We have been working with models for live-recaptures data in recent weeks. You have seen the single-age CJS model and learned how to evaluate models that allow parameters to vary with time, by group, and in accordance with diverse covariate types. We briefly discussed the fact that you can also modify model parameterizations to incorporate age structure, e.g., immature vs adult or continuous age. If you want to learn more about that, you can work your way through chapter 7 of CW to get started. And, the WNC book (ch. 17.) and the primary references it provides will give you detailed information on the possibilities of what can be done with multiple-age models.

Now, we are going to generalize the modeling that can be done for live-recaptures data even further with multistate models. Multistate models allow individuals to move or transition among multiple sites or multiple phenotypic states between sampling occasions in a stochastic way. Multistate models consider the probabilities of transitions between various states. Many interesting questions relate to (1) transition probabilities among states and (2) state-specific vital rates. Examples exist in evolutionary biology (trade-offs), metapopulation biology, and population & habitat management. Imagine being interested if animals are more likely to leave fragmented habitat and to move to intact habitat. Or, imagine animals moving from a breeding state to a non-breeding state in an iteroparous species in which individuals do not necessarily breed each year.

With multistate models, you can estimate state-specific survival rates, detection rates, and transition probabilities. Such information can be important to both applied (e.g., habitat management and prioritizing land protection/acquisition) and basic (e.g., evolutionary ecology questions regarding trade offs associated with being in various phenotypic states). We will use the term state in multistate modeling in the course regardless of whether we are referring to the location of the animal, i.e., site, when it is captured, or its phenotypic state at the time of capture.

#### Transition Matrix

As a starting point for considering the transition probabilities, you need to have a basic understanding of a Markov chain. Here, we will consider a first-order Markov process or Markov chain. In a first-order Markov process, an animal’s state at \(time_{i+1}\) depends only on its state at \(time_i\). That is, its states at earlier times do not influence the probability that it will change states between \(time_i\) and \(time_{i+1}\). (Andrei Markov was a Russian mathematician who was the first to study matrices of transition probabilities.) The probabilities consider all possible combinations of transitions between or among a finite set of states. For example, here is a transition matrix for movement among 3 locations. Notice that the values in a row sum to 1, i.e., the matrix is row stochastic. Also, notice that because the values in a row sum to 1, if you know the values for 2 cells in a row, you can get the remaining value by subtraction, which will come into play when we do multistate modeling in Program MARK.

Here is a transition matrix for 2 phenotypic states.