1. Path analysis of the effects of biotic interactions on fruit production and demographic fates in a neotropical herb
Carol Horvitz
University of Miami, Coral Gables, FL 33124, USA
and
Douglas Schemske,
Michigan State University, East Lansing, MI 48824, USA

2. Effects of biotic interactions on “fitness”?
Within the year: within season fruit production
Between years: demographic fates from t to t+1
Population growth rate (for single time step of 1 year, time-invariant population projection matrix)
Long run growth rate, e.g. stochastic growth rate (for multiple time steps, varying population projection matrix)

3. Regression analysis and causality
No regression model is assumption-free about causality among a set of variables
Only the correlation matrix makes no assumption about the causal relationships among variables
The model of causal relationships comes from biological insights external to the data at hand
Causal relationships are depicted in path diagrams a a b c c b

4. Regression analysis and causality, 2
The “direct effects” leading from one cause to one effect are depicted by straight, single-headed arrows
Unresolved correlations are depicted by curved, two-headed arrows
Correlations in the data set can be decomposed into linear combinations more than one way
Matrix algebra facilitates the process
Both direct and indirect causal effects can thus be quantified

5. Regression analysis and causality, 3
The path coefficients for the direct effects are standardized regression coefficients
They quantify “the average change in standard deviation units of the dependent variable for one standard deviation unit of each independent variable” (Sokal and Rohlf 1981, p. 623)

6. The study system Calathea ovandensis (Marantaceae)
Laguna Encantada, Los Tuxtlas, Veracruz, MX
Natural variation in parameters quantified for individual plants in the field
Two studies
Within year: pollinator visits, antguards, herbivore of reproductive tissues, flowers, initiated and mature fruits (2 yrs)
Between years: size, herbivore damage to leaves, competition, fruits, survival, growth, inflorescence production (5 yr to yr steps)

7. Biotic interactions Antguards An herbivore of reproductive tissues
Pollinators
Size
Herbivory of leaf tissues
Neighbours
Fruits

8. Biotic interactions Antguards An herbivore of reproductive tissues
Pollinators
Size
Herbivory of leaf tissues
Neighbours
Fruits

9. Biotic interactions acting on fruit production within a season
Antguards (many taxa)
Herbivory of reproductive tissues
Eurybia elvina (Riodinidae)
Pollinator visits
Euglossa spp
Eulaema cingulata
Eulaema polychroma
Exaerete smaragdina
Rhathymus sp

10. Conclusions for effects of biotic interactions on fruit production
Ants: direct on flower production, both yrs
Ants: indirect on fruit production, both yrs
Eurybia: direct on flower production, both yrs
Eurybia: direct on fruit production, both yrs
Pollinators: + direct on fruit initiation, one yr only (the year with more abundant high quality visitors)

11. Biotic interactions Antguards An herbivore of reproductive tissues
Pollinators
Size
Herbivory of leaf tissues
Neighbours
Fruits

12. Biotic interactions at time t acting on demographic fates at t+1
Current size (leaf area, cm2)
Herbivory of leaf tissues (% leaf area gone)
Neighbourhood competition (leaf area, cm2)
Fruits produced

13. Biotic interactions at time t acting on demographic fates at t+1
Current size (leaf area, cm2)
Herbivory of leaf tissues (% leaf area gone)
Lema bipsitulata, L. plumbea (Chrysomelidae)
Saliana sp, Podalia sp (Lepidoptera)
Unidentified Orthopterans in rolled leaf
Neighbourhood competition (leaf area, cm2)
conspecifics in area
radius defined by leaf length
Fruits produced
no. of inflorescences and biotic interactions

14. standardized regression analysis
Path diagram for
standardized regression analysis
Size
Herbivory
Neighbours
[Fruits]
Survival

15. standardized regression analysis
Path diagram for
standardized regression analysis
Size
Herbivory
Neighbours
[Fruits]
Relative
growth

16. standardized regression analysis
Path diagram for
standardized regression analysis
Size
Herbivory
Neighbours
[Fruits]
Inflore-
scences

17.  1982 1983 Seedlings Juveniles 1984 Pre-reproductives 1985
Separate analyses for each dependent variable by stage and year (total of 55 analyses!)
1982
1983
1984
1985
1986
Seedlings
Juveniles
Pre-reproductives
Reproductives 

18. Effects on Survival

19. Effects on Survival Fisher’s combined probability statistic,
-2lnP across years for a stage
STB, standardized regression coefficient

20. Effects on Survival P< 0.0001
(Fisher’s combined probability statistic,
-2lnP, across years for a stage)
P< 0.05
(STB, standardized regression coefficient)

21. Results Growth Reproduction Survival
Size: + for seedlings and juveniles
Competition: - for seedlings
Competition temporal pattern: - in 1983 (highest year)
Herbivory temporal pattern: - in 1985 (NOT highest!)

22. Effects on Relative Growth P< 0.0001, P<0.05, P<0.001
(Fisher’s combined probability statistic,
-2lnP, across years for a stage)
P< 0.05
(STB, standardized regression coefficient)

23. Results Survival Size: + for seedlings and juveniles
Herbivory temporal pattern: - in 1985 (NOT highest!)
Competition: - for seedlings
Competition temporal pattern: - in 1983 (highest year)
Growth
Size: - for all stages
Herbivory temporal pattern: - in 1985 (NOT highest!)
Competition: - for juveniles
Competition: - in 1984
Fruit: +
Reproduction

24. Effects on Inflore-scences Produced
(Fisher’s combined probability statistic,
-2lnP, across years for a stage)
P< 0.05
(STB, standardized regression coefficient)

25. Results Survival Size: + for seedlings and juveniles
Herbivory temporal pattern: - in 1985 (NOT highest!)
Competition: - for seedlings
Competition temporal pattern: - in 1983 (highest year)
Growth
Size: - for all stages
Herbivory temporal pattern: - in 1985 (NOT highest!)
Competition: - for juveniles
Competition: - in 1984
Fruit: +
Reproduction
Size: + for pre-reproductives and reproductives
Competition: - only for reproductives in 1983
Fruit: +

26. Conclusions for effects of biotic interactions on demographic fates
Size: important for all stages, improving survival (of smallest ones) and improving reproduction, but slowing down relative growth.
Herbivory: very low in general; it had mysterious negative effects in 1985, not the year it was highest.
Competition: strongest negative impact on seedlings, but also had temporal pattern (partially) consistent with its strength.
Fruit production: positive impacts on future growth and future reproduction.

27. Biotic interactions Antguards An herbivore of reproductive tissues
Pollinators
Size
Herbivory of leaf tissues
Neighbours
Fruits

28. Tying the two studies together plus...
Standardized regression coefficients (path coefficients) are a great tool for summarizing the magnitudes of effects over many analyses
The stage-specific demographic influence of an animal may not be predicted its magnitude
Animals affecting fruit production may also influence future demographic fates in unpredicted ways (beyond “cost of reproduction” sorts of ideas)

29. Tying the two studies together plus...placing them into broader context
Within and between season effects: population projection matrix growth rate sensitivity
Variable environments: a set of matrices appropriately linked and analyzed to determine stochastic growth rate sensitivity

30. References and ongoing collaborations
Schemske, D.W. and C.C. Horvitz, Plant-animal interactions and fruit production in a neotropical herb: a path analysis. Ecology 69:
Horvitz, C.C. and D.W. Schemske, Effects of plant size, leaf herbivory, local competition and fruit production on survival, growth and future reproduction of a neotropical herb. Journal of Ecology 90:
Plant-animal interactions in random environments: ongoing collaborations with Tuljapurkar, Pascarella, Ehrlen and Matlaga