Answer Happy

In an age structured population of Moray eels, each eel lives at most n years. We denote by N.(t) the population density of age k at time t, so the population can be written as N(t) = (N(t),..., N.(t)). The expected number of offspring of each eel of age k is by, which is known to decrease with the age of the eel as byk with B and a > 1. The probability p that an individual aged k > 0 (with k = 0 corresponding to newborns) survives to age k+1 is a constant 0 <p<1. (a) Show that the evolution of the population follows N(t+1) = LN(t), where L is a matrix that you should find. (b) Show that the eigenvalues of matrix L, and its eigenvectors v, satisfy the equations 1 * B Ë@) - (3" v=(1,5 ) ...¢)"). (c) in the scenario that the eels do not have a maximum age limit (n+00), find the condition in terms of Q, Band p for which the population does not collapse. (d) In an alternative scenario in which n = 3, find the conditions in terms of a, B and p for which the population does not collapse. [You may state without proof theorems from the lecture notes.]