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
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