A joint model of species interaction dynamics for multistate processes incorporating imperfect detection

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1 A joint model of species interaction dynamics for multistate processes incorporating imperfect detection Jay E. Jones and Andrew J. Kroll Weyerhaeuser Company, 22 Occidental Avenue S, Seattle, Washington 984 USA Citation: Jones, J. E., and A. J. Kroll. 26. A joint model of species interaction dynamics for multistate processes incorporating imperfect detection. Ecosphere 7(:e477..2/ecs2.477 Abstract. The dynamics of species interactions are critical to understanding community ecology. Although species co- occurrence models have long been available, recent methodological advances support extensions to incorporate temporal population dynamics with variation in the observation process. Contemporary dynamic co- occurrence models often focus on simple presence/absence despite evidence that the influence of interactions may extend beyond species presence to behaviors influencing fitness such as mating success or recruitment. Here, we propose a joint model of species interaction dynamics that extends to multistate processes for each species, while limiting the attendant combinatorial expansion in the number of parameters. In our model, one or more states of a subordinate species are assumed to depend on the state of a dominant species, but a reciprocal dependency does not exist. This model results in a more concise representation of dominant- subordinate species interactions compared with other approaches, which should aid in model setup, fitting, and interpretation. The basic framework we outline can be fit in either Bayesian or maximum- likelihood modes of inference, can include interaction lag effects, and deals efficiently with cases where data are missing for one species but not the other. We evaluate interactions between the northern spotted owl (Strix occidentalis caurina and barred owl (Strix varia the latter species is considered to be dominant using long- term empirical data collected in Oregon, USA. Our results suggest state- specific interactions between the two species where barred owls are associated with declines in spotted owl pairings, but not necessarily occupancy. Key words: community assembly; competition; co-occurrence dynamics; detection probability; state-space models. Received 8 June 26; accepted 2 June 26. Corresponding Editor: D. P. C. Peters. Copyright: 26. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. jay.jones@weyerhaeuser.com Introduction Biotic interactions such as competition, facilitation, and mutualism are important determinants of species turnover and fundamental in shaping community structure (Diamond 975. In addition, quantifying spatial and temporal variation in species interactions has meaningful conservation and management applications, including control of invasive organisms, reduction in hyperabundant populations, and developing species reintroduction efforts (Scott et al. 22, Strayer 22. To evaluate long- standing hypotheses in community ecology, workers have developed methods to evaluate co- occurrence patterns of multiple species (Gotelli and McCabe 22. However, contemporary advances in statistical methodology support development of more realistic and flexible models to quantify how organisms interact while explicitly separating observation error from process error. Questions in community ecology may extend beyond the impact of species interactions on occurrence to effects on behaviors such as mating or reproduction. For example, MacKenzie et al. (29 developed a dynamic model for different states (e.g., a single organism, a pair, or a reproductive pair that accounts for imperfect October 26 v Volume 7( v Article e477

2 species detection. This model estimates initial state probabilities, as well as state- dependent between season transition probabilities which may be represented in a transition probability matrix (TPM. Miller et al. (22 and Yackulic et al. (24 utilized the MacKenzie et al. multistate model to investigate co- occurrence dynamics. In both cases, each possible occupancy combination of two species is considered as a separate state. Similarly, Waddle et al. (2 provided an alternative parameterization of a single- season co- occurrence model under the restriction that occupancy of a dominant species may impact the occupancy status of a subordinate species, but that a reciprocal dependency does not exist. This additional structure allows for a direct parameterization of occupancy and detection for each species as well as the interaction among species. Although the Miller et al. (22 and Yackulic et al. (24 applications did not consider multiple states for an individual species, the basic framework of the MacKenzie et al. (29 model allows for such extensions. For example, if each of two species had three states of interest, the combinations could be modeled as nine potential states using a 9 9 TPM with 72 free parameters. Two potential limitations exist with this approach. First, the number of possible states, and thus parameters, rises quickly with the number of states for each individual species, potentially making both inference and interpretation difficult. Second, this approach may be inefficient with regard to missing data. For example, if data for a site in a given year are missing for one species and not the other, the non- missing data must be discarded because the combined state is unknown. We address these issues with a joint model for species interaction dynamics. In this article, we describe a model for estimating across- season species interaction dynamics in multistate systems for a dominant and subordinate species while incorporating imperfect species detection. Our model combines the dynamic multistate approach presented in MacKenzie et al. (29 with the joint species parameteri zation of Waddle et al. (2. This model results in lower dimensional representations of dominant- subordinate species interactions compared with other approaches, which should aid in model setup, fitting, and