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Keywords:

  • California spotted owl;
  • Lithobates sylvatica;
  • multistate model;
  • patch occupancy dynamics;
  • second-order Markov;
  • Strix occidentalis occidentalis;
  • wood frog

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

1. Dynamic occupancy models are often used to investigate questions regarding the processes that influence patch occupancy and are prominent in the fields of population and community ecology and conservation biology. Recently, multistate occupancy models have been developed to investigate dynamic systems involving more than one occupied state, including reproductive states, relative abundance states and joint habitat-occupancy states. Here we investigate the sensitivities of the equilibrium-state distribution of multistate occupancy models to changes in transition rates.

2. We develop equilibrium occupancy expressions and their associated sensitivity metrics for dynamic multistate occupancy models. To illustrate our approach, we use two examples that represent common multistate occupancy systems. The first example involves a three-state dynamic model involving occupied states with and without successful reproduction (California spotted owl Strix occidentalis occidentalis), and the second involves a novel way of using a multistate occupancy approach to accommodate second-order Markov processes (wood frog Lithobates sylvatica breeding and metamorphosis).

3. In many ways, multistate sensitivity metrics behave in similar ways as standard occupancy sensitivities. When equilibrium occupancy rates are low, sensitivity to parameters related to colonisation is high, while sensitivity to persistence parameters is greater when equilibrium occupancy rates are high. Sensitivities can also provide guidance for managers when estimates of transition probabilities are not available.

4.Synthesis and applications. Multistate models provide practitioners a flexible framework to define multiple, distinct occupied states and the ability to choose which state, or combination of states, is most relevant to questions and decisions about their own systems. In addition to standard multistate occupancy models, we provide an example of how a second-order Markov process can be modified to fit a multistate framework. Assuming the system is near equilibrium, our sensitivity analyses illustrate how to investigate the sensitivity of the system-specific equilibrium state(s) to changes in transition rates. Because management will typically act on these transition rates, sensitivity analyses can provide valuable information about the potential influence of different actions and when it may be prudent to shift the focus of management among the various transition rates.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Occupancy models are often used to investigate questions regarding the processes that influence patch occupancy dynamics and are prominent in the fields of population and community ecology and conservation biology (Simberloff 1969; Hanski 1999; Moilanen 1999; Amarasekare & Possingham 2001; Hill, Witman & Caswell 2004). The development of statistical inference methods that permit estimation of parameters for these models while accounting for imperfect detectability (MacKenzie et al. 2002, 2003, 2006) has led to increased use of patch occupancy models in metapopulation studies (Martin et al. 2009a, 2010; MacKenzie et al. 2011) and large-scale monitoring programmes (e.g. Muths et al. 2005; O’Connell et al. 2006; Collier et al. 2010).

Most applications of occupancy models focus on the presence/absence of a species and the dynamic processes influencing the occupancy status of patches over time. Occupancy dynamics are described through a first-order Markov process, in which the probability a patch is occupied by a target species at time t is dependent on the state of the patch in time t−1 (MacKenzie et al. 2006; Martin et al. 2009b):

  • image(eqn 1)

where ψt represents the probability a patch is occupied at time t and ɛt-1 and γt–1 are conditional, time-specific probabilities of local extinction and colonisation, respectively. This model can also be written in matrix form (MacKenzie et al. 2006; Martin et al. 2009b):

  • image(eqn 2)

where

  • image

and

  • image

The matrix Φt−1 is a transition probability matrix, with elements representing the probabilities of patches remaining within a state from t−1 to t or transitioning to the other state. The transition probability of row m and column n corresponds to the probability that a patch will be in state m in t given it was in state n at t−1; states correspond to rows in the state vector, Πt−1. For example, a patch that is unoccupied at t−1 (i.e. row 2 in Πt-1) will become occupied in t with the probability γt−1 (i.e. row 1, column 2 in Φt−1). Written in matrix form, Martin et al. (2009b) noted the clear parallels with population projection matrices for which sensitivity or elasticity is commonly evaluated (e.g. Crouse, Crowder & Caswell 1987; Brault & Caswell 1993; Doak, Kareiva & Klepetka 1994; Heppell, Walters & Crowder 1994).

