The Ideal Free Distribution: from hypotheses to tests

Transcription

1 The Ideal Free Distribution: from hypotheses to tests Vlastimil Krivan Biology Center and Faculty of Science Ceske Budejovice Czech Republic

2 Talk outline 1. Distribution of a single species among two habitats: Swans and Fish 2. Distribution of a two fish species: A test with fish 3. Population dynamics and distribution of a single population: A test with bacteria growing on two sugars 2

3 1. How does a single population of a fixed size distribute in a heterogeneous space? 3

4 Swan distribution Distribution of 8 swans among two feeding patches % of swan in patch æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ Fish distribution (Milinski, 1979) Time [min] 4

5 The Parker matching principle (Parker 1978) r 1 r 2 m 1 m 2 m i = abundance in the i th patch M = m 1 + m 2 is the total abundance r i = resource input rate in patch i V i = resource input rate animal abundance in the patch = r i m i The population distribution: p i = m i M = r i r 1 +r 2 5

6 The Ideal Free Distribution Definition (Fretwell and Lucas 1969). Population distribution p =(p 1,...,p n ) is called the Ideal Free Distribution if payoffs in the occupied habitats are the same and maximal. V 1 (p 1 )= = V k (p k )=:V? V j (0) for j = k +1,,n. Dual meaning of p =(p 1,...,p n ), p p n =1: 1. Population distribution 2. For a monomorphic population it is a strategy of an individual (p i is the proportion of the lifetime an individual spends in patch i) 6

7 The IFD as a game theoretical concept: The Habitat Selection Game (Cressman and Krivan, 2006) V 1 (p 1 )= = V k (p k )=:V? V j (0) for j = k +1,,n. Proposition. Let payoffs be negatively density dependent. Then the strategy corresponding to the Ideal Free Distribution is the Nash equilibrium of the Habitat selection game. Proposition. Let payoffs be negatively density dependent. Then the strategy correspondingtotheifdisanessofthehabitatselectiongame. 7

8 2. How do two interacting populations of a fixed size distribute in a heterogeneous space consisting of two patches? Gerbillus pyramidum Northern Collared Lemming American Brown Lemming Gerbillus allenbyi Tests with rodents (Rosenzweig 1979,, Morris 1987, ) 8

9 Isolegs (Rosenzweig 1979) and Isodars (Morris 1987) An isoleg is a curve in the species 1-species 2 density phase space that separates regions with qualitatively different species distributions (e.g., species 1 occupies habitat A only, species 2 occupies both habitats and similarly for species B) 9

10 The IFD for two competing species (Krivan and Sirot 2002) m 1 n 1 Fast dispersal m 2 n 2 M = m 1 + m 2 N = n 1 + n 2 R 1 R 2 Patch 1 Patch 2 m i = p i M n i = q i N pecies 1 payoff in habitat i : V i (p, q; M,N) = r i ³ 1 p i M K i pecies 2 payoff in habitat j: W j (p, q; M,N) = s j ³ 1 q j N L j α i q i N K i β j p j M L j i =1, 2 j =1, 2. 10

11 Equal fitness lines for two competing Lotka-Volterra model Equal payoff lines at fixed population abundances: V 1 (p, q; M,N) =V 2 (p, q; M,N) W 1 (p, q; M,N) =W 2 (p, q; M,N) Solid line Dotted line 11

12 TwospeciesESS(Cressman1992) Equal payoff lines at fixed population abundances: V 1 (p, q; M,N) =V 2 (p, q; M,N) W 1 (p, q; M,N) =W 2 (p, q; M,N) Solid line Dotted line Not a 2-species ESS 2-species ESS 2-species ESS 12

