- 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

N ' t K

N'-No (carryingcapacity)

.in#E....E.....

.:.................Allee Effect: whenthere i s a crit ical minimum

i s inevitable