 Integrated Modelling of Habitat and Species Occurrence Dynamics

Basic Idea Habitat state/condition changes as a Markov process. Changes in occupancy state are associated with changes in habitat.  e.g., breeding amphibians in vernal pools. Habitat classified into discrete types (A, B, …) Special case: characterize habitat as suitable or unsuitable (e.g., Lande 1987, 1988).

Basic Idea System state now characterized by proportion of units in each habitat type and proportion of each type of units occupied. In the face of habitat change, proportion of units occupied of a particular habitat type is not an adequate descriptor of system state (e.g., conservation).

Sampling Situations 1. Habitat type can be assessed with certainty at each season. Estimate habitat transitions directly. 2. Habitat state is not known with certainty but is assessed with non-negligible Pr(misclassification). Habitat modelling requires a misclassification parameter, and joint modelling of habitat and occupancy dynamics are more complicated.

Developing a Model Consider the case with 2 habitat types (A or B) and 2 levels of occupancy (presence or absence) 2 approaches for developing a model  single state variable {habitat, occupancy} pair  double state variable habitat and occupancy|habitat

Developing a Model = Pr(unit is of habitat type H in first season) = Pr(unit transitions from habitat H t at time t to H t+1 at t+1| species was present (X=1) or absent (X=0) at time t)

Developing a Model = Pr(unit is occupied|habitat H in first season) = Pr(extinction between t and t+1|habitat changed from H t to H t+1 ) = Pr(colonization between t and t+1|habitat changed from H t to H t+1 )

Developing a Model = Pr(detection in survey j, season t|unit is habitat H)

Example H i = A A B h i =

{Habitat, Occupancy} Pair Consider a unit to be in 1 of 4 possible states: 1. {A, absent} 2. {A, present} 3. {B, absent} 4. {B, present} Initial probability vector:

{Habitat, Occupancy} Pair Transition probability matrix:

{Habitat, Occupancy} Pair - Detection vector examples

{Habitat, Occupancy} Pair Likelihood formulation same as for previous modelling.

Relationship with Multi-state Occupancy TrueHabitat StateOccupancy 0A0 1A1 2B0 3B1

Relationship with Multi-state Occupancy Observed State {Habitat, Occ} True State {Habitat, Occ}0 {A,0}1 {A,1}2 {B,0}3 {B,1} 0 {A,0} {A,1}00 2 {B,0} {B,1}00

Habitat and Occupancy|Habitat Habitat state variable has 2 possible states: A and B Occupancy|habitat has 2 possible states: present and absent Consider the transitions of the state variables separately.

Habitat and Occupancy|Habitat Habitat Occupancy|Habitat

Habitat and Occupancy|Habitat Detection

Habitat and Occupancy|Habitat Likelihood formulation (I think) where

Special Case If habitat A = suitable and B = unsuitable (e.g., Lande 1987, 1988), this implies that: is undefined

Extensions Can (relatively) easily extend the model to allow >2 habitat types, and/or >2 occupancy types. Could also extend occupancy portion of model to accommodate 2 or more species, with potential interactions between them.

Interesting Biology/Conservation Concept of “extinction threshold” (Lande 1987, 1988): when proportion of suitable habitat drops below a certain threshold, then Pr(system-wide extinction) 1. Can model unconditional (regardless of habitat state at t +1) and system-wide Pr(extinction) as a function of proportion of patches that are suitable.

Interesting Biology/Conservation Incorporation of succession and habitat alteration into occupancy modeling (Ellner-Fussmann 2003). Extinction/colonization in response to changes in habitat (e.g., Breinenger, Florida scrub jay) permit much stronger inferences than inferences from: Static habitat-occupancy patterns (incidence function approach). Dynamic approach in which extinction and/or colonization are modeled as functions of habitat at start ( t ) or end ( t +1) of interval.

Interesting Biology/Conservation Multi-species extension:  Different species exhibit different relationships between vital rates and habitat states.  Occupancy by 1 species influences vital rates of other species.  Permits tests of Caswell-Cohen ideas about predator-mediated and disturbance-mediated coexistence of competing species in patch systems.

Example 1 Spotted salamanders (Ambystoma maculatum) in Canaan Valley National Wildlife Refuge, WV. 63 pools defined as small ( 25m 2 ) each year , egg mass detection

Canaan Valley NWR Change in pond size varies annually or is different if salamanders present in previous year occupancy depended upon pond size. Colonization and extinction probabilities depended upon pond size in previous and/or current year.

Canaan Valley NWR 63 models fit to the data. Based upon AIC:  Probabilities of ponds changing size varied annually  Colonization and extinction depend upon pond size in current rather than previous year

Canaan Valley NWR In 2005, approx 50% ponds were small

Canaan Valley NWR 2005 occupancy:  0.18 in small ponds  0.58 in large ponds Colonization  0.10 if small pond in current year  0.39 if large pond in current year Extinction  0.15 if small pond in current year  0.07 if large pond in current year

Example 2 Spotted salamanders at Patuxent Research Refuge Pools defined as ‘suitable’ if standing water, ‘unsuitable’ otherwise

Patuxent RR Early and late spring egg mass surveys, dip net summer survey 2 observers with each survey 3 ‘seasons’ per year 56 pools

Patuxent RR March 2006, probability pond contained standing water = t+1 = 2 April t+1 = 3 June t+1= 4 March t+1 = 5 April t+1 = 6 June t+1 = 7 March t+1 = 8 April t+1 = 9 June 0.30 (0.10) 0 (-)0.83 (0.06)0.25 (0.18)0 (-) 0.81 (0.06) 0.01 (0.01) 0.03 (0.03) 0.99 (0.01) 0.16 (0.08)1.0 (-)0.99 (0.02)0.02 (0.02)1.0 (-) 0.66 (0.10) 0.86 (0.15) 1.0 (-)0.70 (0.09)1.0 (-) 0.24 (0.12)1.0 (-) 0.96 (0.03) 0.99 (0.02)

Patuxent RR March 2006 occupancy of pools containing standing water = t+1 = 2 April t+1= 4 March t+1 = 5 April t+1 = 7 March t+1 = 8 April 0.62 (0.17) 0.47 (0.09) 0 (-) 0.40 (0.07) na † 0.47 (0.11) 0.91 (0.08) 0.05 (0.05) 0 (-) 0.19 (0.10)

Patuxent RR