interpretation. Additionally, Bayesian implementations of this method are efficient with missing data. We begin by describing a joint state- space model for two species. We follow with an expression of the unconditional model likelihood and show that this likelihood can be represented in the form of the MacKenzie et al. (29 dynamic multistate model with suitable restrictions on detection probabilities and transition parameters. Finally, we use the model to evaluate an empirical example of indirect competition between northern spotted (Strix occidentalis caurina and barred owls (Strix varia in Oregon, USA. Basic Sampling Situation The data requirements for estimation with our model are similar to other occupancy- based met hods for multiseason inference (Pollock s robust design, Pollock 982. We assume that a sufficient sample of R conditionally independent units is available from a defined population of units, all owing for generalization to the population. Each of the R units is to be sampled across T primary sampling occasions (usually seasons or years over which interval the population is assumed to be open. Transition parameters are estimated across the primary sampling occasions. Within each primary sampling occasion, a site is visited J times over an interval of sufficiently short duration that the state of the site can be assumed closed for all species included in the model. The number of visits and the number of primary sampling occasions need not be the same across all sampling units. At each primary and secondary sampling occasion, an observer records the observed state of both species to be included in the model, where each species may be characterized by two or more states. In our representation, we assume that ambiguity in the observed state (for each species is unidirectional, extending only to higher states. Consider a three- state example ( = unoccupied, 2 = occupied, and 3 = reproducing. Detection of state 3 allows no uncertainty in the true state. If state is detected, the true state could be, 2, or 3. An observation of state 2 would imply a true state of either 2 or 3. We note that such unidirectional uncertainty 2 October 26 v Volume 7( v Article e477

3 assumptions are not strictly necessary, and more general detection probability models may be used (Miller et al. 23. Species Interaction Model We describe our model for two species of interest, S and D, where each species may be characterized by two or more states. We let vectors Z S and ZD represent the true (possibly latent states of, respectively, species S and species D for site i at primary sampling occasion t. For example, if species S had three possible states, and the true state of site i at time t was 2, we would denote this situation with z S = [ ]. Each component of the vector can be indicated with a subscript, so that in the previous example, z S =, zs = and,3 zs =. In our model, we,2 represent Z S and ZD as random vectors following a multinomial distribution. Similar to Waddle et al. (2, our model is developed for an interaction pattern where we assume that the state of a subordinate species, S, is dependent on a dominant species state in the sense that the state prevalence or transition probabilities for species S vary systematically with the dominant species state. Additionally, we assume that the state of the dominant species, D, is independent of the subordinate species state, such that state and transition probabilities for species D do not depend on the state of species S. This structure leads to a natural factorization of the joint probability distribution of Z S and ZD into a marginal distribution for the dominant species and a conditional distribution ( for the ( subordinate ( species, that is, f Z D,ZS = f Z D f Z S ZD. Separating the models in this way leads to several practical modeling benefits, as described in Waddle et al. (2. The marginal distribution for the dominant species can be modeled using the statespace approach of MacKenzie et al. (29. We let ϕ D define the state probability vector for the initial season, where Z D Mult ( ϕ D,. i, For subsequent seasons (t =, 2,, T, we assume that the random vector Z D follows a multinomial distribu tion of the form Z (Z D ZD Mult D ΦD,, where Φ D is a t t TPM and Z D is the transpose of the state vector for the prior time period. TPM defines the probability of a unit being in each true state at time t, given a state at time t. The matrix element ϕ [m,n] from row m and column n determines the probability of a unit transitioning t from state m at time t to state n at time t. The product of Z D and ΦD returns the multinomial probability vector for the state at time t t given the (imputed state at time t. The state distribution of the subordinate species in the initial season is assumed to follow a multinomial distribution, with a state probability vector that is dependent on the true state of ( species ( D: Z S i, ZD Mult i, ϕ S Z D i, ( ϕ S Z D,, where i, represents the conditional state probability vector. Subordinate species states for subsequent seasons are assumed to depend on both the previous state of the subordinate species and the current state of the dominant ( species: that is, Z (Z S ZS,ZD Mult S ΦS Z D,, where t Z S is the transpose of the subordinate species ( state vector at time t and Φ S Z D is a transition probability matrix with values that depend t on the current state of the dominant species. Alternatively, the model could be specified so that the state distribution for species S depends on the state of species D from the previous year: that is, Z S ZS,ZD Mult (Z S ΦS t ( Z D,. We note that a -yr lag effect on the species interaction could be modeled with this type of dependency. Consistent with the state- space approach of MacKenzie et al. (29, we model the observed state vector y, for either species, as a multinomial random vector, conditional on the true state Z, for example, Y,j p,z Mult (Z p,, where p is a M M matrix with M equal to the number of possible states for the species and Z is the transposed state vector. We denote the probability of observing a territory in state n during survey j of year t, given a true occupancy state m as p [m,n] t,j, where m and n also represent the rows and columns, respectively, of matrix p. Uncertainty in observed states is assumed to be 3 October 26 v Volume 7( v Article e477

