l i n t h i s plane, i n t e r s e c t i n g the B a x i s at B = -A f o r a l l three models. From (1.35), the q u a s i - c h a r a c t e r i s t i c polynomial i s Q(z) = ( z 2 - BTz +CDT 2)exp(z) - ATz (2.1) I n s e r t i n g i y f o r z, equating r e a l and imaginary p a r t s to zero, and s o l v i n g f o r B and CD i n (2.1), we a r r i v e at the p a r t i t i o n equations B = -Acos(y) (2.2) CD = ( y / T ) 2 + ( A / T ) y s i n ( y ) (2.3) These equations d e f i n e the s t a b i l i t y boundary in the (B,CD) plane. Because we d e s i r e to i n v e s t i g a t e the e f f e c t of harvest on the dynamics of the system, we need to f i n d the h a r v e s t i n g t r a j e c t o r i e s i n the (B,CD) pla n e . The t r a j e c t o r i e s do not change with d e l a y . The parameters values f o r the t r a j e c t o r y 46 and the q u a s i - c h a r a c t e r i s t i c equation are d e f i n e d by the d e f i n i t i o n s (1.5), (1.36), and (1.7-1.9) along with the e q u i l i b r i u m p o p u l a t i o n s i z e s , d e r i v e d from the d i f f e r e n t i a l equations presented i n - the previous s e c t i o n . Tables (2.1) through (2.3) d i s p l a y the n o n - t r i v i a l e q u i l i b r i u m f o r the three models, the maximum constant-quota harvest r a t e at which the predator p o p u l a t i o n disappears (y*=0), the q u a s i - c h a r a c t e r i s t i c parameter v a l u e s , the s t a r t i n g p o i n t and the endpoint of the t r a j e c t o r y , and f i n a l l y the slope of the t r a j e c t o r y , which i s constant f o r t h i s type of h a r v e s t i n g . Our comments about the models w i l l focus on the equations i n the t a b l e s and the f i g u r e s presented below. 1. L o t k a - V o l t e r r a model with resource l i m i t a t i o n Both the h a r v e s t i n g t r a j e c t o r y and the D - p a r t i t i o n for t h i s model are p l o t t e d i n F i g u r e (2.1). A few o b s e r v a t i o n s can be deduced from the diagram and the equations i n Table (2.1). F i r s t , i n t h i s example, the s t a b i l i t y l i n e f o r T=0 i s a v e r t i c a l l i n e at B=1. The non-delayed model becomes unstable i f B > 1 at H = Hmax. As T i n c r e a s e s , t h i s p a r t i t i o n l i n e becomes concave p o s i t i v e , d e c r e a s i n g the maximum harvest r a t e f o r which the system i s s t a b l e . T h i s r e s u l t c o i n c i d e s with Brauer's (1977) r e s u l t s . Second, since b i s the magnitude of the slope of the h a r v e s t i n g t r a j e c t o r y , i n c r e a s i n g b w i l l d e s t a b i l i z e the harvested system because the time s c a l e f o r the model i s T1 = l / ( r b ) 1 ^ 2 . The s t a b i l i t y of the model depends on the r e l a t i o n s h i p between T 47 Table ( 2 . 1 ) — E q u i l i b r i a , maximum harvest r a t e , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y endpoints and slope f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with c o n s t a n t -quota prey h a r v e s t i n g . N o n - t r i v i a l model e q u i l i b r i u m x* = b/v y* = (r/u)(1 - b/vK) - Hv/ub (2.1.1) (2.1.2) Maximum ha r v e s t r a t e Hmax = (rb/v)(1 - b/vK) (2.1.3) Qu a s i -c h a r a c t e r i s t i c parameter v a l u e s A = -rb/vK B = Hv/b C = ub/v D = (vr/u)(1-b/vK) - Hv 2/ub (2.1.4) (2.1.5) (2.1.6) (2.1.7) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 B = 0 CD = rb(1-b/vK) (2.1.8) (2.1.9) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax B = r(1-b/vK) CD = 0 (2.1.10) (2.1.11) Slope of h a r v e s t i n g t r a j e c t o r y d(CD)/dB = -b (2.1.12) 49 Table ( 2 . 2 ) — E q u i l i b r i a , maximum harvest rate, D-partition parameters, harvesting trajectory endpoints and slope for Holling's model with a type-II functional response and constant-quota prey harvesting. 50 N o n - t r i v i a l model e q u i l i b r i u m x* = J y* = (r/M)(J+L)(1-J/K) - H(J+L)/MJ (2.2.1) (2.2.2) Maximum harvest r a t e Hmax = rJ(1-J/K) (2.2.3) Quasi-c h a r a c t e r i s t i c parameter valu e s A = -rJ/K B = (r J / K ) ( K - J ) / ( J + L ) + H(1/J-1/(J+L)) CD = sMLJ(r(1-J/K) - H / J ) / ( J + L ) 2 (2.2.4) (2.2.5) (2.2.6) Ha r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 B = rJ(K-J)/K(J+L) CD = s J L M r ( l - J / K ) / ( J + L ) 2 (2.2.7) (2.2.8) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax B = r(1-J/K) CD = 0 (2.2.9) (2.2.10) Slope of h a r v e s t i n g t r a j e c t o r y d(CD)/dB = SLM/[(J+L)(1 - ( J + D / J ) ] (2.2.11) 51 Table ( 2 . 3 ) — E q u i l i b r i a , maximum harvest r a t e , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y endpoints and slope f o r I v l e v ' s model with constant-quota prey h a r v e s t i n g . 52 N o n - t r i v i a l model e q u i l i b r i u m x*=J y* = [rJ(1-J/K)-H] / b O - e x p ( - c J ) ) (2.3.1) (2.3.2) Maximum harvest r a t e Hmax = rJ(1- J / K ) (2.3.3) Quasi-c h a r a c t e r i s t i c parameter valu e s A = -rJ/K B = r d - J / K ) - y*bcexp(-cJ) C = b(1 - e x p ( - c J ) ) D = sbcy*exp(-cJ) (2.3.4) (2.3.5) (2.3.6) (2.3.7) Ha r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 B = r ( 1 - J / K ) [ 1 - c J / ( e x p ( c J ) - 1 ) ] CD = s b c e x p ( - c J ) ( r J ( 1 - J / K ) ) (2.3.8) (2.3.9) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax B = r(1-J/K) CD = 0 (2.3.10) (2.3.11) Slope of h a r v e s t i n g t r a j e c t o r y d(CD)/B = -s b ( 1 - e x p ( - c J ) ) (2.3.12) 53 Figure ( 2 . 1 ) — S t a b i l i t y regions for the Lotka-Volterra with resource l i m i t a t i o n , delay, and constant-quota prey harvesting. The parameter values are r=2, K=1000, u=1, V=0.001, b=0.5, and T=1.0. (• D-partition; , harvesting t r a j e c t o r y ) . 54 / 55 and t h i s time s c a l e T1 (May 1973). I f T exceeds T1 , the system becomes u n s t a b l e ; i n c r e a s i n g b causes T1 t o d e c r e a s e and may d e s t a b i l i z e the e q u i l i b r i u m . The parameters v and K (but not u) a l s o a f f e c t the s t a b l e h a r v e s t i n g l i m i t . I n c r e a s i n g K or v l o w e r s the h a r v e s t i n g l i m i t . The c a r r y i n g c a p a c i t y ' s e f f e c t can be e x p l a i n e d by the 'paradox of enrichment' (Rosenzweig 1971; Brauer e t . a l . 1976), which says t h a t r e s o u r c e enrichment can d e s t a b i l i z e ecosystems. T h i s may appear t o be p a r a d o x i c a l i n t h i s model as w e l l because r e s o u r c e l i m i t a t i o n causes a l a g which r e s u l t s i n d e s t a b i l i z a t i o n but r e s o u r c e enrichment a l s o d e s t a b i l i z e s the system. T h i s d i f f i c u l t y can be r e s o l v e d by c o n s i d e r i n g t h a t the l a g i s a measure of the time over which a f i x e d amount of r e s o u r c e i s a v a i l a b l e . The c a r r y i n g c a p a c i t y can be low a t the same time as the r e c o v e r y r a t e i s h i g h : the two parameters o p e r a t e i n d e p e n d e n t l y . 2. H o l l i n g ' s model w i t h a t y p e - I I f u n c t i o n a l response A D - p a r t i t i o n p l o t and h a r v e s t i n g t r a j e c t o r y f o r H o l l i n g ' s model i s g i v e n i n F i g u r e ( 2 . 2 ) . From e q u a t i o n s (2.2.8) and (2.2.11) i n T a b l e (2.2) and F i g u r e ( 2 . 2 ) , we can see t h a t the t h r e e parameters s, M, and L, which c o n t r o l o n l y the e n d p o i n t of the t r a j e c t o r y a t H=0, w i l l e x e r t an i n f l u e n c e on the s t a b i l i t y of the e q u i l i b r i u m p o i n t i n the d e l a y e d model, but not i n the non-delayed v e r s i o n . To u n d e r s t a n d the c o n t r i b u t i o n s of t h e s e p a r a m e t e r s , we w i l l r e f e r t o the t y p e - I I f u n c t i o n a l response drawn i n F i g u r e ( 2 . 3 ) . From 56 Figure ( 2 . 2 ) — S t a b i l i t y region for Holling's model with delay and constant-quota prey harvesting. The parameters are r=2. K=40.,_J=2C, L=10., s = 1 , M=1, and T=0.5. ( , D-partition;- , harvesting t r a j e c t o r y ) . 57 58 T Q I ^ 6  {2d }" T h e tW-11 f u n c t i o n a l response (from H o l l i n g 1966). The parameter M i s the r a t e of s a t i a t i o n ; L i s t h e h a l f - s a t u r a t i o n c o n s t a n t f o r the c u r v e . o cu <D L . Q. t_ (D Q. >> (D L . 60 e q u a t i o n (1.47) and the graph, we see t h a t L d e t e r m i n e s the s l o p e of the response a t z e r o ( f o r f i x e d M) and M s p e c i f i e s the l i m i t i n g r a t e of prey consumption. These r e l a t i o n s h i p s , a l o n g w i t h the e n d p o i n t e q u a t i o n s (2.2.7) and (2.2.8) i n T a b l e ( 2 . 2 ) , r e v e a l the e f f e c t s of each parameter on the f u n c t i o n a l response and on the s t a b i l i t y of the e q u i l i b r i u m . I f we r a i s e M, then CD a t H=0 i s i n c r e a s e d . A l a r g e M l i n e a r i z e s the f u n c t i o n a l r e s p o n s e , t r a n s f o r m i n g the c u r v e t o a c l o s e r a p p r o x i m a t i o n of a t y p e - I f u n c t i o n a l r e s ponse: a l i n e a r l y s l o p i n g l i n e w i t h a t h r e s h o l d where the number of prey eaten i s independent of the pre y d e n s i t y . A s m a l l M s t a b i l i z e s the d e l a y e d dynamics; t h e r e f o r e , the t y p e - I I response e x e r t s a s t a b i l i z i n g i n f l u e n c e over the t y p e - I l i n e a r r e s p o n s e . T h i s i s not t r u e f o r the non-d e l a y e d case where M has no e f f e c t on s t a b i l i t y . Through the same type of a n a l y s i s , we can show t h a t i n c r e a s i n g L can l e a d t o i n c r e a s e d s t a b i l i t y . S t a b i l i t y i n c r e a s e s w i t h i n c r e a s e d L f o r the non-delayed model as w e l l (see e q u a t i o n ( 1 . 5 1 ) ) . I n c r e a s i n g L f l a t t e n s out the e x p o n e n t i a l i n c r e a s e i n a t t a c k r a t e , damping f l u c t u a t i o n s around the e q u i l i b r i u m a t t a c k r a t e and t h e r e b y s t a b i l i z i n g the e q u i l i b r i u m . In the n u m e r i c a l r e s p o n s e , i n c r e a s i n g s would n e g a t i v e l y a f f e c t s t a b i l i t y : the s l o p e of the h a r v e s t i n g t r a j e c t o r y would become more n e g a t i v e . The parameter s does not appear i n the non-delayed s t a b i l i t y c o n d i t i o n s . From t h i s d i s c u s s i o n , we can c o n c l u d e t h a t the e f f i c i e n c y of biomass c o n v e r s i o n by the p r e d a t o r (s) and the s a t i a t i o n l i m i t on prey consumption (M) a f f e c t s t a b i l i t y i n the d e l a y e d 61 model, but not i n the non-delayed v e r s i o n . Lowering e i t h e r parameter decreases the growth rate of the predator p o p u l a t i o n ; consequently, a low maximum growth r a t e of the predator i s c r u c i a l i f c o n t r o l of prey f l u c t u a t i o n s i s sought i n a system with resource recovery d e l a y . F i g u r e (2.4) i l l u s t r a t e s how M changes s t a b i l i t y i n the delayed model but not i n the non-delayed model. 3 . I v l e v ' s model The a n a l y s i s of t h i s model and the c o n c l u s i o n s r e s u l t i n g from i t are i d e n t i c a l to those of the previous model except that b r e p l a c e s M and 1/c r e p l a c e s L. The model has been i n c l u d e d to t r y to e l i m i n a t e some of the author's b i a s e s i n choosing parameters when s i m u l a t i o n s were run. D i f f e r e n c e s between these two models are i l l u s t r a t e d i n the next chapter. Our d i s c u s s i o n up u n t i l t h i s p o i n t has concerned i t s e l f with the i n f l u e n c e s of parameters on the s t a b i l i t y i n delayed and non-delayed models. We are a l s o concerned with the e f f e c t on h a r v e s t i n g i n these models. With prey h a r v e s t i n g , these connections can be drawn from the geometry of the h a r v e s t i n g t r a j e c t o r y and the D - p a r t i t i o n s t a b i l i t y boundary. Because the D - p a r t i t i o n becomes concave with an in c r e a s e i n T, i t i s p o s s i b l e t h a t , f o r some parameter combinations, there c o u l d e x i s t two s t a b l e h a r v e s t i n g r e g i o n s : one at very high harvest r a t e s and one at low r a t e s with an unstable r e g i o n i n the middle. A necessary c o n d i t i o n f o r t h i s p o s s i b i l i t y i s that B < -A s i n c e t h i s a l l o w s a hi g h - h a r v e s t s t a b i l i t y r e g i o n . 62 Figure ( 2 . 4 ) — _ D e l a y e d and non-delayed p a r t i t i o n s with hiqh ? , n *, s a t i a t i o n l i m i t s f o r H o l l i n g ' s model with r = l , K = 4 0 , L=1 0 , J = 2 0 , s=1, (a) M=1 (low) and (b) M=4 ( h i g h ) : ( i ) Non-delayed p a r t i t i o n s , ( 2 ) Delayed p a r t i t i o n s , T = 1 . 5 . ( p a r t i t i o n ; , h a r v e s t i n g t r a j e c t o r y ) . D-2 CD s t a b l e 1 ( b ) (a) \ \ u n s t a b l e B 0.5 64 CD 2 V unstable stable i \ ( b ) ( a ) \ V i B 0.5 (2) 1 65 Another way of expressing t h i s condition i s that, in the equivalent non-delayed models, harvesting can never d e s t a b i l i z e the equilibrium. It should be understood that this double t r a n s i t i o n cannot occur in the non-delayed models. Constant-effort prey harvesting The quasi-characteristic equation for constant-quota prey harvesting applies to constant-effort harvesting as w e l l — only the parameter B in the quasi-characteristic equation changes value. This s h i f t in B modifies the direction of harvesting trajectory but does not change i t s linear slope. Tables (2.4) through (2.6) give the equilibrium values and trajectory information. In addition, since the quasi-characteristic equation i s degenerate in Lotka-Volterra model with resource l i m i t a t i o n model, we include Q(z) in our tables. For the other two models, the p a r t i t i o n equations (2.2) and (2.3) remain the same. 1. Lotka-Volterra model with resource l i m i t a t i o n The parameters of the quasi-characteristic equation (2.1) are the same except that B=0 for a l l harvesting rates. This changes the non-delayed, s t a b i l i t y condition -A > B to rb/vK > 0 i (2.4) 66 Table (2.4)-- E q u i l i b r i a , maximum harvest r a t e , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y endpoints and s l o p e , and the q u a s i - c h a r a c t e r i s t i c equation f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with c o n s t a n t - e f f o r t prey h a r v e s t i n g 67 N o n - t r i v i a l model e q u i l i b r i u m x* = b/v y* = [ r ( l - b / v K ) - H ] / u (2.4.1) (2.4.2) Maximum harvest r a t e Hmax = r ( 1 - b/vK) (2.4.3) Quasi-c h a r a c t e r i s t i c parameter values A = -rb/vk B = 0 CD = rb(1-b/vK) - Hb (2.4.4) (2.4.5) (2.4.6) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 A = -rb/vK CD = rb(1-b/vK) (2.4.7) (2.4.8) H a r v e s t i n g c o o r d i n a t e s A = -rb/vK CD = 0 (2.4.9) (2.4.10) Quasi-c h a r a c t e r i s t i c equation Q(z) = (z 2+CDT 2)exp(z) - ATz (2.4.11) 68 Table ( 2 . 5 ) — E q u i l i b r i a , maximum harvest rate, D-partition parameters, harvesting trajectory endpoints and slope, and the quasi-characteristic equation for Holling's model with a type-I I functional response and constant-effort prey harvesting. '69 N o n - t r i v i a l model e q u i l i b r i u m x*=J y*=(J+L)[r(1-J/K)]/M - H/M (2.5.1) (2.5.2) Maximum harvest r a t e Hmax = r(1-J/K) (2.5.3) Quasi-c h a r a c t e r i s t i c parameter values A = -rJ/K B = [rJ(1- J / K ) - J H ] / ( J + L ) CD = sJAM[r(1-J/K)-H]/(J+L) 2 (2.5.4) (2.5.5) (2.5.6) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 B = Jr(K-J)/K(J+L) CD = Jr s M L ( K - J ) / K ( J + L ) 2 (2.5.7) (2.5.8) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax B = 0 CD = 0 (2.5.9) (2.5.10) Slope of h a r v e s t i n g t r a j e c t o r y d(CD)/dB = sLM/(J+L) (2.5.11) Quasi-c h a r a c t e r i s t i c equation Q(z) = (z 2-TBz+CDT 2)exp(z) - ATz (2.5.12) 70 Table ( 2 . 6 ) — E q u i l i b r i a , maximum harvest r a t e , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y endpoints and s l o p e , and the q u a s i - c h a r a c t e r i s t i c equation f o r I v l e v ' s model with c o n s t a n t -e f f o r t prey h a r v e s t i n g . 71 N o n - t r i v i a l model e q u i l i b r i u m x* = J y* = [rJ ( 1 - J / K ) -HJ]/b[1-exp(-cJ)] (2.6.1) (2.6.2) Maximum harvest r a t e Hmax = r(1-J/K) (2.6.3) Quasi-c h a r a c t e r i s t i c parameter values B = [ r ( 1 - J / K ) - H ] [ l - J c e x p ( - c J ) / ( 1 - e x p ( - c J ) ) ] CD = c b s e x p ( - c J ) [ r ( 1 - J / K ) - H J ] (2.6.4) (2.6.5) Har v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 B = r ( 1 - J / K ) [ l - J c e x p ( - c J ) / ( 1 - e x p ( - c J ) ) ] CD = s b c e x p ( - c J ) [ r J ( 1 - J/K)] (2.6.6) (2.6.7) Ha r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax B = 0 CD = 0 (2.6.10) (2.6.11) • Slope of h a r v e s t i n g t r a j e c t o r y d(CD)/dB = sbcJexp(-cJ) {1-[Jcexp(-cJ) / O - e x p ( - c J ) ) ]} (2.6.12) Quasi-c h a r a c t e r i s t i c equation Q(z) = (z 2-BTz+CDT 2)exp(z) - ATz (2.6.13) 72 which i s true by d e f i n i t i o n of the parameters. Thus, p r o p o r t i o n a l h a r v e s t i n g never d e s t a b i l i z e s the e q u i l i b r i u m p o i n t i n the non-delayed model. Because B=0, we w i l l p l o t the D - p a r t i t i o n i n the (A ,CD) plane. S u b s t i t u t i n g equation (2.2) i n t o (2.3), the p a r t i t i o n equations are m o d i f i e d to a s i n g l e equation CD = ( l / T 2 ) [ a r c c o s ( - B / A ) ] 2 + (A 2 - B 2 ) l / 2 [ a r c c o s ( - B / A ) ] / T (2.5) Because B=0, we have arccos(-B/A) = (2n+1) ir/2, n=0,± 1,± 2,... (2.6) So CD = [ (2n+1 ) ir/2 ] (A/T) + [ ( 2n+1 ) ir/2T ] 2 (2.7) There are m u l t i p l e s t a b i l i t y regions f o r t h i s model but we are mainly concerned with the p a r t i t i o n with n=0 where CD = rrA/2T + U / 2 T ) 2 (2.8) because t h i s p a r t i t i o n appears in the same area as the h a r v e s t i n g t r a j e c t o r y . As T becomes s m a l l e r , the s t a b i l i t y p a r t i t i o n becomes more v e r t i c a l and i n the l i m i t becomes A=0, as one would expect from the non-delayed s t a b i l i t y c o n d i t i o n . 73 The h a r v e s t i n g t r a j e c t o r i e s i n t h i s model are v e r t i c a l l i n e s to the a b c i s s a . An example of the D - p a r t i t i o n and h a r v e s t i n g t r a j e c t o r y i s shown in F i g u r e (2.5). I t i s obvious from F i g u r e (2.5) that c o n s t a n t - e f f o r t h a r v e s t i n g cannot d e s t a b i l i z e an e q u i l i b r i u m , but i t can counter the d e s t a b i l i z i n g i n f l u e n c e of the delay and s t a b i l i z e the e q u i l i b r i u m . A l s o note t h a t , i f rb/vK > u/2T, then no amount of h a r v e s t i n g w i l l promote s t a b i l i t y . I f the predator p o p u l a t i o n i s removed from t h i s r e l a t i o n s h i p or goes e x t i n c t , then the s t a b i l i t y boundary becomes p a r a l l e l CD a x i s . The c o n d i t i o n f o r s t a b i l i t y i s A > - i r/2T (2.9) The h a r v e s t i n g t r a j e c t o r y i s given by A = H - r (2.10) T h i s equation i s d e r i v e d from the e q u i l i b r i u m c o n d i t i o n x*=K(1-H/r) and the d e f i n i t i o n of A i n equation (1.5). The parameter CD i s zero because y*=0. From equations, (2.9) and (2.10), we conclude that an unstable e q u i l i b r i u m becomes s t a b l e when H > r - ir/2T (2.11) When H=0, t h i s reduces to rT<n/2. T h i s corresponds to the c o n d i t i o n given by Hutchinson (1948) and May (1973). 74 Figure ( 2 . 5 ) - S t a b i l i t y r e gion f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , delay, and c o n s t a n t - e f f o r t prey v-n n n 9 i n S ' 1 ™ ! P a r a * e t e r s v * l u e s are r = 2, K=1000, u=1, t r S r y i : ' ^ T = K ° ' ( ~ ^ " P a r t i t i o n ; h a r v e s t i n g 76 2. H o l l i n g ' s model with a type-II f u n c t i o n a l response U n l i k e the previous model, the q u a s i - c h a r a c t e r i s t i c equation i s not degenerate (B i s not i d e n t i c a l l y zero) but B i s s t i l l m o d i f i e d from constant-quota h a r v e s t i n g as shown i n equation (2.5.5) i n Table (2.5). From equation (2.5.11), we n o t i c e that the h a r v e s t i n g t r a j e c t o r y i s s t i l l a s t r a i g h t l i n e but t h i s time i t has a p o s i t i v e slope i n the (B,CD) plane. At H=0, the t r a j e c t o r y begins at the same po i n t as the cons t a n t -quota t r a j e c t o r y but heads toward the o r i g i n of the plane. The p a r t i t i o n doesn't change between the two types of h a r v e s t i n g . An example from t h i s model i s given i n F i g u r e (2.6). The c o n c l u s i o n s from t h i s model are very much the same as the L o t k a - V o l t e r r a model: p r o p o r t i o n a l h a r v e s t i n g w i l l not d e s t a b i l i z e the e q u i l i b r i u m and can, i n some i n s t a n c e s , s t a b i l i z e the e q u i l i b r i u m . On the other hand, i f the p a r t i t i o n l i n e passes to the l e f t of (B,CD)=(0,0) then no h a r v e s t i n g r a t e can s t a b i l i z e the i n t e r a c t i o n . T h i s crossover occurs when ir|A|/2T = U / 2 T ) 2 (2.12) or | A | = rJ/K = ir/2T, (2.13) as i n the L o t k a - V o l t e r r a model with resource l i m i t a t i o n . The r o l e s of s,M, and L i n the p o s i t i o n i n g of the endpoint of the h a r v e s t i n g t r a j e c t o r y are the same as with c o n s t a n t -quota h a r v e s t i n g . 