13 Condition for the 2-species ESS for the Lotka-Volterra competition model (Krivan and Sirot, 2002; Cressman et al. 2004) ³ Species 1 payoff in habitat i : V i (p, q; M,N) = r i 1 p i M K i ³ Species 2 payoff in habitat j: W j (p, q; M,N) = s j 1 q j N L j α i q i N K i β j p j M L j i =1, 2 j =1, 2. Proposition. Let us assume that the interior Nash equilibrium for the distribution of two competing species at population densities M and N exists. If r 1 s 1 K 2 L 2 (1 α 1 β 1 )+r 1 s 2 K 2 L 1 (1 α 1 β 2 )+r 2 s 1 K 1 L 2 (1 α 2 β 1 )+r 2 s 2 K 1 L 1 (1 α 2 β 2 ) > 0 Then this distributional equilibrium is a 2-species ESS. 13

14 The two species IFD (Krivan and Sirot, 2002) σ = the relative strength of intraspecific competition to interspecific competition σ=0: interspecific competition only; σ=1: intraspecific competition only V i (p, q; M,N) = r i ³ 1 p iσm K i W j (p, q; M,N) = s j ³ 1 q jσn L j α iq i (1 σ)n K i β jp j (1 σ)m L j i =1, 2 j =1, 2. 14

15 σ = 1 (Intraspec.comp.only) N p 1 =1 0 <q 1 < 1 0 <p 1 < 1 0 <q 1 < 1 N p 1 =1 0 <q 1 < 1 0 <p 1 < 1 0 <q 1 < 1 p 1 =1 q 1 =1 0 <p 1 < 1 q 1 =1 p 1 =1 q 1 =1 0 <p 1 < 1 q 1 =1 M M p 1 =1 0 <q 1 < 1 p 1 =1 q 1 =0 p 1 =1 0 <q 1 < 1 p 1 =1 q 1 =0 N p 1 =0 0 <q 1 < 1 p 1 =0 q 1 =1 0 <p 1 < 1 0 <q 1 < 1 0 <p 1 < 1 q 1 =0 N p 1 =0 0 <q 1 < 1 p 1 =0 q 1 =1 Two alternative distributions for any population densities p 1 =1 q 1 =1 0 <p 1 < 1 q 1 =1 p 1 =1 q 1 =1 0 <p 1 < 1 q 1 =1 0 <p 1 < 1 q 1 =0 M M M 15

16 Joint distribution of two fish species (Berec et al. 2006) r 1 r 2 Species M: Minnow (Tanichthys albonubes ) Species D: Danio(Danio aequipinnatus) 16

17 M in n o w fi t n es s : V = λ M p1 R 1 + λ M p 2 R 2 D a n i o fi tn e s s: W = λ D q 1 R 1 + λ D q2 R 2 Joint distribution of two fish species (Berec et al. 2006) Species M: Minnow Species D: Danio R i r i (i =1, 2) : standing food density at patch i (i =1, 2) : rate of feeding in patch i dr 1 dt dr 2 dt = r 1 (λ M p 1 R 1 M + λ D R 1 q 1 D) = r 2 (λ M p 2 R 2 M + λ D R 2 q 2 D) Equal fitness lines: λ M R 1 = λ M R 2 λ D R 1 = λ D R 2 R 1 = R 2 17

18 The IFD 2-species distribution At the resource equilibrium the fish distribution satisfies: r 1 λ M p 1 M + λ D q 1 D = r 2 λ M p 2 M + λ D q 2 D And the corresponding distribution satisfies: (p 1 r 1 r 1 + r 2 )λ M M +(q 1 r 1 r 1 + r 2 )λ D D =0 q 1 p 1 18

19 Distribution of two fish species (Berec et al. 2006) 19

20 M in n o w fi t n es s : V = λ M p1 R 1 + λ M p 2 R 2 D a n i o fi tn e s s: W = λ D q 1 R 1 + λ D q2 R 2 Joint distribution of two fish species (Berec et al. 2006) Observation: 1. Minnows are stronger competitors than Danios, because they move faster (λ M > λ D ). Thus, Minnows is competitively dominant species 2. Minnows quickly distribute following their own single-species IFD, i.e., p i = r i r 1 +r 2, i =1, 2 3. This balances the resources at both patches, i. e., R 1 = R 2 4. Danios distribute 50:50 because both patches are equally profitable for them 20