4 Fig.. Directed graph of the proposed dynamic model of species interaction. The nodes with a Z represent the true state vectors, and nodes with a Y represent the observed state vectors. We indicate probabilistic dependencies with an arrow. Dashed arrows represent optional dependencies for the observation process. In our example, we consider spotted owls to be the subordinate species (indicated with superscript S and barred owls to be the dominant species (superscript D. Arrows in the graph are used to denote the assumption that the state of species S at time t, Z S, depends on both the previous state of species S, Z S, as well as the current state of species D, Z D. Similarly, arrows from Z to Y indicate that the detected species state at time t depends on the true species state at time t. constrained such that p [m,n] = for any n > m, that t,j is, the probability of observing a state higher than the true state is zero. For example, if the true state of territory i in year t was state 2, with three possible states, then the multinomial probability vector [ for the observed ] state at visit j, y i,j, would be, which would p [2,2] t,j p [2,2] t,j be the second row of the 3 3 matrix p. In this case, the probability of observing state 2 would be p[2,2] t,j. We note that the detection probabilities, p [m,n] t,j, may be parameterized to include associations with covariates. In particular, one could parameterize the detection probabilities of one species to be associated with the true state of the other species. The dependency structure outlined in our model allows for convenient representation as a directed graph (Fig.. The illustration highlights the dependence of the subordinate species state at time t on both the subordinate state at time t as well as the dominant species state at time t. As noted above, we could alternately specify the model with the subordinate species state at time t being dependent on the dominant species state at time t, in which case the arrows would be directed from Z D to ZS. The graph also illustrates the potential for a dependency of the subordinate species observed state on both the current true state and the true state of the dominant species (although the form of the dependencies differs. The structure of the subordinate- dominant species interaction model, where the dominant species state is assumed to be independent of the subordinate species state, results in reduced dimensionality compared with unstructured interaction models. Consider our earlier example with two species, with each species characterized by three states. Extending the approach of MacKenzie et al. (29 to this setting results in a 9 9 TPM with 72 free parameters. Assuming that the species follow a subordinate- dominant model (Fig., the transition probabilities for the dominant species can be represented by a 3 3 matrix with six free parameters, and the transition probabilities for the subordinate species can be represented by a array (3 prior states 3 current states 3 dominant species states with 8 free parameters. The dimensionality of the transition probabilities in this example is reduced from 72 parameters to 24 parameters. Similar reductions occur with the detection probability parameters. The extent of dimension reduction increases with the number of states. We describe two methods of estimation for our model. The first is a direct implementation of the state- space model using Bayesian Markov chain Monte Carlo (MCMC methods. The second, after further development, is to maximize the unconditional model likelihood. Bayesian Estimation The state- space representation described above may be fit within a Bayesian framework and allows us to calculate estimators based on the latent state vectors, as well as providing for easier incorporation of random effects into the model. The posterior distribution for the joint species interaction model is readily specified using the directed graph in Fig. and can be expressed as 4 October 26 v Volume 7( v Article e477

5 Pr (p S,p D,Φ S,Φ D,ϕ S,ϕD,ZS,Z S Y S,Y D Pr (Y S,Y D p S,p D,Φ S,Φ D,ϕ S,ϕD,ZS,Z S Pr (p S,p D,Φ S,Φ D,ϕ S,ϕD,ZS,Z S [ = (Pr(Y S i, ps,z S i, Pr (YD i, pd,z D i, i Pr (Z S i, ϕs,zd i, Pr (ZD i, ϕd ( ( Pr (Y S ps,z S Pr (YD pd,z D t Pr (Z S ΦS,Z S,ZD Pr (ZD ΦD,Z D ] Pr (p S,p D,ϕ S,ϕD,ΦS,Φ D where terms are as defined above. We note that detection, transition, and initial state probabilities may all be parameterized to include covariates. This model can be fit using standard MCMC methods (Gelman et al. 24. An added advantage of using a Bayesian framework is the facility with handling missing response data. Posterior predictions for missing response variables are automatically generated in MCMC software such as JAGS or WinBUGS (Spiegelhalter et al. 23, and values are integrated over uncertainty in the model parameters. By factoring the joint species interaction model into a marginal and conditional distribution as described above, we can make use of all available data, even when missing for one species but not the other. For example, suppose that at time period t, some sites are missing response data for species D but not species S. Within the MCMC framework, missing state vectors y D and,j zd would be imputed for time period t and passed to the state submodel for S, which would use the observed vector y S and include all posterior uncertainty in,j the predicted quantities of the dominant species. In contrast, neither the approach of Miller et al. (22, if expanded to multiple states for each species, nor the likelihood approach described below, could use the non- missing data for y S in,j this example. In this way, potential gains in efficiency exist for this approach when data are missing for one species but not the other. Likelihood Estimation The above state- space