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- The Logistic Continuous-time population dynamic mo d e l (Logistic)t a t a n e p i n e a d a s
# 'waternutrientsSpace
populations grow, pegr u n s o u t o f r e s o u r c e s
t o fuel i t s growth
i f birth rates dec l ine
- A s • r e s o u r c e s becomem o r e limitint' {dean
rates increase
di - d t c u
bd£¥\, = b - a n{{slopes-
sensitivity of
per-capita birth,death
rates o npopulation s i z e w
{b's b - A N Exponential population growth
d ' i d t c N
eff-IN So l : N a t s Noertuce,#-
(b-d)| substitute b → b'
d - d '
d¥= (b id) N = [(btu)-Cdtet)]NR I
= ((b-d ) - C a t c ) N] N Y¥d€i n
=({¥§µb-d)-(at c ) N) N
-
Cb-off:#-'II.io/n=cs.asfl-9¥,}w
↳
H A M B Y
1¥-Cb-d)fi- ftp.ajn]N÷.fi#..E......b'-dIzero
✓b - AN?
d#CN'
a , ,
NIV,µaze
b - d = A N ' t C N " {¥¥yB : d ' > b' s t a t e
(attractor)N ' s
bat¥
battle: K N A ¥N ' - K
"÷÷÷?L÷......⇒.E r
Logistic Equation
offer'll-¥)....-N-¥}←
.:¥÷÷÷÷N E K
~
when N E OE.I.it#.....'*" " ⇒
dat't date.irN(P-O)
=dN/dt<0 d¥o$rN
N E O
* r n
" "÷.FI#FIT=:...e,N l t ) (sigmoidal
N I K 7o n populationadf.ge#*gI
hi'
" ' " ' " b r a k e "
growth a s N T
Ferri'll-E)I w o r k
=rN(i-l) ¥ - Id¥I0 a-E) → 0
¥¥÷÷÷⇐i÷÷÷¥..*.:group
0 A : d " > b 'N ' E k
Extinctionµ...÷iµ÷÷÷÷E§÷
÷ii÷i÷:'inonsman
steady s ta te
← • o→A : d ' > b'
B : b ' >d , " §i
C : d ' 7b" • •
stable
¥¥µ¥§§←< Consider benefits o f groups
S TA B L EUNSTABLE