1. Gradient Analysis Approach to Ordination

2. Models of Species Response to Gradients

3. Models of Species Response There are (at least) two models:- Linear - species increase or decrease along the environmental gradient Unimodal - species rise to a peak somewhere along the environmental gradient and then fall again

4. A Theoretical Model

5. Linear

6. Unimodal

7. Alpha and Beta Diversity alpha diversity is the diversity of a community (either measured in terms of a diversity index or species richness) beta diversity (also known as species turnover or differentiation diversity) is the rate of change in species composition from one community to another along gradients; gamma diversity is the diversity of a region or a landscape.

8. A Short Coenocline

9. A Long Coenocline

10. Inferring Gradients from Species (or Attribute) Data

11. Indirect Gradient Analysis Environmental gradients are inferred from species data alone Three methods: Principal Component Analysis - linear model Correspondence Analysis - unimodal model Detrended CA - modified unimodal model

12. PCA - linear model

14. Terschelling Dune Data

15. PCA gradient - site plot

16. PCA gradient - site/species biplot

17. Reciprocal Averaging Site A B C D E F Species Prunus serotina 6 3 4 6 5 1 Tilia americana 2 0 7 0 6 6 Acer saccharum 0 0 8 0 4 9 Quercus velutina 0 8 0 8 0 0 Juglans nigra 3 2 3 0 6 0

18. Reciprocal Averaging Site A B C D E F Species Score Species Iteration 1 Prunus serotina 6 3 4 6 5 1 1.00 Tilia americana 2 0 7 0 6 6 0.63 Acer saccharum 0 0 8 0 4 9 0.63 Quercus velutina 0 8 0 8 0 0 0.18 Juglans nigra 3 2 3 0 6 0 0.00 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site Score

19. Reciprocal Averaging Site A B C D E F Species Score Species Iteration 1 2 Prunus serotina 6 3 4 6 5 1 1.00 0.68 Tilia americana 2 0 7 0 6 6 0.63 0.84 Acer saccharum 0 0 8 0 4 9 0.63 0.87 Quercus velutina 0 8 0 8 0 0 0.18 0.30 Juglans nigra 3 2 3 0 6 0 0.00 0.67 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score

20. Reciprocal Averaging Site A B C D E F Species Score Species Iteration 1 2 3 Prunus serotina 6 3 4 6 5 1 1.00 0.68 0.50 Tilia americana 2 0 7 0 6 6 0.63 0.84 0.86 Acer saccharum 0 0 8 0 4 9 0.63 0.87 0.91 Quercus velutina 0 8 0 8 0 0 0.18 0.30 0.02 Juglans nigra 3 2 3 0 6 0 0.00 0.67 0.66 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score 3 0.60 0.01 0.87 0.00 0.78 1.00

21. Reciprocal Averaging Site A B C D E F Species Score Species Iteration 1 2 3 9 Prunus serotina 6 3 4 6 5 1 1.00 0.68 0.50 0.48 Tilia americana 2 0 7 0 6 6 0.63 0.84 0.86 0.85 Acer saccharum 0 0 8 0 4 9 0.63 0.87 0.91 0.91 Quercus velutina 0 8 0 8 0 0 0.18 0.30 0.02 0.00 Juglans nigra 3 2 3 0 6 0 0.00 0.67 0.66 0.65 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score 3 0.60 0.01 0.87 0.00 0.78 1.00 9 0.59 0.01 0.87 0.00 0.78 1.00

22. Reordered Sites and Species Site A C E B D F Species Species Score Quercus velutina 8 8 0 0 0 0 0.004 Prunus serotina 6 3 6 5 4 1 0.477 Juglans nigra 0 2 3 6 3 0 0.647 Tilia americana 0 0 2 6 7 6 0.845 Acer saccharum 0 0 0 4 8 9 0.909 Site Score 0.000 0.008 0.589 0.778 0.872 1.000

23. Lake Nasser Invertebrates

25. CA - unimodal model + + + + + + + + + + Protozoa Rotifera Cladocera Copepoda Insecta Turbellaria Tardigrada Annelida Nematoda

26. Arches - Artifact or Feature?

27. The Arch Effect What is it? Why does it happen? What should we do about it?

28. From Alexandria to Suez

29. CA - with arch effect (species)

30. CA - with arch effect (sites)

31. Long Gradients ABCD

32. Gradient End Compression

33. CA - with arch effect (species)

34. Detrending by Segments

35. DCA - modified unimodal

36. Making Effective Use of Environmental Variables

37. Direct Gradient Analysis Environmental gradients are constructed from the relationship between species environmental variables Three methods: Redundancy Analysis - linear model Canonical (or Constrained) Correspondence Analysis - unimodal model Detrended CCA - modified unimodal model

38. CCA - site/species joint plot

39. CCA - species/environment biplot

40. Removing the Effect of Nuisance Variables

41. Partial Analyses Remove the effect of covariates variables that we can measure but which are of no interest e.g. block effects, start values, etc. Carry out the gradient analysis on what is left of the variation after removing the effect of the covariates.

42. Testing Significance in Ordination

43. Randomisation Tests

45. Randomisation Example Model: cca(formula = dune ~ Moisture + A1 + Management, data = dune.env) Df Chisq F N.Perm Pr(>F) Model 7 1.1392 2.0007 200 < 0.005 *** Residual 12 0.9761 Signif. codes: 0 *** 0.001 ** 0.01 * 0.05