More recently, dynamic occupancy models have been developed to allow for multiple occupancy states and account for imperfect detection and state misclassification (Nichols et al. 2007; MacKenzie et al. 2009). Multistate occupancy models are versatile and have been applied to dynamic systems involving reproductive states, relative abundance states and joint habitat-occupancy states (MacKenzie et al. 2009; Martin et al. 2010). Changes in occupancy states are modelled using first-order Markov transition rates. If these transition rates are constant over time, the system will reach a stable-state equilibrium distribution (Caswell 2001; Martin et al. 2009b). The equilibrium-state distribution (Π*) can be calculated as the eigenvector associated with the dominant eigenvalue of Φ, with the eigenvector elements scaled to sum to 1 (Caswell 2001). It is natural to profess interest in the sensitivity of elements of this stable-state distribution to changes in system transition rates (Martin et al. 2009b), just as many investigators have noted the utility of sensitivity analyses for the study of population and community dynamics (de Kroon et al. 1986; Heppell, Caswell & Crowder 2000; Caswell 2001; Hill, Witman & Caswell 2004).

In this paper, we give two general examples of dynamic multistate occupancy models. The first example represents a common case where occupied sites are separated into those with and without successful reproduction. We use published estimates of transition rates associated with occupancy and reproduction for California spotted owl Strix occidentalis occidentalis surveys to examine the sensitivity of the probability of successful reproduction to each of the transition probabilities. The second example describes a novel approach for systems that are best described as second-order Markov processes, which may be especially useful for species with unobservable life-history stages. Here, we develop a model describing the dynamics of wood frog Lithobates sylvatica aquatic life-history stages at Patuxent National Research Refuge, Maryland, USA. In both examples, we clearly define an expression for equilibrium occupancy and explore the sensitivities of equilibrium occupancy to each parameter in the transition probability matrix, Φ, and discuss implications for management decisions.

Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Generally, state parameter distributions are functions of the transition probabilities represented as elements of the transition matrix, Φ (see eqn 2 and examples below). In each of our multistate examples, our first step in examining sensitivities for the state parameters was to derive expressions for each in terms of the transition probabilities. For a given transition probability matrix Φ, the eigenvector (w, elements scaled to sum to one) associated with the dominant eigenvalue (λ) represents the stable-state distribution and is defined by the following equation,

  • image(eqn 3)

(Caswell 2001; Hill, Witman & Caswell 2004). For stochastic matrices in which elements (transition probabilities) in each column (or row) sum to one, the dominant eigenvalue (λ) will be one (Caswell 2001) and Φw = w. By multiplying Φ by w and setting the product equal to w, we can write the elements of the stable-state distribution vector in terms of the lower-level parameters that define the transition matrix elements. Sensitivity expressions are then derived for each lower-level parameter (see specific expressions in examples below) by first identifying the equilibrium state of interest and then taking the partial derivative of this quantity with respect to the focal lower-level transition parameter.

Example 1: California spotted owls

The California spotted owl, one of three subspecies of spotted owl, occurs entirely within California and northern Baja California, Mexico (Franklin et al. 2004). Concern over the limited range of the species has led to several studies investigating factors influencing demographics and dynamics (Seamans et al. 2001; Franklin et al. 2004; Seamans 2005). Multistate occupancy models have been applied to assess occurrence and reproductive success at established owl territories in the Sierra Nevada, California (Nichols et al. 2007; MacKenzie et al. 2009, 2011). Here, we use estimates based on the fieldwork of R.J. Gutiérrez and M.E. Seamans and published by MacKenzie et al. (2009) to parameterize the transition matrix in the model presented below.

