d N d t = [ bf ( N ) dg ( N )] N Populations change by immigration, - PowerPoint PPT Presentation

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Chapter 3: Density dependence (a) (b) Percentage cells in division Percentage CD8 + naive T cells d N d t = [ bf ( N ) − dg ( N )] N Populations change by immigration, birth, and death processes, which could all depend on the density of the population itself

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Per capita birth or death rate (a) (b) d N d N d N d N d t > 0 d t < 0 d t > 0 d t < 0 b birth rate F(N) death rate death rate birth rate b d d steady state steady state Population density Population density Population density d N d t = [ b − d f ( N )] N F ( N ) = d + cN = d f ( N ) f ( N ) = 1 + N/k ↔ d N 1 + N h ⇣ ⌘i d t = b − d N k N = k b − d ¯ = k ( R 0 − 1) d

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Production or death rate d N d N d t > 0 d t < 0 production rate total death rate s steady state Population density d N 1 + N ⇣ ⌘ d t = s − d N k p dk ( dk + 4 s ) N = − dk ± ¯ 2 d

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Per capita birth or death rate (a) (b) d N d N d N d N d t > 0 d t < 0 d t > 0 d t < 0 b birth rate F(N) death rate death rate birth rate b d d steady state steady state Population density Population density Population density d N d t = [ bf ( N ) − d ] N F ( N ) = b − cN = bf ( N ) f ( N ) = 1 − N/k ↔ d N 1 − N h ⇣ ⌘ i d t = b − d N k 1 − d 1 − 1 ⇣ ⌘ ⇣ ⌘ ¯ N = k = k b R 0

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Production or death rate d N d N d t > 0 d t < 0 production rate s total death rate steady state Population density d N 1 − N ⇣ ⌘ d t = s − dN k sk ¯ N = dk + s

Logistic growth: d N KN (0) d t = rN (1 � N/K ) , with solution N ( t ) = N (0) + e − rt ( K � N (0)) (a) (b) (c) r r (1 � ( N/K ) m ) r (1 � N/K ) K m=2 N ( t ) m=0.5 0 0 0 0 0 0 K K Time N N

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> d N K Generalized logistic growth: d t = rN (1 − ( N/K ) m ) , with N ( t ) = [1 − (1 − [ K/N (0)] m )e − rmt ] 1 /m (a) (b) (c) r r (1 � ( N/K ) m ) r (1 � N/K ) K m=2 N ( t ) m=0.5 0 0 0 0 0 0 K K Time N N

Human logistic growth Human population in Monroe Country, West Virginia

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> (a) (b) d N 1 + N ⇣ ⌘ d t = s − d N k N ( t ) N ( t ) d N 1 − N ⇣ ⌘ d t = s − dN k 0 0 0 0 Time Time

Density dependent birth Salicornia Corn 4.0 per reproducing individual Average number of seeds 10,000 3.8 Clutch size 3.6 1,000 3.4 3.2 3.0 100 0 2.8 0 10 20 30 40 50 60 70 80 0 10 100 Grizzly bear Seeds planted per m 2 Density of females (a) Plantain (b) Song sparrow

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Non-linear density dependence f ( x ) = max(0 , 1 − [ x/k ] n ) f ( x ) = min(1 , [ x/k ] n ) x n f ( x ) = h n + x n 1 g ( x ) = 1 + ( x/h ) n g ( x ) = e − ln[2] x/h f ( x ) = 1 − e − ln[2] x/h

d N d t = ( bf ( N ) − d ) N , (a) (b) (c) b b b f ( N ) f ( N ) f ( N ) + + + d d d − − − 0 0 0 0 0 0 k k k N N N 1 1 f ( N ) = e − ln[2] N/k f ( N ) = 1 + N/k , f ( N ) = and 1 + [ N/k ] 2

1 1 d N f ( N ) = e − ln[2] N/k f ( N ) = 1 + N/k , f ( N ) = and d t = ( bf ( N ) − d ) N , 1 + [ N/k ] 2 (a) (b) (c) b b b f ( N ) f ( N ) f ( N ) + + + d d d − − − 0 0 0 0 0 0 k k k N N N Function f (0) f ( k ) f ( ∞ ) R 0 Carrying capacity Eq. ¯ f ( N ) = max(0 , 1 − [ N/k ] m ) p 1 0 0 b/d N = k m 1 − 1 /R 0 (3.12) ¯ f ( N ) = 1 / (1 + N/k ) 1 0.5 0 b/d N = k ( R 0 − 1) (3.14) N = k √ R 0 − 1 ¯ f ( N ) = 1 / (1 + [ N/k ] 2 ) 1 0.5 0 b/d (3.14) ¯ f ( N ) = e − ln[2] N/k 1 0.5 0 b/d N = ( k/ ln[2]) ln[ R 0 ] (3.14)

Density dependent death (a) (b) d + δ d [1 + ( N/k ) m ] - d + δ f ( N ) + - + d d 0 0 0 N N d N d N d t = ( b − d [1 + ( N/k ) m ]) N d t = [ b − d − δ f ( N )] N ,