1. Other patterns in communities Macroecology: relationships of –geographic distribution and body size –species number and body size Latitudinal gradients: changes in S with latitude Species-Area relations: Island biogeography and related questions

2. Species-area relationships Islands, either oceanic or habitat Selected areas within continents How is number of species related to area? S A

3. Mathematics S = c A z –S is number of species –A is area sampled –c is a constant depending on the taxa & units of area –z is a dimensionless constant often 0.05 to 0.37

4. Often linearized ln (S ) and ln (A ) ln (S ) = ln (c ) + z ln (A ) –z is now the slope –ln (c ) is now the intercept ln (S ) ln (A )

5. Theory & Hypotheses Area per se hypothesis –why S goes up with A –why S = c A z –why z takes on certain values Habitat heterogeneity hypothesis –why S goes up with A Passive sampling hypothesis –why S goes up with A

6. Area per se large heterogeneous assemblage  log normal distribution of species abundances assume log normal ("canonical log normal") –Abundance class for most abundant species = abundance class with most individuals –constrains variance (s 2 ) of the distribution assume that N increases linearly with A Yield: unique relationship: S = c A z for "canonical" with S > 20: S = c A 0.25 nini SnSn

7. Area per se z varies systematically –larger for real islands vs. pieces of contiguous area z does not take on any conceivable value –if log normal had s 2 = 0.25 (very low) –then z  0.9 … which is virtually unknown in nature –implies constraints on log normal distributions

8. Dynamics of the area per se hypothesis open island of a given area rate of immigration (sp. / time) = I initially high once a species is added, I declines nonlinear: –1st immigrants best dispersers –last are poorest dispersers S I STST

9. Dynamics of the area per se hypothesis rate of extinction (sp. / time) = E initially 0 as species are added, E increases nonlinear: –lower n as S increases –more competition as S increases S E STST

10. Dynamic equilibrium equilibrium when E = I determines S * how are rates related to area? S RATE S*S* E I

11. Effect of area on S * 2 islands equally far from mainland large & small extinction rate greater on small –smaller n’s –greater competition under this hypothesis I is not related to area S RATE S * large E large I E small S * small

12. Area per se Neutral hypotheses vs. Niche hypotheses Neutral hypotheses – presume that biological and ecological differences between species, though present, are not critical determinants of diversity Area per se is a neutral hypothesis –S depends only on the equilibrium between species arrival and extinction –Large A  large populations  low prob. extinction

13. Niche-based hypotheses Niche hypotheses - presume that that biological and ecological differences between species are the primary determinants of diversity Niche differences enable species to coexist stably Does not require equilibrium between extinction and arrival

14. Habitat heterogeneity Niche-based hypothesis Larger islands  more habitats –Why? More habitats  more species –does not require competition –does not require equilibrium –does not exclude competition or equilibrium

15. Passive sampling Larger islands  bigger “target” Neutral hypothesis More immigration  more species –competition & equilibrium not necessary (but possible) –under this hypothesis E is not related to area S RATE S * small E I large S * large I small

16. Processes Interspecific competition

17. Competition Competition occurs when: –a number of organisms use and deplete shared resources that are in short supply –when organisms harm each other directly, regardless of resources –interspecific, intraspecific

18. Resource competition competitor #1 competitor #2 - - competitor #1 competitor #2 resource - - + + Interference competition

19. Competition Interference –Direct attack –Murder –Toxic chemicals –Excretion Resource –Food, Nutrients –Light –Space –Water Depletable, beneficial, & necessary

20. Competition & population Exponential growth dN / dt = r N –r = exponential growth rate –unlimited growth N t = N 0 e rt N t

21. Competition & population Logistic growth: [ K - N ] dN / dt = r N  K r = intrinsic rate of increase K = carrying capacity N t K

22. Carrying capacity Intraspecific competition –among members of the same species As density goes up, realized growth rate (dN / dt) goes down What about interspecific competition? –between two different species

23. Lotka-Volterra Competition N 1 N 2 r 1 r 2 K 1 K 2 [ K 1 - N 1 -   N 2 ] dN 1 / dt = r 1 N 1  K 1 [ K 2 - N 2 -   N 1 ] dN 2 / dt = r 2 N 2  K 2

24. Lotka-Volterra Competition  1 = competition coefficient –Relative effect of species 1 on species 2  2 = competition coefficient –Relative effect of species 2 on species 1 equivalence of N 1 and N 2

25. Effects of N i & N i’ on growth [ K 1 - N 1 -   N 2 ] dN 1 / dt = r 1 N 1  K 1  In the numerator, a single individual of N 2 has a equivalent effect on dN 1 / dt to  2 individuals of N 1