Model development

The California spotted owl system includes three occupancy states including territory sites that are: unoccupied (state = 0), occupied without reproduction (state = 1) and occupied with reproduction (state = 2). MacKenzie et al. (2009) expanded the standard dynamic occupancy model (eqn 2) to include these three states. The probability a territory is in state m at time t is given by inline image and inline image. The probability of a territory (site) being occupied at time t, given that it was in state m at time t−1, is given by inline image. Conditional on occupancy at time t, inline image, is the probability of successful reproduction at the territory (state 2), given it was in state m at time t−1. The probability of a territory supporting successful reproduction in year t, given that it was in state m in year t−1, is the product, inline image (MacKenzie et al. 2009). It is important to note that these inline image parameters are equivalent to inline image described by MacKenzie et al. (2009).

The model can be written in the same matrix form as eqn 2, where

  • image

where T denotes the transpose of a vector, and

  • image

Assuming constant transition rates, the system will eventually reach equilibrium.

Sensitivity expressions

Using the general approach described in the first paragraph of the Methods section, we defined expressions for equilibrium-state parameters (inline image) as  functions of the lower-level transition parameters:

  • image(eqn 4)
  • image(eqn 5)
  • image(eqn 6)

where

  • image

In the California spotted owl multistate system, the probability a territory is occupied and supports reproduction (i.e. π[2]) may be of most interest to managers. Sensitivities of π[2]* (i.e. defined as equilibrium probability of occupancy with reproduction) to small changes in each parameter of interest, θ (θ is simply used to define a lower-level parameter), describe how π[2]* changes as θ changes, in cases where the perturbations preserve the column sums of Φ (true in our case because all matrix elements were written as functions of conditional binomial parameters). Sensitivities are computed as the partial derivative of π[2]* with respect to θ (inline image; Caswell 2001; Martin et al. 2009b). The sensitivities of π[2]* given in eqn 6 to each of the six parameters defining Φ are given by the equations:

  • image(eqn 7)
  • image
  • image(eqn 8)
  • image(eqn 9)
  • image(eqn 10)
  • image(eqn 11)
  • image(eqn 12)
Model parameterization

We used the parameter estimates of MacKenzie et al. (2009) for the above model and examined the sensitivity of π[2]* to changes in each of the six lower-level transition parameters. Sensitivity analysis and equilibrium occupancy distributions rely on the premise of time-constant transition probabilities, but the best fitting model for the California spotted owl data suggested that conditional probabilities of reproduction varied among years, inline image (MacKenzie et al. 2009). Still, we believe the concepts of equilibrium states and associated sensitivities are relevant even when a stationary Markov process does not exist (see similar arguments in Caswell 2001; Martin et al. 2009b). Accordingly, we used the time-constant parameter estimates for occupancy, inline image, and averaged the year-specific estimates of conditional reproduction parameters to obtain the following parameter estimates for our analysis (see Fig. 1 in MacKenzie et al. 2009): inline image = 0·17 (95% CI = 0·11, 0·26), inline image = 0·87 (0·79, 0·92), inline image = 0·91 (0·83, 0·95), inline image = 0·13 (range = 0·00, 0·93), inline image = 0·51 (range = 0·05, 0·85), inline image = 0·50 (range = 0·15, 0·78). Using these parameter estimates as starting values, we systematically varied each parameter individually between zero and one, while holding the other five parameters constant and calculated the sensitivities of π[2]* using eqns 7–12 for each combination of parameter values.

image

Figure 1.  Equilibrium occupancy probabilities for California spotted owls in the Sierra Nevada, California, USA. Multistate occupancy probabilities include: unoccupied, occupied without reproduction and occupied with reproduction. Panels (a–f) show the results of varying a single parameter (x-axis) from 0 to 1, while maintaining the other parameters at constant baseline rates estimated by MacKenzie et al. (2009): inline image = 0·17, inline image = 0·87, inline image = 0·91, inline image = 0·13, inline image = 0·51, inline image = 0·50. Filled circles represent the occupancy rates at these baseline values.

