1. The Markovian Patch- Occupancy (MPO) framework in Omri Allouche Prof. Ronen Kadmon Sep 2008 Community Ecology

2. Photos taken from Life, the science of biology, WH Freeman

3. Semi-universal patterns

4. The Aim Develop a general framework for modeling ecological communities: Elementary community dynamics Simplified representation of communities Analytic solution Able to qualitatively produce known patterns of species-diversity Useful for the study of complex ecological phenomena

5. The importance of Demography Species richness is determined by species extinctions and colonizations Species extinctions and colonizations result from actions of individuals The basic elements of ecological communities are individuals Individuals go through demographic processes of birth, death and migration

6. The MPO Framework Individual-based Island receiving immigrants from a mainland Multiple species Basic demographic processes: Local Reproduction Mortality Immigration + Emigration General Framework for Modeling Ecological Communities

7. In a small-enough time interval only a single event can take place A Markov chain: Implicit space - Global dispersal We wish to find the steady state probability of The MPO Framework

8. Mortality Emigration Decrease in the number of individuals of species k Possible events: Local reproduction Immigration from the mainland Increase in the number of individuals of species k The Master Equation:

9. The transition rates are highly flexible, and can incorporate complex ecological phenomena. We only require that: This holds for many interesting cases, and can be easily checked for any transition rates. Analytic Solution

10. The steady-state probability of state is given by: Analytic Solution Abundance of species 1 (N 1 ) Abundance of species 2 (N 2 )

11. (0,0,0) (1,0,0) (1,1,0) (1,2,0) (1,2,1) Abundance of species 1 (N 1 ) Abundance of species 2 (N 2 )

12. “detailed balance”

13. “If a system is an irreducible and ergodic Markov process and there exists a sequence of non-negative numbers which sum to 1 and satisfy the 'detailed balance' equation of the model, then that sequence of numbers is the steady-state distribution of the system” For the solution to hold, we require that: This holds for many interesting cases, and can be easily checked for any transition rates.

14. Analytic Solution Steady-state distribution: Abundance distribution: Species Richness: Community size distribution:

15. Island of area A Multiple species Rates of: Birth b k Death d k Immigration i k Relative regional abundance Individuals only establish in vacant sites Multispecies community with competition for space

16. Local reproduction: Mortality: Immigration from the regional pool: The solution:

17. The MPO Framework Individual-based Explicit consideration of demographic processes: reproduction, mortality and migration Demographic differences among species Analytically tractable Highly flexible

18. Applications Habitat loss Habitat heterogeneity Habitat preference Non-random dispersal Productivity Allee effect Population-level density dependence Community-level carrying capacity Hubbell’s mainland-island model Independent species Community-level density dependence Fitness equivalence Neutral Models

19. 020406080100 0 20 40 60 80 100 Habitat loss (%) Species richness 1.5 3 5 10 Reproduction A D = The number of destroyed sites Habitat Loss “The greatest existing threat to biodiversity “

20. 1.An island consisting of A sites, divided among H habitats 2.Each species is able to establish and persist in only one habitat 3.Individuals disperse and immigrate to random sites Habitat Heterogeneity and niche partitioning

21. Habitat Heterogeneity 246810 0 20 40 60 80 100 120 140 Heterogeneity (H) Species richness Reproduction = 1.5 3 5 10 Kadmon and Allouche 2007 Am Nat total area and H is the total number of habitats community size in habitat of species k

22. Habitat Preference Offspring arrive in suitable habitats more than in unsuitable habitats due to active site selection or environmental autocorrelation and limited dispersal

23. Non-random Dispersal Offspring tend to arrive to vacant sites more than to occupied sites

24. Productivity R = Productivity 0.20.40.60.81 40 60 80 100 Productivity (R) Species richness b=1.5 5 10 20 Increased reproduction due to more resources

25. Reduced reproduction of small populations due to mate finding, predation, aggregation, environmental modification… the level of Allee effect for species k Allee Effect

26. Population-Level Density Dependence Increased mortality of large populations due to parasites and diseases, predation, environmental modification…

27. Community-Level Carrying Capacity Increased mortality of large communities due to parasites and diseases, predation, environmental modification…

28. The Meaning of Life

29. General Framework for Neutral Models Hubbell’s mainland-island model (Hubbell 2001) where: Independent neutral species (Volkov et al. 2003, 2005 Nature, He 2005 Func Ecol., Etienne et al. 2007 JTB) Community-Level Density-Dependence (Haegeman & Etienne 2008 JTB)

30. 1.Island receiving immigrants from an outer mainland 2.Functionally equivalent species 3.Saturated community – all sites are occupied 4.In each time-step, one individual dies and is immediately replaced by: Immigrant m Local offspring (1-m) 5.Neutral species – individuals are equal in probability of death and replacement, regardless of their species identity Hubbell’s Neutral Theory of Biodiversity

31. Hubbell’s Mainland-Island Neutral Model where: Transition rates: Hubbell 2001, Vallade and Houchmandzadeh 2003 Phys Rev. E, Volkov et al. 2003 Nature, McKane et al. 2004 TPB positive influence negative influence Steady-state distribution: Abundance distribution: i b m J SR d A m J dilution effect

32. Independent Neutral Species Transition rates: Steady-state distribution: Abundance distribution: Community size distribution: Etienne et al. 2007 JTB

33. Community-Level Density- Dependence of Neutral Species Transition rates: Steady-state distribution: Haegeman and Etienne 2008 JTB

34. Are species neutral? (?!) “Life history trade-offs equalize the per capita relative fitness of species in the community, which set the stage for ecological drift” S. Hubbell (2001) Median Annual Growth Annual Survival Defining ecological fitness as birth/death ratio

35. The solution: Extending Hubbell’s Strict Neutrality to Fitness Equivalence

36. The MPO Framework - Summary General framework for modeling ecological communities Individual-based Basic demographic processes Demographic differences among species Analytically tractable A general framework for neutral null models Extends patch-occupancy theory

37. The MPO Framework Highly flexible Useful for the study of complex ecological phenomena Able to qualitatively produce leading patterns of species- diversity Useful for other fields (???)

39. Trees, Pasoh forest, Peninsular Malaysia Neutral models fit empirical data Hubbell 2001

40. Neutral models fit empirical data He 2005 Func. Ecol. Volkov et al. 2005 Nature

41. Neutral models fit empirical data Volkov et al. 2007 Nature

42. Neutral models fit empirical data Hubbell 2006 Ecol. Tropical tree community, Lambir Hills National Park, Borneo

43. Marine planktonic copepods, northeastern Pacific gyre Mixed Mesophytic Forest. Cumberland Plateau, Kentucky Insectivorous birds, Hubbard Brook, New Hampshire Tropical freshwater fish community, Venezuela Neutral models fit empirical data Hubbell 2001

44. Markov processes The Markov property: Time-homogeneous Markov chain: Irreducibility = any state is eventually reachable from any other state.