77 Figure (2.6)-- S t a b i l i t y region for Holling's model with delay and constant-effort harvesting. The parameters are r=2, K=40, J = 20, L=10, s=1, M=1, and T=0.5. ( , D-pa r t i t i o n ; , harvesting t r a j e c t o r y ) . *78 1 2 79 3. I v l e v ' s model An example of t h i s model i s given i n F i g u r e (2.7). The c o n c l u s i o n s from H o l l i n g ' s model apply to t h i s model as w e l l f o r c o n s t a n t - e f f o r t prey h a r v e s t i n g . Prey s t o c k i n g Although the term for negative h a r v e s t i n g , s t o c k i n g , i s d e r i v e d from f i s h e r i e s a p p l i c a t i o n s , the models reviewed in t h i s s e c t i o n p e r t a i n to other e c o l o g i c a l s i t u a t i o n s as w e l l . The i n t r o d u c t i o n of n a t u r a l enemies to c o n t r o l a pest p o p u l a t i o n c e r t a i n l y f a l l s i n t o t h i s category. In a d d i t i o n , s t o c k i n g i s analogous to immigration and d i s p e r s a l i n t o a p o p u l a t i o n . From a mathematical st a n d p o i n t , s t o c k i n g i s innocuous compared to h a r v e s t i n g . Brauer (1979b) showed t h a t with constant non-negative s t o c k i n g r a t e s i n non-delayed models, the p o p u l a t i o n s must e i t h e r approach e q u i l i b r i u m or o r b i t the e q u i l i b r i u m i n s t a b l e l i m i t c y c l e s . Although t h i s r e s t r i c t i o n i s encouraging, i t i s l e s s meaningful e c o l o g i c a l l y . F i r s t , as Brauer et.a_l. (1976) e x p l a i n , when a p o p u l a t i o n becomes too s m a l l , i t may be e x t i n g u i s h e d by s t o c h a s t i c demographic f l u c t u a t i o n s . Second, i n the s t o c k i n g of i n s e c t p r e d a t o r s f o r i n s t a n c e , l i m i t c y c l e s are not c o n s i d e r e d a c c e p t a b l e management r e s u l t s : a low e q u i l i b r i u m pest p o p u l a t i o n or pest e x t i n c t i o n i s sought. So i f one's p e r s p e c t i v e i s s p e c i e s p r e s e r v a t i o n , s t o c k i n g appears to be a s t a b l e s t r a t e g y ; however, f o r a pest 80 Figure ( 2 . 7 ) — S t a b i l i t y region for Ivlev's model with delav and constant-effort harvesting with r=1, K=40 J=20 c-0 I b=4, s=1, and T=!.0. ( D - n a r t i t i A r . v ' C ' 9 ' 1 ' t r a j e c t o r y ) . ' P a r t ^ i o n ; } harvesting 82 manager, s t o c k i n g may be i l l - a d v i s e d as a continuous p o l i c y . From e i t h e r p o i n t of view, boundedness arguments and pro o f s f a l l s h o r t . S t a b i l i t y remains a s i g n i f i c a n t i s s u e along with a c h a r a c t e r i z a t i o n of the t r a n s i e n t behaviour. 1 We s t i l l want to d i s t i n g u i s h between l i m i t c y c l e behaviour and approach to a s t a b l e e q u i l i b r i u m . L i m i t c y c l e behaviour occurs i f the e q u i l i b r i u m i s l o c a l l y unstable and i s u n l i k e l y to occur i f i t i s l o c a l l y s t a b l e (Bulmer 1976). The s t a b i l i t y c o n d i t i o n s d e r i v e d f o r constant-quota prey h a r v e s t i n g s t i l l h o l d f o r s t o c k i n g . The d i f f e r e n c e i s the d i r e c t i o n of the h a r v e s t i n g t r a j e c t o r y ; B decreases and CD in c r e a s e s with s t o c k i n g . The p a r t i t i o n i s i d e n t i c a l i n shape for a l l three models. The h a r v e s t i n g t r a j e c t o r y i n the (B,CD) plane i s extended i n t o the second quadrant as d i s p l a y e d i n Fi g u r e (2.8). T h i s s t a b i l i z e s an unstable e q u i l i b r i u m but cannot d e s t a b i l i z e a s t a b l e one (but see the chapter on smoothed d e l a y ) . C o n c l u s i o n s In t h i s chapter we have d i s c u s s e d the l o c a l s t a b i l i t y of prey h a r v e s t i n g of two types-- constant-quota and constant-e f f o r t . In systems with delay, the s t a b i l i t y c r i t e r i a d i f f e r from non-delayed models i n that the maximum predator consumption r a t e s and the predator c o n v e r s i o n e f f i c i e n c y are key parameters i n determining the s t a b i l i t y of models with f u n c t i o n a l responses with a h y p e r b o l i c shape. For some combinations of these parameters, the constant-quota h a r v e s t i n g 83 Figu r e ( 2 . 8 ) — S t a b i l i t y regions with prey s t o c k i n g i n I v l e v ' s model with delay with r=2, K=35, J=20, b=1, c=0.1, s=1, and T=1.0. ( , D - p a r t i t i o n ; T h a r v e s t i n g t r a j e c t o r y ) . I 84 85 t r a j e c t o r y can pass through two e q u i l i b r i u m r e g i o n s . Although the parameters s and M a f f e c t the h a r v e s t i n g t r a j e c t o r y i n both h a r v e s t i n g f u n c t i o n s , we found that c o n s t a n t - e f f o r t harvest with delay i s much the same as without delay i n that h a r v e s t i n g can s t a b i l i z e an unstable e q u i l i b r i u m , provided the delay i s not too l a r g e . For i n c r e a s i n g d e l a y s , the s t a b i l i z i n g c o n s t a n t - e f f o r t harvest r a t e w i l l i n c r e a s e . F i g u r e (2.9) demonstates the d i f f e r e n c e between c o n s t a n t - e f f o r t h a r v e s t i n g in the delayed and non-delayed models. Prey s t o c k i n g i s most r e l e v a n t to f i s h e r i e s p o l i c y . Dynamics t e l l s us that prey s t o c k i n g i s always a s t a b l e p o l i c y f o r both delayed and non-delayed models i n our study. If a preda t o r - p r e y i n t e r a c t i o n i s u n s t a b l e , then prey s t o c k i n g can s t a b i l i z e i t . T h e o r e t i c a l l y , there i s no l i m i t on s t o c k i n g though, i n r e a l i t y , i t i s l i m i t e d by the p r o d u c t i v i t y of the next lower t r o p h i c l e v e l . LeBrasseur e t . a l . (1978) found that f e r t i l i z a t i o n of a salmon-inhabited lake i n c r e a s e s both the s i z e of the f i s h and the numbers. The success of f e r t i l i z a t i o n experiments r e l i e s to a great extent on the t r o p h i c r e l a t i o n s h i p s between s p e c i e s , e s p e c i a l l y at the lower t r o p h i c l e v e l s ( L a r k i n 1977). S t o c k i n g or enhancement i n f i s h e r i e s i s o f t e n dominated by ge n e t i c c o n s i d e r a t i o n s r a t h e r than dynamical ones. S u c c e s s f u l r e c r u i t m e n t i n t o the c a t c h a b l e a d u l t p o p u l a t i o n depends on the a b i l i t y of the stocked s t r a i n to compete f o r food, a v o i d p r e d a t i o n , and withstand d i s e a s e ( C a l a p r i c e 1969). At l e a s t some of the successes i n s t o c k i n g r e s t on the p a r t i c u l a r s t r a i n 86 Figure ( 2 . 9 ) — (a) Non-delayed and (b) delayed p a r t i t i o n s with c o n s t a n t - e f f o r t prey h a r v e s t i n g . The model i s ' I v l e v ' s with r=1, K=40, J=20, c = 0.1, b=4, s=1, and T=1. ( , D-p a r t i t i o n ; , h a r v e s t i n g t r a j e c t o r y ) . 87 88 of f i s h being introduced (Larkin 1977). This chapter also indicates some c h a r a c t e r i s t i c s of unmanaged systems. The dependence of s t a b i l i t y on the rate of growth parameters s and M in Holling's model has evolutionary implications for the predator and prey. Natural selection w i l l promote increases in e f f i c i e n c y of the predator in exploiting the prey resource by increasing s or M or by decreasing J . In Holling's predator-prey model without delay, increasing M w i l l lower the predator equilibrium but have no e f f e c t on s t a b i l i t y . Increasing s has no effect on s t a b i l i t y or density l e v e l s . Decreasing the maintenance l e v e l of food intake J lowers both the prey and predator equilibrium and decreases s t a b i l i t y (Rosenzweig and MacArthur 1963). Consequently, natural selection on the predator can raise e f f i c i e n c y without a f f e c t i n g s t a b i l i t y by increasing s or M in systems without delay, contrary to the results of Rosenzweig and MacArthur (1963). In contrast, for systems with resource recovery delay, we have found that increasing s or M or , decreasing J de s t a b i l i z e s the predator-prey interaction. This suggests that in such systems selective pressure on the predator w i l l decrease s t a b i l i t y . If selection on the prey species forces a greater reproductive rate r, then in both the delayed and non-delayed models the interaction w i l l be d e s t a b i l i z e d . But looking only at selective pressure on model parameters obscures the actual b i o l o g i c a l mechanisms at work. A strongly selected t r a i t in the face of intense resource competition i s t e r r i t o r i a l i t y . 89 G i l l (1972) c a l l s t h i s kind of s e l e c t i o n o O s e l e c t i o n , as opposed to r - s e l e c t i o n or R - s e l e c t i o n (Southwood 1976). T h i s type of s e l e c t i o n reduces the i n t r i n s i c r a t e of i n c r e a s e and the e f f i c i e n c y of resource e x p l o i t a t i o n due to the maintenance c o s t of a t e r r i t o r y but i n c r e a s e s the c o m p e t i t i v e advantage of the i n d i v i d u a l . T e r r i t o r i a l i t y would e f f e c t i v e l y e l i m i n a t e the l a g term i n our i n t e r a c t i o n s , p rovided an i n d i v i d u a l organism c o u l d maintain a l a r g e enough t e r r i t o r y to i n t e g r a t e over temporal and s p a t i a l changes in the resource. The advantage of t e r r i t o r i a l i t y i s dependent upon the h e t e r o g e n e i t y of the environment as w e l l as the a b i l i t i e s of the organism. In t h i s sense, t e r r i t o r i a l i t y can be thought of as a means of overcoming resource recovery delays and i n c r e a s i n g an organism's c o m p e t i t i v e advantage, although other s e l e c t i v e p r e s s u r e s may act to induce t e r r i t o r i a l behaviour. T e r r i t o r i a l i t y w i l l s t a b i l i z e the predator-prey i n t e r a c t i o n . In delayed systems then, the general o b s e r v a t i o n of Rosenzweig and MacArthur (1963) f o r non-delayed systems that s e l e c t i o n on the predator d e s t a b i l i z e s the i n t e r a c t i o n , while s e l e c t i o n on the prey s p e c i e s s t a b i l i z e s the i n t e r a c t i o n seems to h o l d , s u b j e c t to the c o n d i t i o n s of l i m i t e d h e t e r o g e n e i t y of the resource. S e l e c t i o n on the resource a l s o a f f e c t s the s t a b i l i t y of the o v e r a l l i n t e r a c t i o n . 90 I I I . PREDATOR HARVESTING AND STOCKING AND DELAY Constant~ef f o r t predator harvest ing C o n s t a n t - e f f o r t h a r v e s t i n g of pre d a t o r s i n a predator-prey r e l a t i o n s h i p i s common i n f i s h e r i e s and i n w i l d l i f e management. C o n s t a n t - e f f o r t h a r v e s t i n g decreases the numerical response of the p r e d a t o r to changes i n prey d e n s i t y and, at the same time, allows the e q u i l i b r i u m prey d e n s i t y to i n c r e a s e . The i n t e r p l a y between the i n c r e a s e d consumption by the predator and the decreased r a t e of i n c r e a s e at a given prey d e n s i t y determines the s t a b i l i t y of the e q u i l i b r i u m p o p u l a t i o n . Brauer and Soudack (1978) have shown t h a t , in non-delayed predator-prey models with the same assumptions we have o u t l i n e d , p r o p o r t i o n a l h a r v e s t i n g of the predatdr never d e s t a b i l i z e s the e q u i l i b r i u m and p o s s i b l y s t a b i l i z e s an unstable e q u i l i b r i u m i f h a r v e s t i n g i s great enough. Our past examples have shown the i n c r e a s e d importance of numerical response i n the s t a b i l i t y of delayed models. Because of t h i s , one might expect the c o n c l u s i o n s concerning c o n s t a n t - e f f o r t harvest to be mo d i f i e d i n some way. The D - p a r t i t i o n equations f o r c o n s t a n t - e f f o r t predator h a r v e s t i n g are the same as with c o n s t a n t - e f f o r t prey h a r v e s t i n g (equations (2.2), (2.3), and ( 2 . 7 ) ) . However, because c o n s t a n t - e f f o r t predator h a r v e s t i n g r a i s e s the e q u i l i b r i u m s i z e of the prey p o p u l a t i o n , the D - p a r t i t i o n s move with h a r v e s t i n g . T h i s added dynamic makes i t necessary to view the (B,CD) plane i n s l i c e s at d i f f e r e n t harvest r a t e s f o r H o l l i n g ' s and I v l e v ' s models. In the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , 91 the v a r i a t i o n i n the prey e q u i l i b r i u m i s presented as a change i n the h a r v e s t i n g t r a j e c t o r y i n the (A,CD) plane. As i n the p r e v i o u s chapter, Tables (3.1) through (3.3) d e t a i l the e q u i l b r i u m p o p u l a t i o n s i z e s , the maximum h a r v e s t i n g r a t e , the parameters of the q u a s i - c h a r a c t e r i s t i c equation, and the s t a r t i n g p o i n t and endpoint of the h a r v e s t i n g t r a j e c t o r y fo r the three models with c o n s t a n t - e f f o r t predator h a r v e s t i n g . The slope of the t r a j e c t o r y i s not constant and was t h e r e f o r e omitted from the t a b l e s in t h i s chapter. 1. L o t k a - V o l t e r r a model with resource l i m i t a t i o n Because the q u a s i - c h a r a c t e r i s t i c parameter B=0 f o r Lotka-V o l t e r r a model with resource l i m i t a t i o n , the s t a b i l i t y c o n d i t i o n i n the non-delayed model from equation (1.46) i s - r ( b + H)/vK < 0 (3.1) because the per c a p i t a death r a t e of the predator i s e f f e c t i v e l y b+H, i n s t e a d of b. Equation (3.1) i s s a t i s f i e d f o r a l l H > 0. In other words, the e q u i l i b r i u m i s never u n s t a b l e . T h i s robust s t a b i l i t y would make t h i s h a r v e s t i n g s t r a t e g y most p r a c t i c a l f o r resource management. For the delayed model, the D - p a r t i t i o n regions i n t h i s model are the same as with p r o p o r t i o n a l prey h a r v e s t i n g ; however, the h a r v e s t i n g t r a j e c t o r y i s no longer l i n e a r . As seen from equations (3.1.4) and (3.1.6) i n Table (3.1), A v a r i e s l i n e a r l y with H but CD v a r i e s q u a d r a t i c a l l y with H. An 92 Table ( 3 . D - - E q u i l i b r i a , maximum harvest r a t e , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y endpoints, and the q u a s i -c h a r a c t e r i s t i c equation f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and c o n s t a n t - e f f o r t predator h a r v e s t i n g 93 N o n - t r i v i a l model e q u i l i b r i u m x* = (b+H)/v y* = r ( 1 - (b + H)/vK)/u (3.1.1) (3.1.2) Maximum h a r v e s t r a t e Hmax = vK - b (3.1.3) Q u a s i -c h a r a c t e r i s t i c parameter v a l u e s A = -r(b+H)/vK • B = 0 CD = r(.b+H) (1-(b+H)/vK) (3.1.4) (3.1.5) (3.1.6) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s a t H = 0 A = -rb/vK CD = rb(1-b/vK) (3.1.7) (3.1.8) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax A = - r CD = 0 (3.1.9) (3.1.10) Q u a s i -c h a r a c t e r i s t i c e q u a t i o n Q(z) = ( z 2 + C D T 2 ) e x p ( z ) - ATz (3.1.11) 94 Table ( 3 . 2 ) — E q u i l i b r i a , maximum h a r v e s t r a t e , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y e n d p o i n t s , and the q u a s i -c h a r a c t e r i s t i c e q u a t i o n f o r H o l l i n g ' s model w i t h a t y p e - I I f u n c t i o n a l response and c o n s t a n t - e f f o r t p r e d a t o r h a r v e s t i n g . 95 N o n - t r i v i a l model e q u i l i b r i u m x* = (d+H)L/(Ms-(d+H)) y* = r(x*+L)(1-x*/K) (3.2.1) (3.2.2) Maximum harv e s t r a t e Hmax = [MsK/(K+D] - d (3.2.3) Quasi-c h a r a c t e r i s t i c parameter v a l u e s A = -(r/K)[(d+H)L/[sM-(d+H)] B = rx*(1-x*/K)/(x*+L) CD = sLMB/(x* + L) (3.2.4) (3.2.5) (3.2.6) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 A » -rJ/K B = r J ( K - j ) / K ( J + L ) CD = sJ L M r ( 1 - J / K ) / ( J + L ) 2 (3.2.7) (3.2.8) (3.2.9) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax A = - r B = 0 CD = 0 (3.2.10) (3.2.1 1 ) (3.2.12) Qua s i -c h a r a c t e r i s t i c equation Q(z) = (z 2-BTz+CDT 2)exp(z) -ATz (3.2.13) 96 Table ( 3 . 3 ) — E q u i l i b r i a , maximum h a r v e s t r a t e , D - p a r t i t i o n p arameters, h a r v e s t i n g t r a j e c t o r y e n d p o i n t s , and the q u a s i -c h a r a c t e r i s t i c e q u a t i o n f o r I v l e v ' s model w i t h c o n s t a n t - e f f o r t p r e d a t o r h a r v e s t i n g . 97 N o n - t r i v i a l model e q u i l i b r i u m x* = - l n [ ( e x p ( - c J ) - H/sb]/c y* = rx*(1-x*/K)/b(1-exp(-cx*)) (3.3.1) (3.3.2) Maximum harvest r a t e Hmax = s b [ e x p ( - c J ) - e x p ( - c K ) ] (3.3.3) Quasi-c h a r a c t e r i s t i c parameter values A = -rx*/K B = r(1-x*/K) - y*bcexp(-cx*) CD = s c b 2 e x p ( - c x * ) [ e x p ( - c J ) - e x p ( - c x * ) ] y * (3.3.4) (3.3.5) (3.3.6) Ha r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 A = -rJ/K B = r ( 1 - J / K ) [ 1 - c J / ( e x p ( c J ) - 1 ) ] CD = s b c e x p ( - c J ) [ r J ( 1 - J / K ) ] (3.3.7) (3.3.8) (3.3.9) Ha r v e s t i n g t r a j e c t o r y c o o r d i n a t e s a t H = Hmax A = - r B = 0 CD = 0 (3.3.10) (3.3.11) (3.3.12) Quasi-c h a r a c t e r i s t i c equation Q(z) = (z 2-BTz+CDT 2)exp(z) -ATz (3.3.13) 98 example of the p a r t i t i o n and a t r a j e c t o r y i s g i v e n i n F i g u r e ( 3 . 1 ) . From o b s e r v a t i o n of the h a r v e s t i n g t r a j e c t o r y and i t s en d p o i n t a t H = Hmax ( e q u a t i o n s (3.1.9-3.1.10) i n T a b l e ( 3 . 1 ) ) , we can g i v e a n e c e s s a r y c o n d i t i o n f o r i n s t a b i l i t y : i f r > u/2T, then the p r e d a t o r - p r e y system w i l l become u n s t a b l e f o r some H < Hmax. A consequence of t h i s i n s t a b i l i t y i s t h a t a maximum s u s t a i n a b l e y i e l d p o l i c y w i t h c o n s t a n t - e f f o r t h a r v e s t i n g i s no l o n g e r n e c e s s a r i l y s t a b l e . T h i s i s p a r t i c u l a r l y e v i d e n t i n prey s p e c i e s w i t h h i g h i n t r i n s i c growth r a t e s . In our example, H=0.1, the o p t i m a l h a r v e s t a c c o r d i n g t o the MSY p o l i c y , l e a d s to, an u n s t a b l e e q u i l i b r i u m . Thus, c o n t r a r y t o the c o n c l u s i o n s of May e t ^ . a l . (1979), we c o n c l u d e t h a t the f e a s i b i l i t y of an MSY p o l i c y i n h a r v e s t i n g a t the t o p of the t r o p h i c c h a i n depends l a r g e l y on the time s c a l e s of the lower t r o p h i c l e v e l s . 2. H o l l i n g ' s model w i t h a t y p e - I I f u n c t i o n a l response As i n p r e v i o u s c a s e s , i n the non-delayed model, the s t a b i l i t y c o n d i t i o n i s -A > B (3.21 ) R e a r r a n g i n g t h i s e x p r e s s i o n w i t h ( 3 . 2 . 1 ) , ( 3 . 2 . 4 ) , and (3.2.5) i n T a b l e ( 3 . 2 ) , we a r r i v e a t the i n e q u a l i t y H > MsL (K - 2J - L) / (K + L) ( J + L) (3.3) 99 Figure ( 3 . 1 ) - - S t a b i l i t y region f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and delay with c o n s t a n t - e f f o r t predator h a r v e s t i n g . The parameters are r= 2 , K=2000 , u=1, v= 0 . 0 0 2 , b= 0 . 2 , T=1. ( , D - p a r t i t i o n ; , h a r v e s t i n g t r a j e c t o r y ) . 101 Thus, i f K - 2J - L < 0, h a r v e s t i n g w i l l never d e s t a b i l i z e a s t a b l e e q u i l i b r i u m i n the non-delayed case. I f K - 2J - L > 0, the unharvested e q u i l i b r i u m i s unstable and an i n c r e a s e i n harvest can s t a b i l i z e the e q u i l i b r i u m . In the delayed model, the dynamics of A f o r c e s the s t a b i l i t y boundary to move toward the CD a x i s as H i n c r e a s e s . C o n c u r r e n t l y , the h a r v e s t i n g t r a j e c t o r y moves toward the o r i g i n . The i n t e r p l a y between these two movements d e f i n e s the ha r v e s t s f o r which the e q u i l i b r i u m i s s t a b l e . As shown i n Fi g u r e (3.2), t h i s can le a d to m u l t i p l e t r a n s i t i o n s between s t a b l e and unstable e q u i l i b r i a with i n c r e a s i n g harvest r a t e , i n c o n t r a s t to the non-delayed model which has at most one t r a n s i t i o n . One can see t h i s phenomenon even more c l e a r l y i n F i g u r e (3.3) where the s t a b i l i t y boundary i s p l o t t e d i n the (H,T) p l a n e . 