representation of this model is fit readily within a Bayesian framework, but the latent state parameters may pose estimation and computational challenges within a frequentist framework. However, it is possible to integrate analytically over the latent states and specify an unconditional likelihood for this model. Our development of the unconditional model likelihood assumes familiarity with the likelihood approach of MacKenzie et al. (29, as a similar specification is used here. Before specifying the joint likelihood, we consider the marginal likelihood of the dominant species D and the conditional likelihood of the subordinate species S. As species D is assumed independent of species S, the marginal parameterization is identical to that described in MacKenzie et al. (29. We let p D represent the h, state- dependent detection probability vector for history h of species D at location i and time t. Each element of the vector corresponds to the probability of the detection history h for a different true state. For example, if species D was characterized by three states (, 2, and 3, the detection probability vector would be defined as: Pr (h z D, = p D h, = Pr (h z D,2 = Pr (h z D,3 = where z D is the jth element of the true state vector,j z D, which is equal to if the true state is j, or zero otherwise. Additional parameters used to specify the likelihood include, ϕ D and ΦD, the initial state probability vector and TPM, respectively, for species D. The former is a vector of length equal to the number of possible states of species D, with elements defining the initial probabilities of a site belonging to each state. The TPM represents the state- dependent transition probabilities from season t to t. For example, the TPM for a species with three possible states could be written as [ ϕ D[,] ϕ D[,2] ϕ D[,3] ] Φ D = ϕ D[2,] ϕ D[2,2] ϕ D[2,3] ϕ D[3,] ϕ D[3,2] ϕ D[3,3] where rows represent the true state at time t, and columns represent the true state at time t. Each row in the matrix must sum to one, so M free parameters will exist for each row in a matrix 5 October 26 v Volume 7( v Article e477

6 of size M. For example, in this parameterization, ϕ D[3,2] represents the probability of a site transitioning from state 3 in season t to state 2 in season t. For the subordinate species S, we define similar quantities, with the exception that the parameter values are assumed to depend on the state of species D. With a slight abuse of notation, we let p S D=j represent the state- dependent detec- h, tion probability vector for species S when species D is in state j. Similarly, we let ϕ S D=j and Φ S D=j represent the initial state probability vector and TPM for species S when species D is in state j. To develop the unconditional joint likelihood, we consider each combination of species states as a separate state. For example, if each species was represented by two states (, 2, then four combined states exist: {, 2, 2, 22}. Likewise, if each species is represented by three states (, 2, 3, then nine combined states exist: {, 2, 3, 2, 22, 23, 3, 32, 33}. The number of states for each species need not be the same. As noted above, the detection probability vector for species S depends on both the true state of species S and D. We can represent the detection probabilities for species S across all the states of D with an expanded vector denoted p S D h,, where M D is the total number of states for species D: p S D h, = p S D= h, p S D=2 h, p S D=M D h,. The state- dependent detection probability vector for the combined histories of D and S is the product of the state- dependent detection probabilities for each species. For example, if two states (, 2 characterized both species, then the joint detection history vector can be expressed as Pr (h D z D, = Pr (h S z S, =,zd, = Pr (h D z D, p DS = Pr (h S z S,2 (h =,zd, = D z (h D,2 = S z S, =,zd,2 =. h = Pr Pr (h D z D,2 = Pr Pr (h S z S,2 =,zd,2 = This vector can be represented compactly in matrix notation using the detection probability vectors for each individual species p DS = ( Diag ( p D h h IMS p S D, h where denotes the Kronecker product and I MS is a diagonal identity matrix of size equal to the number of possible states for species S, M S. To illustrate, suppose that species D and S were both characterized with two states ( = unoccupied, 2 = occupied with detection histories h D = and h S = 22. Then, [ ] p D, = ( p D[2,2] 3 and [ ] p S D=j = ( 22, p S D=j[2,2] p S D=j[2,2] p S D=j[2,2], where, for species S, the value of p S D = j[2,2] will differ depending on the unknown state of species D. The joint detection probability vector for this example would then be as follows: p DS,22, ( p S D=[2,2] p S D=[2,2] p S D=[2,2] = ( p D[2,2] 3. ( p D[2,2] 3 ( p S D=2[2,2] p S D=2[2,2] p S D=2[2,2] The joint initial state probability vector is developed similarly, in terms of the individual species marginal and conditional state probability vectors, which for the two- state case is specified: Pr (z D, = Pr (z S, = zd, = Pr (z D, = Pr (z S,2 (z = zd, = (z D,2 = S, = zd,2 = ϕ DS = Pr Pr = ϕ D[] ϕ D[] ϕ D[2] ϕ D[2] (z D,2 = Pr ϕ S D=[] ϕ S D=[2] ϕ S D=2[] ϕ S D=2[2] Pr (z S,2 = zd,2 =. As with the detection probability vector, the initial joint state probability vector can be represented in matrix notation using the marginal and conditional initial state probability vectors 6 October 26 v Volume 7( v Article e477

