1. Niches, distributions… and data Miguel Nakamura Centro de Investigación en Matemáticas (CIMAT), Guanajuato, Mexico nakamura@cimat.mx Warsaw, November 2007

3. “Data” Presences Environmental layers

4. Niche and distribution concepts Ecological theory conceived in Nature realized in Niche models inferred using Data useproduces Theoretical niche defines Distribution

5. Premise #1: an observation is the result of at least two, multi-factor processes  Biology: the fundamental niche, biotic conditions, sink populations, etc.  Humans: the collector introduces bias, methods used determine detection, etc.

6. Premise #2: randomness involved  If sites 1 and 2 both have equal conditions X as far as we can see, it does NOT necessarily follow that “species present at site 1 implies species present at site 2”  Reason: apart from conditions X, there may be other, non-visualized conditions, Z, that also influence presence. These may differ between site 1 and site 2.  Refer to probabilities of presence at a site having conditions X, instead of a deterministic statement, “species is present at a site”.

7. “Probability trees”  Graphical devices for tracking random experiments, especially when sequential processes or stages are involved.  Probabilities can be assigned to branches, for calculations.  Example: die is cast to observe number of spots (N), then N coins are tossed and number of heads counted.

8. Die 1 2 Begin 3 4 5 6 1/6 # Heads 1 0 1 0 2 1 0 2 3.50.25.50.125.375.125 etc. Probability of this branch=(1/6)×(.50) Probability of this cluster=sum of probabilities of individual branches

9. Species present? Site visited? Species detect? True presence True absence A site Elementary probability tree for describing occurrence data False absence False presence False absence False presence False absence

10. Abiotic OK? Site visited? Species detect? Biotic OK? Species moved? More-elaborate probability tree: “biological presence” has been expanded Presence-only data: the probability of this branch is product of all probabilities in its path. This is data niche models will use.

11. Abiotic OK? Site visited? Species detect? Biotic OK? Species moved? Filling-in probabilities in the tree A B C D E A×B×C×D×E

12. Interpreting the probabilities Abiotic OK? Site visited? Species detect? Biotic OK? Species moved?A B C D E A×B×C×D×E Motility of species (history, barriers, dispersal capacities, etc.) Suitability of biotic conditions (competitors, predators, mutualists, etc.) Suitability of abiotic conditions (resistance to temperature extremes, water stress, etc.) Sampling bias (accessibility, roads, etc.) Probability of detection (methods and effort of collection)

13. Spatial sampling bias Occurrence Data (reptiles)

14. Spatial sampling biasEnvironmental sampling bias ? e1 e2 Geographical spaceEnvironmental space

15. Abiotic OK? Site visited? Species detect? Biotic OK? Species moved? A pet example 1 1.20 1 Prob=.20×.80=.16.80.32.50 Prob=.32×.50=.16 Important conclusion: Factors can combine in different ways and still produce the same observed presence rate! Two different species Two different sampling schemes

16. Issue raised by pet example  Distribution of presence-only data is a function of all factors in the tree. Factors can combine in different ways and still produce the same observed presence rate!  Since observed data is probabilistically identical, any method that uses observed data only, is unable to discern between Species #1 and Species #2.  Sampling bias and other conditions become crucial.

17. Abiotic OK? Site visited? Species detect? Biotic OK? Species moved? In general, areas of distribution≠data A B C D E Data=A×B×C×D×E Occupied area=A×B×C Colonizable area=B × C Abiotically suitable area=C

18. MotilityBioticAbioticSamplingDetectionData Occupied area Colonizable area Abiotically suitable 0 General case ABCDEABCDEABCBCC 1 Full motility, biotic irrelevant, well-sampled, sure detection 11C11CCCC 2 Full motility, well-sampled, sure detection 1BC11BCBCBCC 3 Full motility, abiotic irrelevant, well-sampled, sure detection 1B111BBB1 4 Partial motility, biotic irrelevant A1C11ACACCC 5 Well-sampled, sure detection ABC11ABCABCBCC 6 Full motility, biotic irrelevant, sampling bias 11CD1CDCCC 7 Cosmopolitan species 111DEDE111 Some special cases

19. Conclusions  One thing is distribution of species, and another issue is distribution of observed data. Relationship between data and the niche must be understood.  Previous tree diagram is far more complicated: Interactions. Interactions. Sink populations. Sink populations. Grid resolution (more on this shortly). Grid resolution (more on this shortly). Recording errors, classification errors. Recording errors, classification errors.  Some special cases allow for simplifications: Uniform sampling. Uniform sampling. Sure detection. Sure detection. Unrestricted species motility. Unrestricted species motility.

20. Conclusions  Algorithms use observed data. They will all try to fit observed data to environmental variables.  This may or may not produce what you are interested in. It may if you are willing to make some assumptions regarding data.  It is your responsibility to determine if these assumptions are met and to interpret results accordingly. A modeling algorithm will not know better.  It is useful to think of “data” as including operational assumptions, not merely “numbers”.

21. Probability trees used to understand changes in grid resolution 1km 2km

22. Merging two sites Site 1 Site 2 Site 1-2 A1A1A1A1 B1B1B1B1 C1C1C1C1 D1D1D1D1 E1E1E1E1 A2A2A2A2 B2B2B2B2 C2C2C2C2 D2D2D2D2 E2E2E2E2 A 12 B 12 C 12 D 12 E 12

23. Is there a relationship between A 1, B 1, C 1, D 1, E 1, A 2, B 2, C 2, D 2, E 2 and A 12, B 12, C 12, D 12, D 12 ?  If new probabilities are derived from the pair of old sets, then merged tree is function of components. Will show that this cannot be done coherently.

25. To produce coherent interpretations for the new tree, the “biotic” probability in the new tree must necessarily depend on accessibility, biotic, and abiotic terms of the original trees. Since this interpretation is senseless, the conclusion is that a change in resolution implies a new description of niches/distributions.