Chapter 9: Competition From: Gause 1934 Competitive exclusion and - PowerPoint PPT Presentation

Chapter 9: Competition From: Gause 1934

Competitive exclusion and co-existence Asterionella formosa Synedra ulna Together

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Competitive exclusion: several consumers using 1 resource Closed system with fixed amount of resource K : n d N i R 0 i = b i K X F = K − e i N i , d t = N i ( b i F − d i ) , for i = 1 , 2 , . . . , n , d i i Since for each species ¯ F = d i /b i = K/R 0 i they have to exclude each other d 1 b i d 1 b i > b 1 b i ¯ F − d i > 0 or b i − d i > 0 or > 1 or , (9.3) b 1 d i b 1 d i d 1

Competitive exclusion: several consumers using 1 resource Closed system with fixed amount of resource K : n d N i X F = K − e i N i , d t = N i ( b i F − d i ) , for i = 1 , 2 , . . . , n , i Carrying capacity of one species: N i = K − d i /b i = K (1 − 1 /R 0 i ) K i = ¯ F = K − e i ¯ ¯ with N i = e i e i

Nullclines for 2-D closed system n d N i X F = K − e i N i , d t = N i ( b i F − d i ) , for i = 1 , 2 , . . . , n , (9.1) i F = K - e 1 N 1 - e 2 N 2 N 2 = K − d 1 /b 1 − e 1 N 1 = K (1 − 1 /R 0 1 ) − e 1 N 2 = K (1 − 1 /R 0 2 ) − e 1 N 1 and N 1 , (9.4) e 2 e 2 e 2 e 2 e 2 e 2

Nullclines for 2-D closed system n d N i X F = K − e i N i , d t = N i ( b i F − d i ) , for i = 1 , 2 , . . . , n , (9.1) i F = K - e 1 N 1 - e 2 N 2 N 2 = K − d 1 /b 1 − e 1 N 1 = K (1 − 1 /R 0 1 ) − e 1 N 2 = K (1 − 1 /R 0 2 ) − e 1 N 1 and N 1 , (9.4) e 2 e 2 e 2 e 2 e 2 e 2 N1 N2 ● ● ● K 2 Density N 2 ● ● ● ● ● ● K 1 N 1 Time

<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Competitive exclusion when birth rate is saturated (closed) ✓ b i F n ◆ d N i X F = K − e i N i , d t = N i h i + F − d i i Carrying capacity of one species, and the corresponding steady state for F : N i = K ( R 0 i − 1) − h i h i ¯ ¯ with F = e i ( R 0 i − 1) R 0 i − 1 Thus the consumer with the lowest h i over R 0 -1 ratio depletes the resource most. _ At the lowest F the other species cannot invade: b j ¯ F h j ¯ F > d j or F > h j + ¯ R 0 j − 1

Competition in open systems (one resource) ✓ b i c i R n ◆ d R d N i X = s − dR − R c i N i with d t = N i h i + c i R − d i or d t i =1 ✓ b i R n ◆ d R c i N i d N i X = s − dR − R with d t = N i h i + R − d i or d t h i + R i =1 ✓ b i c i R n ◆ d R d N i X = rR (1 − R/K ) − R c i N i with d t = N i h i + c i R − d i or d t i =1 ✓ b i R n ◆ d R c i N i d N i X = rR (1 − R/K ) − R with d t = N i h i + R − d i , d t h i + R i =1 Exclusion because h i /c i h i R 0 i = b i R ⇤ R ⇤ i = or i = R 0 i − 1 , where , R 0 i − 1 d i

(c) (d) ● ● ● ● ● N2 N2 ● ● ● ● ● ● N1 N1 (0,0,0) (0,0,0) % % % R ∗ R ∗ % R R 1 1 — R R ∗ R ∗ 2 2 — N 1 ● ● ● — N 2

Quasi steady state to reveal interactions: resource with source ✓ b i c i R n ◆ d R d N i X = s − dR − R c i N i with d t = N i h i + c i R − d i d t i =1 n ✓ ◆ s ˆ d + P c i N i R = , d N i b i s β i ⇣ ⌘ ⇣ ⌘ s + ( h i /c i )( d + P c j N j ) � d i 1 + P N j /k j d t = N i = N i � d i K i = s � d s � d ⇣ ⌘ R 0 i � 1 = h i c i c i R ∗ c i i