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Results

The equilibrium distribution of occupancy states using the transition rate estimates reported in MacKenzie et al. (2009) was π* = [0·396 0·324 0·280], suggesting that at equilibrium, ∼ 60% of owl territories were occupied (inline image) and 28% supported successful reproduction (π[2]*). The equilibrium probability that a territory is occupied and supports reproduction, π[2]*, was most sensitive to changes in thez ϕ[m] parameters, with inline image = 0·666, inline image = 0·652 and inline image = 0·572, and was less sensitive to changes in the conditional reproduction rates (Figs 1 and 2). Intuitively, at low colonisation rates (inline image = 0·17), we would expect π[2]* to be most sensitive to ϕ[0] because there are more unoccupied patches (π[0]* = 0.396) for ϕ[0] to act upon. If colonisation rates remain low, π[2]* is relatively insensitive to variation in R[0] because few unoccupied territories are colonised. Because the probability of local extinction is low ((inline image) = 0·13, (inline image) = 0·09), π[2]* is more sensitive to R[1] and R[2], than R[0] (inline image = 0·302, inline image = 0·273, inline image = 0·072).

image

Figure 2.  Sensitivity of equilibrium probability of occupancy with reproduction (π[2]*) to changes in the values of parameters, ϕ[m] and R[m], for California spotted owls in the Sierra Nevada, California, USA. Sensitivities are calculated for each parameter identified on the x-axis using eqns 7–12, varying the focal parameter (x-axis) from 0 to 1 and holding all other parameters fixed at values as estimated by MacKenzie et al. (2009): inline image = 0·17, inline image = 0·87, inline image = 0·91, inline image = 0·13, inline image = 0·51, inline image = 0·50. Filled circles represent the sensitivities at these baseline values.

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In general, sensitivities of π[2]* to the parameters are all positive and follow predictable patterns, assuming that only one parameter changes at a time, though the magnitudes are dependent on the combinations of parameter values (Fig. 2). Increases in ϕ[0] result in a decline in inline image because, as more territories become occupied, there are fewer unoccupied territories that may become colonised. Similarly, as owl persistence, ϕ[1] or ϕ[2], increases, the sensitivity of π[2]* to each of these parameters increases because more territories become occupied and, therefore, more territories support reproduction.

Example 2: Wood Frog second-order Markov model

The wood frog is one of the most widely distributed frog species in North America, ranging from the southern Appalachians to Alaska (Martof & Humphries 1959). Wood frogs rely on small, temporary, predator-free vernal pools for breeding (Berven 1982, 1990; Hopey & Petranka 1994; Lichko & Calhoun 2003) but reside in adjacent upland habitats during the non-breeding season (Berven & Grudzien 1990; Regosin, Windmiller & Reed 2003; Rittenhouse & Semlitsch 2007). With human development as an ever increasing threat to vernal pool habitat, federal lands may become isolated refugia for many amphibian species. However, even in protected areas, climate change may jeopardize the existence of amphibian species that rely on vernal pools, and active management may be necessary to maintain viable populations of these species. Unfortunately, little is known about which actions land managers could, or should, take to prevent, reverse or stop the loss of vernal pools and their associated fauna. Based on wood frog life history in the Mid-Atlantic region, we developed a multistate occupancy model to describe the dynamics of wood frog breeding and successful metamorphosis and determined what parameter(s) are most influential in increasing the probability of successful metamorphosis to guide management efforts.

Model development

The occurrence of breeding frogs at a pool at time (year) t is likely to be influenced by two processes: (1) the return of breeding adults from the previous year (t−1) and (2) first time (female) breeders that successfully metamorphosed 2 years prior (t−2) and returned to their natal pond. Because the occurrence of breeding wood frogs may depend on a pond’s occupancy state in each of the previous 2 years, and because the 1-year-old female class is not available for sampling, we developed a multistate occupancy model, where the occupancy states reflect this second-order Markov process. Specifically, six mutually exclusive occupancy states, inline image, represent a combination of egg mass (denoted r) state in the current year and late-stage tadpole (denoted s) state in the current and previous years, t and t−1. Thus, the occupancy state vector Πt in our model can be written as:

  • image

For example, inline image represents the probability that a pool had egg masses but no metamorphs in year t (rt = 1, st = 0) and failed to produce metamorphs in year t−1 (st−1 = 0). We use the presence of late-stage tadpoles to signify successful metamorphosis, and our model assumes that ‘occurrence’ of eggs and metamorphs is assessed during both periods (breeding and metamorphosis) each year.

