Go To Markov Martrix Data Sheet

HABITAT CHOICE BY HERMIT CRABS: Animals select their habitat according to a variety of criteria where natural selection is expected to favor efficient selection patterns. Here we shall examine habitat choice by hermit crabs. Hermit crabs [see Figure] are anomuran decapods whose chitinous exoskeleton is not well-calcified. To protect the soft abdomen from predation as well as exposure to dessication and thermal stress, the crab inserts this part of its anatomy into a suitable empty gastropod shell. Suitability depends on the ability of the crab to withdraw its entire body into the shell so that the large claw, or cheliped, serves as a kind of opeculum, sealing off the crab's vulnerable parts from intrusions (Radinovsky and Henderson 1974).
 
 

Hermit crabs show preferences for particular types of habitats (e.g. high vs low intertidal). Within a habitat, they make further decisions concerning which gastropod shell to occupy. Crabs grow larger in size and shells deteriorate with time so hermit crabs need new shells. Hermit crabs do not obtain their new shells from live snails. They often appear to be attracted to sites where predation on live snails is occurring and a scuffle for the newly-empty shell can be observed (Rittschof 1980, Hazlett and Hernnkind 1980). [How might they find or orient to such predation sites?] The crabs also show distinct preferences for certain species of gastropod shells and within a species, preferences are noted for particular sizes of shell. The crab compares information concerning its own shell and that of the new shell to make its decision on whether to move into the new shell (Elwood and Stewart 1985). Because large shells are scarce, there can be tough competition for good shells. We will see aggressive behavior and posturing between crabs of all sizes (Hazlett 1966) as they challenge others for their shells, attempting to pry the inhabitant out. In many communities of hermit crabs, the availability of suitable empty shells is a limiting resource. Small shells tend to be more abundant than larger shells. The scarcity of large shells has been shown to be a limiting factor in population growth (Vance 1972).
 
 

VACANCY CHAINS: Hence, resource acquisition by a hermit crab is quite different from that considered in other ecological circumstances. Instead of being independent events, it is a series of interconnected events, where the acquisition of a ‘better’ resource unit by one individual depends on the prior acquisition events by other individuals (Weissburg et al. 1991). In this case, the resource unit is a vacant shell and acquisition is via a vacancy chain. The vacancies are causal; they must exist first before individuals can move (Chase 1991). Resource allocation is controlled by pairwise interactions between individuals or contests occurring simultaneously among all participants (i.e. interference and exploitatitve competition). In these systems where resources are distributed by vacancy chains, an initial resource introduction allows a succession of individuals to gain new units as the vacancy flows through a population of users. These individuals often vary in some attribute such as status or size. In contrast, in a non-vacancy chain system, only a single individual benefits from the resource introduction. The resource also is re-usable until its quality is nil. Weissburg et al. (1991) showed that, given these attributes of the vacancy chain system, a first-order embedded Markov model, which describes a set of events in terms of their associated probabilities, fits their data well. Vacancy chain models organize the consequences of resource introduction by determining the probability that any given individual will gain a new resource via the chain created by the initial introduction. They demonstrate how this model then could be used to examine broad patterns of resource selection including questions concerning life history evolution and conservation biology.
 
 

LABORATORY EXERCISE

In this laboratory, we will go out to Flax Pond to make observations of hermit crabs kept there in the sea tables. These crabs were collected from West Meadow Beach at low tide last week. The best times to collect these crabs is in the warm summer months.
 
 

1. Observation of behavior without any empty shells available: For the first part of this lab, we shall make observations of the interactions between crabs of different size classes. For this section, please make a list of the responses that you have been able to observe that make up the behavioral repertoire of a hermit crab. Although you will see much aggressive behavior, focus on the inspection process as it will be important that you develop a search image for distinguishing different types of inspection processes and the likelihood of a shell exchange.

B. Vacancy chains: For the rest of the lab period, we will offer the hermit crabs empty shells and observe the movement of their resource, the empty shell or vacancy, in a chain of interactive events. We will document the number of crabs involved.
 