7 ϕ DS = ( Diag ( ϕ D IMS ϕ S D where ϕ S D is an expanded vector of all of the conditional initial state vectors for species S: ϕ S D = ϕ S D= ϕ S D=2 ϕ S D=M D. The TPM for the joint model expresses the probabilities of moving between any possible combinations of states. For example, in the running case of two possible states for each species, the matrix below describes the possible transitions from one season to the next, where rows represent the true state at time t, and the columns represent the true state at time t:,,,,2, 2,, 2,2,2,,2,2,2 2,,2 2,2 2,, 2,,2 2, 2, 2, 2,2. 2,2, 2,2,2 2,2 2, 2,2 2,2 The quantity in the first row and column therefore represents the transition from state (, at time t to state (, at time t. The quantity in the third row and fourth column represents the transition from state (2, at time t to state (2,2 at time t. If the marginal and conditional transition probabilities for species D and S, respectively, are defined as above, then the joint TPM for the dominant- subordinate relationship for the two- state case can be specified as: transition probabilities are independent of species S. For the general case of M D dominant species states and M S subordinate species states, we define an expanded conditional TPM for species S as a block diagonal matrix of the state- specific TPM: Φ S D = From which the joint TPM may be calculated as: where Φ D is the marginal TPM for species D, denotes the Kronecker product, and I MS is an identity matrix of size M S. Defining each unique combination of subordin ate and dominant species states as a separ ate joint state, but with the added restrictions detailed above, allows us to fit this species interaction model within the framework of the MacKenzie et al. s (29 multi state occupancy dynamic model. The unconditional probability of detection history h i across all seasons for site i can be expressed using the same matrix notation as in MacKenzie et al. (29: Pr ( h i θ =ϕ DS Φ S D= Φ S D=2 Φ S D=M D Φ DS = ( Φ D I MS Φ S D, T t= [ ( Diag p DS. Φ DS t ] p DS i,t ϕ D[,] ϕ S D=[,] ϕ D[,] ϕ S D=[,2] ϕ D[,2] ϕ S D=2[,] ϕ D[,2] ϕ S D=2[,2] Φ DS ϕ = D[,] ϕ S D=[2,] ϕ D[,] ϕ S D=[2,2] ϕ D[,2] ϕ S D=2[2,] ϕ D[,2] ϕ S D=2[2,2] ϕ D[2,] ϕ S D=[,] ϕ D[2,] ϕ S D=[,2] ϕ D[2,2] ϕ S D=2[,] ϕ D[2,2] ϕ S D=2[,2] ϕ D[2,] ϕ S D=[2,] ϕ D[2,] ϕ S D=[2,2] ϕ D[2,2] ϕ S D=2[2,] ϕ D[2,2] ϕ S D=2[2,2] ( where Diag where ϕ D[m,n] represents the probability of species D transitioning from state m to state n, and ϕ S D = j[m,n] represents the probability of species S transitioning from state m to state n, when species D is in state j. In this example, a total of 2 parameters, and six free parameters, exist. Without the subordinate- dominant restriction, the TPM above would have 6 total, and 2 free parameters (three per row, the difference coming from the assumption that species D p DS is a diagonal matrix with p DS on the diagonal, θ denotes all model parameters, and all other terms are as defined above. The distinction between this model and that of MacKenzie et al. (29 lies in the construction of the parameters ϕ DS, pds, and ΦDS to reflect the t restrictions implied by the dominant- subordinate species interaction assumption. The joint model likelihood across all sites and seasons is expressed below, and 7 October 26 v Volume 7( v Article e477

8 maximum- likelihood estimates may be obtained using general optimization algorithms: Northern spotted owls and barred owls: data We apply the above model to data from visits to 47 northern spotted owl sites from March to August on Weyerhaeuser s Millicoma Tree Farm, Coos Bay, Oregon, USA. Data were collected following survey protocols developed by the United States Fish and Wildlife Service (99, 2, 22 and The Pacific Southwest Forest and Range Experiment Station (Miller 99. A complete visit to a site either located owls or covered all potential habitats within one mile of the most recent activity center. A single visit may include multiple outings spread over several days and nights. During each visit, technicians recorded one of three possible states for both spotted and barred owls: no owls detected, a single owl detected, and a pair of owls detected. We emphasize that the survey protocol is designed to detect spotted owls: barred owls are detected opportunistically during these surveys. Northern spotted owls and barred owls: models and estimation We considered three states for the northern spotted owl (unoccupied, single, and pair and two states for the barred owl (unoccupied, occupied. For the barred owl occupancy process, we let Z D denote barred owl initial i, occupancy status for site i, and Z D denote the occupancy status at a subsequent time period t. Following Royle and Kery (27, we assumed that the initial occupancy state follows a Bernoulli distribution Z D Bern ( ψ D, and subsequent states depend on the prior state and local colonization (γ and extinction parameters (ε: Pr (h θ = L (θ h = Z D ZD Bern (( Z D R Pr ( h i θ. Common colonization and extinction parameters were used over all seasons. We modeled the observation process as a quadratic function of Julian data and an indicator i= γ+z D ( ε. variable for whether the survey was conducted during the day or night. We allowed the detection parameters to vary across years via random effects, assuming that parameters may be similar, but not identical, across seasons: y (Z D,j Bern D pd and,j ( ( logit p D =α D,j +ad,i + α D +ad,i ( + α D 2 +ad 2,i JD,j JD 2,j + ( α D 3 +ad 3,i Night,j where (,σ ad N 2 q,i q for covariates q =,, 2, 3. We assumed that the northern spotted owl is subordinate to the barred owl, in a manner consistent with our model. We parameterized the NSO TPM (Φ S in terms of occupancy probability (ψ and the probability of a spotted owl pair conditional on occupancy (θ, similar to the parameterization in Nichols et al. (27: ψ,s ψ,s ( θ,s ψ,s θ,s Φ S = ψ 2,S ψ 2,S ( θ 2,S ψ 2,S θ 2,S ψ 3,S ψ 3,S ( θ 3,S ψ 3,S θ 3,S To satisfy our dominant- subordinate model assumptions, we allowed the occupancy and pair probabilities in each row of the TPM to vary with the imputed barred owl occupancy status: ( logit ( logit ψ m,s θ m,s =β m,s =δ m,s +β m,s Z D and +δ m,s Z D. The parameters β m,s and δ m,s are the log- odds ratio for the association of barred owl occupancy with spotted owl occupancy probability and pair probability, respectively, given prior spotted owl state m. We modeled spotted owl detection probabilities as functions of Julian date, barred owl status, and an indicator for night surveys. Following Kroll et al. (26, we allow the covariate effects to vary across seasons by incorporating random effects via a multilogit transformation to ensure that row probabilities sum to one. Specifically, we assume for occupied territories with true state m and observed state n that 8 October 26 v Volume 7( v Article e477