Next, we define lower-level transition probability parameters:

inline image = probability that a pool is occupied with egg masses (state = 1) in year t given that it was in metamorph state s in year t−2 and egg state r in year t−1.

δt = probability that a pool is in state = 1 in year t, given that it is in state = 1 in year t (i.e., probability that pool produces metamorphs, given it had eggs).

Under this model, a pool with no metamorphs in year t−2 and both egg masses and metamorphs in year t−1 (probability associated with such a pool denoted inline image) could transition to one of three mutually exclusive states in year t, with corresponding transition probabilities:

(1) Support wood frog breeding (egg masses) but not have successful metamorphosis:

Pr (rt = 1 and st = 0 |st−2 = 0 and rt−1 = 1) = inline image

(2) Support wood frog breeding (egg masses) and have successful metamorphosis:

Pr (rt = 1 and st = 1 |st−2 = 0 and rt−1 = 1) = inline image

(3) Support no wood frog breeding and, thus, no metamorphs:

Pr (rt = 0 and st = 0 |st−2 = 0 and rt−1 = 1) = inline image

By defining similar transition probabilities for all six occupancy states, we used combinations of the parameters above to construct our time-constant transition probability matrix.

By expanding the number of states and associated transition probabilities, we have accommodated the second-order Markov dynamic process into a multistate occupancy model with the familiar form, Πt = Φ × Πt−1, as Brownie et al. (1993) and Pradel (2005) did with second-order Markov processes in capture–recapture situations.

  • image
Sensitivity expressions

Because we are interested in the equilibrium probability that a pool produces metamorphs, we defined a relevant equilibrium occupancy, π*, as:

  • image(eqn 13)

where π011* and π111* are the equilibrium occupancy probabilities for pools with successful breeding and metamorphosis (i.e. where rt = st = 1). Expanding the expression to include system dynamics and writing the state parameters in terms of the underlying lower-level transition parameters yield:

  • image(eqn 14)

where

  • image

Sensitivities of π* to each of the five parameters underlying the transition rates are given by the following equations:

  • image(eqn 15)
  • image(eqn 16)
  • image(eqn 17)
  • image(eqn 18)
  • image(eqn 19)
Model parameterization

We obtained parameter estimates for the model described above and examined the sensitivity of π* to each of the model parameters using wood frog occupancy data collected at Patuxent Research Refuge (PRR). PRR consists of mainly upland hardwood forest and is located approximately 30 km from Washington, DC. Vernal pools were located using a dual-frame sampling design (see Van Meter, Bailey & Grant 2008). Occupancy surveys for breeding wood frogs (i.e. egg masses) and late-stage tadpoles have been conducted at 55 vernal pools at PRR since 2006. Visual encounter surveys, involving multiple independent observers, were used to detect egg masses in early spring (generally early-March), and dip-net surveys were conducted in late-May to early-June to target late-stage tadpoles (Mattfeldt, Bailey & Grant 2009).