3. I v l e v ' s model Des p i t e the apparent s i m i l a r i t y to H o l l i n g ' s model i n the s t r u c t u r e of the f u n c t i o n a l response term, our t r i a l s with I v l e v ' s model d i d not i n d i c a t e any m u l t i p l e s t a b i l i t y t r a n s i t i o n s . Even with the analogous parameter values ( i . e . L = 1/c, b=M), the maximum s t a b l e delay decreased m o n o t o n i c a l l y with i n c r e a s i n g h a r v e s t . T h i s i s the reason f o r i n c l u d i n g I v l e v ' s model i n t h i s study even though i t i s f u n c t i o n a l l y e q u i v a l e n t to H o l l i n g ' s model. An example of the D - p a r t i t i o n i s given i n Fi g u r e (3.4) along with an H-T p l o t i n F i g u r e (3.5). 102 Figure (3.2)-- S t a b i l i t y region for Holling's model with delay and constant-effort predator harvesting. CD , H=0.01: the equilibrium i s unstable; (T) , H=0.07: the equilibrium i s stable; (S) , H=0.11: the equilibrium is unstable. The model parameters are r=l, K=40, L=iO, J=20, s=l, M=1, and T=1.67. (» , D-partition; , harvesting t r a j e c t o r y ) . 104 Figure ( 3 . 3 ) — Maximum allowable delay for Holling's model g i v e „ c ° „ n s ^ u ; r u " , p r e d a t o r h a r v e s t f n 9- Th* -« 105 106 Figure ( 3 . 4 ) — S t a b i l i t y region for Ivlev's model with delay and constant-effort predator harvesting. For t h i s t r i a l , r=2, K=30, J=20, b=1, c=0.1, s=1, and T=0.5. The equilibrium is stable for a l l harvesting rates. ( , D-partition; harvesting t r a j e c t o r y ) . ' ' 108 Figure ( 3 . 5 ) — Maximum allowable delay for Ivlev's model. The parameters are given in Figure (3.4). 110 A p o s s i b l e e x p l a n a t i o n f o r t h i s d i f f e r e n c e between the two models i s t h a t the type-II f u n c t i o n a l response i n H o l l i n g ' s model i s s l i g h t l y more d e s t a b i l i z i n g than the ex p o n e n t i a l form i n I v l e v ' s model, at l e a s t f o r the parameter ranges surveyed. In t h i s c ontext, 'more d e s t a b i l i z i n g ' means that f l ( x ) / x has a gr e a t e r negative slope at e q u i l i b r i u m . Constant-quota predator h a r v e s t i n g As mentioned p r e v i o u s l y , the s t r a t e g y of constant-quota predator h a r v e s t i n g adds a e x t r a dimension of complexity to our a n a l y s i s because the e q u i l i b r i u m c o n d i t i o n no longer r e q u i r e s that F=0 i n the q u a s i - c h a r a c t e r i s t i c equation. A c c o r d i n g l y , the c o n c l u s i o n s of t h i s s e c t i o n are drawn from i n d i v i d u a l t r i a l s r a t her than from geometries. To p r o v i d e a template f o r comparison, we w i l l f i r s t a nalyze the non-delayed model. The q u a s i - c h a r a c t e r i s t i c equation f o r constant-quota predator h a r v e s t i n g was presented i n equation (1.16). I f we l e t T=0 i n that equation, the r e s u l t i s Q ( s ) = S 2 - ( B + F ) s - A s + A F + C D + B F (3.4) The s t a b i l i t y c o n d i t i o n s of equation (3.4) are d e r i v e d from the Routh-Hurwitz c r i t e r i o n (See Schwarz and F r i e d l a n d , pp.401-413). These are B < -A - F and CD > -F(B + A) 1 1 1 (3.,5) (3.6) I f F i s set to zero, these c o n d i t i o n s reduce to the simpler set d e r i v e d p r e v i o u s l y . In the (B,CD) plane, the f i r s t c o n d i t i o n r e p r e s e n t s a p a r t i t i o n of a v e r t i c a l l i n e at B = -A - F. The second c o n d i t i o n i s s a t i s f i e d f o r a l l the models which we w i l l be studying f o r one of the e q u i l i b r i u m p o i n t s and are v i o l a t e d f o r the other. T h e r e f o r e , a l l the models have one p o t e n t i a l l y s t a b l e p o i n t or focus and one saddle p o i n t . The second s t a b i l i t y c o n d i t i o n corresponds to the statement that the slope of the prey i s o c l i n e without h a r v e s t i n g i s l e s s than the slope of the predator i s o c l i n e with h a r v e s t i n g r a t e H chosen so that the predator i s o c l i n e passes through (0,P) where P i s the maximum predator d e n s i t y f o r which the p o p u l a t i o n can e s t a b l i s h i t s e l f from a small i n i t i a l p o p u l a t i o n (Brauer and Soudack 1979). Only the upper, c o n d i t i o n a l l y s t a b l e e q u i l i b r i u m i s analyzed i n t h i s c h a pter. Brauer e t . a l . (1976) showed t h a t , i n H o l l i n g ' s model with a type-II f u n c t i o n a l response and i n I v l e v ' s model, constant-quota h a r v e s t i n g may s t a b i l i z e an unstable e q u i l i b r i u m . I f there i s a l i m i t c y c l e , the l i m i t c y c l e may decrease i n amplitude u n t i l a c r i t i c a l h a rvest r a t e i s reached at which the system c o l l a p s e s . They g i v e no examples for the L o t k a - V o l t e r r a model with resource l i m i t a t i o n . For constant-quota predator h a r v e s t i n g and s t o c k i n g , 1 12 Tables (3.4) and (3.5) supply i n f o r m a t i o n about the e q u i l i b r i u m p o p u l a t i o n s i z e s ; the maximum h a r v e s t i n g and s t o c k i n g r a t e s ; the parameter values f o r q u a s i - c h a r a c t e r i s t i c equation; and the t r a j e c t o r y p o i n t s at H=0, H=Hmax, and H=-Smax f o r H o l l i n g ' s model with a type-II f u n c t i o n a l response and L o t k a - V o l t e r r a model with resource l i m i t a t i o n . For I v l e v ' s model, a n a l y t i c d e r i v a t i o n of the e q u i l i b r i u m was imp o s s i b l e ; t h e r e f o r e , numerical e s t i m a t i o n was used. 1. L o t k a - V o l t e r r a model with resource l i m i t a t i o n Since B=0 i n t h i s model, the s t a b i l i t y c o n d i t i o n f o r the non-delayed model reduces to -A > F with A = -rx*/K and F = vx* - b. S i m p l i f y i n g the s t a b i l i t y c o n d i t i o n , we see that x* < b / (v - r/K) (3.9) Since x* i n c r e a s e s with h a r v e s t i n g , the e q u i l i b r i u m becomes uns t a b l e , stays u n s t a b l e , or stays s t a b l e i n the presence of h a r v e s t i n g . If v < r/K, t h i s c o n d i t i o n i s not s a t i f i e d f o r f e a s i b l e x* (x > 0). An example of a complete t r a j e c t o r y with h a r v e s t i n g and s t o c k i n g (H < 0) i s given i n F i g u r e (3.6). (3.7) (3.8) 113 Table ( 3 . 4 ) — E q u i l i b r i a , maximum h a r v e s t i n g and s t o c k i n g r a t e s , D - p a r t i t i o n parameters, h a r v e s t i n g t r a j e c t o r y endpoints, and the q u a s i - c h a r a c t e r i s t i c equation f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and c o n s t a n t -quota predator h a r v e s t i n g and s t o c k i n g . 114 N o n - t r i v i a l model e q u i l i b r i u m x* = [(b/v+K) ± [ ( b / v - K ) 2 - 4 K ( u H / v r ) 1 / 2 ] / 2 y* = r d - x * / K ) / u (3.4.1) (3.4.2) Maximum harvest r a t e Hmax = r ( b - v K ) 2 / 4Kuv (3.4.3) Maximum s t o c k i n g r a t e Smax = rb/u (3.4.4) Quasi-c h a r a c t e r i s t i c parameter values A = -rx*/K B = 0 CD = vrx*(1-x*/K) F = vx* - b (3.4.5) (3.4.6) (3.4.7) (3.4.8) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 A = -rb/vK CD = rb(1-b/vK) F = 0 (3.4.9) (3.4.10) (3.4.11) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s H = Hmax A = -r(1 + b/vK)/2 CD = r [ ( v K ) 2 - b 2 ] F = (Kv-b)/2 (3.4.12) (3.4.13) (3.4.14) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at S = Smax A = 0 CD = 0 F = -b (3.4.15) (3.4.16) (3.4.17) 115 Table ( 3 . 5 ) — E q u i l i b r i a , maximum harvesting and stocking rates, D-partition parameters, harvesting trajectory endpoints, and the quasi-characteristic equation for Holling's model with a type-II functional response and constant-quota predator harvesting and stocking. 116 N o n - t r i v i a l model e q u i l i b r i u m x* = [(K+J) ± [ ( K - J ) 2 - 4 K ( J + L ) H / r s L ] 1 / 2 ] / 2 y* = r(x*+L)(1-x*/K)/M (3.5.1) (3.5.2) Maximum h a r v e s t i n g r a t e Hmax = srL(K-J.) 2/4K(J+L) (3.5.3) Maximum st o c k i n g r a t e Smax = rsLJ/(J+L) (3.5.4) Quasi-c h a r a c t e r i s t i c parameter values A = -rx*/K B = r(1-x*/K)(x*/(x*+L)) CD = srMLx*(K-x*)/K(x*+L) 2 F = Ms[x*/(x*+L) - J / ( J + U ] (3.5.5) (3.5.6) (3.5.7) (3.5.8) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = 0 A = -rJ/K B = Jr(K-J)/K(J+L) CD = srML J ( K - J ) / K ( J + L ) 2 F = 0 (3.5.9) (3.5.10) (3.5.11) (3.5.12) Ha r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at H = Hmax A = -r(J+K)/2K B = r(K 2-J 2)/2K(K+J+2L) CD = srML(K 2-J 2)/K(K+J+2L) 2 F = Ms[(J+K)/(J+K+2L) - J/(J+L)] (3.5.13) (3.5.14) (3.5.15) (3.5.16) H a r v e s t i n g t r a j e c t o r y c o o r d i n a t e s at S = Smax A = 0 B = 0 CD = 0 F = -MsL/(J+L) (3.5.17) (3.5.18) (3.5.19) (3.5.20) 1 1 7 Figure ( 3 . 6 ) — Constant-quota harvesting and stocking trajectory for the Lotka-Volterra model with resource l i m i t a t i o n . The parameters are r=1, K=2000, u=1, V=0.0015, b=0.2. The lower equilibrium is always unstable and i s not plotted on thi s figure. 1 19 A s u r p r i s e o c c u r s when d e l a y i s added t o the system. F i r s t , the s t a b i l i t y p a r t i t i o n shows b i f u r c a t i o n s i n the r e g i o n of the h a r v e s t i n g t r a j e c t o r y . As T i n c r e a s e s , the p a r t i t i o n moves toward the A a x i s . T h i s i s s i m i l a r t o h a r v e s t i n g p r e y a t c o n s t a n t - e f f o r t which d e v e l o p s m u l t i p l e t r a n s i t i o n s as T i s i n c r e a s e d . The b i f u r c a t i o n a l l o w s the p o s s i b i l i t y of more t r a n s i t i o n s between s t a b l e and u n s t a b l e r e g i o n s than would occur i n the absence of d e l a y . An example of t h i s i s shown i n F i g u r e (3.7) where, a t H=0.0533, the t r a j e c t o r y e n t e r s the u n s t a b l e r e g i o n . 2. H o l l i n g ' s model w i t h a t y p e - I I f u n c t i o n a l response A sample h a r v e s t i n g t r a j e c t o r y f o r H o l l i n g ' s model i s g i v e n i n > F i g u r e ( 3 . 8 ) . The s t a b i l i t y c o n d i t i o n f o r the non-d e l a y e d system i s , as f o r the o t h e r models i n t h i s s e c t i o n , B < -A - F (3.10) From the c o n d i t i o n s (3.5.5-3.5.8), e q u a t i o n (3.11) changes t o x* 2 - (w + (K - L ) / 2 ) x * + wJ > 0 (3.12) where w = MsLK / 2 r ( L + J) > 0. T h i s becomes 120 Figure ( 3 . 7 ) — P a r t i t i o n s l i c e s for the Lotka-Volterra model with resource l i m i t a t i o n , delay, and constant-quota predator harvesting. The parameters for this t r i a l are r=l, K=3000, u=1, v=0.0002, b=0.2, and T=3.5. The cross + marks the point on the harvesting trajectory for the p a r t i c u l a r s l i c e . The following figures are: (a) p a r t i t i o n s l i c e at H=0, (b) p a r t i t i o n s l i c e at H=0.0133, (c) p a r t i t i o n s l i c e at H=0.0267, (d) p a r t i t i o n s l i c e at H=0.0400, (e) p a r t i t i o n s l i c e at H=0.0533. 121 J L> m m m o i II II Q U ru ru i ru i m i I ru i i 124 125 126 Figure (3.8)-- Constant-quota h a r v e s t i n g and s t o c k i n g s-t Jnd°M-l H o l l i n 9 ' s m o d e l w i t h r = 1 ? K=40, L=10, , 0.25 0.5 128 x* > [0.5(K-D + w - [ ( 0 . 5 ( K - L ) - w ) 2 - 4 w j ] 1 ' 2 ]/2 (3.13) or x* < [0.5(K-D + w + [ ( 0 . 5 ( K - L ) - w ) 2 - 4 w j ] 1 / 2 ] / 2 H a r v e s t i n g i n c r e a s e s x*; t h e r e f o r e , c o n d i t i o n s (3.13) r e v e a l that h a r v e s t i n g can s t a b i l i z e an unstable e q u i l i b r i u m and can a l s o d e s t a b i l i z e a s t a b l e one. The l a t t e r t r a n s i t i o n a r i s e s when M i s l a r g e enough to allow F to inc r e a s e q u i c k l y with h a r v e s t i n g r e l a t i v e to -A. For example, with the parameters r=1, K=40, J=17, M=6, s=1, the e q u i l i b r i u m d e s t a b i l i z e s at H=0.3. Because M s i g n i f i e s the number of prey eaten i n a time i n t e r v a l ( a t t a c k r a t e times the time a v a i l a b l e f o r e n c o u n t e r s ) , a system with a vo r a c i o u s predator i s more l i k e l y to be d e s t a b i l i z e d by h a r v e s t i n g , even without d e l a y . For moderate values of M, the h a r v e s t i n g t r a j e c t o r y can t r a v e r s e the s t a b i l i t y boundaries twice: s t a r t i n g as a focus or node, becoming unstable at int e r m e d i a t e harvest r a t e s , and then r e s t a b i l i z i n g at high h a r v e s t i n g r a t e s . An instance of t h i s type turns up when M=4.5 r e p l a c e s M=6 i n the above example. The f i r s t t r a n s i t i o n occurs at H=0.6; the second at H=1.2. The maximum harvest r a t e i s H=1.225. As these m u l t i p l e t r a n s i t i o n s happen at only very s e l e c t i v e parameter values, one would expect that o b s e r v a t i o n s of such would be r e l a t i v e l y r a r e f o r the non-delayed model. For small M, the e q u i l i b r i u m i s not d e s t a b i l i z e d . An in c r e a s e i n the con v e r s i o n e f f i c i e n c y of the predator a l s o causes the above s t a b i l i t y t r a n s i t i o n s . The geometry of the delayed model d i f f e r s l i t t l e from the non-delayed one. The change i n cu r v a t u r e of the p a r t i t i o n 129 curve i n the (B,CD)-plane i s i d e n t i c a l to that of the p revious h a r v e s t i n g s t r a t e g i e s , except that both A and F change with h a r v e s t i n g r a t e s . T h i s c u r v a t u r e i n c r e a s e s the chance of double t r a n s i t i o n s across the s t a b i l i t y boundary. F i g u r e (3.9) shows a t y p i c a l example with one t r a n s i t i o n from s t a b i l i t y to i n s t a b i l i t y . As i n the non-delayed model, constant-quota predator h a r v e s t i n g tends to s t a b i l i z e the e q u i l i b r i u m . Delay g e n e r a l l y e nlarges the harvest r a t e at which the e q u i l i b r i u m s t a b i l i z e s . The maximum p o s s i b l e number of t r a n s i t i o n s i n the delayed case should be l a r g e r than in the non-delayed case because of the a d d i t i o n of a degree of freedom with the parameter T. I n v e s t i g a t i n g t h i s p o t e n t i a l would i n v o l v e an enormous number of t r i a l s without any d e f i n i t i v e c o n c l u s i o n being reached. Our s i m u l a t i o n s have not shown more than two t r a n s i t i o n s so that the t r a n s i t i o n a l behaviour of the delayed model appears to be s i m i l a r to the non-delayed. However, i t s t i l l h olds that the delayed model i s g e n e r a l l y l e s s s t a b l e f o r the same parameter values because of the movement of the D - p a r t i t i o n . Our g l o b a l s i m u l a t i o n t r i a l s c o n f i r m t h i s o b s e r v a t i o n . 3. I v l e v ' s model The e q u i l i b r i u m from t h i s model are o btained form the e q u i l i b r i u m c o n d i t i o n s sby*[exp(-cJ) - exp(-cx*)] = H y* = r x * ( ! - x*/K) / b d - exp(-cx*)) (3.14) (3.15) 130 Figure (3.9)-- P a r t i t i o n s l i c e s for Holling's model with delay and constant-quota predator harvesting. The parameter values are r=1, K=60, J=20, L=10, M=1, s=l and T=0.5. The figures are: (a) p a r t i t i o n s l i c e at H=0, (b) p a r t i t i o n s l i c e at H=0.444, (c) p a r t i t i o n s l i c e at H=0.889, (d) p a r t i t i o n s l i c e at H=1.333, (e) p a r t i t i o n s l i c e at H=1.778. 1 3 1 H Vi § 6 9 6 H II u i- < U_ • ru ro ru T - l i i ru i ru i a I 132 D 6 -I D II ll II •I- < U_ • ru 11 ru ru i m i i ru i m 133 O O • I D II II ll t- < Ix. m nj • m nj i T 4 I nj i I 134 135 136 These equations cannot be s o l v e d a n a l y t i c a l l y but we used numerical e s t i m a t i o n to f i n d the e q u i l i b r i u m v a l u e s . The e s t i m a t i o n method i s M u l l e r ' s method which was i n c o r p o r a t e d i n t o a prepared program at the U n i v e r s i t y of B.C. Computing Center. Once the e q u i l i b r i u m p o p u l a t i o n s x* and y* are found, the other parameters may be computed i n a s t r a i g h t f o r w a r d a n a l y t i c a l f a s h i o n . F i g u r e (3.10) shows an example of the h a r v e s t i n g t r a j e c t o r y and F i g u r e (3.11) diagrams the D-part-ition r e g i o n s . As can be seen from these graphs, the c o n c l u s i o n s obtained f o r H o l l i n g ' s model apply here as w e l l . Predator s t o c k i n g As s t o c k i n g of p r e d a t o r s i s c o n f i n e d to the same quadrant of the (B,CD) plane, i t i s a n a t u r a l extension to our h a r v e s t i n g i n v e s t i g a t i o n . Predator s t o c k i n g i n t r o d u c e s a g r e a t e r complexity to the p r e d a t o r - p r e y i n t e r a c t i o n as s t a b i l i t y i s not guaranteed and the s t a b i l i t y p a r t i t i o n s are dynamic as i n constant-quota predator h a r v e s t i n g . However, i n s t a b i l i t y i m p l i e s l i m i t c y c l e behaviour under t h i s s t r a t e g y and not e x t i n c t i o n of e i t h e r the predator or the prey i n non-delayed models. 1. L o t k a - V o l t e r r a model with resource l i m i t a t i o n An example of the h a r v e s t i n g t r a j e c t o r y f o r L o t k a - V o l t e r r a model with resource l i m i t a t i o n with s t o c k i n g i s i n c l u d e d i n 137 Fi g u r e ( 3 . 1 0 ) — C o n s t a n t - e f f o r t h a r v e s t i n g and s t o c k i n g t r a j e c t o r y f o r I v l e v ' s model with r=2, K=35, J=20, b=1, c=0.1, s=1 . ' 139 Figure (3.11)— P a r t i t i o n s l i c e s for Ivlev's model with delay and constant-quota predator harvesting. With r=2, K=35, J=20, b=1, c=0.1, s=1, and T=0.9. The figures are: (a) p a r t i t i o n s l i c e at H=0, (b) p a r t i t i o n s l i c e at H=0.1, (c) p a r t i t i o n s l i c e at H=0.3, (d) p a r t i t i o n s l i c e at H=0.6, (e) p a r t i t i o n s l i c e at H=0.9. 1 4 1 ru i i 143 1 45 F i g u r e (3.6). The s t o c k i n g s e c t i o n of the t r a j e c t o r y extends the h a r v e s t i n g s e c t i o n to the o r i g i n . As with h a r v e s t i n g , the non-delayed model i s always s t a b l e (Brauer and Soudack 1981a). With small d e l a y s , there i s g e n e r a l l y no d i f f e r e n c e between the delayed and non-delayed behaviours. As delay i n c r e a s e s , the s t a b i l i t y p a r t i t i o n b i f u r c a t e s , c ausing i n s t a b i l i t y i n the second quadrant of the (B,CD) plane. An example of the p a r t i t i o n f o r t h i s model are shown in F i g u r e (3.12) for T=1.0. Thus, i f the delay i s l a r g e enough, i n s t a b i l i t y may r e s u l t . We w i l l d i s c o v e r i n our g l o b a l s t a b i l i t y t r i a l s whether t h i s l o c a l i n s t a b i l i t y extends to the g l o b a l s i t u a t i o n . 2. H o l l i n g ' s model with a type-II f u n c t i o n a l response The h a r v e s t i n g t r a j e c t o r y with s t o c k i n g ends with B=0, CD=0, as can be seen in F i g u r e (3.8). Both with and without d e l a y s , there can be m u l t i p l e t r a n s i t i o n s between s t a b l e and unstable e q u i l i b r i a . F i g u r e (3.13) g i v e s an example with and without d e l a y . In both, a l a r g e enough s t o c k i n g l e v e l causes the e q u i l i b r i u m to become s t a b l e , although t h i s t r a n s i t i o n may occur at very low prey e q u i l i b r i u m l e v e l s , thereby exposing the prey to the danger of e x t i n c t i o n by exogenous p e r t u r b a t i o n s . I t i s d i f f i c u l t to d i s s e c t the r o l e of the v a r i o u s parameters i n causing m u l t i p l e t r a n s i t i o n s with predator s t o c k i n g but one can say that i f -A B at H = 0, then m u l t i p l e t r a n s i t i o n s are more l i k e l y . That i s to say, i f the system without s t o c k i n g i s c l o s e to i n s t a b i l i t y , then a small 146 Figure (3.12)-- P a r t i t i o n s l i c e s f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , delay, and predator s t o c k i n g with r=1, K=3000, u=1, v=0.0002, b=0.2, and T=1. The f i g u r e s a r e : (a) p a r t i t i o n s l i c e at S=0, (b) p a r t i t i o n s l i c e at S=0.04, (c) p a r t i t i o n s l i c e at S=0.16. 147 II II I- U. crj • rn ru ro ru I ru i rn i ru i m m i 148 • rn i 149 150 Figure (3.13)-- P a r t i t i o n s l i c e s f o r H o l l i n g ' s model with and without d e l a y and with predator s t o c k i n g . The parameter values are r = l , K=40, J=20, L=10, M=1, s=1, T=1. The delayed p a r t i t i o n i s shown as' ; the non-delayed p a r t i t i o n i s given by a dashed l i n e , . The f i g u r e s a r e : (a) p a r t i t i o n s l i c e at S=0, (b) p a r t i t i o n s l i c e at S=2.67, (c) p a r t i t i o n s l i c e at S=4.00. 151 152 153 154 amount of s t o c k i n g w i l l induce l i m i t c y c l e behaviour and a d d i t i o n a l enhancement may r e s t a b i l i z e the e q u i l i b r i u m . 