9 p S[m,n],j exp (α [m,n] = Pr (y S,j,n = zs,m = =,t m for < n m, and p S[m,] itj + l=2 l=2 exp (α [m,l],t +α [m,n],t +α [m,l],t JD ij +α [m,n] 2,t JD ij +α [m,l] 2,t = Pr (y S itj, = zs it,m = = m + exp (α [m,l] +α [m,l] JD,t,t ij +α [m,l] 2,t JD 2 ij +α[m,n] 3,t JD 2 ij +α[m,l] 3,t JD 2 ij +α[m,l] 3,t z D +α[m,n] Night 4,t itj z D +α[m,l] Night 4,t itj z D +α[m,l] Night 4,t itj with normal random effects for the ( intercept, linear, and quadratic terms: α [m,n] N μ [m,n] q,t q,σ 2[m,n] q for covariates q =,,, 4. We specified prior distributions for the random effects mean and variance as N(,2 and Gamma(,.5, respectively. Temporal trends in state counts were estimated from the posterior distribution using linear contrasts (Kroll et al. 26. Such estimates allow for inference on long- term declines or increases in territory counts by state, without imposing such structured assumptions on the fitted model. We fit the model using JAGS (Plummer 23 called from R (R Development Core Team 2 using package R2jags (Su and Yajima 22. We fit four chains each of length, with a burn- in of and / thinning. Convergence was assessed using a.5 threshold with the Gelman rubin statistic (Gelman et al. 24 and visual inspection of the chains. JAGS code for the model and code for fitting a simplified version of this model via maximum- likelihood appear in Data S. Northern spotted owls and barred owls: results Estimates and 9% posterior credible intervals (Kroll et al. 25 of β m,s and δ m,s, the parameters used to describe the association between barred owl presence and spotted owl transitions to occupied and paired states, respectively, are shown in Table. In general, low precision characterized all of the estimates, reflecting the amount of information contained in imperfect observations from 47 territories. The posterior intervals for most of the transition parameters contained zero, indicating uncertainty around the direction of the association. However, the results suggest a negative association between barred owl occupancy and the probability of a spotted owl territory transitioning from a single state to a paired state ( δ 2,S = 3., 9% CRI = 5.4 to.9. The estimate of 3. on the log- odds scale indicates an expected 2- fold lower odds of having a paired territory follow a single territory when a barred owl is present compared to when a barred owl is absent. In contrast, our estimates for the association between barred owl presence and the probability of a paired state persisting from one year to the next ( δ 3,S =.2, 9% CRI =.4 to.8 or the probability of transitioning from an unoccupied to a paired state ( δ 2,S =., 9% CRI = 3.7 to 2. did not provide a clear indication of a consistent trend. The estimated associations between barred owl presence and spotted owl occupancy were similarly ambiguous (β m,s in Table. Estimated - yr temporal trends indicated a sharp rise in barred owl territory counts, suggesting that counts doubled over the - year span of this data set, with an average increase in 2. occupied territories per year (Fig. 2, Table 2. During the same time period, the results indicate a decline in the number of territories with spotted owl pairs, with an estimated loss of approximately.3 paired territories per year over the ten years examined in this analysis. The trend Table. Estimates of the coefficients for barred owl associations with northern spotted owl occupancy and pair probabilities, given spotted owl prior state, Coos Bay, Oregon, USA, Prior state β m,s (Occupancy δ m,s (Pair Estimate 9% CRI Estimate 9% CRI Unoccupied. 2.2, , 2. (m = Single.5 2.3, ,.9 (m = 2 Pair (m = 3., ,.8 Notes: Posterior medians and 9% posterior credible limits are shown. All estimates are on the logit scale. Negative estimates indicate a lower probability of transitioning to an occupied or paired state when barred owls are present. 9 October 26 v Volume 7( v Article e477

10 Fig. 2. Estimated territory counts (9% posterior credible intervals for barred owls (BDOW and spotted owls (SPOW by occupancy state, Coos Bay, Oregon, USA, Each point represents the posterior mean territory count based on imputed state values. estimates are positive for both unoccupied and single territories; however, in both cases, posterior credible intervals contained zero, indicating uncertainty in the trend direction. Given the observational nature of this data set, we cannot attribute causal explanations to the apparent decline in the number of territories with spotted owl pairs. However, model results for the association between barred owl presence and probabilities of transitioning to a paired state (δ m,s in Table, as discussed above, are suggestive that the decline in paired territories could be due to a barred owl influence on spotted owl pairing, with no clear evidence of an association with spotted owl occupancy (β m,s in Table. Table 2. Estimated linear - yr trends in territory counts by species (barred or spotted owl and occupancy state, Coos Bay, Oregon, USA, State Mean 9% CRI Barred owl: occupancy 2..5, 2.6 Spotted owl: unoccupied.6.5,.5 Spotted owl: single.8.2,.8 Spotted owl: pair.3 2.,.6 Notes: Mean estimates represent the expected change in territory count for a -yr interval. Uncertainty in the estimates is represented with 9% posterior credible intervals. Population mean estimates of detection probability trends indicate variation associated with both by date and by state, with evidence of higher combined detection for paired territories than for single territories (Fig. 3. If we had ignored state- dependent differences in detection (e.g., by combining single and pair territories, we would have missed this difference in detection rate and may have calculated biased estimates of detection probability and occupancy probability. We note that the dip in the estimated probability of detecting a single owl when the true state is pair coincides with the peak probability of detecting an owl pair, because these detection probabilities (along with the probability of detecting no owl must sum to one. The impact of barred owl occupancy on spotted owl detection was not clear from this analysis. We estimated the probability of detecting a single owl to be lower when barred owls were present, and we note that sufficient uncertainty existed in these estimates that we could not preclude negligible or even positive effects. Discussion The model we describe here represents a reparameterization and extension of the species October 26 v Volume 7( v Article e477

