MATH0300 Problem Sheet 7
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MATH0300 Problem Sheet 7
1. The period r of a Leslie matrix, L, is defined by:
r (L) = GCD{n|bk > 0}
if r (L) > 1, then L is said to be periodic. Most biologically realistic examples of periodic Leslie matrices arise for species which breed once, after a fixed number of years, say p 2 2, and then die. One examples is the periodic cicada, which breeds once after a juvenile period of exactly 17 years in the northern USA (12 years in the southern USA).
(a) Show that p = r
(b) Show that the eigenvalues of the Leslie matrix of such a species all lie on a circle in the complex plane
(c) Show that the asymptotically attracting age-structure is cyclic of period r .
(d) Under which condition the population explodes?
2. For a population of T age-classes and population vector N
z) e RT with class survival probabilities ti and fertility rates ki :
(a) Show that the total population, s = k(T)=1 Nk satisfies the dynamics, s(z + 1) = Ω R
(z)、s(z)
where 
(z) = 
(z)/s(z) is the age-structure at time z, and
Ω R
(z)、= k(T)=1 (kk + tk )Xk
(b) Show that if the Leslie matrix is aperiodic then Ω R
(z)、 → A+ when z → o
where A+ is the leading eigenvalue matrix.
(c) Show that the age structure dynamics follow
k+1(z + 1) = 
(1 s n < p)
Why is an additional equation for X1 (z + 1) not required?
3. In the case in which there are 3 distinct age classes and surviving probabilities p0 = p1 = p2 = p, and per capita number of offspring b1 = 0, b2 = b > 0, and b3 = 
b, with 
e (0; 1):
(a) Discuss the ecological scenario studied
(b) Write the Leslie Matrix and the general expression for the evolution of the pop- ulation vector 
(t + 1) as a function of 
(t)
(c) Given the age structure dynamics in terms of the two independent variables x = X2 and y = X3 , show that the stable age-structure (
; y¯) lies on the ellipse
2 + 
y¯ + y¯2 _ y¯ = 0
4. An annual plant lives for one year (=1 age-period), at the end of which it produces nan seeds, and then dies. In contrast, a perennial plant may live for many years, and produces npn seeds at the end of each year it is alive. The survival probability for both annuals and perennials during the first (and only, in the case of annuals) year of life is p0 . For perennials, the per-year survival probability after the first year is pk = ppn = consant; Ak 2 1.
(a) Show that the principal eigenvalues of annuals and perennials are 
an = p0nan , and 
pn = p0 bpn + ppn .
(b) Under what circumstance does the annual plant population grows faster than the perennial one?
2023-05-15