We analysed the data from PRR using the robust design occupancy model in Program MARK (White & Burnham 1999) to obtain estimates for the transition rate parameters, inline image and δ. We estimated inline image parameters using covariates, assuming known (i.e. p* = 1) metamorph, s, and egg mass, r, states (1 =  present/occupied, 0 = unoccupied) and modelled both inline image and ε (i.e. extinction probability between egg and metamorph stages, 1−δ) as time-varying or constant. Because pools were surveyed by multiple independent observers, we could estimate the overall probability of observing egg masses or metamorphs at least once at occupied pools and found it to be high (inline image = 0·992) providing justification for the use of observed states as covariates, though this approach may not be appropriate for cases in which p* is not close to 1. Similar to the results for the California spotted owl, the top model for the PRR wood frog data suggested that inline image estimates were time-constant, but δ estimates varied among years (range: 0·35–0·79). Because of the time-constancy implied by sensitivity analyses and equilibrium occupancy rates, we used the model that included a time-constant estimate of δ. Estimates, with standard errors, are as follows: inline image = 0·06 (0·03), inline image = 0·47 (0·13), inline image = 0·26 (0·21), inline image = 0·84 (0·11), inline image = 0·42 (0·11).

We varied each parameter from zero to one while holding the other parameters constant using the estimates presented above and calculated the sensitivity of π* to small changes in each individual parameter using eqns 15–19. Because ϕ00 appeared to be the most influential parameter in our current system (see Fig. 3a) and because we believed that it could be influenced via management, we repeated the exercise using medium (0·20) and high (0·50) levels of ϕ00 to investigate how sensitivities would change if this parameter was successfully increased via management. Hereafter, we refer to ϕ00 as colonisation, as it represents the probability of successful breeding given the pool was unoccupied by eggs and metamorphs in the preceding 2 years. We also calculated equilibrium occupancy for all combinations of colonisation, ϕ00, and δ (hereafter, probability of metamorphosis), to examine the simultaneous influence of these two parameters on equilibrium occupancy. Most conceivable management actions would likely influence both colonisation and metamorphosis probabilities (e.g. enhancing or enlarging pools, pool construction, translocation of eggs, addition of water to pools).

image

Figure 3.  Sensitivity of equilibrium occupancy (π*) to changes in the parameter values of inline image and δ based on estimated values for wood frogs at Patuxent Research Refuge, Maryland, USA. Sensitivities are calculated using eqns 15–19; each line represents the sensitivity of π* to the designated transition parameter (x-axis), being varied from 0 to 1, while all other parameters are fixed at estimated values: inline image = 0·47, inline image = 0·26, inline image = 0·84, inline image = 0·42 and ϕ00, where ϕ00 is low representing current conditions (a, inline image = 0·06). We also present sensitivities where the baseline colonisation rate, ϕ00, is raised via management to ϕ00 = 0·20 (b) or ϕ00 = 0·50 (c). Filled circles represent the sensitivities at these baseline values.

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Results

Parameter estimates from PRR revealed that only 5·4% of pools produced metamorphs at equilibrium (π* = 0·054, Π* = [0·847 0·057 0·041 0·024 0·017 0·013]). Most pools do not support wood frog breeding (e.g., π000* = 0·85). Thus, even though colonisation probabilities are low (inline image = 0·06), this transition rate is over four times more influential than metamorphosis probabilities, the parameter to which π* is next most sensitive (inline image = 0·76, Sδ = 0·16, Fig. 3a). Of the remaining transition rates, only ϕ01 significantly influenced equilibrium occupancy and only when values greatly exceeded current estimates at PRR (Fig. 3a).

If colonisation probabilities were increased (e.g. via management) to ϕ00 = 0·20, equilibrium occupancy would be π* = 0·133 and most sensitive to colonisation rate (inline image = 0·41) followed closely by probability of metamorphosis (Sδ = 0·37; Fig. 3b). In systems with high colonisation probabilities (ϕ00 = 0·50), equilibrium occupancy increased to 21·3% and was most sensitive to probability of metamorphosis (Sδ = 0·54) and only somewhat sensitive to ϕ01 (inline image = 0·21) and colonisation (inline image = 0·17; Fig. 3c). Interestingly, the transition rates, ϕ10 and ϕ11, roughly corresponding to persistence of breeding adults, had very little influence on equilibrium occupancy for any of the scenarios we considered (Fig. 3).