3. I v l e v ' s model The e q u i l i b r i u m f o r t h i s model i s found i n the same way that the h a r v e s t i n g e q u i l i b r i u m i s found the e q u i l i b r i u m of t h i s model i s found through the numerical e s t i m a t i o n of the e q u i l i b r i u m from equations (3.14) and (3.15), as i n the constant-quota h a r v e s t i n g case. The maximum s t o c k i n g r a t e , at which x*=0 i n equation (3.14), i s Smax = s b d - e x p ( - c J ) ) y * . (3.16) Since l i m y* = r/bc (3.17) x*->0 by L ' H o s p i t a l ' s r u l e , Smax i s given by Smax = sr(1 - e x p ( - c J ) ) / c (3.18) which i s analogous to the equation from H o l l i n g ' s model with a type-II f u n c t i o n a l response. Fig u r e (3.10) d i s p l a y s a h a r v e s t i n g t r a j e c t o r y f o r t h i s model which i s s i m i l a r to that of H o l l i n g ' s model with a type-II f u n c t i o n a l response. In F i g u r e (3.14), we provide an example of the s t a b i l i t y p a r t i t i o n s . The behaviour of t h i s 155 Figure (3.14)-- P a r t i t i o n s l i c e s for Ivlev's model with delay and predator stocking. The parameter values are r=2, K=35, J=20, b=1, c=0.1, s=1, and T=0.9. The figures are: (a) p a r t i t i o n s l i c e at S=0, (b) p a r t i t i o n s l i c e at S=6., (c) p a r t i t i o n s l i c e at S=8., (d) p a r t i t i o n s l i c e at S=16.0. 156 nj i * m i •158 : D'BOO -D-4543 160 model mimics that of the p r e v i o u s one almost e x a c t l y . C o n c l u s i o n s In two of our three models, we found that c o n s t a n t - e f f o r t h a r v e s t i n g can d e s t a b i l i z e a s t a b l e e q u i l i b r i u m , c o n t r a r y to the p r e d i c t i o n s of the non-delayed models. The g e o m e t r i c a l complexity of the D - p a r t i t i o n causes t h i s d i s s i m i l a r i t y . With the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , we showed that t h i s dynamical change can l e a d to an unstable MSY h a r v e s t i n g p o l i c y . Constant-quota s t o c k i n g and h a r v e s t i n g appear to have the same number of t r a n s i t i o n s i n the delayed and non-delayed cases. However, because the p a r t i t i o n i s moved toward the CD a x i s , delay makes the i n t e r s e c t i o n of the p a r t i t i o n and the h a r v e s t i n g - s t o c k i n g t r a j e c t o r y more l i k e l y , and thereby i n c r e a s e s the chances of m u l t i p l e t r a n s i t i o n s . Our numerical s i m u l a t i o n s c o n f i r m these o b s e r v a t i o n s . A l t e r n a t i v e h a r v e s t i n g s t r a t e g i e s , such as bang-bang h a r v e s t i n g (Clark 1976b) and s t r a t e g i e s more s e n s i t i v e to the l i m i t i n g food resource (Noy-Meir 1975), might be c o n s i d e r e d i n delayed systems where these f i x e d h a r v e s t i n g r a t e s become un s t a b l e . Future r e s e a r c h e r s might c o n s i d e r these a l t e r n a t i v e s w i t h i n t h i s m u l t i s p e c i e s or delayed r e s o u r c e - l i m i t a t i o n framework. An i n t e r e s t i n g system which might be used to t e s t some of the dynamical p r e d i c t i o n s made by the delayed models i s the a l e w i f e - lake t r o u t system i n the Great Lakes. An economically 161 v i t a l forage f i s h , the a l e w i f e , feeds s e l e c t i v e l y on the l a r g e r s p e c i e s of zooplankton. Wells (1970) r e p o r t s that the zooplankton community changes with f l u c t u a t i o n s i n a l e w i f e numbers. When a l e w i f e abundance i s hig h , the smal l e r s p e c i e s predominate; when i t i s low, the l a r g e r s p e c i e s have a chance to r e c o v e r . Even though the recovery time f o r any s p e c i e s may be s m a l l , the recovery time f o r the l a r g e r plankton community may be much l a r g e r . Wells observed t h i s sequence of events a f t e r the major 1967 d i e o f f when a l e w i f e d e n s i t y plunged f o r unknown reasons. Kohler and Ney (1981) c o n f i r m that the r e l a t i v e abundance of zooplankton s p e c i e s s h i f t s f o l l o w i n g a d i e o f f . In the s p r i n g and summer a f t e r a d i e o f f i n a V i r g i n i a r e s e r v o i r , a midsized c l a d o c e r a n Diaphanosoma leuctenbergianum was a major component of the zooplankton community but then d e c l i n e d markedly i n abundance the f o l l o w i n g summer. The most abundant zooplankter i n the year a f t e r the d i e o f f was a small c l a d o c e r a n Bosmina l o n g i r o s t r i s . The lengths of the four most abundant z o o p l a n k t e r s were s i g n i f i c a n t l y l a r g e r immediately f o l l o w i n g the d i e o f f . In c o n j u n c t i o n with these changes i n ,the resource base, the f e c u n d i t y of the a l e w i f e i s l a r g e (Norden 1967), magnifying the e f f e c t of any resource recovery d e l a y s . While these changes i n the resource base do not e s t a b l i s h the e x i s t e n c e of a resource recovery d e l a y , that p o s s i b i l i t y i s c e r t a i n l y a t e s t a b l e h y p o t h e s i s . From a resource management p e r s p e c t i v e , s i n c e a l e w i f e i s preyed upon by lake t r o u t stocked i n the Great Lakes, i t would 1 62 be most u s e f u l to f i n d out how p r e d a t i o n by the lake t r o u t or h a r v e s t i n g by the fisherman a f f e c t these f l u c t u a t i o n s . T h i s i s p a r t i c u l a r l y c r u c i a l because the salmonid-stocking program i n the Great Lakes has t r i g g e r e d i n t e r e s t i n s t o c k i n g alewives i n r e s e r v o i r s that l a c k a s u i t a b l e forage base f o r sport f i s h (Kohler and Ney 1981) . 1 63 IV. COEXISTENCE REGIONS AND GLOBAL STABILITY I n t r o d u c t i o n As s t r e s s e d i n our i n t r o d u c t i o n to l o c a l s t a b i l i t y , the n o n l i n e a r nature of e c o l o g i c a l systems demands g l o b a l a n a l y s i s of s t a b i l i t y r e g i o n s . In t h i s chapter, we w i l l map out g l o b a l s t a b i l i t y regions a r i s i n g from constant-quota h a r v e s t i n g of the predator and prey s i m u l t a n e o u s l y to d i s c o v e r how delays change the shape of these r e g i o n s . T h i s work p a r a l l e l s that of Brauer and Soudack (1981b) with the analogous non-delayed systems. Methods The c o n c e n t r a t i o n i n t h i s chapter i s on the h a r v e s t - r a t e plane rather than on the phase plane i n order to emphasize the r e l a t i o n s h i p between delay and h a r v e s t i n g that has been s t r e s s e d i n the p r e v i o u s c h a p t e r s . Brauer and Soudack (1981b) compare the s t a b i l i t y boundary p r e d i c t e d by l o c a l a n a l y s i s , which they c a l l sigma (and which we w i l l c a l l p ) , and the c o e x i s t e n c e boundary, h. The re g i o n between the p and h boundaries i s l o c a l l y unstable and h a r v e s t i n g r a t e s w i t h i n t h i s r e g i o n l e a d to l i m i t c y c l e behaviour. They concluded that the r e g i o n of c o e x i s t e n c e was o f t e n only a small p a r t of the region of f e a s i b l e e q u i l i b r i a (x*,y* > 0). We w i l l use the same terminology i n d i v i d i n g the H-G plane i n t o these d i f f e r e n t types of r e g i o n s . The regions of f e a s i b l e e q u i l i b r i a w i l l be bounded by the l i n e s l a b e l l e d R. While the theorems that they 164 proved to a i d the d e f i n i t i o n of the boundaries do not t e c h n i c a l l y apply to the delayed equations, we s h a l l see that the theorem that the h boundary w i l l not enter the t h i r d quadrant of the H-G plane appears to h o l d i n our s i m u l a t i o n s . The magnitude of delays i n the s i m u l a t i o n s were chosen to be of the same order as the time s c a l e of prey growth. As we s h a l l d i s c o v e r , there i s a continuous v a r i a t i o n of the s t a b i l i t y r e g i o n s with the delay so that the magnitude of the delay i s not a c r u c i a l f a c t o r i n our s i m u l a t i o n s . The i n t e g r a t i o n method employed was a m o d i f i e d Runga-Kutta a l g o r i t h m , from Grafton (1972) who a p p l i e d t h i s method to the L i e n a r d equation with d e l a y . T h i s a l g o r i t h m i s o u t l i n e d in Appendix B along with an e v a l u a t i o n of e r r o r . For each of the three models, we give f i v e examples. F i g u r e s (4.1)-(4.5) give the r e s u l t s f o r H o l l i n g ' s model with a type-II f u n c t i o n a l response; F i g u r e s (4.6)~(4.10) f o r I v l e v ' s model; and F i g u r e s (4.11)-(4.15) f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n . We w i l l r e f e r to these examples in the f o l l o w i n g d i s c u s s i o n s of g l o b a l s t a b i l i t y r e s u l t s . R e s u l t s 1. H o l l i n g ' s model with a type-II f u n c t i o n a l response A. Time delay e f f e c t s To get a general idea of the e f f e c t s of delay on the shape 165 Figure (4.1)-- H-G plane for Holling's model with r=2, K=40, J=20, L=10, M=1., s=1, and T=0.75. 166 167 Figure ( 4 . 2 ) — H-G plane for Holling's model with r=l, K=60 J=20, L=10, M=1., s=1, and T=1.5. 168 169 Figure ( 4 . 3 ) — H-G plane for Holling's model with r=1, K=40, J=20, L=10, M=1., s=0.5, and T=1.5. 170 171 Figure ( 4 . 4 ) — H-G plane for Holling's model with r=1, K=40, J=20, L=15, M=1., s=1, and T=1.5. 172 o va CM — i — C D i Q_ » 173 Figure (4.5)-- H-G plane for Holling's model with r=l, K=40, J=20, L=10, M=3., s=0.33, and T=1.5. 175 Figure ( 4 . 6 ) ~ H-G plane for Ivlev's model with r=2, K=35f J=20, b=1, c=0.1, s=1, and T=0.9. 176 177 Figure (4.7)-- H-G plane for Ivlev's model with r=1, K=40 J=30, b=1, c=0.1, s=1, and T=1.5. 179 Figure (4.8)-- H-G plane for Ivlev's model with J-20, b=3, c=0.1, s=0.33, and T=1.5. 180 I 0 I Figure (4.9)-- H-G plane for Ivlev's model with r=l, K=40, J=20, b=1, c=0.05, s=1, and T=1.5. 183 Figure (4.10)— H-G plane for Ivlev's model with r=2, K=40, J=20, b=l, c=0.1, S=1, and T=0.9. 184 185 Figure (4.11)-- H-G plane f o r resource l i m i t a t i o n with r=2, and T=1.1 the L o t k a - V o l t e r r a model with K=2000, U=0.015, V=0.001, b=0.5, 186 187 Figure (4.12)-- H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r=1, K=3000, U=0.015, V=0.001, b=0.5 and T=1.5 ' 189 Figure ( 4 . 1 3 ) — H-G plane f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with r=4, K=2000, U=0.015, V=0.001, b=0.5, and T=1 .5 191 Figure ( 4 . 1 4 ) — H-G plane f o r resource l i m i t a t i o n with r=1 and T=1.5 ' the L o t k a - V o l t e r r a model with K=2000, U=0.015, V=0.001, b=0.3, 193 Figure ( 4 . 1 5 ) — H-G plane for the Lotka-Volterra model with resource l i m i t a t i o n with r=1, K=2000, u=0.06, V=0.001, b=0 5 and T=1.5 194 195 of the s t a b i l i t y r e g i o n , we w i l l r e f e r to our l o c a l r e s u l t s . S t a r t i n g with constant-quota prey h a r v e s t i n g , F i g u r e (4.16) p r e s e n t s a p l o t of the a l l o w a b l e time d e l a y a g a i n s t the h a r v e s t i n g r a t e f o r the f i r s t t r i a l of t h i s model ( F i g u r e ( 4 . 1 ) ) . Up to a c e r t a i n delay of approximately T=0.75, the e q u i l i b r i u m i s s t a b l e . Past t h i s d e l a y , the higher harvest r a t e s become un s t a b l e , then the lower ones d e s t a b i l i z e . T h i s i s expected f o r our e x p l a n a t i o n s concerning the p a r t i t i o n because, at H=0, the t r a j e c t o r y i s c l o s e to the o r i g i n i n the (B,CD) plane f o r t h i s example and t h e r e f o r e d e s t a b l i z e s at a g r e a t e r delay than p o i n t s on the h a r v e s t i n g t r a j e c t o r y where H i s near Hmax. Moving to l a r g e r d e l a y s , the e n t i r e p o s i t i v e H a x i s becomes u n s t a b l e . Once t h i s occurs, there i s s t i l l a constant prey s t o c k i n g r a t e which can s t a b i l i z e the e q u i l i b r i u m . Predator h a r v e s t i n g behaves d i f f e r e n t l y from prey h a r v e s t i n g under the i n f l u e n c e of d e l a y . The geometry of the delayed model d i f f e r s l i t t l e from the non-delayed one. As i n prey h a r v e s t i n g , the higher h a r v e s t i n g r a t e s become unstable f i r s t , then the lower r a t e s . However, in t h i s case, there i s not a t h r e s h o l d delay below which there i s t o t a l s t a b i l i t y of the a x i s . T h i s i s because the l o c a l s t a b i l i t y boundary p i s n e a r l y orthogonal to the G a x i s i n the h a r v e s t i n g plane. C o l l e c t i n g these o b s e r v a t i o n s , we draw the approximate l o c a l s t a b i l i t y curves f o r d i f f e r e n t delay v a l u e s i n F i g u r e (4.17). The i n f l u e n c e of time delay on the l o c a l s t a b i l i t y curve i s to push i t up and to the l e f t i n the H-G plane, 196 Figure ( 4 . 1 6 ) — Allowable time delay f o r H o l l i n c ' s model t r i a l 1 ( F i g u r e 4.1) 198 F i g u r e ( 4 . 1 7 ) — L o c a l s t a b i l i t y c u r v e s i n the H-G p l a n e f o r H o l l i n g ' s model w i t h T=0, T=0.75, and T=0.85 w i t h r=2, K=40, A=10, s=1, and M=1. 199 X 200 e n l a r g i n g the domain f o r l i m i t c y c l e t r a j e c t o r i e s . In a d d i t i o n , n o t i c e that F i g u r e s (4.1) and (4.3) are s i m i l a r i n shape, although the delay i s doubled. T h i s s i m i l a r i t y a r i s e s because the i n t r i n s i c r a t e of growth, an important c o n t r o l l i n g f a c t o r of the i n f l u e n c e of delay, i s halved. B. The h p a r t i t ion As Brauer and Soudack observed f o r the non-delayed models, the h p a r t i t i o n f o r the delayed t r i a l s does not enter the and G f o r constant-quota predator h a r v e s t i n g . F i g u r e (4.18) shows the h d i v i d e r f o r the non-delayed model with parameters from t r i a l two; F i g u r e (4.2) d i s p l a y s the delayed v e r s i o n . Because of t h i s bounding of the h d i v i d e r , i t appears t h a t delay has a small impact on the shape of t h i s g l o b a l l y u nstable r e g i o n s i n c e delay tends to s h i f t the boundaries i n t o the t h i r d quadrant of the H-G plane. S e v e r a l of our s i m u l a t i o n s , however, show an upward s h i f t i n t h i s d i v i d e r with delay. The d i f f e r e n c e between the e f f e c t of delay on these boundaries e n l a r g e s the area of l i m i t c y c l e behaviour, at l e a s t f o r the range of delay we e x p l o r e d . Our t r i a l s , t h e r e f o r e , i n d i c a t e t h at delay i n c r e a s e s the l i k e l i h o o d of p e r i o d i c o s c i l l a t i o n s . C. L i m i t c y c l e behaviour The shape of the l i m i t c y c l e s i n the delayed model d i f f e r from those observed i n the non-delayed model. F i g u r e (4.19) 201 Figure (4.18)-- H-G plane for Holling's model without delay (Figure 4.2). 203 shows the l i m i t c y c l e which occurs with zero h a r v e s t i n g f o r H o l l i n g ' s model, t r i a l two (Figure 4.2). F i g u r e (4.20) d i s p l a y s the same t r a j e c t o r y with no delay. There i s a b i f u r c a t i o n between these two cases where, f o r some delay, the two-point l i m i t c y c l e of the non-delayed model becomes the f o u r - p o i n t l i m i t c y c l e of the delayed model. S i m i l a r l y , f o r dec r e a s i n g predator h a r v e s t i n g , there i s a b i f u r c a t i o n between the two-point l i m i t c y c l e a t G=-2 with delay and G=-1 (Figure (4.21) and (4.22), r e s p e c t i v e l y ) . The b i f u r c a t i o n a l s o showed up i n t r i a l 3 (F i g u r e ( 4 . 2 ) ) . The only o u t s t a n d i n g f e a t u r e i n t r i a l 2 was the l a r g e c a r r y i n g c a p a c i t y . The c a r r y i n g c a p a c i t y s t a b i l i z e s the system, opposing the delay which d e s t a b i l i z e s i t . Consequently, one suspects t h a t the c a r r y i n g c a p a c i t y p l a y s a par t i n the b i f u r c a t i o n s of the l i m i t c y c l e . In s i m u l a t i o n s i n v o l v i n g the p e r t u r b a t i o n of K, we observed t h a t , at K=40, a two-point l i m i t c y c l e e x i s t e d f o r G=-1 and H=0. As the c a r r y i n g c a p a c i t y was r a i s e d to K=55, the two-point c y c l e d i v i d e d i n t o a d i s t i n c t f o u r - p o i n t c y c l e f o r the same s t o c k i n g r a t e . We c o n t i n u e d incrementing K and e v e n t u a l l y the f o u r -p o i n t c y c l e r e v e r t e d to a two-point c y c l e (at K=150). The f o u r - p o i n t c y c l e re-emerged at G=-0.25, H=0. E v i d e n t l y , with l a r g e r c a r r y i n g c a p a c i t i e s , the b i f u r c a t i o n boundaries approach the o r i g i n i n the H-G plane. T h i s i s expected because the o r i g i n and the negative G-axis bound the h l i n e , near which b i f u r c a t i o n s would a r i s e . We con t i n u e d our a n a l y s i s by v a r y i n g G while h o l d i n g the 204 Figure (4.19)-- L i m i t c y c l e f o r H o l l i n g ' s model with delay and no h a r v e s t i n g . The time delay f o r t h i s example i s T=1.5. (X i s the prey d e n s i t y ; Y i s the predator d e n s i t y ) 206 Figure (4.20)-- L i m i t c y c l e f o r H o l l i n g ' s model without d e l a y and no h a r v e s t i n g . (X i s the prey d e n s i t y ; Y i s the p r e d a t o r d e n s i t y ) 207 8 IB IB 9 9 in 208 Figure ( 4 . 2 1 ) — L i m i t c y c l e f o r H o l l i n g ' s model with d e l a y and predator s t o c k i n g , G=-2. The time delay f o r t h i s example i s T=1.5. (X i s the prey d e n s i t y ; Y i s the predator d e n s i t y ) 209 210 Figure (4.22)— L i m i t c y c l e f o r H o l l i n g ' s model with delay and predator s t o c k i n g , G=-1. For t h i s example, the time delay i s T=1.5. (X i s the prey d e n s i t y ; Y i s the predator d e n s i t y ) 211 R B B 2 9 212 c a r r y i n g c a p a c i t y c o n s t a n t . At. G=-0.1, H=0, a s i x - p o i n t c y c l e appeared. L i and Yorke (1975) proved that period-3 i m p l i e s chaos and guarantees the e x i s t e n c e of p e r i o d i c o r b i t s of a l l p e r i o d s and a p e r i o d i c o r b i t s . Period-3 c y c l e s should immediately precede period-6 c y c l e s as G i s decreased, although t h i s was not observed. Chaos comes before period-3 c y c l e s . Guckenheimer e_t.al (1977) emphasized that the parameter space for which chaos i s present can be n e g l i g i b l y s m a l l . Whether or not t h i s holds i n our case i s d i f f i c u l t to a s c e r t a i n because of l i m i t a t i o n s on the l e n g t h of s i m u l a t i o n s . In a d d i t i o n to the b i f u r c a t i o n s , we n o t i c e d two general trends i n the data as the c a r r y i n g c a p a c i t y was i n c r e a s e d . F i r s t , the s i z e of the l i m i t c y c l e i n c r e a s e d with K. At K=50 in the above example with G=-2, the amplitude of the prey l i m i t c y c l e was about 55 and the maximum amplitude of the predator c y c l e was approximately 15. At K=65, the maximum amplitude of the predator c y c l e shot up to 54. While these numbers are r e l a t i v e , an 18-percent increment i n the c a r r y i n g c a p a c i t y caused a 260-percent i n c r e a s e i n the amplitude of the predator c y c l e s . In c o n j u n c t i o n with an i n c r e a s e i n amplitude, the c y c l e s approach zero d e n s i t y at t h e i r lowest p o i n t as K i s in c r e a s e d . These two trends u n d e r l i n e the dangers of enhancement of the c a r r y i n g c a p a c i t y by f e r t i l i z a t i o n to achieve i n c r e a s e d p r o d u c t i v i t y . D. Other parameters The l i t e r a t u r e (Hutchinson 1948; Wangersky and Cunningham 213 I957a,b; May 1973) has emphasized the r o l e of the i n t r i n s i c growth r a t e , r, i n connection with time d e l a y s , i n determining the s t a b i l i t y of the e q u i l i b r i u m . Our s i m u l a t i o n s support t h i s c o n t e n t i o n . F i g u r e (4.23) r e p r e s e n t s the H-G plane f o r r=1, K=50, L=10, J=20, s=1, M=1, T=1.5. Comparing t h i s to F i g u r e (4.1) where r = 2, K=40, L=10, J=20, s=1, M= 1 , T=0.75, the match i s c l o s e i f F i g u r e (4.1) i s shrunk by a f a c t o r of 2, n o r m a l i z i n g the diagram with maximum harvest r a t e s . Notice that the product rT i s the same in both diagrams. For the c a r r y i n g c a p a c i t y , Brauer and Soudack (1981b) show for non-delayed systems that i n c r e a s i n g the c a r r y i n g c a p a c i t y i n c r e a s e s the region of l o c a l and g l o b a l i n s t a b i l i t y . F i g u r e s (4.2) and (4.23) show the same t r e n d . I n c r e a s i n g parameter L in our s i m u l a t i o n s pushed the i n s t a b i l i t y l i n e s , both h and p, upward (Figure (4.4) and (4.23)). T h i s induces i n s t a b i l i t y with pure prey h a r v e s t i n g . 