Based on «Modelling Population Dynamics» by André M. de Roos, University of Amsterdam, The Netherlands

Transcription

1 Based on «Modelling Population Dynamics» by André M. de Roos, University of Amsterdam, The Netherlands David Claessen CERES ERTI & Labo «Ecologie & Evolution» UMR 7625 CNRS UPMC ENS

2 Plan Lecture 1: Intraspecific competition & population regulation Logistic growth (no explicit resources) Explicit resource dynamics: 1 consumer and 1 resource Phase plane method Lecture 2: Interspecific competition Lotka Volterra competition (no explicit resources) Two species extension of logistic growth Tilman s model (explicit resources) Two consumers & two resources

3 Lotka Volterra compe11on model No explicit resources in the model Presence of competitor reduces net population growth Reduce reproduction Increase mortality Equivalent of logistic growth (but for 2 species) Parameters r i K i β ij

4 Phase plane method Isoclines for N1 and N2 Steady states = intersection of N1 and N2 isoclines Stability of equilibrium? Isoclines Solve dn1/dt = 0 Solve dn1/dt = 0

5 Case I

6 Add the arrows, and the steady-states Case I

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8 Equilibria:

9 Outcome of competition: For the special case K1=K2, coexistence requires: In words, coexistence is possible if interspecific competition < intraspecific competition

10 Explicit resources Consumer resource model Tilman (1980)

11 Func1onal response

12 Equilibrium Steady state resource concentration. Solve dn/dt = 0 Steady state consumer population Tilman (1980, 1981, 1982) The critical quantity for outcome of competition is not N* but R* Tilman s theory is called «R* theory»

13 Two consumers, one resource Extension of previous model to two consumers Critical resource concentration for species 1 and 2 R 1 * and R 2 * If R 1 *< R 2 * then species 2 will go extinct Species 1 can sustain a population at a resource level too low for species 2

14 Compe11ve exclusion Generalisation: multiple species: p consumers for the same resource

15 Two resources Extension of the same basic model Two essential resources! (versus substitutable) Liebig s law of the minimum

16 Zero net growth isoclines (ZNGI) dn1/dt=0 growth decline decline

17 Steady state of system Two methods: Solve equations (dr1/dt=0, dr2/dt=0, dn1/dt=0) Graphically: Supply vector Consumption vector

18 To find the consumption vector Q 1 : Consider the consumption rates for both resources = (second term in dr i /dt)

19 To find the supply vector S: Consider the supply rates for both resources = (first term in dr i /dt)

20 Steady state The direction of Q 1 is independent of R 1, R 2, and N 1 Steady state: Q 1 and supply vector must be in opposite directions

21 Interspecifc compe11on Tilman 1980

22 ZNGI for both species Coexistence possible only if ZNGI intersect Intersection = equilibrium And only if supply point in region III, IV, or V ZNGI species 1 ZNGI species 2

23 Supply point Supply point in region I: Both consumers extinct ZNGI species 1 ZNGI species 2

24 Supply point Supply point in region II: Consumer 1 persist Consumer 2 extinct ZNGI species 1 ZNGI species 2

25 Supply point in region VI: Consumer 1 extinct Consumer 2 persist ZNGI species 1 ZNGI species 2

26 Coexistence The combined consumption vector is a linear combination of Q1 and Q2 Hence only supply points in region IV can lead to stable coexistence ZNGI species 1 ZNGI species 2

27 Regions III and V These regions can support both species in isolation Region III: species 2 steady state is on vertical ZNGI Species 2 steady state ZNGI species 1 ZNGI species 2

28 Regions III and V These regions can support both species in isolation Region III: species 2 steady state is on vertical ZNGI Species 1 can invade, new steady state, species 2 extinct ZNGI species 1 ZNGI species 2

29 Stable coexistence in region IV?

30 Opposite relation of consumption vectors Same results for regions I, II, III, V, VI Region IV : competitive exclusion dependent on initial conditions compare LV competition Coexistence equilibrium exists but it is a saddle point

31 David Tilman: R* theory

32 Experimental tests Diatom phytoplankton Competing for two resources PO 4 (phosphate) SiO 2 (silicate) Essential resources Asterionella formosa vs Cyclotella meneghiniana

33 : Asterionella dominant : coexistence observed 3: Asterionella should dominate 4: coexistence predicted 5: Cyclotella should dominate : Cyclotella dominant

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35 Model calibra1on Asterionella formosa Fragilaria crotonensis Synedra filiformis Tabellaria flocculosa

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39 First observations & predictions: Asterionella formosa and Fragilaria crotonensis: (almost) identical R* values coexistence when competing with each other same predictions when competing with others Tabellaria flocculosa very high R* values: outcompeted by all other species Synedra filiformis has lowest R* for PO 4 : outcompete all other species in low PO 4 Sf has higher R* for SiO2 than Af and Fc Sf outcompeted at low SiO2 supply Coexistence of Sf AF ans As Fc possible

40 Compe11on between Asterionella formosa and Synedra filiformis Low SiO 2 Observed Model Both high Low PO 4 Tilman (1981)

41 Compe11on between Asterionella formosa and Tabellaria flocculosa Low SiO 2 Low SiO 2 «Both Coexistence high» Low PO 4

42 Asterionel la formosa Fragilaria crotonensis - R* for SiO2 = almost identical, - -R* for PO4 = only very small difference

43 Synedra filiformis Fragilaria crotonensis

44 Synedra filiformis Tabellaria flocculosa

45 Fragilaria crotonensis Tabellaria flocculosa

46 Predic1ons & observa1ons Asterionella formosa and Fragilaria crotonensis: (almost) identical R* values coexistence when competing with each other same predictions when competing with others Tabellaria flocculosa very high R* values: outcompeted by all other species Synedra filiformis has lowest R* for PO 4 : outcompete all other species in low PO 4 Sf has higher R* for SiO2 than Af and Fc Sf outcompeted at low SiO2 supply Coexistence of Sf AF ans As Fc possible

47 All this was for essential resources What about substitutable resources? Isoclines no longer perpendicular but smooth Four possible cases

48 Lotka-Volterra competition model (2 species, no explicit resource) Tilman s competition model (2 resources and & 2 consumers)