1 / 58

Other patterns in communities

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. S. A. Species-area relationships.

trevet
Download Presentation

Other patterns in communities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  3. Mathematics S = c Az • 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. ln (S ) ln (A ) 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

  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. Sn ni 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 (s2) of the distribution • assume that N increases linearly with A • Yield: unique relationship: S = c Az • for "canonical" with S > 20: S = c A0.25

  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 s2 = 0.25 (very low) • then z  0.9 … which is virtually unknown in nature • implies constraints on log normal distributions

  8. I ST S 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

  9. E ST S 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

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

  11. Esmall Elarge RATE I S*small S S*large 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 isnotrelated to area

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

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

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

  20. N t Competition & population • Exponential growth • dN / dt = r N • r = exponential growth rate • unlimited growth • Nt= N0ert

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

  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 N1 N2 r1 r2 K1 K2 • [ K1 - N1- a2 N2 ] • dN1/dt = r1 N1 • K1 • [ K2 - N2- a1 N1 ] • dN2/dt = r2 N2 • K2

  24. Lotka-Volterra Competition • a1 = competition coefficient • Relative effect of species 1 on species 2 • a2 = competition coefficient • Relative effect of species 2 on species 1 • equivalence of N1 and N2

  25. Effects of Ni & Ni’ on growth • [ K1 - N1- a2 N2 ] • dN1/ dt = r1 N1 • K1 • In the numerator, a single individual of N2 has a equivalent effect on dN1 / dtto a2 individuals of N1

  26. Competition coefficients: a’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 • a2 > 1  impact of sp. 2 on sp. 1 greater than the impact of sp. 1 on itself • a2 < 1  impact of sp. 2 on sp. 1 less than the impact of sp. 1 on itself • a2 = 1  impact of sp. 2 on sp. 1 equals the impact of sp. 1 on itself

  27. total population growth dNi / dt = riNi [Ki-Ni-ai’Ni’]/Ki per capita population growth dNi/ Nidt = ri[Ki-Ni-ai’Ni’]/Ki Notation dNi / dtvs. dNi / Nidt

  28. Lotka-Volterra equilibrium • at equilibrium • dN1 / N1dt = 0 & dN2 / N2dt= 0 • also implies dN1 / dt = dN2 / dt= 0, so... • 0 = r1N1[ (K1-N1-a2N2)/ K1] • 0 = r2N2[ (K2-N2-a1N1)/ K2] • true if N1 = 0 or N2 = 0 or r1= 0 or r2 = 0

  29. Lotka-Volterra equilibrium • for 0 = r1N1[ (K1-N1-a2N2)/ K1] • true if 0 = (K1-N1-a2N2)/ K1 • if N2 = 0, implies N1 = K1 (logistic equilibrium) • as N1  0, implies a2N2=K1 or N2 = K1 / a2 • plot as graph of N2vs. N1

  30. N2 K1/a2 dN1/dt<0 dN1/dt>0 N1 K1 Equilibrium • dNi / dt = 0 for both species • K1 - N1 -a2N2 = 0 and K2 - N2 -a1N1 = 0 Zero Growth Isocline (ZGI) for species 1

  31. N2 dN2 /N2 dt < 0 K2 Zero Growth Isocline (ZGI) dN2 /N2 dt = 0 dN2 /N2 dt > 0 0 N1 K2/a1 Zero growth isoclinefor sp. 2

  32. N2 dN1 / N1 dt< 0 K1 /2 Zero Growth Isocline (ZGI) dN1/N1dt = 0 dN1 /N1 dt> 0 0 N1 K1 Zero growth isocline for sp. 1

  33. dN1 / N1dt Isocline in 3 dimensions r1 K1 0 N1 K1 / 2 Zero Growth Isocline... dN1/N1dt = 0 N2

  34. K1 0 N1 K1 / 2 Zero Growth Isocline ... dN1/N1dt = 0 N2 Isocline in 3 dimensions

  35. N2 K1 / 2 Zero Growth Isocline... dN1/N1dt = 0 K1 0 N1 Isocline

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

  37. N2 K1/a2 dN1 / N1dt = 0  K2  dN2 / N2dt = 0  0 K1 N1 K2/a1 Species 1 “wins” Lotka-Volterra Zero Growth Isoclines • K1 / a2 > K2 • K1 >K2 / a1 • Region  dN1/N1dt>0 & dN2/N2dt>0 • Region  dN1/N1dt>0 & dN2/N2dt<0 • Region  dN1/N1dt<0 & dN2/N2dt<0

  38. N2 K2  dN2 / N2dt = 0 K1/a2  dN1 / N1dt = 0  0 N1 K1 K2/a1 Species 2 “wins” Lotka-Volterra Zero Growth Isoclines • K2> K1 / a2 • K2 / a1>K1 • Region  dN1/N1dt>0 & dN2/N2dt>0 • Region  dN1/N1dt<0 & dN2/N2dt>0 • Region  dN1/N1dt<0 & dN2/N2dt<0

  39. Competitive Asymmetry • Competitive Exclusion • Suppose K1 K2. What values of 1 and a2 lead to competitive exclusion of sp. 2? • a2 < 1.0 (small) and a1 > 1.0 (large) • effect of sp. 2 on dN1/ N1dt less than effect of sp. 1 on dN1/ N1dt • effect of sp. 1 on dN2/ N2dt greater than effect of sp. 2 on dN2/ N2dt

  40. K1/a2 Stable coexistence dN1 / N1dt = 0   K2 dN2 / N2dt = 0   0 K1 K2/a1 N1 Lotka-Volterra Zero Growth Isoclines N2 • K1 / a2 > K2 • K2 / a1>K1 • Region  both species increase • Regions  &  one species decreases & one species increases • Region  both species decrease

  41. Stable Competitive Equilibrium • Competitive Coexistence • Suppose K1 K2. What values of 1 and a2 lead to coexistence? • a1 < 1.0 (small) and a2 < 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. Lotka-Volterra Zero Growth Isoclines N2 K2 Unstable two species equilibrium • K2 > K1 / a2 • K1 > K2 / a1 • Region  both species increase • Regions  &  one species decreases & one species increases • Region  both species decrease dN2 / N2dt = 0  K1/a2   dN1 / N1dt = 0  0 K2/a1 K1 N1

  43. Unstable Competitive Equilibrium • Exactly at equilibrium point, both species survive • Anywhere else, either one or the other “wins” • Stable equilibria at: • (N1 = K1 & N2 = 0) • (N2 = K2 & N1 = 0) • Which equilibrium depends on initial numbers • Relatively more N1and species 1“wins” • Relatively more N2 and species 2 “wins”

  44. Unstable Competitive Equilibrium • Suppose K1 K2. What values of 1 and lead to coexistence? • a1> 1.0 (large) and a2 >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) • r1, r2, K1, K2, a1, a2 are constants • competition is expressed as a linear decline in per capita growth (dN / N dt ) with increasing N1or N2 • Some proportional relationship exists between the effects of N1 and N2 on per capita growth • If violated  model (our hypothesis) is wrong

More Related