To summarize, when colonisation rates are less than 0·20, equilibrium occupancy is most influenced by colonisation probabilities, unless the probability of metamorphosis is also low (i.e. δ<0·34) (Figs. 4 and S1 in Supporting Information). At current transition probabilities, once colonisation probabilities exceed 0·217, equilibrium occupancy (with metamorphs) responds more to the conditional probability of metamorphosis and the effects of increasing the colonisation rate quickly become negligible (Fig. 4).

image

Figure 4.  The relationship between equilibrium occupancy (π*; probability a pool produces post-metamorphic wood frogs) and both colonisation rate (ϕ00) and the conditional probability of metamorphosis (δ). Increasing colonisation rates when they are <0·20 results in a larger influence on equilibrium occupancy, while equilibrium occupancy is most influenced by probability of metamorphosis at colonisation rates >0·15. Other parameters are fixed at estimated values: inline image = 0·47, inline image = 0·26 and inline image = 0·84.

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Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Our perturbation analysis of multistate occupancy models provided various insights about associated dynamics that can be generalized to other multistate systems. We have described two common scenarios for the use of multistate occupancy models. The California spotted owl example represents a typical multistate system involving two occupied states, whereas the wood frog example illustrates how a second-order Markov process may be modified to fit a multistate framework. Few studies have examined second-order Markov processes (but see Hestbeck, Nichols & Malecki 1991; Brownie et al. 1993; Ehrlen 2000; Picard, Bar-Hen & Guédon 2003; Pradel 2005), and none has involved occupancy as a state variable. However, we believe many species have breeding populations composed of returning adults and maturing juveniles (e.g. neotropical migrant birds). If juveniles (e.g. the stage between metamorphs and breeding adult wood frogs) can be observed, then modelling such a system as a first-order process with age/stage structure is a reasonable approach; however, when intermediate stages or year classes are unobservable, then models such as that presented above may be useful. More generally, such models could be used in systems where a time-lag is expected or where the effects of management actions or stochastic events may not be immediate.

In standard occupancy scenarios, the probabilities of a patch being occupied or unoccupied are complementary, and therefore, the resulting equilibrium occupancy equation (eqn 1) and sensitivity metrics are simple expressions (Martin et al. 2009b). However, in the case of multistate occupancy dynamics, the stable-state distribution is not always so simply expressed. Additionally, the state of interest (e.g. equilibrium occupancy, π*) in dynamic multistate occupancy models depends on the interests of the investigator. For standard dynamic occupancy models, the definition of equilibrium occupancy is straightforward (i.e., ψ*); however, with dynamic multistate models, equilibrium occupancy rate can be defined by one state or a sum of multiple states. In our California spotted owl example, we chose to focus on the sensitivity of equilibrium patches at which breeding occurs (π[2]), but if the probability of a territory being occupied (regardless of breeding status) was of interest, equilibrium occupancy could be redefined as π[1]*[2]*. In the case of successful wood frog metamorphosis at PRR, we defined equilibrium occupancy state as π* = π011* + π111*, representing the equilibrium probability a pool produces metamorphs. For systems in which the goal is to reduce occupancy rates (e.g. systems involving disease or invasive species), sensitivities to the unoccupied state may be most relevant to questions associated with treatment. It is important that equilibrium occupancy be defined on a case-by-case basis, to reflect focal occupancy state(s) and appropriate sensitivities. As usual, relevance of any inference about sensitivity is determined by the larger programme of science or management within which the inference is embedded.

Despite subtle differences in calculating sensitivities and stable-state distributions between standard and multistate models, there are several analogous relationships between sensitivities and equilibrium occupancy. As in Martin et al. (2009b), we found that when colonisation probabilities (i.e. ϕ[0] and ϕ00 for our two examples, respectively) are small and the majority of patches (or sites) are unoccupied, the system is strongly influenced by small changes in the colonisation probability. This is because of the larger number of unoccupied patches for colonisation to act upon. Martin et al. (2009b) found that when ψ* < 0·50, ψ* was more sensitive to γ (local colonisation); likewise, ψ* was more sensitive to ε (local extinction) when ψ* > 0·50. In our wood frog example, if colonisation levels were increased above approximately 0·22, sensitivity of π* to the parameter that influences successful metamorphosis increased (i.e. δ).