 

Shell types: In the West Meadow Beach habitat, hermit crabs occupy the shells of four species of snails: Littorina littorea, Ilyanassa obsoleta, Urosalpinx cinerea, Nassarius trivittatus.
 
 

States: The attribute of size can designate the state of the shell. Other possible attributes to consider for shell state include: quality, species. [Can you think of other bases for different states (see Bertness 1981)?] Weissburg et al. (1991) separated the shells by size (State 1 = >2g, State 2= 2-1.201 g, State 3= 1.2 – 0.701 g, State 4 = 0.7-0.301 g, State 5 = <=0.3 g) determined by plotting all shell weights used in the different chains. Hence, although weight is a continuous variable, discrete states could be designated by examining the size distribution of shells. They also created different states based on quality:

Quality 1) Shells with no physical blemishes such as holes or missing chips.

Quality 2) Shells with one or two very small holes (< 1mm diameter).

Quality 3) Shells with up to 2 or 3 small holes (from 2-4 mm diameter)

Quality 4) Shells with several larger holes or large areas peeled back from shell margin and/or the opening largely blocked by Crepidula planar.

>>>>> In our experiments, we will simply use size to designate each state. Divide empty shells in different sizes. There need to be from 3-5 states with 10 observations for each probability estimated.
 
 

Initiation of vacancy chain: Drop starting shell into pool. Collect crab that took starting shell. Place in cup 1.
 
 

Continuation of vacancy chain: Collect next crab that took old shell of first crab. Repeat until observation period terminates.
 
 

Chain termination: a) Collect empty shell abandoned by last crab taking part in the chain. b) Collect full shell taken by 'naked' crab. c) End after 30 minutes.
 
 

Other measurements: Remove crabs from shells by immersing in warm (35 degree) tap water. Weigh crabs and measure length of shields (hard portion of carapace). Shake out as much water as possible, blot dry, and weigh shells now (though they should be allowed to dry before weighing). Provide crabs with their shells and release them.
 
 

Constructing the transition matrix [see attached]: Divide data randomly in half. Use first half to get matrix of observed transition frequencies (designate as Table 1). Use the second half to provide empirical estimates of the vacancy chains parameters with which to compare the predictions of the Markov models developed from the first data set.
 
 

Maximum likelihood estimates of the Markov transition matrix can be calculated from the frequency counts (in Table 1) normalized by row sums (divide counts by row sums and designate as Table 2). Use Equation 1 [put into Table 3] and Equation 2 [put in table 4] to calculate the expected chain lengths and the average multiplier effects (MEs; for example, ME₁ = sum of row 1 in Table 3). List these values along with the observed chain lengths into a Table 4. Compare the observed mean chain length to the expected mean chain length in terms of an associated z-score [need to calculate Equation 3], standardized normal deviates (Sokal and Rohlf 1981) and assess significance. Compare goodness of fit for chains having different numbers of states.
 
 

Results noted in Weissburg et al. (1991):

1) Vacancies flowed from larger to smaller shells with fairly large numbers of within state transitions for the smaller states.

2) The probability of absorption varied with shell size. Vacancies in state 1 shells were rarely absorbed. Vacancies on state 5 shells were more likely to be absorbed.

3) Few chains began with bad shells or very small shells.

4) Chains begun in all states are equally likely to end with abandoned shells since many chains, regardless of starting state, eventually move down to small, low quality shells.
 
 

A Markov model describes a set of inter-related events in terms of their associated probabilities.

Requirements:

1) resource is re-usable

2) resource units differ in some attribute indicative of quality, so that each unit can be assigned to a state (class) based on that attribute

3) animals have a recurrent need for resources and continually seek out new resource units of a higher quality than that presently utilized.
 
 

Hermit crab-snail shell system:

1) shells are re-usable

2) shells may be assigned to size categories so that resource state corresponds to a particular size ranges of shells (shell quality [holes and other evidence of damage] also can be ranked).

3) as individual crabs grow, they must find larger shells to inhabit.
 