2. I v l e v ' s model The H-G diagrams f o r t h i s model resemble those of the p r e v i o u s model; the models are f u n c t i o n a l l y e q u i v a l e n t . In s p i t e of the absence of c o n t r a s t s , F i g u r e s (4.6) to (4.10) serve to c o n f i r m our c o n c l u s i o n s about H o l l i n g ' s model with a type-II f u n c t i o n a l response. F i g u r e (4.9) shows that d e c r e a s i n g c (analogous to i n c r e a s i n g A) d e s t a b i l i z e s pure predator h a r v e s t i n g . The t r i a l s of F i g u r e (4.6) and (4.10) d i f f e r i n only a s m a l l amount i n the c a r r y i n g c a p a c i t y yet the shape of the 214 Figure (4.23)-- K-G Diane for Holling's model with r=l, K=40, J=20, L=10, s=1, M=1, T=1.5. 215 216 s t a b i l i t y r e g i o n s i s completely d i f f e r e n t . T h i s i s c h i e f l y a r e s u l t of the parameter chosen; the r e s u l t s simply u n d e r l i n e the f r a g i l i t y of e c o l o g i c a l models with the warning that adjustment of the c a r r y i n g c a p a c i t y i s not without i t s s u r p r i s e s , or perhaps d i s a s t e r s . 3. L o t k a - V o l t e r r a model with resource l i m i t a t i o n As one might expect, the behaviour of the L o t k a - V o l t e r r a model with resource l i m i t a t i o n i s somewhat simpler than the f i r s t two models. The t r a j e c t o r i e s are l e s s complex, with fewer c r o s s i n g s and turns i n the phase plane (predator versus prey d e n s i t i e s ) ; the approach path to e q u i l i b r i u m i s asymptotic or r e g u l a r l y o s c i l l a t o r y . The l i m i t c y c l e s observed i n t h i s model e x h i b i t e d no b i f u r c a t i o n s and were e l l i p t i c a l i n shape, i n c o n t r a s t with F i g u r e s (4.19) and (4.20). Time delays d e s t a b i l i z e d the model by moving the p and h boundaries l e f t . As a b a s i s f o r comparison, the prey h a r v e s t i n g r a t e s f o r which the e q u i l i b r i u m becomes l o c a l l y u n s t a b l e , when no delay e x i s t s , are H=250., 83.3, 500, 45, 125, f o r F i g u r e s (4.11) through (4.15), r e s p e c t i v e l y . Under predator h a r v e s t i n g , the e q u i l i b r i u m can become unstable without delay although predator s t o c k i n g never d e s t a b i l i z e s the e q u i l i b r i u m . T h e r e f o r e , the shape of the curve i s not m o d i f i e d by delay although the s t a b l e t h r e s h o l d h a r v e s t i n g r a t e s f o r prey and predator are decreased. The l a r g e s t change i s i n F i g u r e (4.13). The i n t r i n s i c r a t e of growth i s extremely high i n t h i s case and t h e r e f o r e T should have a r e l a t i v e l y l a r g e r 217 i n f l u e n c e . Other parameters were t e s t e d f o r t h e i r e f f e c t s on s t a b i l i t y . The r e s u l t s agreed with those from non-delayed models. I n c r e a s i n g K or d e c r e a s i n g b d e s t a b i l i z e d the model, i n the sense that the area of s t a b i l i t y i n the f i r s t quadrant was decreased. I t should be noted that i n c r e a s i n g v a l s o d e s t a b i l i z e s the model. The ra t e of e f f e c t i v e search u has no e f f e c t on s t a b i l i t y . C o nclusions The d i f f e r e n c e s between the s t a b i l i t y r e g i o n s of the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and the other two models demonstrate the e s s e n t i a l nature of the f u n c t i o n a l and numerical responses to predator-prey dynamics. To s e t t l e on an i n t e r a c t i o n term f o r want of i n f o r m a t i o n can l e a d to c r i t i c a l d i f f e r e n c e s i n behaviour, e s p e c i a l l y with systems i n c l u d i n g delay or some s o r t of resource dynamics. The area of l i m i t c y c l e behaviour i s not enlarged i n the L o t k a - V o l t e r r a model with resource l i m i t a t i o n except i f r i s very l a r g e , whereas i n the other models, the enhancement i s s i g n i f i c a n t . Changes i n the l i m i t c y c l e c l a s s e s were shown f o r H o l l i n g ' s model. We conclude that b i f u r c a t i o n s are more l i k e l y to a r i s e f o r higher c a r r y i n g c a p a c i t y ; i n the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , l i m i t c y c l e s were always of p e r i o d 2. In c o n j u n c t i o n , the p e r i o d i c i t y of the l i m i t c y c l e v a r i e s with the predator s t o c k i n g r a t e . The work presented here can be repeated f o r constant-218 e f f o r t h a r v e s t i n g of the prey and predator, though c o n s t a n t -e f f o r t s t o c k i n g i s not u s u a l l y a r e a l i s t i c p o s s i b i l i t y . 219 V. SMOOTHED DELAYS I n t r o d u c t i o n As mentioned in our a n a l y s i s of the s i n g l e - s p e c i e s , l o g i s t i c model, smoothing d i s t r i b u t e s the d e n s i t y dependence of the model c o n t i n u o u s l y over past p o p u l a t i o n . d e n s i t i e s . The r e l a t i v e c o n t r i b u t i o n of these past p o p u l a t i o n s to d e n s i t y -dependent growth i s governed by a weighting f u n c t i o n or k e r n a l . For d i s c r e t e d e l a y , the k e r n a l i s a d e l t a f u n c t i o n which weights only the prey d e n s i t y at time t-T. No other p o p u l a t i o n a f f e c t s the density-dependent growth f a c t o r ( 1 - x ( t - T ) / K ) . I f we modify the weighting f u n c t i o n to a r e c t a n g u l a r k e r n a l , the past 2T p o p u l a t i o n d e n s i t i e s are e q u a l l y represented i n the density-dependent term. Mathematically, t h i s weighted d e n s i t y dependence i n v o l v e s i n t e g r a t i n g over the product of the delay ke r n a l and the past d e n s i t i e s . T h i s type of i n t e g r a l i s c a l l e d a c o n v o l u t i o n i n t e g r a l . In past s t u d i e s , r e s e a r c h e r s have complained that d i s c r e t e delay i s u n r e a l i s t i c (May 1973; Cushing 1977; MacDonald 1977) and should be r e p l a c e d with a continuous d e l a y . S e v e r a l experimental s t u d i e s support t h i s c o n t e n t i o n . Caperon (1969) found that h i s chemostat p o p u l a t i o n time s e r i e s was b e t t e r approximated by a continuous l a g k e r n a l of r e c t a n g u l a r or e r r o r - f u n c t i o n shape than by a d i s c r e t e delay. Curry e t . a l . (1978) developed a s t o c h a s t i c model of development f o r the b o l l w e e v i l , Anthonomus g r a n d i s , with a smoothed development l a g . Rather than t a k i n g the mean delay, the authors more reasonably 220 determine the a c t u a l development time d i s t r i b u t i o n . Johnson and K a r l s s o n (1972) g i v e experimental evidence f o r a smoothed delay i n a model f o r s t u d y i n g the b i o l o g i c a l rhythms in p l a n t s . In view of these r e s u l t s , we w i l l now turn to the dynamic d i f f e r e n c e s between systems with d i s c r e t e delay and systems with continuous delay. We w i l l emphasize q u a l i t a t i v e d i f f e r e n c e s between D - p a r t i t i o n shapes and l i m i t s which are s u f f i c i e n t to c h a r a c t e r i z e l o c a l s t a b i l i t y d i f f e r e n c e s . We have chosen to compare four types of smoothed delay; each has d i f f e r e n t shaped k e r n a l s . The four k e r n a l s are shown in F i g u r e (5.1). i Research i n v e s t i g a t i n g the r e l a t i o n s h i p between d i s c r e t e and smoothed lags has been scanty. MacDonald (1978) found g e n e r a l l y t h a t , d i s c r e t e l a g . i s l e s s d e s t a b i l i z i n g than continuous forms. Cushing (1977) compared two types of c o n t i n u o u s l y d i s t r i b u t e d l a g s (3 and 4 i n F i g u r e ( 1 ) ) . He found that delay (3) i s more unstable i n the s i n g l e - s p e c i e s l o g i s t i c model, where the e q u i l i b r i u m i s unstable f o r T > 2/r. If delay (4) i s used, the e q u i l i b r i u m remains a s y m p t o t i c a l l y s t a b l e but the approach to e q u i l i b r i u m ^ i s s t i l l slower than i n the non-delayed model. May (1973) used delay (3) i n h i s a n a l y s i s of delayed p o p u l a t i o n growth equations but d i d not compare the r e s u l t s with a d i s c r e t e d e l a y . 221 Figure (5.1)-- The four smoothed delays examined in t h i s study: (1) rectangular, (2) triangular, (3) 'strong', and (4) 'weak' . 222 223 Method Three types of q u a s i - c h a r a c t e r i s t i c equations have s u r f a c e d i n our a n a l y s e s . The f i r s t and s i m p l e s t i s the equation from the L o t k a - V o l t e r r a model with resource l i m i t a t i o n with c o n s t a n t - e f f o r t prey h a r v e s t i n g and s t o c k i n g : Q(s) = s 2 + CD - Asexp(-sT) (5.1) The second a r i s e s from c o n s t a n t - e f f o r t prey h a r v e s t i n g and s t o c k i n g i n the other models, constant-quota h a r v e s t i n g i n the three models, and c o n s t a n t - e f f o r t predator h a r v e s t i n g i n a l l the c a s e s : Q(s) = s 2 - Bs + CD - Asexp(-sT). (5.2) The t h i r d equation i s d e r i v e d from constant-quota h a r v e s t i n g and s t o c k i n g of the p r e d a t o r : Q(s) = s 2 - (F+B)s + CD + BF - A(s-F)exp(-sT) (5.3) In each of these e x p r e s s i o n s the term exp(-sT) i s the t r a n s f o r m of the delay k e r n a l , 6 ( t ~ T ) . To s u b s t i t u t e d i s t r i b u t e d l a g s , we need only r e p l a c e exp(-sT) by the t r a n s f o r m of the a p p r o p r i a t e k e r n a l . T h i s s u b s t i t u t i o n i s p o s s i b l e because the weighting f u n c t i o n i s a l i n e a r c o n v o l u t i o n i n t e g r a l . A f t e r t h i s , we can use the t r a n s f o r m a t i o n iy=sT to f i n d the D-p a r t i t i o n . 224 Results 1. Rectangular Delay This kernal has a width of 2T and a height of 1/2T to normalize the delay. The equation for the kernal is k(t) = (1/2T)(u(t) - u(t-2T)) (5.4) where u(t) i s the unit step function. Transforming t h i s kernal, we get K(s) = (1/2Ts)[l - exp(-2sT)] (5.5) Inserting K(s) into the prey harvesting Q(s) (equation (5.1)) for exp(-sT) and finding the p a r t i t i o n l i n e by l e t t i n g s=iy, gives us CD = y 2 + A(1 - cos(2Ty))/2T (5.6) and (sin(2Ty))/2Ty = 0 (5.7) for the Lotka-Volterra model with resource l i m i t a t i o n and constant-effort prey harvesting. Equation (5.7) implies y = iir/2T for i = . . . - 1 , + 1 ,+2 (5.8) For i=1, the p a r t i t i o n is 225 CD = tr 2/4T 2 +• A/T (5.9) The next p a r t i t i o n s are CD=TT 2/T 2 and CD=9JT 2/4T 2+A/T f o r i = 2 and 3, r e s p e c t i v e l y . Thus, the p a r t i t i o n s are two se t s of p a r a l l e l l i n e s : one h o r i z o n t a l and the other d i a g o n a l . The p a r t i t i o n f o r i=1 i s shown i n F i g u r e (5.2). Comparing t h i s with the d i s c r e t e p a r t i t i o n ( F i g u r e 2.5), we see t h a t the shape of the p a r t i t i o n i s the same. However, the l i m i t on A has been i n c r e a s e d by a f a c t o r of n/2 by the d i s t r i b u t i o n of the l a g . T h i s i n c r e a s e s the area of s t a b i l i t y by a f a c t o r of n/2 a l s o . With the smoothed delay, i f A > JT 2/4T, then no amount of h a r v e s t i n g w i l l promote s t a b i l i t y . For the second q u a s i - c h a r a c t e r i s t i c equation (5.2), we have, a f t e r the i n c o r p o r a t i o n of the smoothed delay and l e t t i n g s=iy, the f o l l o w i n g p a r t i t i o n e q u a t i o n s : As the a n a l y t i c a n a l y s i s of t h i s delay i s d i f f i c u l t , we w i l l r e s o r t to s i m u l a t i o n s f o r our comparisons. Our s i m u l a t i o n r e s u l t s show that the r e c t a n g u l a r delay s t a b i l i z e s the model r e l a t i v e to the d i s c r e t e d e l a y . A comparison of the de l a y s f o r T=1, A=-1 i s shown i n F i g u r e (5.3). As T i s i n c r e a s e d , the s e p a r a t i o n between the p a r t i t i o n s i n c r e a s e s . For l a r g e r values of A, the s e p a r a t i o n i n c r e a s e s . E c o l o g i c a l l y , A i s the i n t r i n s i c r a t e of growth times the depression of the prey B = - A s i n (2Ty)/2Ty (5.10) CD = y 2 + (A/2T)[1-cos(2Ty)] (5.11) 226 Figure ( 5 . 2 ) — D - p a r t i t i o n f o r the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , r e c t a n g u l a r delay, and c o n s t a n t - e f f o r t h a r v e s t i n g . The delay i s T=1 i n t h i s f i g u r e . 228 Figure ( 5 . 3 ) — Comparison of D-partitions for (a) rectangular and (b) discrete delay with T=1.0. 229 230 e q u i l i b r i u m by the predator (Beddington et.al.. 1978). The smoothing of the delay w i l l have a l a r g e e f f e c t i f the i n t r i n s i c growth r a t e of the prey s p e c i e s i s h i g h and the predator i s r e l a t i v e l y i n e f f e c t i v e at l i m i t i n g the prey d e n s i t y . 2. T r i a n g u l a r Delay The t r i a n g u l a r delay i s a l o g i c a l choice to use to compare smoothed and d i s c r e t e d e l a y s . By i n c r e a s i n g the width of the t r i a n g l e , we can spread out the d e l a y . As the width approaches zero, the t r i a n g u l a r kernal approaches a d e l t a f u n c t i o n and the d i s c r e t e d e lay. Therefore, we have a continuous comparison between d i s c r e t e and smoothed d e l a y with t h i s k e r n a l . We d e f i n e the width of the t r i a n g l e by a and normalize the h e i g h t so that the t o t a l area of the t r i a n g l e i s one. The Laplace transform of the t r i a n g u l a r kernal i s I n s e r t i n g t h i s i n t o the q u a s i - c h a r a c t e r i s t i c polynomial f o r prey h a r v e s t i n g and s e t t i n g s=iy, the p a r t i t i o n l i n e i s given by the equation: K(s) = ( 2 / a 2 s 2 ) e x p ( - s T ) ( c o s h ( s a ) - l ) (5.12) B = - 2 A / y 2 a 2 [ c o s ( y T ) ( c o s ( a y ) - l ) ] (5.13) CD = y 2 . _ 2 A [ s i n ( y T ) ( c o s ( a y ) - 1 ) ] / y a 2 (5.14) In the L o t k a - V o l t e r r a model with resource l i m i t a t i o n and 231 c o n s t a n t - e f f o r t h a r v e s t i n g , B=0, so that e i t h e r cos(yT)=0 or cos(ya)=1. Since the c h o i c e of a i s a r b i t r a r y , cos(yT)=0, and y=± IT/2T,± 3IT/2T,... (5.15) Th e r e f o r e , f o r y<rr/T, CD = U / 2 T ) 2 - 4 A T [ c o s ( a T r / 2 T ) - 1 ] / i r a 2 (5.16) fo r the primary boundary. Because the width of the delay i s c o n t r o l l e d by the parameter a, we want to f i n d out how the p a r t i t i o n changes with a. As a approaches 0 and as the t r i a n g u l a r f u n c t i o n becomes a d e l t a f u n c t i o n , the D - p a r t i t i o n equation (5.16) i s transformed i n t o CD = U / 2 T ) 2 + A(tr/2T), (5.17) which i s the p a r t i t i o n i n the d i s c r e t e delay model. So the t r i a n g u l a r d e l a y ' s p a r t i t i o n c o l l a p s e s to the d i s c r e t e d e l a y ' s p a r t i t i o n , as one would expect. At the maximum width, a=T, equation (5.16) becomes CD = U / 2 T ) 2 + 4A/irT, (5.18) which shows that the s l o p e of the D - p a r t i t i o n has decreased because ir/2 > 4 /tr. For intermediate a v a l u e s between 0 and T, 232 the slope changes m o n o t o n i c a l l y toward that i n the above equ a t i o n . T h i s change i n slope expands the s t a b l e area, promoting s t a b i l i t y . In t h i s case, only i f i s the system incapable of being s t a b i l i z e d with h a r v e s t i n g . In the s i n g l e - s p e c i e s system, t h i s means that s i n c e x*=K. Ther e f o r e , the smoothing i n the s i n g l e - s p e c i e s case e f f e c t i v e l y cubes the l i m i t i n g value of »r/2. With the second q u a s i - c h a r a c t e r i s t i c equation, one can check that as a->0, the p a r t i t i o n converges to the d i s c r e t e case. Our s i m u l a t i o n s show that as a->T, a knot develops i n the p a r t i t i o n . T h i s i s shown i n F i g u r e (5.4) f o r v a r i o u s a v a l u e s . The b i f u r c a t i o n develops as a i n c r e a s e s f o r t h i s example. T h e r e f o r e , we know that the b i f u r c a t i o n i s a r e s u l t of smoothing. The b i f u r c a t i o n was found to occur only with l a r g e d e l a y s . As T i n c r e a s e s , the width f o r which the knot develops decreases. The knot changes the shape of the s t a b i l i t y r e g i o n so that s t o c k i n g of the prey can be un s t a b l e . The t r a j e c t o r y f o r prey s t o c k i n g i s l i n e a r with i n c r e a s i n g CD and d e c r e a s i n g B. Thus, smoothing with t r i a n g u l a r delay changes our previous c o n c l u s i o n about the robust s t a b i l i t y of prey s t o c k i n g . With t h i s delay, a l i m i t develops on the -A > ir 3/8T (5.19) rT > (ir/2) 3 (5.20) 233 Figure (5.4)-- Bifurcation in the D-partition for the triangular delay of various widths. The parameter a i s the width of the triangular delay. The time delay i s 2.5. 234 235 stocked number. The b i f u r c a t i o n only appeared with t r i a n g u l a r delay; t h e r e f o r e , the r e s u l t may be an anomaly. 3. 'Strong' Delay T h i s i s the delay suggested by May (1973) and Cushing (1977) as a s u b s t i t u t e f o r d i s c r e t e delay. I t i s a t t r a c t i v e because i t s transform y i e l d s a simpler f u n c t i o n than the other k e r n a l s or d i s c r e t e delay. But i t i s not r e a d i l y comparable to d i s c r e t e delay because the width of the delay i s not changeable, as i n the t r i a n g u l a r d e l a y . The peak i n t h i s kernal occurs at T. The equation f o r the k e r n a l i s k ( t ) = ( t / T ) e x p ( - t / T ) (5.21 ) and the transform i s given by K(s) = 1 / O + s T ) 2 . (5.22) The q u a s i - c h a r a c t e r i s t i c equation (5.2) becomes Q(s) = s 2 - Bs + CD - As/(1+ST) 2. (5.23) F i n a l l y , our p a r t i t i o n equations are CD = y 2 + 2 y 2 T A / ( ! + y 2 T 2 ) 2 (5.24) B = - A ( 1 - y 2 T 2 ) / ( 1 + y 2 T 2 ) 2 . (5.25) 236 In the case where B=0, which i m p l i e s t h a t y=l/T ( f o r y > 0 ) , the p a r t i t i o n becomes CD = ( 1 / T ) 2 + A/2T (5.26) T h i s d e l a y changes both t h e i n t e r c e p t and s l o p e of the p a r t i t i o n l i n e . S i n c e the t r a j e c t o r y l i n e s a re v e r t i c a l , a good measure of the r e l a t i v e s t a b i l i t y of the d e l a y i s the area i n the s t a b i l i t y r e g i o n i n the (B,CD) p l a n e . In t h i s c a s e , a r e a = ( 2 / T ) ( l / T 2 ) = 2/T 3 (5.27) For the d i s c r e t e d e l a y , area = T T 3 / 8 T 3 = 3.9/T 3 (5.28) For the o t h e r smoothed d e l a y s , area = T T V 1 6 T 3 = 6.1/T 3, (5.29) f o r the r e c t a n g u l a r d e l a y , and a r e a = r r 5/32T 3 = 9.6/T 3 (5.30) f o r the t r i a n g u l a r d e l a y . E q u a t i o n s (5.27) t h r o u g h (5.30) are d e r i v e d from the i n t e r c e p t s on the CD and A axes by the p a r t i t i o n e q u a t i o n s ( 5 . 9 ) , ( 5 . 1 7 ) , ( 5 . 1 8 ) , and ( 5 . 2 6 ) . At l e a s t 237 i n t h i s case, the 'strong' delay appears to be l e s s s t a b l e than the d i s c r e t e delay, while the other smoothed d e l a y s increase s t a b i l i t y , i n the sense given above. For the other 2 models, B i s given by We simulated s e v e r a l p a r t i t i o n s of t h i s type and found that t h i s 'strong' delay decreased the s t a b l e area i n the f i r s t quadrant. As T inc r e a s e d and A decreased, t h i s d i f f e r e n c e was a m p l i f i e d . T h i s means that f o r c o n s t a n t - e f f o r t prey harvest i n H o l l i n g ' s and I v l e v ' s models, the s t a b i l i z i n g h arvest r a t e w i l l i n c r e a s e . F i g u r e (5.5) compares these delays f o r T=1,A=-1. Another d i s t i n c t i v e f e a t u r e of t h i s d elay i s t h a t the p a r t i t i o n l i n e i s c l o s e r to the CD-axis i n the second quadrant than f o r the d i s c r e t e d e lay. T h i s e f f e c t i v e l y enables s t a b i l i t y at a lower s t o c k i n g r a t e . 4. 'Weak' Delay T h i s delay was chosen by Cushing (1977). I t has q u a l i t a t i v e l y d i f f e r e n t f e a t u r e s than the other smoothed d e l a y s : i t s peak value i s at zero. The kernal equation i s B = - A ( 1 - y 2 T 2 ) / ( 1 + y 2 T 2 ) 2 . (5.31 ) k(t)=(1/T)exp(-t/T) (5.32) The Laplace transform of t h i s e x p r e s s i o n i s 238 Figure ( 5 . 5 ) — Comparison of D-partitions for (a) 'strong' and (b) discrete delays. The delay is 1. 