Qualitative modelling in ecology

Examples o Applied Quali a i e Techniques. Chap e 6 Quali a i e modelling in ecology Miguel To o 1 In oduc ion In his example a non-linea sys em is examined. I can be used as a model o he g ow h o wo species. The model ies o cap u e, in he simples way, dependencies o g ow h a e on ood supply and nega i e e ec s o o e c owding. In he same way, dependencies o he g ow h a e o p eda o s on he quan i y o p eys a e cap u ed. Ins ead o explici o mulas o he equa ions, ce ain quali a i e assump ions a e made ega �ing he o m o he equa ions. These assump ions show he quali a i e k.nowledge abou he sys em ha should be aken. Quali a i e analysis and bi u ca ion heo y can be applied o ob ain he long- un quali a i e beha iou o he ajec o ies o he sys em. 2 Speci ying a model The sys em o model, is composed by p eys and p eda o s. The p eys depend on he en i onmen and p eda o s depend on he p eys. E e y popula ion has a bi h a e and a dea h a e. The g ow h a e is de ined as he bi h a e minus he dea h a e. In he ollowing x s ands o l? ey and y s ands o p eda o s. The k.nowledge abou he a iables in he model and ela ions be ween hem a e gi en by he ollowing assump ions: The g ow h a e o p eys depends only on he popula ion o p eys. Le l(x) be he g ow h a e o he p ey. I is plausible o assume ha he e is a limi ing popula ion x 1 such ha l(xl)=O, l(x)<O, x>xl. The g ow h a e is assumed o be posi i e o small popula ions l(O)=a>O. Take in o accoun posi i e social phenomena, o example, a medium size popula ion may be be e o ganised o ob ain ood han a small one. This implies ha he e is a alue xO o he popula ion whe e l has a local maximum. So see ha l(xO)=b, O<xO<xl, b>a. Le us s udy he consequences in he beha iou o he aspec o l. The p ope ies o l a e: { O < xO < x l,O < a < b l = l(O) = a, l(xl) = O l(xO) = b, l'(xO) = O The p ey popula ion is he o al ood supply o he p eda o . The g ow h a e o he p eda o s depends on he p ey popula ions. Le 2(x) be he g ow h a e o p eda o s. Le us assume ha he e is a alue x2 o x such as 2(x2) =O and ha 2 is a mono one inc easing unc ion. I is known ha o la ge sizes o p ey popula ion, he g ow h a e has a alue b. So 2( oo )=b. The p ope ies o 2 a e: Examples o App!ied Quali a i eTechniques Chap e 6 {x2 > O,b > O /2 = 2(x2) = O /2(00) = b Le us assume ha he numbe o p ey killed by each p eda o is a mono one inc easing unc ion o p ey numbe . Le D(x) be his numbe . Then D(O)=O, D(oo)=d and D(x) = e x o small and medium alues o x. { d > 0,e > O 3(x) = /3(0) = 0,/3(00) = d 3(x)-= ex a small x O he de ails ega ding ecological model can be ound in [ART93]. 3 Quali a i e model Ideas om [KUI86] a e he e used o ep esen he quali a i e k.nowledge o he sys ems. This k.nowledge is ep esen ed by a se o a iables, a se o impo an alues o hese a iables and se o unc ional cons ain s be ween hem. Mono one and nonmono one unc ional cons ain s a e .used, and he de i a i e ope a o and a i hme ic ope a o s do. A lis o couples o poin s o speci y he p ope ies o unc ional cons ain s is he e g1 en. The quali a i e model o he sys em is a>O b>a,c>O,d>O O< xO,xO < xl 1: (-00,-00 ), (0,a ), (xO, b ), ( xl,O), ( +00,-00) 2: (-00,-00 ), ( x2,0), ( +oo, e) 3: (-00,-00 ), (0,0), ( +oo,d) dx -=x l(x)- 3(x) y d dy -= 2(x)y dx 109 s • 5 ' s 8 I í o ' s ^ . o a I 8 B 2 S 3 O ^ 3 e n o ^ ^ o í o p ^ P B a - a B o n > 3 . c u o > i 5 2 g ^ B 2 ^ p ^ o e n P P a P l ^ u l i I i 3 ^ 2 - a ^ ^ < ^ g l ^^ c ^ ^ C / l ^ . ( T i • / * ^ B ^ 2 . í ? p . 