6 ✓ ◆ X Quasi steady state to reveal interactions: logistic resource ✓ b i c i R n ◆ d R d N i X = rR (1 − R/K ) − R c i N i with d t = N i h i + c i R − d i d t i =1 ✓ ◆ ✓ ◆ 1 � 1 ˆ X R = K c i N i r b i ( r � P c j N j ) d N i ⇣ ⌘ ( h i /c i )( r/K ) + r � P c j N j d t = N i � d i N i = r 1 � R ∗ ⇣ ⌘ ¯ i c i K

(a) (b) Lotka-Volterra competition model 1 A 12 ● ● ● ● ● ● 1 1 N 2 ● ● ● ● ● ● n 1 d N i ⇣ ⌘ A 12 X d t = r i N i 1 � A ij N j ● ● ● ● ● ● j =1 1 1 1 1 A 21 A 21 (c) (d) 1 − A 11 1 N 2 = N 1 = (1 − N 1 ) 1 A 12 A 12 A 12 A 12 ● ● ● ● ● ● 1 − A 21 1 1 N 2 N 2 = N 1 = (1 − A 21 N 1 ) A 22 A 22 ● ● ● ● ● ● 1 A 12 ● ● ● ● ● ● 1 1 1 1 A 21 A 21 N 1 N 1

Several consumers on two substitutable resources P j c ij R j d N i d R j ⇣ ⌘ X d t = N i , d t = s j � d j R j � c ij N i R j β i � δ i h i + P j c ij R j i Consumer nullcline depends on resources only: c i 2 ( R 0 i � 1) � c i 1 h i R 2 = R 1 Straight line with slope - c i1 / c i2 c i 2 where R 0i = β i / δ i 0 i i i h i , R ∗ ij = Starting and ending at critical resource density: c ij ( R 0 i − 1) = R ∗ � c i − R i 2 � c i 1 R 2 = R ∗ c i 2 R 1 Simplified nullcline:

Several consumers with same diet c i1 and c i2 . (a) (b) R ∗ 31 N1 N2 N3 R ∗ Tilman diagram QSSA 22 R ∗ N3 12 - c i1 / c i2 ● ● ● R 2 ● ● ● (0,0,0) N2 — N 1 0 N1 — N 2 0 R ∗ R ∗ R ∗ — N 3 11 21 31 ● ● ● R 1 h 1 < h 2 < h 3

(a) (b) Several consumers having N1 N1 N2 N2 N3 different diets c i1 and c i2 . R ∗ K 2 Tilman diagram QSSA 32 N 2 R ∗ 22 ● ● Generically only one intersection R 2 ● point between all nullclines: K 1 R ∗ R ∗ R ∗ 11 31 21 maximally two co-existing R 1 N 1 species on two resources. (c) (d) N1 N2 N3 N3 Lowest intersection not invadable QSSA QSSA K 3 K 3 by other consumers N 3 N 3 (but no guarantee that this is a steady state). ● K 1 K 2 N 1 N 2

Essential resources Several consumers: d N i c ij R j d R j ⇣ ⌘ Y X d t = N i , d t = s j − d j R j − c ij N i R j β i − δ i h ij + c ij R j j i Two consumers using two resources: d N 1 c 11 R 1 c 12 R 2 ⇣ ⌘ = N 1 β 1 − δ 1 d t h 11 + c 11 R 1 h 12 + c 12 R 2 d N 2 c 21 R 1 c 22 R 2 ⇣ ⌘ = N 2 β 2 − δ 2 d t h 21 + c 21 R 1 h 22 + c 22 R 2

Essential resources (a) (b) N1 N1 N2 N2 N3 N3 s 2 s 2 ● ● d 2 d 2 d N 1 c 11 R 1 c 12 R 2 ⇣ ⌘ = N 1 β 1 − δ 1 R 2 R 2 d t h 11 + c 11 R 1 h 12 + c 12 R 2 d N 2 c 21 R 1 c 22 R 2 ⇣ ⌘ = N 2 β 2 − δ 2 d t h 21 + c 21 R 1 h 22 + c 22 R 2 ● ● R ∗ R ∗ 32 32 R ∗ R ∗ 22 12 R ∗ R ∗ 12 22 Asymptotes defined by letting s 1 s 1 R ∗ ↑ R ∗ R ∗ ↑ R ∗ 21 31 11 31 d 1 d 1 R ∗ R ∗ R 1 R 1 11 21 or R 1 ! 1 or R 2 ! 1 , i (c) (d) N1 N1 N2 N2 QSSA of (a) QSSA of (b) K 2 K 2 t c 11 > c 12 , c 22 > c 21 and c 31 ' c 32 , N 2 N 2 r two on resource two, and consumer th ● ● Local steepness defines stability K 1 K 1 N 1 N 1