239 240 K(s) = 1/O+sT). (5.33) I n s e r t i n g t h i s i n t o the q u a s i - c h a r a c t e r i s t i c equation (5.2) and f i n d i n g the p a r t i t i o n l i n e s with s=iy, we o b t a i n For the L o t k a - V o l t e r r a model with resource l i m i t a t i o n , because B=0, the s t a b i l i t y c o n d i t i o n becomes A < 0, which i s true by d e f i n i t i o n (1.5) so that the e q u i l i b r i u m i s always s t a b l e no matter how l a r g e T may be. For B not equal to zero, both B and CD are bounded by the axes, r e s t r i c t i n g the p a r t i t i o n to the f i r s t quadrant. Sim u l a t i o n s f o r t h i s delay were s i m i l a r to the strong d e l a y . The weak delay a l s o had a smaller s t a b l e area i n the f i r s t quadrant i n s e v e r a l s i m u l a t i o n s . However, f o r low A v a l u e s and high T v a l u e s , the weak delay appeared more s t a b l e , because the p a r t i t i o n i s bounded by the CD a x i s . The bound enables us to e a s i l y d e f i n e the number of stocked i n d i v i d u a l s needed f o r s t a b i l i t y . F i g u r e (5.6) compares t h i s delay and the d i s c r e t e one f o r T=1,A=-1. Constant-quota predator h a r v e s t i n g and s t o c k i n g The t h i r d q u a s i - c h a r a c t e r i s t i c equation a p p l i e s to constant-quota predator h a r v e s t i n g and s t o c k i n g f o r H o l l i n g ' s and I v l e v ' s models. While changes i n the shape of the D-CD = y 2 + Ay 2T/(1+y 2T 2) (5.34) B = - A / ( 1 + y 2 T 2 ) . (5.35) 241 Figure ( 5 . 6 ) — Comparison of D-partitions for (a) 'weak' and (b) discrete delays. The delay i s 1.5. 242 0. 0.5 243 p a r t i t i o n among the s e v e r a l delay k e r n a l s remains s i m i l a r to the prey h a r v e s t i n g p a r t i t i o n , t h i s s i t u a t i o n c o n t a i n s s e v e r a l unique f e a t u r e s . F i r s t , the h a r v e s t i n g t r a j e c t o r y i s c o n f i n e d to the f i r s t quadrant of the (B,CD) plane; t h i s r e s t r i c t s our a t t e n t i o n to t h i s quadrant. Second, the p a r t i t i o n moves with changes i n the h a r v e s t i n g r a t e . For i n s t a n c e , the base of the p a r t i t i o n i s set by the parameter A. The magnitude of A, |A|, i n c r e a s e s with predator h a r v e s t i n g and decreases with predator s t o c k i n g . As i l l u s t r a t e d i n Chapter I I I , t h i s dynamical a d d i t i o n complicates our i n v e s t i g a t i o n by r e q u i r i n g s l i c e s of the parameter plane at d i f f e r e n t h a r v e s t i n g r a t e s . In s p i t e of t h i s handicap, we w i l l attempt to p o i n t out some g e n e r a l i z a t i o n s . The p a r t i t i o n equations f o r t h i s h a r v e s t i n g type are B = -F - (A/y)[yRe[W(iy)] - FTIm[W(iy)]] (5.36) CD = ( y / T ) 2 - BF - (A/T)[ylm[W(iy)] + FTRe[W(iy)]] (5.37) where W( • ) i s the transform of the delay k e r n a l , Im[W(iy)] i s the imaginary part of the parameterized k e r n a l , and Re[W(iy)] i s the r e a l p a r t . The r e a l and imaginary p a r t s f o r the parameterized k e r n a l s are d e r i v e d by l e t t i n g iy=sT in the transformed k e r n a l s (5.5), (5.12), (5.22), and (5.33), then s e p a r a t i n g the k e r n a l i n t o r e a l and imaginary p a r t s : 244 1. Rectangular Re[W(iy)] Im[W(iy)] s i n ( 2 y ) / 2 y -[1 - co s ( 2 y ) ] / 2 y (5.38) (5.39) 2. T r i a n g u l a r Re[W(iy)] Im[W(iy)] ( 2 T 2 / ( y 2 a 2 ) ) c o s ( y ) ( c o s ( a y / T ) - 1 ) (5.40) ( 2 T 2 / ( y 2 a 2 ) ) s i n ( y ) ( c o s ( a y / T ) - 1 ) (5.41 ) 3. 'Strong' Re[W(iy)] Im[W(iy)] d - y 2 ) / ( l + y 2 ) 2 - 2 y / ( 1 + y 2 ) 2 (5.42) (5.43) 'Weak' Re[W(iy)] Im[W(iy)] 1 / d + y 2 ) - y / O + y 2 ) (5.44) (5.45) By i n s e r t i n g these k e r n a l s i n t o the p a r t i t i o n equations, we can run s i m u l a t i o n s to map the l o c a l s t a b i l i t y boundaries f o r constant-quota predator h a r v e s t i n g and s t o c k i n g . We can a l s o d i s c e r n d i f f e r e n c e s i n the shape and p o s i t i o n of the p a r t i t i o n which would e f f e c t v a r i a t i o n s i n s t a b i l i t y . F i g u r e (5.7) diagrams the p a r t i t i o n s f o r one example. As the f i r s t quadrant of the (B,CD) plane i s our primary concern, a c r u c i a l determinant of s t a b i l i t y i s the i n t e r c e p t of the p a r t i t i o n with the CD a x i s . To begin our a n a l y s i s , we can s i m p l i f y the general s i t u a t i o n by r e q u i r i n g t h a t F=0, then check our r e s u l t s v i a s i m u l a t i o n s f o r F not equal to zero. 245 Figure ( 5 . 7 ) — P a r t i t i o n s f o r H o l l i n g ' s model with c o n s t a n t -quota predator h a r v e s t i n g . The parameter values are r = l , K=40, J=20, L=10, s=1, and M=1. The h a r v e s t i n g r a t e i s G=0.5. (d, d i s c r e t e delay; r, r e c t a n g u l a r delay; t , t r i a n g u l a r d e l a y ; s , ' s t r o n g ' delay; w,'weak' d e l a y ) . 246 1 247 For F=0, the i n t e r c e p t , CD, f o r the d i f f e r e n t d e l ays from equation (5.27) are 1. Rectangular CD = U / 2 T ) 2 + A/T (5.45) 2. T r i a n g u l a r CD = U / 2 T ) 2 + 2A/TTT (5.46) 3. ' Strong' CD = ( 1 / T ) 2 + A/2T (5.47) The 'weak' delay has no i n t e r c e p t ; i t approaches the CD a x i s a s y m p t o t i c a l l y . The i n t e r c e p t f o r the d i s c r e t e delay i s CD = U / 2 T ) 2 + irA/2T (5.48) F i g u r e (5.8) diagrams t h i s r e l a t i o n s h i p between A and CD f o r the d i f f e r e n t d e l a y s . As i n the f i r s t q u a s i - c h a r a c t e r i s t i c equation, the d i s c r e t e , r e c t a n g u l a r , and t r i a n g u l a r d e l a y s f o l l o w a p a t t e r n : the A- a x i s i n t e r c e p t s are separated i n d i s t a n c e by a f a c t o r of n/2. The 'strong' delay does not f a l l i n t o t h i s p a t t e r n . The d i f f e r e n c e s in l o c a l s t a b i l i t y among the v a r i o u s smoothed d e l a y s depend to the magnitude of the parameter A. I t i s e vident from F i g u r e (5.8) t h a t , f o r small |A|, the d i f f e r e n c e s are small because the p a r t i t i o n s i n t e r c e p t the CD 248 Figure ( 5 . 8 ) - - The CD i n t e r c e p t as a f u n c t i o n of A f o r the smoothed delay k e r n a l s . (d, d i s c r e t e delay; r, r e c t a n g u l a r delay; t , t r i a n g u l a r d e l a y ; s , ' s t r o n g ' d e l a y ) . 249 250 a x i s at approximately the same p o i n t , except f o r the 'weak' and 'strong' k e r n a l s . For l a r g e r values of |A|, the d i f f e r e n c e s are exaggerated. Because |A| i s d e f i n e d as the i n t r i n s i c growth r a t e of the prey times the predator d e p r e s s i o n (x*/K), we can a s c e r t a i n t h a t , f o r l a r g e growth r a t e s or high d e p r e s s i o n d e n s i t i e s , the shape of the delay and, t h e r e f o r e , the dynamics of the resource i s c r u c i a l i n determining the s t a b i l i t y of the e q u i l i b r i u m . As s t o c k i n g of the predator lowers the depression and h a r v e s t i n g i n c r e a s e s i t , the h a r v e s t i n g of predators should emphasize the d i f f e r e n c e s between the delay k e r n a l s . However, t h i s i s only p a r t i a l l y c o r r e c t s i n c e the parameter F has an e f f e c t on the p a r t i t i o n , e s p e c i a l l y at h i g h harvest or s t o c k i n g r a t e s and with l a r g e d e l a y s . To i n v e s t i g a t e these suggestions, we diagrammed the H-G plane with l o c a l s t a b i l i t y boundaries f o r the d i f f e r e n t delay k e r n a l s f o r one example of H o l l i n g ' s model with d e l a y s of 1.5 and 2.5 i n F i g u r e (5.9). These d i v i d e r s were drawn from s i m u l a t i o n s along the axes. From the diagram, we see that s t a b i l i t y d i f f e r e n c e s between the smoothed k e r n a l s i n c r e a s e s with i n c r e a s e d h a r v e s t i n g as a n t i c i p a t e d by our d i s c u s s i o n . For T=1.5, the 'strong' delay i s the most unstable k e r n a l , f o l l o w e d by d i s c r e t e , r e c t a n g u l a r , 'weak', and t r i a n g u l a r d e l a y s . In t h i s c o n t e x t , more s t a b l e means a smaller area of l o c a l i n s t a b i l i t y between the R and p boundaries. T h i s order i s changed f o r T=2.5 where the 'strong' delay moves i n s i d e the d i s c r e t e delay and the 'weak' delay moves o u t s i d e the 251 Figure ( 5 . 9 ) — Local s t a b i l i t y boundaries in the H-G plane for Holling's model with the smoothed delays. The parameters are r=1, K=40, J=20, L=10, s=1, and M=1. The delay i s T=1.5 in (a) and 2.5 in (b). (d, discrete delay; r, rectangular delay; t, triangular delay; s,'strong' delay; w, 'weak' delay). (b)T=2.5 253 r e c t a n g u l a r delay. Note that the order of the d i s c r e t e , r e c t a n g u l a r , and t r i a n g u l a r delays does not change with delay. The p a r t i t i o n f o r the 'strong' delay remains c l o s e to the CD a x i s so t h a t , f o r l a r g e d e l a y s , i t w i l l be more s t a b l e than the d i s c r e t e d e l a y , even though i t i s bounded by the CD a x i s . O v e r a l l , we see that small changes i n the shape of the delay k e r n a l can enormously modify the l o c a l s t a b i l i t y of a managed ecosystem. In a d d i t i o n , a change i n the shape of the delay may modify the shape of the t r a n s i e n t response, as Caperon (1969) observed. C o n c l u s i o n s Smoothed delays a f f e c t the l o c a l s t a b i l i t y of a model by changing the shape and p o s i t i o n of the D - p a r t i t i o n l i n e s . We have found t h a t two of our d e l a y s s t a b i l i z e the model while two others cause mixed e f f e c t s . At the very l e a s t , t h i s p o i n t s to the importance of determining the shape of the delay k e r n a l where p o s s i b l e . On the whole, smoothed delays do not change our c o n c l u s i o n s about m u l t i p l e t r a n s i t i o n s between s t a b l e and unstable r e g i o n s ; the shape of the p a r t i t i o n s remain concave p o s i t i v e most of the time. The exception i n the t r i a n g u l a r delay which develops a knot f o r widths comparable to the delay f o r l a r g e d e l a y s with c o n s t a n t - h a r v e s t i n g of prey. T h i s knot occurs only with the t r i a n g u l a r delay k e r n a l . From a mathematical p o i n t of view then, i t i s important that the types of delay and t h e i r e f f e c t s be more thoroughly i n v e s t i g a t e d . E c o l o g i c a l l y , determining the dynamics of the 254 food resource can be c r i t i c a l i n p r e d i c t i n g the success of a p a r t i c u l a r management s t r a t e g y . 255 VI. TIME DELAYS IN CHEMOSTATS I n t r o d u c t i o n To v a l i d a t e our models with a c t u a l h a r v e s t i n g data i s d i f f i c u l t , c o n s i d e r i n g both the sampling i n t e r v a l s used i n most f i e l d s t u d i e s and exogenous i n f l u e n c e of environmental v a r i a t i o n . We may, however, t e s t the hypothesis that time delay promotes m u l t i p l e t r a n s i t i o n s i n harvested p o p u l a t i o n s with phytoplankton chemostat experiments. Chemostats serve as c u l t u r e s f o r i n v e s t i g a t i n g the uptake k i n e t i c s for a p a r t i c u l a r l i m i t i n g n u t r i e n t by a m i c r o s c o p i c organism. Growth k i n e t i c s are coupled to the uptake k i n e t i c s v i a a biochemical pathway. T h i s pathway c r e a t e s a delay between growth and uptake. At the same time, the d i l u t i o n of the c u l t u r e h a r v e s t s the algae i n a f i x e d p r o p o r t i o n , analogous to c o n s t a n t - e f f o r t h a r v e s t i n g . Thus, continuous c u l t u r e experiments i n c l u d e both time delay and h a r v e s t i n g . In nature, the s i n k i n g r a t e a c t s as a h a r v e s t i n g mechanism while u p w e l l i n g r e p r e s e n t s s t o c k i n g . Most experiments i n chemostat dynamics are aimed at d e r i v i n g the k i n e t i c c o n s t a n t s : the maximum uptake r a t e , h a l f -s a t u r a t i o n c o n s t a n t , and quota or y i e l d , through s t e a d y - s t a t e e q u i l i b r i a (Caperon 1968; Caperon and Meyer 1972a,b; Droop 1973,1975). These values are then r e l a t e d to e c o l o g i c a l f a c t o r s such as s p e c i e s s i z e or environment to deduce the e c o l o g i c a l s i g n i f i c a n c e of such the f a c t o r s (Eppley e t . a l . 1969; Maclsaac and Dugdale 1969). E q u a l l y c r u c i a l to the s u r v i v a l of a s p e c i e s i s i t s response to p e r t u r b a t i o n s i n 256 n u t r i e n t a v a i l a b i l i t y caused by upwelling or seasonal changes. Only the t r a n s i e n t response of a p o p u l a t i o n to p e r t u r b a t i o n s can r e v e a l t h i s r e s i l i e n c e ( H o l l i n g 1973). Caperon (1969) i n v e s t i g a t e d the t r a n s i e n t response of I s o c h y s i s qalbana to step changes i n the d i l u t i o n r a t e of a n i t r a t e - l i m i t e d c u l t u r e . He f i t a delayed Monod model to h i s p e r t u r b a t i o n data. Cunningham and Maas (1978) e s s e n t i a l l y repeated t h i s procedure with Chlamydomonas r e i n h a r d i i , a l s o f i t t i n g a delayed model to the data. W i l l i a m s (1971) found a delayed r e l a t i o n s h i p between uptake and growth, a r r i v i n g at a time delay of 14.1 hours f o r C h l o r e l l a pyrenoidosa Chick by approximating the d e l a y by the p e r i o d of o s c i l l a t i o n of the t r a n s i e n t response d i v i d e d by 4.4. T h i s approximation i s suggested by Wangersky and Cunningham (1957a). S i m i l a r experiments have been performed with b a c t e r i a l c u l t u r e s (Powell 1967). We w i l l look at Caperon's model i n t h i s chapter and i n v e s t i g a t e s t a b i l i t y t r a n s i t i o n phenomena with changing d i l u t i o n r a t e . Caperon's continuous c u l t u r e experiment provides r e l a t i o n s h i p s between i n t e r n a l n u t r i e n t c o n c e n t r a t i o n , the quota, and the d e n s i t y of the a l g a l p o p u l a t i o n s . No r e l a t i o n s h i p was found r e l a t i n g e x t e r n a l n i t r a t e c o n c e n t r a t i o n s and d e n s i t i e s (but see H a r r i s o n e t . a l 1976). Caperon's model i s based on the r e l a t i o n s h i p between growth and r e s e r v o i r n u t r i e n t c o n c e n t r a t i o n v = r(q-qO)/(a+(q-qO)) (6.1) 257 where » i s the growth r a t e , q i s the quota or the i n t e r n a l c o n c e n t r a t i o n of n u t r i e n t per u n i t biomass, and qO i s the quota at zero-growth r a t e . Caperon renamed the q u a n t i t y q-qO the r e s e r v o i r n u t r i e n t c o n c e n t r a t i o n s. By i n c o r p o r a t i n g t h i s r e l a t i o n s h i p i n t o a p a i r of d i f f e r e n t i a l equations f o r the c e l l d e n s i t y and r e s e r v o i r c o n c e n t r a t i o n and d e l a y i n g the response between uptake and growth, Caperon a r r i v e d at the equations dn/dt = r n s ( t - T ) / ( A + s ( t - T ) ) - wn (6.2) ds/dt = Bw/n - ( q O + s ) ( r s ( t - T ) / ( A + s ( t - T ) ) (6.3). where n = a l g a l c e l l d e n s i t y ( c e l l s / m l x l 0 " 3 ) , s = r e s e r v o i r n u t r i e n t c o n c e n t r a t i o n (»»»gat/cell) w = d i l u t i o n r a t e (1/hr) A = h a l f - s a t u r a t i o n constant ( n n g a t / c e l l ) , r = i n t r i n s i c growth r a t e ( 1 / h r ) , qO = quota at zero-growth r a t e , B = the net supply of n u t r i e n t to the growth chamber (j/ n g a t / c e l l ) . Nyholm (1977) f i t a s i m i l a r model to a p h o s h a t e - l i m i t e d chemostat where a c o n v e r s i o n r a t e constant r e p l a c e d the maximal growth r a t e r . M a t h e m a t i c a l l y , the models are e q u i v a l e n t . Caperon (1968) f i t r , A and qO u s i n g s t e a d y - s t a t e r e s u l t s . The delay between uptake and growth was f i t with the t r a n s i e n t response d a t a . From equations (6.2) and (6.3), the e q u i l i b r i u m r e s e r v o i r 258 and c e l l d e n s i t y are s* = wA/(r-w) (6.4) n* = B/(qO+s*). (6.5) Non-delayed model F i r s t , we w i l l i n v e s t i g a t e the l o c a l s t a b i l i t y of the same equations without delay. L i n e a r i z i n g the equations (6.2) and (6.3) with e(t)=n*-n(t) and f ( t ) = s * - s ( t ) , we f i n d that de/dt = r n * A / ( A + s * ) 2 f ( t - T ) - we(t) (6.6) d f / d t = [Bw/n* 2]e(t) - [ r s * / ( A + s * ) ] f ( t ) - [ ( q O + s * ) ( r A / ( A + s * ) 2 ) ] f ( t - T ) (6.7) Let C D E F rn*A/(A+s*) 2 Bw/n*2 -(qO+s*)(rA/(A+s*) 2) = -w, (6.8) (6.9) (6.10) (6.11) The Jacobian of the l i n e a r i z e d equations i s J = (6.12) 259 The non-delayed model i s l o c a l l y s t a b l e when the t r a c e of the Jacobian i s negative and the determinant i s p o s i t i v e : CD > 0 (6.13) and E + F < 0 (6.14) For p o s i t i v e n* and s* and from (6.8-6.11), these c o n d i t i o n s h o l d f o r a l l w>0. T h e r e f o r e , the non-delayed model i s l o c a l l y s t a b l e f o r a l l d i l u t i o n r a t e s . Delayed model The l i n e a r i z e d equations f o r the delayed model are dx/dt = Cy(t-T) (6.15) dy/dt = -Dx(t) - F y ( t ) - Ey ( t - T ) (6.16) where x=(n(t)-n*) and y = ( s ( t ) - x * ) . Taking the Laplace transform of (6.15) and (6.16), the q u a s i - c h a r a c t e r i s t i c equation i s given by Q(s) = s 2 - sEexp(-sT) - sF + CDexp(-sT). (6.17) L e t t i n g iy=sT and equating the r e a l and imaginary p a r t s i n (6.17), we o b t a i n CD = y 2 / c o s y + Eytan y/T (6.18) 260 F = -Ecos y - TCDsin y/y (6.19) These equations•are the D - p a r t i t i o n equations f o r the model. The D - p a r t i t i o n s f o r the chemostat model were found with Caperon's (1968) data. We d i s c o v e r e d t h a t , f o r these and other parameter values i n the same range, the D - p a r t i t i o n had a constant s l o p e . T h i s was due to the r e l a t i v e l y small values of CD and F. For s m a l l y, the p a r t i t i o n equations ( 6 . 1 8 ) and ( 6 . 1 9 ) become CD = 0, (6.20) F = -E - TCD. (6.21 ) s i n c e l i m s i n ( y ) / y = 1 . (6.22) y->0 Thus, the slope of the p a r t i t i o n l i n e depends l i n e a r l y on T. T h i s approximation holds f o r smoothed delays as w e l l . To see the s i g n i f i c a n c e of the p a r t i t i o n , we must f i n d the h a r v e s t i n g (or, more p r o p e r l y , the d i l u t i o n ) t r a j e c t o r y . T h i s t r a j e c t o r y i s p l o t t e d i n F i g u r e (6.1) with a p a r t i t i o n . I t i s evident from the p a r t i t i o n equation t h a t i f T i s l a r g e enough, the p a r t i t i o n l i n e w i l l i n t e r s e c t the h a r v e s t i n g t r a j e c t o r y at two p o i n t s . For a l l d i l u t i o n r a t e s , there i s some delay at which i n s t a b i l i t y o ccurs. T h i s t r a n s i t i o n a l delay i s lowest f o r i n t e r m e d i a t e d i l u t i o n r a t e s because the t r a j e c t o r y p o i n t s 261 Figure ( 6 . 1 ) — Harvesting trajectory and D-partition for Caperon's model. 262 263 c o r r e s p o n d i n g to those r a t e s are near the maximum of the t r a j e c t o r y . I f delay i s l a r g e enough, the t r a j e c t o r y i s d i v i d e d i n t o a s t a b l e low d i l u t i o n r e g i o n , an unstable i n t e r m e d i a t e r e g i o n , and another s t a b l e r e g i o n at a high d i l u t i o n r a t e . Thus, i t i s c e r t a i n l y p o s s i b l e that delay can e f f e c t m u l t i p l e t r a n s i t i o n s . The q u e s t i o n remains whether observed delays are l a r g e enough. The two values f o r delay from Caperon (1968) and Cunningham and Maas (1978) are 14 hours and 10 hours, r e s p e c t i v e l y . Without f u r t h e r i n f o r m a t i o n , i t i s impossible to determine whether or not these are r e p r e s e n t a t i v e of uptake-growth delays i n phytoplankton. For the purposes of t h i s d i s c u s s i o n , we w i l l assume that the values are t y p i c a l . In F i g u r e (6.2), we have p l o t t e d the time delay necessary f o r i n s t a b i l i t y over the range of d i l u t i o n r a t e s f o r n i t r o g e n l i m i t a t i o n . T r a n s i t i o n delays f o r other n u t r i e n t s such as s i l i c o n (Paasche I973a,b) and phosphorus (Fuhs 1 9 6 9 ; Nyholm 1976, 1977) are i n c l u d e d in F i g u r e s (6.3) and (6.4). The d e l a y s f o r a l l n u t r i e n t s are i n the same range as i n the case of n i t r o g e n l i m i t a t i o n . The a c t u a l delays are f a r below those r e q u i r e d to induce l o c a l i n s t a b i l i t y at i n termediate d i l u t i o n r a t e s . However, i t should be kept i n mind that l o c a l s t a b i l i t y may be l i m i t e d to very small p e r t u r b a t i o n s from e q u i l i b r i u m . We t e s t e d the g l o b a l s t a b i l i t y of Caperon's model with a smoothed r e c t a n g u l a r delay by p e r t u r b i n g c e l l d e n s i t y and r e s e r v o i r about the e q u i l i b r i u m . F i g u r e (6.5) diagrams the s t a b l e area f o r low d i l u t i o n r a t e s ; F i g u r e (6.6) f o r higher 264 1 F i g u r e ( 6 . 