11 Fig. 3. Population- average estimates of mean spotted owl detection probabilities by Julian date, occupancy state, and barred owl status (present or absent, Coos Bay, Oregon, USA, Vertical bars represent 9% posterior credible intervals at median Julian date (jittered to prevent overplotting. Panels show detection probabilities for three possible cases: detecting a single owl when the true state was a single owl ( Pr(y = 2 z = 2 ; detecting a single owl when the true state was an owl pair ( Pr(y = 2 z = 3 ; and detecting a pair of owls when the true state was an owl pair ( Pr(y = 3 z = 3. Vertical tick marks above the x- axis indicate the sample minimum, 25th percentile, median, 75th percentile, and maximum Julian date across all years of study. interaction models discussed in Miller et al. (22 and Yackulic et al. (24 to the case of multi state species descriptions with an assumed dominant- subordinate species interaction. Although other options for restricted parameterizations exist (Miller et al. 22, the dominant- subordinate assumption, when appropriate, simplifies conceptualization of model dynamics and substantially reduces model dimensionality compared with an unstructured model of all possible state combinations. A special case of the model we describe, where each species is described by two states, has been implemented in program MARK (White and Burnham 999. Dugger et al. (26 used this model in a demographic study of northern spotted owls and found support for a negative association between spotted owl occupancy and barred owl presence. Our model is described for an interaction pattern where the state probability for the subordinate species depends on current state of the dominant species, as well as the previous state of the subordinate species. As with other approaches (Kroll et al. 2, Dugger et al. 2, this specification may be broadened to include lag effects such as where subordinate species transition probability from time t to time t depends on the dominant species state at time t or potentially other time lags. Information- theoretic approaches have been used in the literature to select among different choices on interaction lags (Kroll et al. 2, Dugger et al. 2. As our example shows, the extension to multistate species interactions provided by our model can provide a more detailed description of these interactions. In our example, information about reproductive state (e.g., presence of nestlings would have provided additional insight into the relationship between barred and spotted owls, but this information was not available for our analysis. The empirical example also serves to illustrate the need for sufficiently large samples in order to obtain estimates with reasonable precision. Our example included observations from 47 owl circles, and even though our model was relatively modest with three potential spotted owl states and two potential barred owl states our parameter estimates showed considerable posterior uncertainty, particularly for parameters related to less frequent states. The number October 26 v Volume 7( v Article e477

12 of sites required to reach a desired level of precision will increase with the number of potential states (MacKenzie et al. 29 and therefore the number of parameters in the model. We reiterate that dimension reduction was a motivation for our model while emphasizing, no matter which model is considered, that a sufficient amount of data must be available to estimate parameters with an acceptable level of precision. A priori simulation studies may assist in helping to determine data requirements to reach target levels of precision (Kroll et al. 25. Our model may not be appropriate for all species interactions. The parameterization of our model derives from the assumption that dominant species state dynamics are independent of the subordinate species, which may not always occur. For example, in a predator/prey scenario, a collapse in the prey population could lead to local extinction of the predator. In other cases, temporal changes in habitat quality may impact state dynamics independent of any interactions. Such influences may confound estimated interaction effects if not properly accounted for in the model setup. Alternatively, one may wish to test the assumption of independence using a more general model. As we show, the proposed model may be fit within a Bayesian framework or via maximum likelihood after integrating over the latent state variables. Although either approach may be seen to have advantages, a Bayesian approach allows for the calculation of estimands based on state parameters (see Fig. 2 or Table 2. The unconditional model likelihood, used for maximumlikelihood estimates, also has utility for model comparison in a Bayesian framework. WAIC is a particularly attractive AIC- like approach for comparison of Bayesian models (Watanabe 2, Hooten and Hobbs 25. However, the WAIC penalty term may be unreliable for hierarchical models when group size goes to (Vehtari et al. 26, as is typically the case with occupancy models using latent state parameters. This problem may be circumvented by specifying the log predictive density, used in the WAIC calculation, in terms of the unconditional likelihood, even if the model has been fit using a state- space representation. We developed our model to assess long- term co- occurrence dynamics of spotted and barred owls, given associative evidence for negative effects of barred owls on spotted owl occupancy dynamics (Olson et al. 25, Kroll et al. 2, Dugger et al. 2. Also, previous modeling efforts have provided incomplete solutions to the problem of how barred owls influence the detection process for spotted owls, given that barred owls are only detected incidentally during surveys for spotted owls. By fully accounting for the barred owl detection process, our model addresses this latter issue. Waddle et al. (2 identify other potentially suitable patterns of cooccurrence compatible with this structure, such as predator prey relationships and parasitism. Our model would be appropriate for such interactions when added interest exists in interannual dynamics as well as multiple states for each species. Literature Cited Diamond, J. M Assembly of species communities. Pages in M. L. Cody and J. M. Diamond, editors. Ecology and evolution of communities. Harvard University Press, Cambridge, Massachusetts, USA. Dugger, K. M., R. G. Anthony, and L. S. Andrews. 