Despite potential uncertainty about estimates of transition probabilities, our findings, as well as those by Martin et al. (2009b), suggest that knowledge of the state distribution itself can inform management decisions. Assuming the system is near equilibrium, if few patches are occupied, then focusing management activities on improving colonisation, or transitions from unproductive to productive states, is likely to increase the overall equilibrium occupancy as defined in our examples. We acknowledge that many populations may not be near equilibrium, but we believe equilibrium concepts and associated sensitivities based on stationary dynamic processes are relevant and useful for examining the relative effects of different management alternatives or assessing uncertainties related to sampling error (see Martin et al. 2009b and Caswell 2007 for similar arguments for simple occupancy matrices and population projection matrices). It would be much wiser to base management decisions on estimates of both dynamic parameters and current-state distributions. As seen in the wood frog example (Fig. 3), the ability to change transition rates via management may influence future management decisions because sensitivities of the equilibrium-state distribution to each of the transition rates may change. In other words, if management is successful at influencing dynamic parameters and, thus, occupancy states, then optimal management actions in future time steps may change. In our examples, if management successfully increased colonisation to a consistent level above 0·2, then the future management actions should shift to efforts to improve the probability of successful reproduction/metamorphosis conditional on occupancy (i.e. R[m] and δ). Management actions that target both colonization and reproduction, given occupancy, simultaneously should be favoured. These recommendations assume that the costs associated with changing the different parameters are roughly equal. If this is not the case, results may be misleading (Link & Doherty 2002; Nichols & Hines 2002; Baxter et al. 2006), and alternative sensitivity expressions incorporating relative costs of changing parameters could be developed and used to explore the sensitivities of the stable equilibrium to changes in dynamic parameters that are scaled to management actions per unit cost (see Martin et al. 2009b).

In closing, we make two points about the computation of the sensitivity values. First, our approach was to first express the stable-state distribution in terms of transition parameters and to then compute the partial derivatives of the focal-state equilibrium value(s) with respect to the various transition parameters. Caswell (2001) provided a general expression for the computation of sensitivities for scaled eigenvectors (such as our equilibrium-state distribution) that may be more easily implemented than our approach. Indeed, we checked the results of our computations with this alternative approach. Second, the rows or columns (in our case) of our stochastic transition matrices must sum to 1; thus, any perturbation of one matrix element (e.g., a transition parameter) must be compensated by corresponding changes to other matrix elements. This issue requires computation of a total, rather than a partial, derivative and specification of an approach to compensation (Caswell 2001; Hill, Witman & Caswell 2004). In both of our examples, transition matrix elements were written in terms of conditional binomial parameters. For example, with the spotted owls, a site was either occupied (inline image) or not (inline image) and, conditional on occupancy, reproduction either was (inline image) or was not (inline image) successful. Compensation occurs naturally under this parameterization; e.g., a small increase in inline image is naturally compensated by an equivalent decrease in inline image. When more general multinomial parameterizations are used, we recommend that the reader consults Caswell (2001) for various approaches to maintaining the stochastic matrix constraint.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

We thank Julien Martin, the Wagar 113 Superpopulation and participants of the structured decision-making workshop for their comments and discussions. We thank David Miller for useful discussions that improved the paper. U.S. Geological Survey Amphibian Research and Monitoring Initiative (ARMI) provided funding for this project, and this represents contribution number 374 for this programme. We also thank Evan Grant and the field technicians who collected the occupancy data at Patuxent Research Refuge.

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  7. References
  8. Supporting Information
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Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Methods
  5. Discussion
  6. Acknowledgements
  7. References
  8. Supporting Information

Figure S1. Shifts between two parameters with the greatest influence (sensitivity) on wood frog equilibrium occupancy using current, estimated transition rates.

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