4-dimensional Jacobian d R 1 = s 1 − d 1 R 1 − c 11 N 1 R 1 − c 21 N 2 R 1 , d t d R 2 = s 2 − d 2 R 2 − c 12 N 1 R 2 − c 22 N 2 R 2 , d t et c 11 > c 12 and c 22 > c 21 , d N 1 c 11 R 1 + c 12 R 2 ⇣ ⌘ = N 1 , β 1 − δ 1 d t h 1 + c 11 R 1 + c 12 R 2 d N 2 c 21 R 1 + c 22 R 2 ⇣ ⌘ = N 2 , β 2 − δ 2 d t h 2 + c 21 R 1 + c 22 R 2   � d 1 � c 11 ¯ N 1 � c 21 ¯ � c 11 ¯ � c 21 ¯   N 2 0 R 1 R 1 ∂ R 1 R 0 ∂ N 2 R 0 . . . 1 1 � d 2 � c 12 ¯ N 1 � c 22 ¯ � c 12 ¯ � c 22 ¯ . ... 0 N 2 R 2 R 2 .   J =  =   .    Φ 1 c 11 Φ 1 c 12 0 0   ∂ R 1 N 0 ∂ N 2 N 0 . . . 2 2 Φ 2 c 21 Φ 2 c 22 0 0 where β 1 h 1 ¯ β 2 h 2 ¯ N 1 N 2 Φ 1 = and Φ 2 = ( h 1 + c 11 ¯ R 1 + c 12 ¯ ( h 2 + c 21 ¯ R 1 + c 22 ¯ R 2 ) 2 R 2 ) 2   0 − ρ 1 − γ 11 − γ 21 λ 4 + a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 0 − ρ 2 − γ 12 − γ 22   J =   0 0 φ 11 φ 12   0 0 φ 21 φ 22 a 0 = ( γ 11 γ 22 − γ 12 γ 21 )( φ 11 φ 22 − φ 12 φ 21 )

4-dimensional Jacobian: essential resources ¯ ¯ R 2 R 1 ! d N 1 c 11 R 1 c 12 R 2 Φ 1 Φ 1 ✓ ∂ R 1 N 0 ◆ ✓ φ 11 ◆ ⇣ ⌘ ∂ R 2 N 0 φ 12 = N 1 β 1 � δ 1 1+ ¯ 1+ ¯ R 1 /H 11 R 2 /H 12 1 1 d t h 11 + c 11 R 1 h 12 + c 12 R 2 = = ¯ ¯ R 2 R 1 ∂ R 1 N 0 ∂ R 2 N 0 φ 21 φ 22 d N 2 c 21 R 1 c 22 R 2 Φ 2 Φ 2 ⇣ ⌘ 2 2 = N 2 β 2 � δ 2 1+ ¯ 1+ ¯ R 1 /H 21 R 2 /H 22 d t h 21 + c 21 R 1 h 22 + c 22 R 2 where H ij = h ij /c ij and β 1 ¯ β 2 ¯ N 1 N 2 Φ 1 = and Φ 2 = R 2 ) . ( H 11 + ¯ R 1 )( H 12 + ¯ ( H 21 + ¯ R 1 )( H 22 + ¯ R 2 )   0 − ρ 1 − γ 11 − γ 21 λ 4 + a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 0 − ρ 2 − γ 12 − γ 22   J =   0 0 φ 11 φ 12   a 0 = ( γ 11 γ 22 − γ 12 γ 21 )( φ 11 φ 22 − φ 12 φ 21 ) 0 0 φ 21 φ 22 et c 11 > c 12 and c 22 > c 21 , φ 11 φ 22 − φ 12 φ 21 Unknown sign: If negative, steady state will be unstable.