"Time-dependent" colonization probabilities

The assumption of constant colonization probabilities extensively used in Unit 4 and 5 has been questioned (ter Braak at al. 1998). Among other factors, metapopulation size, strong distance dependence on dispersal and rescue effect can cause fluctuation in the C_i values (Moilanen 1999). In this cases, both connectivity and colonization probability can be modelled as "time-dependent" by computing S_i ad any time t based on the occupancy \(\delta_i(t)\) , that is by substituting \(\delta_i(t)\) to \(\delta_i\) in Eq. (4.4) to obtain S_i(t) , and S_i(t) in Eq. (4.5) to obtain the probability that a site which is empty at time t will be occupied at time t+1:

As a matter of fact, the expression "time-dependent" is misleading, as it suggests the giving out of the homogeneity property discussed in Unit 2. This is not the case, and we try to clearly explain this concept.

In Unit 4 we introduced the IFM by considering one patch at a time: in each patch the dynamics evolves according to an homogeneous Markov chain with some initial probability and the transition matrix defined by Eq. (4.2) and Eq. (4.5). The extinction probability as expressed in Eq. (4.2) only depends on patch-specific features (the patch area). However, the colonization probability as expressed in Eq. (4.5) depends on the occupancy of all the other patches (and also on other patches features, like area). That is, Eq. (4.5) expresses the probability that a patch which is empty this year will be occupied the next year conditional to the knowledge of the occupancies of all the other patches in a given point in time. Equation (6.1) further specifies that the probability that a patch which is empty this year will be occupied the next year depends on the knowledge of the occupancies of all the other patches this year. This means that we cannot consider each patch as independent from the other patches, but the dynamics at all the patches should be considered jointly (metapopulation dynamics). The argument holds true if Eq. (4.2) is modified by Eq. (5.2).
In mathematical terms, this means that the stochastic process we are considering, (X_t)_t≤1, is a multivariate Markov chain, that is

• X_t  is the vector (X_t(1), X_t(2), ..., X_t(n))  of the occupancies of all the n patches at time t.
• E_i expresses P(X_t+1(i) = 0| (X_t(1), ..., X_t(i) = 1, ..., X_t(n) )  as a function of A_i only, that is P(X_t+1(i) = 0| (X_t(1), ..., X_t(i) = 1, ..., X_t(n) ) = P(X_t+1(i) = 0| X_t(i) = 1)  = (A₀ /A_i )^x and this transition does not depend neither on the occupancy of the other patches nor on time.
• C_i expresses P(X_t+1(i) = 1| (X_t(1), ..., X_t(i) = 0, ..., X_t(n) )  as a function of all the other occupancies X_t(k). That is, equal configurations of patch occupancies (X_t(1), ..., X_t(i) = 0, ..., X_t(n)) yield the same colonization probability, regardless of the transition time t. This concept is exemplified in Example 6.1.

Once this has been pointed out, it should be clear that we are in the same framework of Unit 2. Therefore, we could write the transition matrix of the multivariate Markov chain (X_t) _t≥1. Unfortunately, the state space of this multivariate Markov Chain is the set {0, 1}ⁿ (set of all the vectors with n elements   each of which is either 0 or 1) that is a huge space (2ⁿ elements) even for small values of the metapopulation size, n, and therefore it is impossible to use a matrix representation. However, we can concisely and easily write any element of the transition matrix as in Eq. (6.2).

(6.2) \(P(X_{t+1} = (x_1, ... x_n) | X_t = (w_1, ..., w_n )) = \prod_{i=1}^n \begin{cases} C_i(t), & if\:x_i=1 \:\&\:w_i=0\\ 1-C_i(t), & if\:x_i=0 \:\&\:w_i=0\\ E_i(t), & if\:x_i=0 \:\&\:w_i=1\\ 1-E_i(t), & if\:x_i=1 \:\&\:w_i=1\\ \end{cases} \)

As explained before, this probability does not depend on t and then the multivariate process is homogeneous. This justifies the use of inverted commas around “time-dependent” in the title of this section. From the point of view of modelling, the basic model described in Unit 4 is slightly different from (6.2)  as in some sense it refers to some available data set. The colonization probabilities are defined according to the evidence from that data by using either Eq. (4.4) or Eq. (5.1). This data set then determines the model-predicted, long-term probability of patch occupancy J_i.  To the best of our knowledge, to find the stationary distribution associated to Eq. (6.2), if any, is analytically impossible, and therefore the distribution of X_t should be computed, in some way, by the general formula (3.2) for some initial distribution.

In our opinion, if the previously mentioned data set is some “historical” data set, including a lot of information about the species biology and dynamics, and this information allows us to reasonably assume that the colonization probability is constant, the basic model of Units 4 and 5 is certainly the most sound to be used, as it strongly simplifies estimation and dynamics simulation. Unfortunately, such historical data are seldom available and therefore the current data become the basis for model implementation. In this situation, the uncertainty about all the assumptions, the biological as well as the statistical ones, should be carefully considered.

Finally, the transition probabilities can be modified to take into account the rescue effect:

(6.3) \(P(X_{t+1} = (x_1, ... x_n) | X_t = (w_1, ..., w_n )) = \prod_{i=1}^n \begin{cases} C_i(t), & if\:x_i=1 \:\&\:w_i=0\\ 1-C_i(t), & if\:x_i=0 \:\&\:w_i=0\\ [1-C_i(t)]E_i(t), & if\:x_i=0 \:\&\:w_i=1\\ 1-[1-C_i(t)]1-E_i(t), & if\:x_i=1 \:\&\:w_i=1\\ \end{cases} \)