Erin Conlisk, Michael Bloxham, John Conlisk, Brian Enquist, and John Harte. 2007. A new class of models of spatial distribution. Ecological Monographs 77:269–284.

Appendix B. Derivation of Theorem 2.

Theorems. 2.1 and 2.2. Explained in the text.

Theorem 2.3. Using the Sequence 2 viewpoint, consider any cell of the index i grid. Its abundance n is acquired from its predecessor cell in the index i–1 grid, hence

Since the predecessor cannot pass on more individuals than it holds, the sum begins at q = n; and, since the predecessor could be holding all n₀ individuals, the sum ends at q = n₀. The summand probability on the right can be broken into a product of two probabilities, yielding

On the left is . On the right are and , where the latter equals F(a,n)F(a,q–n)/F(2a,q) by Theorem 1.3. Filling in this notation yields Theorem 2.3.

Theorem 2.4. Summing out all but one cell from the multivariate PDF, as was done for the single-division model, appears intractable. Hence we instead solve the recursion in 2.3. If the index i univariate probabilities are arrayed as a row vector

the recursion can be expressed in vector-matrix form as

where is the univariate probability matrix (for c = 2) defined above Theorem 1 and characterized in Theorem 1.4. If the fixed indices are suppressed, this version of the recursion takes the compact form

which can be recognized as the state probability equation of a Markov chain. The chain, with index i, is the sequence of cells from the initial area at i = 0 to the cell of interest at the final value of i. The states are the possible numbers of individuals in a cell (a number from 0 to n₀). P_i arrays the state probabilities as a row vector; Q is the transition probability matrix; and (B.1) governs the evolution of P_i. Back substitution yields the solution P_i = P₀Q_i. The initial condition is P₀ = (0,...,0,1), which states that the chain starts with all n₀ individuals in the whole area at index i = 0. Thus, P_i = P₀Q_i can be written P_i = [last row of Q_i]. By Theorem 1.4, . Writing out the right side and invoking the property of U that UU = I yields and thus . Substituting from the definitions of U and yields the vector P_i, whose nth element (starting from n = 0) is Theorem 2.4.

Theorem 3. The derivations of Theorem 3 closely follow those of Theorem 2.