2 ) — T r a n s i t i o n a l delays i n n i t r a t e - l i m i t e d chemostats. (a) The parameter v a l u e s were given f o r I .galbana i n Caperon (1 969) : r = 0.0422, a=0.0074, b=l0.89, and g0=0.0328. The observed delay was 14 hours. (b) The parameter values are from Cunningham and Maas (1978) f o r C . r e i n h a r d i i ; r = 0.094, a=0.36, and q0=0.25. The observed delay was 10 hours. 15 4 0 c o c CU o (a) 0 0 2 0.04 - I Dilution rate (hr ) 8 0 cu T3 nal 4 0 g -«-» c CU L. H 0 (b) 0.05 _( 0.1 Dilution ra te (hr ) 266 F i g u r e ( 6 . 3 ) — T r a n s i t i o n a l d e lays in s i l i c o n - l i m i t e d chemostats. (a) The parameter values f o r T.pseudonana are r=0.115, a=1.39, and q0=0.67; (b) f o r S.costatum, r=0.069, a=0.8, and q0=0.32 (data from Paasche 1973b). The observed delay was not p u b l i s h e d . 268 figure ( 6 . 4 ) — T r a n s i t i o n a l delays in phosphorus-limited :hemostats. For C.pyrenoidosa, r=0.0850, a=0.l9, q0=0.00l (data from Nyholm 1976). 270 Figure ( 6 . 5 ) — Perturbation boundary for Caperon's model with d i l u t i o n rate, w=0.009 per hour. The parameter values are given in Figure (6.2(a)). CO I o -P 400 x £ 00 u 00 c U 500 300 200 100 0 equilibrium stable • 0.01 0.02 0.03 0.04 Cell reservoir (jj|jgat/cel!) 272 r a t e s . The l o w - d i l u t i o n e q u i l i b r i u m i s more f r a g i l e than the higher, e s p e c i a l l y i n terms of p e r t u r b a t i o n s i n the r e s e r v o i r ; The l i m i t i n g value f o r a r e t u r n to e q u i l i b r i u m at the low r a t e s was approximately 0.01 M p - g a t / c e l l . As the p h y s i c a l c o n s t r a i n t on the s i z e of the pool i s g r e a t e r than 0.0334, the peak experimental value of the r e s e r v o i r , v t h e r e l a t i v e f r a g i l i t y of the e q u i l i b r i u m cannot be a t t r i b u t e d to a p h y s i o l o g i c a l l i m i t a t i o n . The c e l l may, however, l i m i t the suddenness of any f l u c t u a t i o n . i n the r e s e r v o i r by a d j u s t i n g the uptake r a t e . In e f f e c t , u n t i l we e s t a b l i s h the r e l a t i o n s h i p between r e s e r v o i r or quota and e x t e r n a l s u b s t r a t e c o n c e n t r a t i o n , we w i l l remain in the dark about the r e s i l i e n c e of the a l g a l f l u c t u a t i o n s to p e r t u r b a t i o n s i n n u t r i e n t s h i f t s . What we have found i s that a high immigration r a t e can be absorbed without causing a c o l l a p s e of t h a t p o p u l a t i o n . Nyholm (1977) presented the t r a n s i e n t response of C h l o r e l l a pyrenoidosa to step p e r t u r b a t i o n s in d i l u t i o n r a t e of phosphorus ( i n h i s F i g u r e 11(b)). We attempted to f i t both Nyholm's model with smooth and d i s c r e t e delay to t h i s data as p e r i o d i c f l u c t u a t i o n s with a p e r i o d of approximately 38 hours was observed. I t i s reasonable to suspect that the o s c i l l a t i o n s are induced by d e l a y . In c o n t r a s t to t h i s h y p o t h e s i s , only small o s c i l l a t i o n s were observed i n the model s i m u l a t i o n s . In a d d i t i o n , the f i r s t o s c i l l a t i o n to emerge was an overshoot of the e q u i l i b r i u m , which was not observed. We t h e r e f o r e conclude that the o s c i l l a t i o n s i n the data were not caused by d e l a y . Two p o s s i b l e a l t e r n a t i v e mechanisms to 273 F i g u r e ( 6 . 6 ) — P e r t u r b a t i o n boundary f o r Caperon's model w i t h d i l u t i o n r a t e , w=0.02 per hour. The parameter v a l u e s a r e the same as i n the p r e v i o u s f i g u r e . *274 • * « 0.01 0.02 0.03 0.04 Cell reservoir (uugat/cell) 275 produce the o s c i l l a t i o n s are environmental or experimental noise and synchronized r e p r o d u c t i o n . I f a delay was present, i t was smaller than 12 hours, the l a g at which the overshoot arose. No delays have been f i t t e d to t r a n s i e n t response of experiments with l i m i t a t i o n of s i l i c o n . Davis e_t.al. ( 1978) formulated a model f o r s i l i c o n uptake and growth with a n u t r i e n t pool but p e r t u r b a t i o n experiments were i n c o n c l u s i v e about the presence of d e l a y . To conclude from these f i n d i n g s that delays do not e f f e c t c o n s i d e r a b l e d e v i a t i o n s i n the t r a n s i e n t response of chemostat p o p u l a t i o n s from non-delayed responses would be i n c o r r e c t . F i g u r e (6.7) shows s i m u l a t i o n s of growth with no delay, d i s c r e t e delay, and smoothed delay from Caperon's (1969) data. The delay imparts o s c i l l a t i o n s i n the t r a n s i e n t response. With a smoothed, r e c t a n g u l a r delay, the damping of these o s c i l l a t i o n s i s g r e a t e r . The p e r i o d i c o s c i l l a t i o n s agree with those found i n Caperon ( 1 9 6 9 ) . Cunningham and Nisbet (1980) observed n o n - p e r i o d i c o s c i l l a t i o n s i n t h e i r s t u d i e s of the t r a n s i e n t response of Chlamydomonas r e i n h a r d i i . W i l l i a m s (1971) i n t e r p r e t e d t h i s as s y n c h r o n i z a t i o n of r e p r o d u c t i o n w i t h i n the c u l t u r e , which adds a second harmonic to the p e r i o d of o s c i l l a t i o n . D i s c u s s i o n Although the time delays observed i n the growth of n i t r a t e - l i m i t e d chemostat p o p u l a t i o n s are not l a r g e enough to produce i n s t a b i l i t y , they are of primary concern i n 276 Figure (6.7)-- Chemostat simulations with Caperon's model and r=0.0422, a=0.0074, q0=0.0328, B=10.89. The d i l u t i o n rate i s set at w=0.01095. The figures are (a) run without delay; (b) run with discrete delay of 14 hours; (c) run with rectangular delay of 16 hours. r n I DENSITY (CELLS/ML X 1 0 - 3 ) H i—i I 8 3 8-at 8 * 8 • 8 s § 9 8 8 8 8 8 LLZ I 279 280 i n t e r p r e t i n g dynamic f l u c t u a t i o n s i n phytoplankton d e n s i t i e s . Because of the importance of delay i n dynamics, measurement of delay i s d e s i r a b l e . Both Caperon (1969) and Cunningham and Nisbet (1980) estimated by the delay by a l e a s t - s q u a r e s f i t of the t r a n s i e n t data. C l e a r l y , an independent estimate of t h i s parameter would be more d e s i r a b l e . T h i s would i n v o l v e study of the p h y s i o l o g i c a l causes of the delay. Grenney e t . a l . (1973) developed a model of n i t r a t e uptake and growth by d i v i d i n g the n i t r o g e n content of the growth chamber i n t o 4 p a r t s : the c o n c e n t r a t i o n of n i t r a t e i n the environment, i n o r g a n i c n i t r o g e n i n the c e l l , organic i n t e r m e d i a t e s , and p r o t e i n . In e f f e c t , they uncoupled the uptake process and the u t i l i z a t i o n process. They were then able to f i t the model t o Caperon's t r a n s i e n t data without i n c l u d i n g any e x p l i c i t d e l a y s . Although the data f i t t i n g does not imply model c o r r e c t n e s s , there are reasons to b e l i e v e that t h i s i s a step i n the r i g h t d i r e c t i o n . F i r s t , Caperon (1969) found that a smoothed delay between quota and growth f i t s the data b e t t e r than a s t r a i g h t d e l a y . A smoothed delay would i n d i c a t e that the organism has the c a p a c i t y to i n t e g r a t e over past quotas. A simple b i o l o g i c a l mechanism f o r i n t e g r a t i o n i s p o o l i n g intermediate products between the uptake of i n o r g a n i c n i t r o g e n and r e p r o d u c t i o n . The pool a c t s as a b u f f e r between the two p r o c e s s e s . Thomas and Krauss (1955) r e p o r t e d that Scenedesmus accumulates a p p r e c i a b l e amounts of organic n i t r o g e n w i t h i n the c e l l . A second i n d i c a t o r i s t h a t of the c h a r a c t e r i s t i c time 281 s c a l e of the organic i n t e r m e d i a t e s . May and Anderson (1978) found that adding an intermediate d i f f e r e n t i a l equation f o r i n f e c t i v e stages of a p a r a s i t e to a coupled set of host-p a r a s i t e equations mimicked the behaviour of the same model with a d e l a y . They found that the c h a r a c t e r i s t i c time s c a l e f o r changes i n the number of i n f e c t i v e s was t h e o r e t i c a l l y equal to the de l a y . For the model of Grenney e t . a l . , the c h a r a c t e r i s t i c time s c a l e f o r changes in the c o n c e n t r a t i o n of orga n i c i n t e r m e d i a t e s i s given by the r e c i p r o c a l of AH a . i , , = _ K 2*v2/(K2+N2*/N3*) 2 - K4 - Q (6.23) where 3 ^  V «K . N N2* = the e q u i l i b r i u m c o n c e n t r a t i o n of organic i n t e r m e d i a t e s , N3* = the e q u i l i b r i u m c o n c e n t r a t i o n of p r o t e i n (assumed p r o p o r t i o n a l to c e l l d e n s i t y ) . K2 = the h a l f - s a t u r a t i o n constant f o r uptake of in o r g a n i c n i t r o g e n , V2 = the rate constant f o r uptake of i n o r g a n i c n i t r o g e n . K4 = e x c r e t i o n r a t e Q = d i l u t i o n r a t e . I f we use t h e i r data: N3* = 5.0 ngat/1, 282 N2* = 1.4 ngat/1, K2 = 2.51, V2 = 3.57 per day, K4 = 0.2 per day, Q = 0.2 per day. To e v a l u a t e the c h a r a c t e r i s t i c time s c a l e , the time s c a l e i s 0.64 days or about 15 hours. T h i s i s extremely c l o s e to Caperon's value of 14 hours f o r the time delay. For d i l u t i o n r a t e s of 0.4, 0.6, and 0.8, the time s c a l e i s between 15 and 16 hours. T h i s agreement confirms our c l a i m that delay i s a f i r s t - a p p r o x i m a t i o n to a t h i r d coupled d i f f e r e n t i a l e q uation. The d i f f e r e n t i a l equation a l s o e s t a b l i s h e s that the delay i s a d j u s t a b l e by changes i n the uptake k i n e t i c s of i n o r g a n i c n i t r o g e n and the e x c r e t i o n r a t e . The above c a l c u l a t i o n of the time s c a l e , however, i n d i c a t e that the delay i s not modified by metabolic adjustment. Further s t u d i e s are necessary to t e s t t h i s o b s e r v a t i o n . T h i s a n a l y s i s would not be complete without o u t l i n i n g the caveats of the uptake-growth model. There i s some q u e s t i o n about the v a l i d i t y of the Caperon model at high and low d i l u t i o n r a t e s . H a r r i s o n £t-al. (1976) and Conway e t . a l . (1976) have suggested that a l t e r n a t i v e r e g u l a t o r y mechanisms govern uptake at the ends of the growth rate spectrum. At the low d i l u t i o n r a t e , the quota i n c r e a s e s with d e c r e a s i n g d i l u t i o n r a t e . More data i s necessary b e f o r e t h i s phenomenon can be i n t e r p r e t e d although H a r r i s o n et.aJL. (1976) hypothesize that 283 a n o t h e r p h y s i o l o g i c a l s t a t e e x i s t s a t these low r a t e s . At the o t h e r end of the spectrum, a t h i g h d i l u t i o n r a t e s , a l a r g e change i n the d i l u t i o n r a t e produces o n l y a s m a l l change i n the q u o t a . A p o s s i b l e e x p l a n a t i o n here i s t h a t the c e l l i s a d j u s t i n g i t s maximal growth r a t e t o the improved n u t r i e n t c o n d i t i o n s t h rough a s e r i e s of i n c r e a s e s i n RNA, DNA, p r o t e i n s y n t h e s i s , and, f i n a l l y , growth r a t e . Whatever the mechanisms i n v o l v e d , the d a t a o b t a i n e d a t t h e s e h i g h e r growth r a t e s would s i g n i f i c a n t l y d e c r e a s e the d e l a y n e c e s s a r y t o d e s t a b i l i z e the e q u i l i b r i u m . S i n c e t h i s ' s h i f t - u p ' c o u l d double t h e maximal growth r a t e ( H a r r i s o n e t . a _ l . 1976), the d e l a y n e c e s s a r y f o r i n s t a b i l i t y would drop by a p p r o x i m a t e l y a f a c t o r of two. Goldman and McCarthy (1978) have q u e s t i o n e d the v a l i d i t y of the ' s h i f t - u p ' as t h e i r e x p e r i m e n t s w i t h T h a l l a s s i o s i r a pseudonana showed no such t r e n d . C o u n t e r b a l a n c i n g t h i s p o t e n t i a l i n c r e a s e i n t h e maximal growth r a t e , the growth c o n d i t i o n s i n the environment are s u b o p t i m a l w h i l e chemostat c u l t u r e s t e s t i n g n u t r i e n t l i m i t a t i o n a r e grown at o p t i m a l t emperature and l i g h t i n t e n s i t y . P r o d u c t i o n r e l a t i o n s w i t h l i g h t i n t e n s i t y and t e m p e r a t u r e a r e d i s c u s s e d by E p p l e y (1972). Laws and B a n n i s t e r (1980) d e v e l o p e d a model of l i g h t - l i m i t e d and n u t r i e n t - l i m i t e d uptake and growth f o r chemostat c u l t u r e s . B e s i d e s the maximal growth r a t e , o t h e r parameters assumed t o be c o n s t a n t i n Caperon's model a c t u a l l y v a r y w i t h change i n the d i l u t i o n r a t e or growth r a t e . McCarthy and Goldman (1979) found an i n v e r s e r e l a t i o n s h i p between the maximum uptake r a t e 284 and the growth r a t e . They hypothesized that t h i s feedback mechanism, which accounts f o r the phenomenon of luxury consumption (Droop 1973), e x p l a i n s the growth of oceanic phytoplankton i n environments where the ambient n u t r i e n t c o n c e n t r a t i o n i s below the measurable l i m i t . The h a l f -s a t u r a t i o n constant decreases i n the t r a n s i t i o n from n u t r i e n t r i c h to n u t r i e n t - p o o r h a b i t a t s (Maclsaac and Dugdale 1969; Perry 1976). In our time s c a l e equation, both these trends cause a decrease in the delay between uptake and growth as the d i l u t i o n r a t e decreases. Because o s c i l l a t i o n s i n the time s e r i e s f i t t e d by the delayed model occur at the lower d i l u t i o n r a t e s , the delay parameter i s r e p r e s e n t a t i v e of the time s c a l e at these low r a t e s . T h i s means that the delay r i s e s as the d i l u t i o n r a t e i n c r e a s e s . Consequently, i t i s s t i l l p o s s i b l e that n u t r i e n t l i m i t a t i o n d i c t a t e s the l i m i t s on the uptake r a t e and h a l f - s a t u r a t i o n constant at i n t e r m e d i a t e d i l u t i o n r a t e s . The t r a n s f o r m a t i o n between delay and i n t e r n a l n u t r i e n t p o o l s e l u c i d a t e s the p o s s i b l e e v o l u t i o n a r y advantages of n u t r i e n t storage from the p e r s p e c t i v e of dynamics. Delays are inherent i n b i o c h e m i c a l p r o c e s s i n g of absorbed n u t r i e n t s . If a l a r g e amount of p r o c e s s i n g of the absorbed n u t r i e n t s i s necessary, the delay may be l a r g e . If r e p r o d u c t i o n i s l a r g e l y uncoupled from uptake and n u t r i e n t c o n c e n t r a t i o n and the n u t r i e n t c o n c e n t r a t i o n i s l i m i t i n g , the r e p r o d u c t i o n may endanger the s u r v i v a l of the i n d i v i d u a l and i t s o f f s p r i n g . The d e s t a b i l i z i n g e f f e c t of d e l a y s are minimized with lowered 285 maximal growth r a t e s . A p r e l i m i n a r y comparison of the maximal growth r a t e s and the observed d e l a y s from the data of Caperon (1968), W i l l i a m s (1971), Nyholm (1976) i n d i c a t e s an inverse r e l a t i o n s h i p between observed growth r a t e s and d e l a y . Future s t u d i e s , along the l i n e s i n d i c a t e d by Calow (1977), may strengthen t h i s s u g g e s t i o n . Besides s t a b i l i z a t i o n through parameter v a r i a t i o n , the amplitude of the o s c i l l a t i o n s can be damped by smoothing delay through storage of intermediate products as we observed i n the s i m u l a t i o n s i n F i g u r e (6.7). Pools smooth d e l a y s between uptake and r e p r o d u c t i o n but a l s o moderate the e f f e c t of environmental f l u c t u a t i o n s . The g r e a t e r the s i z e of the p o o l , the l e s s l i k e l y an organism i s to be n u t r i e n t s t a r v e d . However, an organism must be c l o s e l y aware of changing n u t r i e n t c o n d i t i o n s i f i t s o f f s p r i n g are to s u r v i v e . Evidence suggests that there i s a feedback l i n k between the processes of uptake and growth, although the p r e c i s e mechanism by which t h i s feedback operates i s u n c l e a r . T h e o r e t i c a l l y , the longer the feedback loop i n the pathway, the g r e a t e r the s t a b i l i t y of the system. Rhee (1978) argues that the c o n c e n t r a t i o n of amino a c i d s r e g u l a t e s the uptake rate e i t h e r d i r e c t l y or i n d i r e c t l y . He d i s c o v e r e d an i n v e r s e r e l a t i o n s h i p between these two v a r i a b l e s f o r n i t r a t e uptake in Scenedesmus sp. Solomonson and Spehar (1977) propose a model fo r r e g u l a t i o n of n i t r a t e a s s i m i l a t i o n i n which amino a c i d s and/or ammonia i n h i b i t s n i t r a t e uptake. A feedback loop from amino a c i d pools to uptake i s almost as long as the pathway 286 i t s e l f . In c o n t r a s t to these r e s u l t s , Passera and F e r r a r i (1975) demonstrated that the s u l f a t e ion and not i t s me t a b o l i t e s r e g u l a t e s the t r a n s p o r t of a d d i t i o n a l s u l f a t e a c r o s s the c e l l membrane i n C h l o r e l l a v u l g a r i s . At low e x t r a c e l l u l a r s u l f a t e c o n c e n t r a t i o n s , s u l f a t e a c t s as a co o p e r a t i v e e f f e c t o r of t r a n s p o r t , i n c r e a s i n g the uptake r a t e . At h i g h c o n c e n t r a t i o n s , s u l f a t e e x h i b i t s d i f f e r e n t e f f e c t s depending on past c o n c e n t r a t i o n s of the s u b s t r a t e to which the c e l l has been exposed. The feedback loop i n t h i s case appears to be short compared to the l e n g t h of the bioche m i c a l pathway. Biochemical mechanisms r e g u l a t i n g uptake may d i f f e r among environments as w e l l . McCarthy and Goldman (1979) i n j e c t e d ammonium i n t o c u l t u r e s of T h a l l a s s i o s i r a pseudonana clones from oceanic and c o a s t a l environments. The oceanic clone r a p i d l y a d j u s t e d i t s maximal uptake r a t e i n response to n u t r i e n t d e p r i v a t i o n ; the c o a s t a l c l o n e d i s p l a y e d a much slower response. The d i f f e r e n c e between the c l o n e s i n d i c a t e s a p o s s i b l e v a r i a t i o n i n feedback mechanisms. In a d d i t i o n , i t appears that the feedback mechanism depends on the s p a t i a l h e t e r o g e n e i t y i n n u t r i e n t s . Whatever the mechanisms, feedback r e g u l a t i o n should be i n c o r p o r a t e d i n t o the uptake growth model as i t can have a s i g n i f i c a n t impact on s t a b i l i t y . C o n c l u s ion In t h i s chapter we showed that time delays can t h e o r e t i c a l l y produce two s t a b i l i t y t r a n s i t i o n s with changing d i l u t i o n r a t e s . From c u r r e n t experimental evidence, we 287 concluded that a c t u a l d e l a y s are not l a r g e enough to cause these t r a n s i t i o n s , although delays do cause s i g n i f i c a n t d i f f e r e n c e s i n the behaviour of the t r a n s i e n t response. In the d i s c u s s i o n , we r e l a t e d the e x i s t e n c e of time delays to i n t e r n a l storage pools and b r i e f l y c o n s i d e r e d the e v o l u t i o n of such p o o l s . 