26. Competition coefficients:  ’s Proportional constants relating the effect of one species on the growth of a 2nd species to the effect of the 2nd species on its own growth –  2 > 1  impact of sp. 2 on sp. 1 greater than the impact of sp. 1 on itself –  2 < 1  impact of sp. 2 on sp. 1 less than the impact of sp. 1 on itself –  2 = 1  impact of sp. 2 on sp. 1 equals the impact of sp. 1 on itself

27. total population growth dN i / dt = r i N i [K i -N i -  i’ N i’ ]/K i Notation per capita population growth dN i / N i dt = r i [K i -N i -  i’ N i’ ]/K i dN i / dt vs. dN i / N i dt

28. Lotka-Volterra equilibrium at equilibrium –dN 1 / N 1 dt = 0 & dN 2 / N 2 dt = 0 –also implies dN 1 / dt = dN 2 / dt = 0, so... 0 = r 1 N 1 [ (K 1 -N 1 -   N 2 )/ K 1 ] 0 = r 2 N 2 [ (K 2 -N 2 -   N 1 )/ K 2 ] true if N 1 = 0 or N 2 = 0 or r 1 = 0 or r 2 = 0

29. for 0 = r 1 N 1 [ (K 1 -N 1 -   N 2 )/ K 1 ] true if 0 = (K 1 -N 1 -   N 2 )/ K 1 if N 2 = 0, implies N 1 = K 1 (logistic equilibrium) as N 1  0, implies   N 2 =K 1 or N 2 = K 1 /   plot as graph of N 2 vs. N 1 Lotka-Volterra equilibrium

30. Equilibrium dN i / dt = 0 for both species K 1 - N 1 -  2 N 2 = 0 and K 2 - N 2 -  1 N 1 = 0 N2N2 K1/K1/ dN 1 /dt<0 N1N1 K1K1 dN 1 /dt>0 Zero Growth Isocline (ZGI) for species 1

31. Zero growth isocline for sp. 2 N2N2 N1N1 0 K2/K2/ K2K2 dN 2 /N 2 dt > 0 dN 2 /N 2 dt < 0 Zero Growth Isocline (ZGI) dN 2 /N 2 dt = 0

32. Zero growth isocline for sp. 1 N2N2 N1N1 0 K1K1 K 1 /  2 dN 1 /N 1 dt > 0 dN 1 / N 1 dt < 0 Zero Growth Isocline (ZGI) dN 1 /N 1 dt = 0

33. Isocline in 3 dimensions N2N2 N1N1 0 K1K1 K 1 /  2 Zero Growth Isocline... dN 1 /N 1 dt = 0 r1r1 dN 1 / N 1 dt

34. Isocline in 3 dimensions N2N2 K 1 /  2 N1N1 0 K1K1 Zero Growth Isocline... dN 1 /N 1 dt = 0

35. Isocline N2N2 K 1 /  2 N1N1 0 K1K1 Zero Growth Isocline... dN 1 /N 1 dt = 0

36. Two Isoclines on same graph May or may not cross Indicates whether two competitors can coexist For equilibrium coexistence, both must have –N i > 0 –dN i / N i dt = 0

37. Lotka-Volterra Zero Growth Isoclines K 1 /   > K 2 K 1 > K 2 /   Region  dN 1 /N 1 dt>0 & dN 2 /N 2 dt>0 Region  dN 1 /N 1 dt>0 & dN 2 /N 2 dt<0 Region  dN 1 /N 1 dt<0 & dN 2 /N 2 dt<0 N2N2 N1N1 0 K2/K2/ K2K2 dN 2 / N 2 dt = 0 K1/K1/ K1K1 dN 1 / N 1 dt = 0    Species 1 “wins”

38. Lotka-Volterra Zero Growth Isoclines K 2 > K 1 /   K 2 /    >  K 1 Region  dN 1 /N 1 dt>0 & dN 2 /N 2 dt>0 Region  dN 1 /N 1 dt 0 Region  dN 1 /N 1 dt<0 & dN 2 /N 2 dt<0 N2N2 N1N1 0 K2/K2/ K2K2 dN 2 / N 2 dt = 0 K1/K1/ K1K1 dN 1 / N 1 dt = 0    Species 2 “wins”

39. Competitive Asymmetry Competitive Exclusion Suppose K 1  K 2. What values of  1 and  2 lead to competitive exclusion of sp. 2?   1.0 (large) effect of sp. 2 on dN 1 / N 1 dt less than effect of sp. 1 on dN 1 / N 1 dt effect of sp. 1 on dN 2 / N 2 dt greater than effect of sp. 2 on dN 2 / N 2 dt

40. Lotka-Volterra Zero Growth Isoclines N2N2 K 1 /   > K 2 K 2 /    >  K 1 Region  both species increase Regions  &  one species decreases & one species increases Region  both species decrease N1N1 0 K2/K2/ K2K2 dN 2 / N 2 dt = 0 K1K1 dN 1 / N 1 dt = 0    K1/K1/  Stable coexistence