5 e n P B e n o w o j A 3 e n O e n PP e n P e n 3 O B - I - + ; P B e n b ^ 2 O O 3 g : p b. 3 e n ' - O o u ^ ^ a 3 . w a - c c . e n 2 - B ^ p ^ . 2 a . B 3 - a ; p a 3 c a 3 3 o o O O 3 e n a O 3 B e n 3 c e e n 0 3 e n e n c ^ • o í e n i " 03 O e n e n p P O 1 - 1 e n o B e n e n e n X 3 X e n 03 O I p e n P o b ^ U - 3 S a ¡ I ^ p ^ 3 * e n B 3 g O > - " < o ^ o e n 3 - & g E S . o p o ^ o s ^ 3 2 ^ 2 . e n o í 2 . U a 3 ^ 1 1 / 1 a p S g 2 p B o < P . ^^ 2 ^ l C l ^ < o ; e n O ^ ó ; ^ P * a p o < - + o o e n c e e n " p ^ 3 3 e n o s S ^ g ^ B ^ í ^ - ^ h p x . ^ ^ 2 . ^ B m ^ ^ - ^ ^ x o x i . ^ ^ p ^ ^ ^ B ^ ^ p g ^ ^ x B % & ^ 3 . b ^ - g ^ a - p e n ^ . R ^ o e n p i ! 5 ' 2 i ! 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B í H a ^ ^ a - g - c b ^ p & S ^ i . e n B ^ < B n o 2 . 3 B B O P a 8 * 5 ' ^ ^ e n a - ^ ' C L e ^ B í * i g - ^ 3 3 3 e n i 0 0 e n ^ i - , C T " ? ^ B 3 . ^ e n e n 3 0 3 O O X II O J O o í o II 0 Q ^ < o x ü n n I x ( B C T n p ' o o B e n C u B p ' e i l i ? o > - o e n o i y 1 + 1 S o " H g * o ^ , H e n ' 3 Ü . x 3 ' e n o II o o 0 3 O ^ o T O 3 ^ p o e n ^ < P < 3 o o O V O o o p e n o o ' p ^ a o j O O S 5 ' e s e , ^ p & B ^ ^ 5 2 - - P K 3 p p P C Z ) P . O P s | o " a 3 o ^ ^ ^ a E L 2 a " 1 — e B I 1 I ' ^ i " ^ a - - B * K 8 a . o s a i x 5 3 a I e ^ P O B B - B . ^ P o p o p , p - . O P P O B ^ O P 3 g " B p P < T P . ^ . ^ P s ^ B ^ C ^ ^ p * c p l — x • 3 g O T p o B e n B S 3 B c ñ S < * - e n P ^ P ' ^ i e n o 3 I 1 a a o " e ? B " . 3 > ^ ^ a 0 3 1 — ' C s J O - S O S O K S O - J C i Q . Q J e / i P ^ p " e . a ^ ^ C J - ) e s . S ; ^ ' ^ . o q q a . P O C n O a P O B ^ ^ 3 P 5 5 " O Q ü . p ^ - o e 4 x ^ ^ e 3 ^ ^ 1 e p o - ^ s > í i . 5 i p - o ' > p d i p T w b a - . ; - O P T ^ ^ ^ 3 o c P - C 3 p ¡ ^ C U J — o ' — ' p _ — 1 q i . i " ! c / i s ^ q j S O T a - 3 - 5 ' ¡ I ^ S i . o s a o ^ - e n X x e • - < 1 x _ X X X X o c T I o q " i I e n i on a s i OQ OQ 3 n e i g h b o u m h e 0 p B 0 ' ? i 3 p 5 S o : 0 3 0 - P e n • < ; e n p 3 ^ e n O • - ) 3 . p s a j ee T h e p a e g i o 3 o i ee o i e c o s s - ^ . p F o c — * C 9 0 — 1 > a s i n p . a i n e c e n ' O g u e p i l B ^ | s h o ^ p o m a ( / > - i p o i n w i h u i o n V V 0 5 S J 9 J 1 m ' h e - B n g e n - P e n q ¡ a y p - p 0 p ' c < ; • > i - o o h e i i - 1 — * - p m u s b b 0 - - ; i n n e p o du V e n e m 0 b a i a l g e u. I n 0 ~ B . O P P B " - . ^ . p e a n g l a c u i 0 1 ? p ee n p o d u c b e w h i s < - + > g n o e n h e l u d e c o n c - O i b l e 0 e n E n " c e - * p c a s 0 p p i i e . ] p o s i e n : S O h i y a p h . T h e < : e c 0 d i é p 3 ' 0 c 0 p a c e e n p c c e p 5 " V i 0 o n s . m e g i e g i F o i e n e 1 i s i ^ - * a ss o c i a e d o p - i a . l i li b E T p ? p < c p 5 2 p ^ p . w Í 3 • e g i o p 1 — ^ ^ . i h a e i é 3 . g a j o - p B - o ^ ^ o P p P O s ^ e n p ^ j P P P ^ ^ - 3 . ^ p . o s y s e e n ^ ; i - P 3 P C T Q P . K >