21 3. How does a single population distribute in a heterogeneous space when it undergoes demographic changes? 21

22 Bacterialgrowthontwosubstrates Diauxie (J. Monod): microbial cells consume two or more substrates in a sequential pattern, resulting in two separate growth phases (phase I and II). During the first phase, cells preferentially metabolize the sugar on which it can grow faster (often glucose). Only after the first sugar has been exhausted do the cells switch to the second. At the time of the diauxic shift, there is often a lag period during which cells produce the enzymes needed to metabolize the second sugar. Lac operon: Molecular mechanism that regulates diauxic growth (F. Jacob and J. Monod, Nobel prize 1965) Adaptation: Evolution should result in optimal timing of the diauxic switch so that bacterial fitness maximizes Question: Is the lac operon evolutionarily optimized? 22

23 Michaelis-Menten batch population kinetics C u 1 u 2 S 1 S 2 ds 1 dt ds 2 dt dc dt = 1 Y 1 S 1 K 1 + S 1 u 1 C = 1 S 2 u 2 C Y 2 K 2 + S µ 2 μ1 S 1 = u 1 + μ 2S 2 u 2 C K 1 + S 1 K 2 + S 2 S 1, S 2 - sugar concentration (eg. glucose and lactose) C - bacterial population u i -bacterial preference for the i th (u 1 + u 2 =1,i =1, 2) sugar 23

24 Optimal bacterial strategy Fitness= per capita bacterial population growth rate i.e., 1 C dc dt = μ 1S 1 u 1 + μ 2S 2 u 2 7 max K 1 + S 1 K 2 + S 2 u i The optimal strategy: μ 1 S 1 K 1 + S 1 > μ 2S 2 K 2 + S 2 = u 1 =1 μ 1 S 1 K 1 + S 1 < μ 2S 2 K 2 + S 2 = u 1 =0 24

25 Michaelis-Menten batch population kinetics with optimal sugar switching ds 1 dt ds 2 dt dc dt = 1 Y 1 S 1 K 1 + S 1 u 1 C = 1 S 2 u 2 C Y 2 K 2 + S µ 2 μ1 S 1 = u 1 + μ 2S 2 u 2 C K 1 + S 1 K 2 + S 2 u 1 = ( 1 when μ 1 S 1 K 1 +S 1 > μ 2S 2 K 2 +S 2 0 when μ 1 S 1 K 1 +S 1 < μ 2S 2 K 2 +S 2 μ 1 S 1 K 1 +S 1 = μ 2S 2 K 2 +S 2 25

26 Growth rate parameters of Klebsiela oxytoca on single substrate (Kompala et. al. 1986) glucose μ K YY arabinose fruktose xylose lactose

27 Predicted and observed switching times Conclusion: There is no significant difference between observed times of switching and predicted times of switching. Thus, bacteria switch between different sugars at times at which their fitness is maximized. This shows that the lac operon is evolutionarily optimized. 27

28 Bacteria Sugar 1 Sugar 2 28

29 References Abrams, P., Cressman, R., Krivan, V The role of behavioral dynamics in determining the patch distributions of interacting species. American Naturalist 169: Cressman, R., Krivan, V Migration dynamics for the Ideal Free Distribution. American Naturalist 168: Cressman, R., Krivan, V Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds. Journal of Mathematical Biology, DOI /s Krivan, V Dynamic ideal free distribution: effects of optimal patch choice on predator-prey dynamics. American Naturalist 149: Krivan, V., Sirot, E Habitat selection by two competing species in a two-habitat environment. American Naturalist 160: Krivan, V Dispersal dynamics: Distribution of lady beetles. European Journal of Entomology 105: Krivan, V., Sirot, E Habitat selection by two competing species in a two-habitat environment. American Naturalist 160: Krivan, V., Cressman, R., Schneider, C The Ideal Free Distribution: A review and synthesis of the game theoretic perspective. Theoretical Population Biology 73: More at 29