41. Stable Competitive Equilibrium Competitive Coexistence Suppose K 1  K 2. What values of  1 and   lead to coexistence?   < 1.0 (small) and   < 1.0 (small) effect of each species on dN/Ndt of the other is less than effect of each species on its own dN/Ndt Intraspecific competition more intense than interspecific competition

42. N1N1 0 K2/K2/ K2K2 dN 2 / N 2 dt = 0 K1K1 dN 1 / N 1 dt = 0    Lotka-Volterra Zero Growth Isoclines K1/K1/ N2N2  K 2 > K 1 /   K 1 > K 2 /   Region  both species increase Regions  &  one species decreases & one species increases Region  both species decrease Unstable two species equilibrium

43. Unstable Competitive Equilibrium Exactly at equilibrium point, both species survive Anywhere else, either one or the other “wins” Stable equilibria at: –(N 1 = K 1 & N 2 = 0) –(N 2 = K 2 & N 1 = 0) Which equilibrium depends on initial numbers –Relatively more N 1 and species 1 “wins” –Relatively more N 2 and species 2 “wins”

44. Unstable Competitive Equilibrium Suppose K 1  K 2. What values of  1 and lead to coexistence?   > 1.0 (large) and   >1.0 (large) effect of each species on dN/Ndt of the other is greater than effect of each species on its own dN/Ndt Interspecific competition more intense than intraspecific competition

45. Lotka-Volterra competition Four circumstances –Species 1 wins –Species 2 wins –Stable equilibrium coexistence –Unstable equilibrium; winner depends on initial N’s Coexistence only when interspecific competition is weak Morin, pp. 34-40

46. Competitive Exclusion Principle Two competing species cannot coexist unless interspecific competition is weak relative to intraspecific competition What makes interspecific competition weak? –Use different resources –Use different physical spaces –Use exactly the same resources, in the same place, at the same time  Competitve exclusion

47. Model assumptions All models incorporate assumptions Validity of assumptions determines validity of the model Different kinds of assumptions Consequences of violating different kinds of assumptions are not all the same

48. Simplifying environmental assumption The environment is, with respect to all properties relevant to the organisms: –uniform or random in space –constant in time realistic? if violated  need a better experimental system

49. Simplifying biological assumption All the organisms are, with respect to their impacts on their environment and on each other: –identical throughout the population clearly must be literally false if seriously violated  need to build a different model with more realistic assumptions

50. Explanatory assumptions What we propose as an explanation of nature (our hypothesis) –r 1, r 2, K 1, K 2,  ,   are constants –competition is expressed as a linear decline in per capita growth (dN / N dt ) with increasing N 1 or N 2 –Some proportional relationship exists between the effects of N 1 and N 2 on per capita growth If violated  model (our hypothesis) is wrong

51. Interspecific competition: Paramecium George Gause P. caudatum goes extinct Strong competitors, use the same resource (yeast) Competitve asymmetry Competitive exclusion

52. P. caudatum & P. burseria coexist P. burseria is photosynthetic Competitive coexistence Apparently stable Interspecific competition: Paramecium

53. Mechanism of coexistence Paramecium caudatum –nonphotosynthetic; feeds on yeasts only –must be near surface (O 2 ) Paramecium burseria –endosymbiotic algae; photosynthesis; produce O 2 –can feed on yeasts at the bottom of the test tube Two species used different resources –weak interspecific competition; coexistence

54. Experiments in the laboratory Gause’s work on protozoa Flour beetles (Tribolium) Duck weed (Lemna, Wolffia) Mostly consistent with Lotka Volterra No clear statement of what causes interspecific competition to be weak

55. Alternative Lotka-Volterra competition Absolute competition coefficients dN i / N i dt = r i [1 –  ii N i -  ij N j ] equivalent to: dN i / N i dt = r i [K i - N i -  j N j ] / K i = r i [K i /K i - N i /K i -  j N j /K i ] = r i [1- (1/K i )N i – (  j /K i )N j ]

56. Absolute Lotka-Volterra N1N1 0 1/  21 1/  22 dN 2 / N 2 dt = 0 1/  11 dN 1 / N 1 dt = 0    1/  12  Stable coexistence N2N2

57. Competitive effect vs. response Effect: impact of density of a species –Self density (e.g.,  11 ) –Other species density (e.g.,  21 ) Response: how density affects a species –Self density (e.g.,  11 ) –Other species’ density (e.g.,  12 ) Theory: effects differ (  11 >  21 ) Experiments: responses (  11,  12 )

58. Absolute Lotka-Volterra N1N1 0 1/  21 1/  22 dN 2 / N 2 dt = 0 1/  11 dN 1 / N 1 dt = 0    1/  12  Stable coexistence N2N2

59. Not ecological models No mechanisms of competition in the model –Phenomenological Environment not explicitly included Mechanistic models of Resource competition