 

Interconnected events: Introduce an unutilized (vacant) resource unit of state i into a population of users. An individual takes that shell, leaving a resource unit (vacant shell) of state j. Hence, the vacancy has flowed from state i to j, while the individual has made the opposite move from a resource unity in j to a unit in i. At this point, the vacancy resides in resource unit in state j, which then may be taken by another individual, and so forth.
 
 

Transition matrix: Repeatedly introduce resource units and observe vacancy movements and movements into absorption state (see Termination). Summarize observed moves in a matrix [Table 1] where aij is the number of vacancy movements from resource units in state i to units in state j. [Note this is an asymmetrical m x s matrix since vacancies move to, but not from, the absorption state.] By normalizing each element in T by dividing by appropriate row sum, or tij = a ijSaij , [summed from j = 1 to j = s (the last state)], over all values of i, the matrix is transformed into a matrix of maximum likelihood estimates, T, where every element tij is the probability of a vacancy moving from i to j, such that Stij = 1, for all values of i (Billingley 1961). T is called the transition matrix [Table 2] and forms the basis for predicting the characteristics of the vacancy chain process.

Causality: Vacancies are causal, moves of individuals are resultant. No move can occur until a vacancy is created. A series of resource acquisition events occur as the vacancy flows through a population of users.
 
 

Transient states i and j: As chain moves towards termination, these states are occupied temporarily by a vacancy.
 
 

Termination: Vacancy chain ends with movement of vacancy into an absorption state (a state which does not in turn liberate a vacant resource unit such as a destroyed shell or a shell taken by a naked crab).
 
 

Assumptions:

1) Probabilities governed by a first-order process where tij is dependent only on the resource state i and j, and is blind to previous vacancy moves.

2) Vacancy chains are embedded where time is not explicit and is measured over an ecologically relevant time frame. The resource base remains relatively static.

3) The model is unaffected by resource selection behavior of individual animals. The model does NOT track individual choices but follows vacancies as they move through a population of users. There may be individual biases, but model considers the average, aggregate consequences of vacancy movement.
 
 

Uses of Transition Matrix:

1. Predictor of average vacancy chain length = The number of moves in an average chain beginning with the first move from the initial resource unit and ending with the final move to absorption. This is contingent on state of initial vacant resource. This also estimates the #animals acquiring new units which is distinct from the #moves. This is called the Multiplier Effect.
 
 

Take only the elements of the matrix T that includes transition (no non-absorbing states) for form fundamental matrix N (Table 3), where each element nij equals the expected #times a vacancy starting out in state i will be in resource units in state j before it is absorbed. This is calculated as:
 
 

N = (I-Q)-1 Equation (1)
 
 

where Q is the m x m submatrix of T and I is an m x m identity matrix.
 
 

MEi, the expected chain length resulting from a vacancy initially in state i, can be computed as MEi = Snij , [summed from j = 1 to j = s (the last state)], and
 
 

n = N1 Equation (2)
 
 

where n is a column vector of MEi's. The variances of these expected chain lengths (for Table 4) are calculated as
 
 

s^{2 = (2N-I) n-n2 Equation (3)}
 
 

^{2. Probability for absorption in each different absorption state as a function of the state of the initial vacancy: If the absorptive state occurs when a 'naked' animal takes the final resource unit, animal ME = vacancy ME. If the last unit is abandoned or destroyed, the vacancy is considered to make one last move outside the system, without a corresponding acquisition event by the animal, and the animal ME = ME -1. This can be predicted from the Markov model [see Weissburg et al. 1991) by creating a submatrix R containing only the transition probabilities into the absorption states.}
 
 

^{3. Predictor of #individuals moving from resource unit of state i to state j: Calculates the aggregate effects of a number of resource introductions.}
 
 

References:

Billingsley, P. 1961. Statistical Inference for Markov Processes. University of Chicago Press, Chicago.

Chase, ID. 1991. Vacancy Chains. Ann. Rev. Sociol. 17: 133-154.

Weissburg, MJ, L. Roseman, and ID Chase. 1991. Chains of opportunity: a Markov model for acquisition of reusable resources. Evolutionary Ecology 5: 105-117.