2. Transient dynamics of invasive competition: barred owls, spotted owls, habitat, and the demons of competition present. Ecological Applications 2: Dugger, K. M., et al. 26. The effects of habitat, climate, and Barred Owls on long- term demography of Northern Spotted Owls. The Condor: Ornithological Applications 8:57 6. Gelman, A., J. B. Carlin, H. S. Stern, and D. B. rubin. 24. Bayesian data analysis. Second edition. Chapman & Hall/CRC, Boca Raton, Florida, USA. Gotelli, N. J., and D. J. McCabe. 22. Species co- occurrence: a meta- analysis of J.M. Diamond s assembly rules model. Ecology 83: Hooten, M. B., and N. T. Hobbs. 25. A guide to Bayesian model selection for ecologists. Ecological Monographs 85:3 28. Kroll, A. J., T. L. Fleming, and L. L. Irwin. 2. Site occupancy dynamics of northern spotted owls in the eastern Cascades, Washington, USA, Journal of Wildlife Management 74: Kroll, A. J., T. Garcia, J. Jones, K. Dugger, B. Murden, J. Johnson, S. Peterman, B. Brintz, and M. Rochelle. 25. Evaluating multi- level models to test occupancy state responses of plethodontid salamanders. PLoS One :e October 26 v Volume 7( v Article e477

13 Kroll, A. J., J. Jones, A. Stringer, and D. Meekins. 26. Multistate models reveal long- term trends of Northern Spotted Owls in the absence of a novel competitor. PLoS One :e MacKenzie, D. I., J. D. Nichols, M. E. Seamans, and R. J. Gutiérrez. 29. Modeling species occurrence dynamics with multiple states and imperfect detection. Ecology 9: Miller, G. 99. Standards and guidelines for establishment and implementation of demographic and density study areas for spotted owls. United States Geological Survey, Oregon Cooperative Wildlife Research Unit, Oregon State University, Corvallis, Oregon, USA. Miller, D. A. W., C. S. Brehme, J. E. Hines, J. D. Nichols, and R. N. Fisher. 22. Joint estimation of habitat dynamics and species interactions: Disturbance reduces co- occurrence of non- native predators with an endangered toad. Journal of Animal Ecology 8: Miller, D. A. W., J. D. Nichols, J. A. Gude, L. N. Rich, K. M. Podruzny, J. E. Hines, and M. S. Mitchell. 23. Determining occurrence dynamics when false positives occur: estimating the range dynamics of wolves from public survey data. PLoS One 8:e6588. Nichols, J. D., J. E. Hines, D. I. MacKenzie, M. E. Seamans, and R. J. Gutiérrez. 27. Occupancy estimation and modeling with multiple states and state uncertainty. Ecology 88: Olson, G. S., R. G. Anthony, E. D. Forsman, S. H. Ackers, P. J. Loschl, J. A. Reid, K. M. Dugger, E. M. Glenn, and W. J. Ripple. 25. Modeling of site occupancy dynamics for northern spotted owls, with emphasis on the effects of barred owls. Journal of Wildlife Management 69: Plummer, M. 23. JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling. Pages 2 22 in K. Hornik, F. Leisch, and A. Zeileis, editors. 3rd International Workshop on Distributed Statistical Computing (DSC 23. Austrian Association for Statistical Computing (AASC, Vienna, Austria. Pollock, K A capture- recapture design robust to unequal probability of capture. Journal of Wildlife Management 46: R Development Core Team. 2. R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Royle, J. A., and M. Kéry. 27. A Bayesian statespace formulation of dynamic occupancy models. Ecology 88: Scott, J. M., P. J. Heglund, M. L. Morrison, J. B. Haufler, M. G. Raphael, W. A. Wall, and F. B. Samson, editors. 22. Predicting species occurrences: issues of accuracy and scale. Island Press, Covelo, California, USA. Spiegelhalter, D. J., A. Thomas, N. G. Best, and D. Lunn. 23. WinBUGS version.4 user manual. MRC Biostatistics Unit, Cambridge, UK. Strayer, D. L. 22. Eight questions about invasions and ecosystem functioning. Ecology Letters 5: Su, Y.-S., and M. Yajima. 22. R2jags: a package for running Jags from R. package=r2jags United States Fish and Wildlife Service. 99. Protocol for surveying proposed management activities that may impact northern spotted owls, revised 7 March 992. U.S. Fish and Wildlife Service, Portland, Oregon, USA. United States Fish and Wildlife Service. 2. Protocol for surveying proposed management activities that may impact northern spotted owls. U.S. Fish and Wildlife Service, Portland, Oregon, USA. United States Fish and Wildlife Service. 22. Protocol for surveying proposed management activities that may impact northern spotted owls. U.S. Fish and Wildlife Service, Portland, Oregon, USA. Vehtari, A., A. Gelman, and J. Gabry. 26. Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC Waddle, J. H., R. M. Dorazio, S. C. Walls, K. G. Rice, J. Beauchamp, M. J. Schuman, and F. J. Mazzotti. 2. A new parameterization for estimating co- occurrence of interacting species. Ecological Applications 2: Watanabe, S. 2. Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research : White, G. C., and K. P. Burnham Program MARK: survival estimation from populations of marked animals. Bird Study 46:S2 S38. Yackulic, C. B., J. Reid, J. D. Nichols, J. E. Hines, R. Davis, and E. Forsman. 24. The roles of competition and habitat in the dynamics of populations and species distributions. Ecology 95: Supporting Information Additional Supporting Information may be found online at: ecs2.477/supinfo 3 October 26 v Volume 7( v Article e477