288 CONCLUSION Time d e l a y s a r e i n h e r e n t i n i n c o m p l e t e d e s c r i p t i o n s of e c o l o g i c a l models. The a d d i t i o n of a d e l a y s u b s t i t u t e s f o r a more p r e c i s e d e f i n i t i o n of a d y n a m i c a l l i n k between an u n r e p r e s e n t e d p r o c e s s and the p r o c e s s e s i n the model. In e c o l o g i c a l s i t u a t i o n s w i t h a s l o w l y growing r e s o u r c e , a r e s o u r c e r e c o v e r y d e l a y mimics the dynamics of the r e s o u r c e . In c h e m o s t a t s , a d e l a y c a p t u r e s the g r o s s f e a t u r e s of the i n t e r n a l s t o r a g e mechanisms of the a l g a l c e l l . By e x p l o r i n g the outcome of combining p r e d a t o r - p r e y models and r e s o u r c e r e c o v e r y d e l a y , we a r e , i n e f f e c t , t a k i n g a s t e p toward the g e n e r a l u n d e r s t a n d i n g of a 'three-body problem' i n e c o l o g y : a system w i t h r e s o u r c e , p r e y , and p r e d a t o r . In f a c t , some of t h e m a t h e m a t i c a l methods employed here were o r i g i n a l l y d e v e l o p e d by L a p l a c e t o a n a l y z e the three-body problem of p l a n e t a r y motion ( K l i n e 1980). De l a y e d models and non-delayed ones are c o n t i n u o u s i n the sense t h a t the t r a j e c t o r i e s change c o n t i n u o u s l y f o r s m a l l d e l a y s . T h i s remark i m p l i e s t h a t the r e s o u r c e must be s l o w l y growing t o e f f e c t q u a l i t a t i v e changes i n the dynamics of the system. T h i s remark agrees w i t h H o l l i n g ' s ( H o l l i n g , p e r s . comm.) o b s e r v a t i o n about the r e l a t i o n s h i p between ' f a s t ' and 'slow' v a r i a b l e s . In h i s s i m u l a t i o n s t u d i e s , he found t h a t 'slow' v a r i a b l e s s e t the time s c a l e f o r the f a s t e r v a r i a b l e s i n the spruce budworm system, f o r e s t f i r e ecosystems, and semi-a r i d g r a z i n g savannas. We have d e t e r m i n e d t h a t the p r e s ence of a slow v a r i a b l e s t i m u l a t e s c y c l i n g and l o c a l i n s t a b i l i t y of the 289 e q u i l i b r i u m , e s p e c i a l l y w i t h i m m i g r a t i o n or s t o c k i n g . For systems w i t h 'slow' v a r i a b l e s , we wanted t o e x p l o r e the e f f e c t s of management m a n i p u l a t i o n s and c o n t r a s t t h e r e s u l t i n g dynamics t o the dynamics of systems w i t h ' f a s t ' r e s o u r c e v a r i a b l e s . L o c a l s t a b i l i t y a n a l y s i s i s our p r i m a r y t o o l f o r d i s c e r n i n g t h e s e e f f e c t s . L o c a l s t a b i l i t y a n a l y s i s d i f f e r e n t i a t e s between p o p u l a t i o n s which approach an e q u i l i b r i u m d e n s i t y and those which do n o t . We performed l o c a l a n a l y s i s f o r c o n s t a n t - q u o t a and c o n s t a n t - e f f o r t h a r v e s t i n g p o l i c i e s of b o t h the prey and p r e d a t o r s p e c i e s as w e l l as c o n s t a n t s t o c k i n g f o r b o t h . The m a t h e m a t i c a l language of D-p a r t i t i o n s was employed t o i l l u s t r a t e our c o n c l u s i o n s ; i t c o n t r a s t s the s t a b i l i t y of the d e l a y e d and non-delayed models. The models were l i m i t e d t o t h r e e : H o l l i n g ' s model w i t h a t y p e -I I f u n c t i o n a l r e s p o n s e , I v l e v ' s model, and L o t k a - V o l t e r r a model w i t h r e s o u r c e l i m i t a t i o n . T h i s procedure can e a s i l y be extended t o o t h e r t w o - s p e c i e s or t w o - v a r i a b l e models, as was demonstrated i n the c h a p t e r on ch e m o s t a t s . In d e l a y e d systems w i t h p r e y h a r v e s t i n g , the s t a b i l i t y c r i t e r i a d i f f e r from non-delayed models i n t h a t the maximum p r e d a t o r consumption r a t e s and p r e d a t o r c o n v e r s i o n e f f i c i e n c y a r e p r e s e n t i n the c o n d i t i o n s . The same parameters have no i n f l u e n c e on the s t a b i l i t y of the non-delayed models w i t h h a r v e s t i n g . T h i s i n d i c a t e s t h a t , i f a f i s h i s t o be s t o c k e d or a n a t u r a l enemy i n t r o d u c e d i n t o a system w i t h a 'slow' r e s o u r c e , one s h o u l d pay c a r e f u l a t t e n t i o n t o i t s i n t e r a c t i o n w i t h the prey or p e s t i n terms of consumption and growth of the 290 p r e d a t o r . A v o r a c i o u s p r e d a t o r w i t h a h i g h c o n v e r s i o n e f f i c i e n c y w i l l be more l i k e l y t o d e s t a b i l i z e the system. For p r e y s t o c k i n g , the d e l a y e d and non-delayed systems s t a y s t a b l e or become s t a b l e . The two t y p e s of p r e d a t o r h a r v e s t i n g are q u i t e d i f f e r e n t m a t h e m a t i c a l l y and d y n a m i c a l l y . For c o n s t a n t - e f f o r t h a r v e s t i n g , we found t h a t the h a r v e s t i n g can d e s t a b i l i z e a s t a b l e e q u i l i b r i u m , c o n t r a r y t o non-delayed p r e d i c t i o n s . The i n t e r p l a y between i n c r e a s e d consumption by the p r e d a t o r w i t h i n c r e a s e d h a r v e s t i n g and the d e c r e a s e d r a t e of i n c r e a s e a t a g i v e n prey d e n s i t y p e r m i t s t h i s d e s t a b i l i z a t i o n and m u l t i p l e t r a n s i t i o n s between s t a b l e and u n s t a b l e e q u i l i b r i a . Once a g a i n , the consumption and p r e d a t o r growth r a t e a r e c r u c i a l . For a h i g h l i m i t i n g consumption r a t e or h i g h c o n v e r s i o n e f f i c i e n c y , the d e s t a b i l i z a t i o n i s more l i k e l y because a g r e a t e r range of e q u i l i b r i u m d e n s i t i e s i s a l l o w e d . C o n s t a n t - q u o t a p r e d a t o r h a r v e s t i n g and s t o c k i n g c r e a t e s a complex d y n a m i c a l s i t u a t i o n . More t r a n s i t i o n s from s t a b l e t o u n s t a b l e r e g i o n s a r e o b s e r v e d w i t h t h i s p o l i c y than w i t h c o n s t a n t - q u o t a h a r v e s t i n g of the p r e y . Both d e l a y e d and non-d e l a y e d models show the c a p a c i t y f o r moving from s t a b i l i t y t o i n s t a b i l i t y t o s t a b i l i t y as p r e d a t o r h a r v e s t i n g i s d e c r e a s e d ( n e g a t i v e h a r v e s t i n g i s s t o c k i n g ) . The 'slow' r e s o u r c e v a r i a b l e o n l y enhances the l i k e l i h o o d of the system t o undergo th e s e t r a n s i t i o n s . To complement our l o c a l a n a l y s i s , we performed g l o b a l a n a l y s i s t h r o u g h n u m e r i c a l i n t e g r a t i o n . Our s i m u l a t i o n s showed 291 that delay i n c r e a s e s the chances that a managed system w i l l undergo o s c i l l a t i o n s , e s p e c i a l l y with s t o c k i n g . We a l s o concluded that the c a r r y i n g c a p a c i t y p l a y s a v i t a l r o l e i n d e f i n i n g the shape of the l i m i t c y c l e s . The s i z e of the l i m i t c y c l e s i n c r e a s e d with i n c r e a s i n g c a r r y i n g c a p a c i t y as d i d s u s c e p t i b i l i t y to p e r t u r b a t i o n s , as measured by d i s t a n c e from zero d e n s i t i e s . In a d d i t i o n , changes in the c a r r y i n g c a p a c i t y generated b i f u r c a t i o n s i n the l i m i t c y c l e as w e l l f o r d i f f e r e n t s t o c k i n g r a t e s . O v e r a l l , our s i m u l a t i o n study r e i n f o r c e d the importance of c o r r e c t l y determining the f u n c t i o n a l response of the predator-prey system. Because some re s e a r c h e r s have ob j e c t e d to the use of d i s c r e t e d e l ays as u n r e a l i s t i c , we checked our l o c a l s t a b i l i t y r e s u l t s with d i s c r e t e delay a g a i n s t those with smoothed d e l a y s . Although smoothing of the delay d i d not c o n t r a d i c t most of our c o n c l u s i o n s , delay with a t r i a n g u l a r k e r n a l allowed prey s t o c k i n g to d r i v e the system unstable fo r l a r g e d e l a y s ; no other delayed kern a l showed t h i s behaviour f o r the range of d e l a y s t e s t e d . The d i f f e r e n t k e r n a l s of continuous delay, however, vary g r e a t l y i n the h a r v e s t i n g ranges f o r which the e q u i l i b r i u m i s l o c a l l y u n s t a b l e . In our f i n a l chapter, we a p p l i e d our mathematical techniques to chemostat experiments. We found that delays between the uptake of n u t r i e n t s and the growth of the c u l t u r e can e f f e c t m u l t i p l e t r a n s i t i o n s between s t a b i l i t y r e g i o n s : s t a b i l i t y at low d i l u t i o n r a t e s , i n s t a b i l i t y at intermediate d i l u t i o n r a t e s , and s t a b i l i t y at high d i l u t i o n r a t e s . We then 292 t e s t e d whether the observed d e l a y s were l a r g e enough to cause these t r a n s i t i o n s . We concluded from data taken from the l i t e r a t u r e that they c o u l d not. The delays observed were the same as the c h a r a c t e r i s t i c time s c a l e f o r organic i n t e r m e d i a t e s from the model of Grenney e t . a l . (1973). Although delay was assumed to be constant i n t h i s study, the reader should remember that the delay i s a mathematical approximation f o r resource dynamics. A measure of the delay i s the c h a r a c t e r i s t i c time s c a l e of the resource. In our models, we assume that the delay i s c o n s t a n t over v a r i a t i o n s i n prey and predator biomass. But over a longer time frame, the delay may change, p o s s i b l y because of density-dependent resource growth or because of the i n f l u e n c e of another v a r i a b l e i n the system. The treatment of a constant delay i s analogous to the examination of f a s t v a r i a b l e s as slow v a r i a b l e s are h e l d constant (see Ludwig e t . a l . (1978) f o r d e t a i l s ) . May (1977a) summarizes other systems which behave in t h i s way. The e a s t e r n spruce budworm system i s a case where resource recovery delay changes. The system behaves two d i f f e r e n t ways: (1) l o n g - l a s t i n g i n f e s t a t i o n s occur at frequent to moderate i n t e r v a l s , (2) l o n g - l a s t i n g i n f e s t a t i o n s occur at i r r e g u l a r i n t e r v a l s (McNamee et.al.. 1981). We w i l l o u t l i n e the f i r s t c l a s s of behaviour i n terms of d e l a y . I f the budworm d e n s i t y i s much l e s s than the maximum supportable d e n s i t y by the f o l i a g e (the c a r r y i n g c a p a c i t y ) , then the delay i s c l o s e l y approximated by the c h a r a c t e r i s t i c time s c a l e of the f o l i a g e . The delay changes because the 293 f o l i a g e resource grows i n a density-dependent f a s h i o n . The behaviour of the system f o l l o w s a c y c l i c p a t t e r n . As the budworm d e f o l i a t e s the f o r e s t from the f o l i a g e c a r r y i n g c a p a c i t y , the time s c a l e on which the f o l i a g e grows i n c r e a s e s , i n c r e a s i n g the d e l a y . F u r t h e r d e f o l i a t i o n i n c r e a s e s the delay to a p o i n t where the e q u i l i b r i u m becomes unstable and the budworm d e n s i t y explodes to the outbreak l e v e l . Complete d e f o l i a t i o n ensues and the budworm d e n s i t y c o l l a p s e s . At the end of the outbreak, budworm d e n s i t y i s low and f o l i a g e biomass i s low but i n c r e a s i n g (McNamee e t . a l . 1981). The budworm d e n s i t y i s kept low by p r e d a t i o n while the f o l i a g e regrows. The c y c l e then begins once again. The second c l a s s of behaviour i s s l i g h t l y more complex s i n c e the f o r e s t dynamics enter i n t o the system behaviour. Our a n a l y s i s only e x p l a i n s p a r t of the budworm c y c l e and that p a r t i n the mathematical shorthand of delay. The analyses of H o l l i n g e t . a l . (1977) and Ludwig e t . a l . ( 1 9 7 8 ) are a t t r a c t i v e because the dynamical p i c t u r e i s f u l l y e x p l a i n e d i n terms of changing b i o l o g i c a l q u a n t i t i e s and not mathematical a b s t r a c t i o n s . However, such i n t e n s i v e and e x t e n s i v e s t u d i e s are r a r e . To more q u i c k l y e v a l u a t e the t e n a b i l i t y of the m u l t i p l e e q u i l i b r i u m h y p o t h e s i s , the delay shorthand might be of great v a l u e . Measurement of delayed density-dependence g i v e s an i n d i c a t i o n of the time s c a l e of the slow v a r i a b l e s of the system and the degree to which resource l i m i t a t i o n a f f e c t s the dynamics of the slowly changing resource without r e q u i r i n g a c t u a l measurement of the resource. 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Time delays and chaos in two competing s p e c i e s . Math. B i o s c i . 51:1 99-21 1 . Shoemaker, C. 1973. O p t i m i z a t i o n of a g r i c u l t u r a l pest management. I l l : r e s u l t s and e x t e n s i o n s of a model. Math. B i o s c i . 18:1-22. Slobodkin, L.B. 1954. P o p u l a t i o n dynamics i n Daphnia obtusa Kutz. E c o l . Monographs 24:69-88 . Solomonson, L.P. and A.M. Spehar. 1977. Model f o r r e g u l a t i o n of n i t r a t e a s s i m i l a t i o n . Nature 265:373-375. Southwood, T.R.E. 1976. Bionomic s t r a t e g i e s and p o p u l a t i o n parameters. I_n May, R.M. , ed. T h e o r e t i c a l Ecology: P r i n c i p l e s and A p p l i c a t i o n s . W.B. Saunders: P h i l a d e l p h i a . S t e e l e , J.H. 1974. The s t r u c t u r e of marine ecosystems. B l a c k w e l l S c i e n t i f i c : Oxford. S t e e l e , J.H. 1976. T h e o r e t i c a l models i n ecology. J . Theor. B i o l . 63:443-451 . T a y l o r , C.E., and R.R. Sokal. 1976. O s c i l l a t i o n s i n h o u s e f l y p o p u l a t i o n s i z e s due to time l a g s . Ecology 57:1060-1067. 308 Thomas, W.H., and R.W. Krauss. 1955. Nitrogen l i m i t i t a t i o n i n Scenedesmus as a f f e c t e d by environmental changes. P l a n t . P h y s i o l . 30:113-122. van Emden, H.F., and M.J. Way. 1973. Host p l a n t s i n the po p u l a t i o n dynamics of i n s e c t s . I_n Royal E n t o m o l o g i c a l Soc. Symposia: 6. I n s e c t / P l a n t R e l a t i o n s h i p s , pp. 181-200. B l a c k w e l l : London. V o l t e r r a , V. 1931. V a r i a t i o n s and f l u c t u a t i o n s in the numbers of c o e x i s t i n g animal s p e c i e s . I_n Scudo, F.M., and Z i e g l e r , J.R. 1978. The golden age of t h e o r e t i c a l ecology, 1923-1940: a c o l l e c t i o n of workc by V. V o l t e r r a , V.A. K o s t i t z i n , A.J. Lotka, and A.N. Kolmogoroff. pp. 65-236. Lectur e Notes i n Biomathematics. v.22. 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I_n Anderson, R.M., Turner, B.D., and T a y l o r , L.R. Po p u l a t i o n Dynamics, pp. 207-222. B l a c k w e l l S c i e n t i f i c : Oxford. 309 W i l l i a m s , F.M. 1971. Dynamics of m i c r o b i a l p o p u l a t i o n s . I_n Patten, B.C., ed. Systems A n a l y s i s and S i m u l a t i o n i n Ecology. V o l . 1. Academic P r e s s : N.Y. 310 APPENDIX A The D - p a r t i t i o n method The method of D - p a r t i t i o n s or s t a b i l i t y p l a t e a u s (Pinney 1958) i s d e r i v e d from the Cauchy-Goursat Theorem: If Q(s) has poles as i t s only s i n g u l a r i t i e s i n s i d e a c l o s e d contour C ( t r a v e r s e d i n the c o u n t e r - c l o c k w i s e d i r e c t i o n ) and i s not i n f i n i t e or zero at any p o i n t on the contour,/ then where N i s the number of zeros and P i s the number of p o l e s of Q(s) w i t h i n the contour ( P h i l l i p s 1951). By choosing the contour C i n t e l l i g e n t l y , t h i s theorem can be a p p l i e d to the s t a b i l i t y of c l o s e d - l o o p feedback systems with the Nyquist C r i t e r i o n or to d e f i n i t e i n t e g r a t i o n . For s t a b i l i t y i n v e s t i g a t i o n s , we are i n t e r e s t e d i n whether or not any zeros of the q u a s i - c h a r a c t e r i s t i c equation l i e in the r i g h t - h a n d s i d e of the s plane where the o r d i n a t e i s Re(s) and the a b s c i s s a i s Im(s). An a p p r o p r i a t e contour would be one which e n c l o s e s the e n t i r e r i g h t - h a n d s i d e of the s plane. T h i s can be represented by a contour which f o l l o w s the imaginary a x i s and winds around the plane along a s e m i c i r c l e of i n f i n i t e r a d i u s . T h i s contour i s shown in F i g u r e ( A . 1 ( a ) ) . If p o l e s or zeros l i e along the imaginary a x i s the contour can be m o d i f i e d so that the theorem can s t i l l be used. If a zero e x i s t s i n s i d e t h i s contour, then the contour i n t e g r a l given above w i l l have a residue and the d i f f e r e n c e N-P w i l l be non-Q'(s)/Q(s) ds = N-P (A. 1 ) 311 Fig u r e ( A . 1 ) — (a) A c l o s e d contour encompassing the r i g h t -hand s i d e of the s plane. (b) A mapping i n the Q(s) plane for Q(s)=s+a. 313 zero. T h i s c o n d i t i o n can be transformed to a g r a p h i c a l r e s u l t by the f o l l o w i n g means. If we l e t C be the contour d e f i n e d i n F i g u r e (A. 1(a)) and l e t Q_(s) = Rexp(iM) (A.2 ) then l o g Q = iM + l o g R (A.3) and d ( l o g Q) = idM + d ( l o g R) (A.4) But Q'(s)/Q(s) = d ( l o g Q)/ds (A.5 ) So from ( A . 1 ) and ( A . 5 ) , ( 1 / 2 i r i ) ^ d ( l o g Q) = d / 2 r r i ) ^d(log R) + ( l / 2 t r ) ^"dM (A.6) Since C i s c l o s e d , ( l / 2 i r i ) ^ d ( l o g R) = 0 (A.7) And from ( A . 6 ) , t h e r e f o r e , ( 1 / 2 J T ) ^ d M = N-P. (A.8 ) T h i s i n t e g r a l s t a t e s that the number of times the map of the contour c i r c l e s the o r i g i n i n the Q plane i n the counter-314 c l o c k w i s e d i r e c t i o n i s the d i f f e r e n c e between the number of zeros and poles (Schwartz and F r i e d l a n d 1965, p. 428). I f the q u a s i - c h a r a c t e r i s t i c equation has no p o l e s , then the number of e n c i r c l e m e n t s i s simply the number of zeros of Q(s) i n the ri g h t - h a n d plane. By graphing the mapping from s to Q ( s ) , we can f i n d the number of e n c i r c l e m e n t s of the o r i g i n . A simple example i l l u s t r a t e s t h i s p o i n t . Let Q(s) = s + a (A.9) For s=iy, the imaginary a x i s i n the s plane, R e [ Q ( i y ) ] = a (A.10) Im[Q(iy)] = y (A.11) For s=Rexp(iM), the mapping forms a s e m i c i r c l e on the r i g h t -hand s i d e of the Q(s) plane. As R-*-oo, the r a d i u s of the s e m i c i r c l e i n the Q(s) plane becomes i n f i n i t e . The mapping i s a displacement of the contour i n the s plane. A diagram of the mapping i s shown i n F i g u r e (A.1). The c o n d i t i o n f o r i n s t a b i l i t y i n t h i s case i s a<0. Thus, the value of a determines the s t a b i l i t y of the q u a s i - c h a r a c t e r i s t i c e quation. We c o u l d map t h i s boundary on a s i n g l e - l i n e graph of a, d i v i d i n g t h i s parameter l i n e i n t o unstable and s t a b l e r e g i o n s at a=0. As an example of a t r a n s c e n d e n t a l q u a s i - c h a r a c t e r i s t i c , 315 F i g u r e (A.2) presents the mapping of the equation Q(s) = s 2 + bs + exp(-s) (A.12) The mapping e n c i r c l e s the o r i g i n when b=0 and b=0.5, but not when b=1. To d i s c o v e r the reason f o r t h i s , we f i n d the r e a l and imaginary p a r t s of the mapping of the imaginary-s a x i s with s=iy R e [ Q ( i y ) ] = cos(y) - y 2 Im[Q(iy)] = by - s i n ( y ) (A.13) (A.14) The c r i t i c a l b f o r which a zero e n t e r s the ri g h t - h a n d plane occurs when both the imaginary and r e a l p a r t s are zero. By r e w r i t i n g the above equation, we o b t a i n the D - p a r t i t i o n b = s i n ( y ) / y (A.15) y 2 = cos(y) (A.16) These equations are s a t i s f i e d f o r y=0.834 and b=0.888. T h i s d i v i d e s the parameter b i n t o a set of s t a b l e and unstable v a l u e s as determined by the D - p a r t i t i o n e q u a t i o n s . I f two parameters were v a r i a b l e , we c o u l d d i v i d e a parameter plane using the p a r t i t i o n equations. T h i s i s e x a c t l y our procedure in t h i s t h e s i s . 316 Figure ( A . 2 ) — The mapping in the Q(s) plane for Q(s) = s 2 + bs + exp(-s). 317 318 APPENDIX B Numerical i n t e g r a t i o n a l g o r i t h m The method of numerical i n t e g r a t i o n employed i n t h i s t h e s i s was a m o d i f i e d f o u r - s t e p Runga-Kutta method. For the systems, we are studying dx/dt = x ( t ) f ( x ( t ) , x ( t - T ) , y ( t ) ) ( B . I ) dy/dt = y ( t ) g ( x ( t ) , y ( t ) ) (B.2) Let T be the s i z e of the d e l a y . From time [~T,0], we s p e c i f y a constant i n i t i a l f u n c t i o n (For i n f l u e n c e s of other i n i t i a l f u n c t i o n s see Jones 1962a,b). Given a s p e c i f i e d s t e p s i z e h, we choose R=T/h and N=T/h to be i n t e g e r s . Then x( t - T ) = x((N-R)h) (B.3) Let x((N-R)h) = u(N-R) (B.4) y(Nh) = w(N) (B.5) The delayed x i s evaluated a t d i s c r e t e i n t e r v a l s a constant number of steps behind the c u r r e n t value of x. K(1,1) = f(u(N),u(N-R),w(N)) (B.6) K(1,2) = f(u(N)+hK(1,1)/2,(u(N-R)+u(N-R+1))/2, w(N)+hK(2,1)/2) (B.7) K(1,3) = f(u(N)+hK(1,2)/2,(u(N-R)+u(N-R+1))/2, 319 w(N)+hK(2,2)/2) (B.8) K(1,4) = f(u(N)+hK(1,3),u(N-R+1),w(N)+hK(2,3)) (B.9) K(2, 1 ) = g ( u ( N ) , w ( N ) ) (B.10) K(2,2) = g(u(N)+hK(1,1)/2,w(N)+hK(2,1)/2) (B.11) K(2,3) = g(u(N)+hK(1,2)/2,w(N)+hK(2,2)/2) (B.12) K(2,4) = g(u(N)+hK(1,3).,w(N)+hK(2,3)) (B.13) The approximation formulas are U(N+1) = u ( N ) + h(K(1,1)+2K((1,2)+2K(1,3)+K(1,4))/6 (B.14) W(N+1) = w(N) + h(K(2,1)+2K((2,2)+2K(2,3)+K(2,4))/6 (B.15) T h i s a l g o r i t h m was t e s t e d by comparison to the a n a l y t i c a l s o l u t i o n of dx/dt = -x - y ( t - 1 ) + 10 (B.16) dy/dt = x (B.17) with i n i t i a l c o n d i t i o n s x(M) = cos(M) (B.18) dx(M)/dt = sin(M), -1 < M < 0, (B.19) suggested by Banks and Kappel (1979). With h=0.00l, the maximum e r r o r i n the range [0.25,2] of t was 10"°. The e r r o r s are w i t h i n the range necessary f o r our a n a l y s i s . 

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