seasonal interaction factor

[giulia.santulli]

Hello,
I have a question that I think can be useful for others working with the two-species multi-season model. As suggested by Darryl, I've used a transition probability matrix to obtain seasonal occupancy (viewtopic.php?f=11&t=2566#p8127). What I got is the probability of the four possible states of occupancy of a cell (both species present, only A, only B, none) over the studied time period (2000 - 2011). Here my results:
AB Ab aB ab
2000 0.12 0.47 0.52 0.13
2001 0.14 0.47 0.45 0.16
2002 0.16 0.48 0.41 0.19
2003 0.16 0.50 0.37 0.21
2004 0.15 0.51 0.34 0.23
2005 0.15 0.52 0.32 0.24
2006 0.14 0.54 0.30 0.25
2007 0.14 0.55 0.28 0.26
2008 0.13 0.56 0.27 0.27
2009 0.13 0.58 0.26 0.27
2010 0.12 0.59 0.25 0.28
2011 0.12 0.60 0.24 0.28

Occupancy of species A increases over time, while occupancy of species B decreases. Now, it would be great to derive the species interaction factor (SIF) over time from this table with the formula phi= psiAB/psiA x psiB, but I'm not sure if it is conceptually and mathematically correct, and if it make sense to hypothesize that the interaction between to competitive species changes over time, in relation to changes in occupancy.
any advice would be very appreciated!

Giulia

[darryl]

Hmm, something is not quite right here as each row should sum to 1.... Which parameterisation are you using?

[giulia.santulli]

Sorry,
the vector I used is
[ psiAB psiA-psiAB psiB-psiAB 1-psiA-psiB+psiAB]
and the matrix is the one you reported in the previous post.

I used the psiBa/rBa AND the phi/delta parametrization,
where the estimates obtained from my best models are (psiBa/rBa par.):
psiA = 0.4675
psiBA = 0.249
psiBa =0.7637

and (phi/delta par.):

psiA 0.4675
psiB 0.5231
phi 0.4761

Probably I got lost with the meaning of those parameters. Shall I use psiB from the phi/delta parameterization, and psiBA from the psiBa/rBa parametrization?
I calculated psiAB from the phi estimates as psiAB= psiA*psiB*phi and then calculate phi back with phi=psiAB/psia*psiB...is it the right way?
This is the corrected table (sum of rows[1:4]=1), where last column is phi value, calculated as reported:

AB Ab aB ab phi
2000 0.116 0.351 0.407 0.126 0.476
2001 0.129 0.361 0.360 0.150 0.540
2002 0.132 0.373 0.325 0.169 0.572
2003 0.131 0.386 0.299 0.184 0.588
2004 0.126 0.400 0.278 0.196 0.594
2005 0.121 0.413 0.261 0.205 0.595
2006 0.116 0.426 0.246 0.212 0.593
2007 0.111 0.438 0.233 0.218 0.589
2008 0.107 0.450 0.222 0.222 0.585
2009 0.103 0.460 0.212 0.225 0.580
2010 0.099 0.471 0.203 0.228 0.576
2011 0.096 0.480 0.195 0.229 0.572

Values of phi indicates that the species have smaller probability of co-occuring than expected under a hypothesis of independence, all over the studied time period.
Is it correct?
Thank you

Giulia

[darryl]

I'd assumed that you'd used the psiBa/rBa parameterization. The transition matrix I gave you is only correct for that parameterisation. I'm not sure exactly how the parameters are defined for the other parameterisations, Jim Hines might chip in on that one.

Using the psiBa/rBa parameterisation, then the initial state vector would be
[psiA*psiBA psiA*(1-psiBA) (1-psiA)*psiBa (1-psiA)*(1-psiBa)]

You can calculate phi in the manner you describe if you want to.

Cheers
Darryl