E4.19 Leslie population model. The Leslie model is used in population ecology to model the changes in a population of organisms over a period of time; see the original reference (Leslie, 1945) and a comprehensive text (Caswell, 2006). In this model, the population is divided into n groups based on age classes; the indices i are ordered increasingly with the age, so that i = 1 is the class of the newborns. The variable x_i(k), i ? {1, ..., n}, denotes the number of individuals in the age class i at time k; at every time step k the x_i(k) individuals
• produce a number ?_i x_i(k) of offsprings (i.e., individuals belonging to the first age class), where ?_i ? 0 is a fecundity rate, and
• progress to the next age class with a survival rate ?_i ? [0, 1].
If x(k) denotes the vector of individuals at time k, the Leslie population model reads
x(k + 1) = Ax(k) = [?_1 ?_2 ... ?_{n-1} ?_n; ?_1 0 ... 0 0; 0 ?_2 ... 0 0; ? ? ? ? ?; 0 0 ... ?_{n-1} 0] x(k), (E4.4)
where A is referred to as the Leslie matrix. Consider the following two independent sets of questions. First, assume ?_i > 0 for all i ? {1, ..., n} and 0 < ?_i ? 1 for all i ? {1, ..., n - 1}.
(i) Prove that the matrix A is primitive.
(ii) Let p_i(k) = x_i(k) / ?_{i=1}^n x_i(k) denote the percentage of the total population in class i at time k. Call p(k) the population distribution at time k. Compute lim_{k??} p(k) as a function of the spectral radius ?(A) and the parameters (?_i, ?_i), i ? {1, ..., n}.
Hint: Obtain a recursive expression for the components of the right dominant eigenvector of A
(iii) Assume ?_i = ? > 0 and ?_i = ?/n for i ? {1, ..., n}. What percentage of the total population belongs to the eldest class asymptotically, that is, what is lim_{k??} p_n(k)?
(iv) Find a sufficient condition on the parameters (?_i, ?_i), i ? {1, ..., n}, so that the population will eventually become extinct.
Second, assume ?_i ? 0 for i ? {1, ..., n} and 0 ? ?_i ? 1 for all i ? {1, ..., n - 1}.
(v) Find a necessary and sufficient condition on ?_1, ..., ?_n, and ?_1, ..., ?_{n-1} so that the Leslie matrix A is irreducible.
(vi) For an irreducible Leslie matrix (as in the previous point (v)), find a sufficient condition on the parameters (?_i, ?_i), i ? {1, ..., n}, that ensures that the population will not go extinct.
