comes freely av R T = R + cN , w d N h + R N = b ( R T cN ) N d t - PowerPoint PPT Presentation

Chapter 5: Killing and Consumption comes freely av R T = R + cN , w d N h + R − δ N = b ( R T − cN ) N d t = bRN h ⇣ ⌘ h + R T − cN − δ N = bN 1 − − δ N , h + R T − cN

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Bacteria in a chemostat: birth rate proportional to consumption aR d R d t = s − wR − aRN d N d t = caRN − ( w + d ) N = caRN − δ N , N = sc δ − w g R = δ ¯ ca, a heir R 0 = ca ¯ R = cas δ w > δ aR − w s R = δ nullclines: R ’=0: N = and: N ’=0: N = 0 or a ca

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Bacteria in a chemostat: birth rate proportional to consumption aR (a) (b) (c) ¯ N Density N N ¯ N ● ¯ R ● ● s δ w ca R R Time d R d N d t = s − wR − aRN d t = caRN − ( w + d ) N = caRN − δ N , aR − w s R = δ N = nullclines N = 0 or a ca

(a) d R d t = s − wR − aRN d N d t = caRN − ( w + d ) N = caRN − δ N , N ¯ N ● ● s δ w ca R ✓ � w � a ¯ � a ¯ ✓ � w � a ¯ ✓ ∂ R f ◆ ◆ ◆ ✓ � α ◆ ∂ N f N R N � δ /c � β J = = = = ca ¯ ca ¯ ca ¯ ∂ R g ∂ N g N R � δ N 0 + γ 0 | ( ¯ R, ¯ N ) ( x, tr = � α < 0, , det = 0 � � βγ = βγ > 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> Saturated consumption in a chemostat, birth rate proportional to consumption d R d t = s − wR − aRN d N d t = caRN h + R − ( w + d ) N = caRN and h + R − δ N . h + R ivial N = h cas e R 0 = ca an R 0 = δ . T or δ ( wh + s ) e fact that the h δ h d N ¯ R = ca − δ = d t = 0 gives R 0 − 1

<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> d R d t = s − wR − aRN d N d t = caRN h + R − ( w + d ) N = caRN and h + R − δ N . h + R (a) (b) (c) R N ¯ N Density N N − w ¯ N a R ● ¯ R ● ● s h w R 0 − 1 R R Time ⇣ s s � wR = aRN h + R $ N = ( h + R )( s � wR ) = h + R d R ⌘ R � w d t = 0 gives aR a

<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 R d t = s − wR − aRN d N d t = caRN h + R − ( w + d ) N = caRN and h + R − δ N . h + R (a) (b) (c) R N ¯ N Density N N ¯ N ● ¯ R ● ● s h w R 0 − 1 R R Time ✓ ◆ ✓ − α ◆ ∂ R f ∂ N f − β J = = ∂ R g ∂ N g + γ 0 | ( ¯ R, ¯ 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> <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> Replicating resource: Lotka-Volterra model, birth rate proportional to consumption d R d N d t = rR (1 � R/K ) � aRN , d t = caRN � δ N . R = δ d N ¯ gives d t = 0 ca d R N = r a (1 − R K ) gives d t = 0 axis, the nullc ⇣ δ ca, r δ h i⌘ ( ¯ R, ¯ N ) = (0 , 0) , ( K, 0) and 1 − Steady states: a caK

d R d N d t = rR (1 � R/K ) � aRN , d t = caRN � δ N . (a) (b) (c) r a ¯ Density N ¯ N ● N = r a (1 − R N N K ) ¯ axis, the nullc R ● ● ● K δ ca R R Time

d R d N d t = rR (1 � R/K ) � aRN , d t = caRN � δ N . (a) (b) (c) r a ¯ Density N ¯ N ● N N ¯ R ● ● ● K δ ca R R Time K ¯ R − a ¯ − a ¯ ✓ r − 2 r − r δ ✓ ∂ R f ◆ ◆ ✓ ◆ ✓ − α ◆ ∂ N f N R − δ /c − β J = = = caK = ca ¯ ca ¯ ca ¯ ∂ R g ∂ N g N R − δ + γ 0 N 0 | ( ¯ R, ¯ 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> <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> Generalized Lotka-Volterra model, birth rate proportional to consumption (a) (b) r r a a N N ● ● K K δ δ ca ca R R N = r d R d N a (1 − ( R/K ) m ) d R d t = rR (1 − ( R/K ) m ) − aRN , d t = caRN − δ N , d t = 0 gives d R d R d t = [ f ( R ) − aN ] R N = f ( R ) /c gives d t = 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> d R d N d t = rR (1 − ( R/K ) m ) − aRN , d t = caRN − δ N , (a) (b) r r a a N N ● ● K K δ δ ca ca R R ✓ ◆ ✓ − α ◆ ∂ R f ∂ N f − β J = = ∂ R g ∂ N g + γ 0 | ( ¯ R, ¯ 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> <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 R d N d t = rR (1 − ( R/K ) m ) − aRN , d t = caRN − δ N , (a) (b) r r a a ✓ ◆ − α − β N N J = + γ 0 ● ● h K K δ δ h R 0 − 1 ca ca R 0 − 1 R R  β R � d N aR R d t = N = [ β f ( R ) − δ ] N h + R − δ g ( aR ) = � H + aR = � gives h + R