Final Project
Population Biology
Wagner Magalhães
Age-Structured Populations with clonal reproduction
The structure of population by age class is very common and useful for investigating future
demographics. The first age class is made up of newborns and as time progresses, these
individuals either move up into subsequent age classes or perish. Several species reproduce
age-dependently, periodical cicadas spend many years in the nymphal stage and reproduce
once in their life time; humans and other mammals do not reproduce during their first years
and then their reproductive success is age-dependent. Leslie (1945) introduced matrix models
that have discrete age classes with synchronous reproduction and this model has been
explained in detail in text books. However, the classical methods of demography presented in
basic ecology texts were developed for use on organisms without clonal reproduction, the
presence of which in a life cycle creates complications for traditional demographic analysis (see
discussion in Caswell, 1985). Thus, this project aims to improve the material covered in class
about age-structured models by providing information about models suitable for clonal
organisms.
The problem of clonal reproduction can be explained in three different depictions of life cycles:
i) ramets classifiable by age; ii) ramets classifiable by size; and iii) genets. This problem will be
approached herein by using life-cycle graphs.
Ramets classifiable by age
Characteristic equation:
Eigenvectors:
𝑊1 = 1 −
𝑃1𝑃2𝐹𝜆−(α+β) 𝑃2𝑉𝜆−β
1=
−
𝜆 − 𝑃3
𝜆 − 𝑃3
𝑃2𝑉𝜆−𝛽
𝑃1𝑃2𝐹𝜆−(𝛼+𝛽)
=
, 𝑉1 = 𝑊1
𝜆 − 𝑃3
𝜆 − 𝑃3
𝑊2 = 𝑃1𝜆−𝛼
𝑊3 =
𝑃1𝑃2𝜆−(𝛼+𝛽−1)
𝜆−𝑃3
𝑉2 =
𝑃2𝐹𝜆−𝛽
𝜆−𝑃3
𝐹
𝑉3 = 𝜆−𝑃3
In this life cycle, sexual and clonal reproduction appear as two distinct modes of reproduction.
α time units are required to go from n1 to n2, and β units to go from n2 to n3. A good example is
the marine polychaete Raphidrilus hawaiiensis that reproduces both by internal gestation and
by fragmentation (Magalhães et al. 2011). Adult worms break off into several fragments; each
fragment regenerates a head end and tail end. If the embryos require 4 weeks to develop to
maturity and the fragments require 2 weeks, the mature individuals produced by the two
pathways can be lumped into a single class; therefore, for this species α = 4 and β = 2. The
survival probabilities from n1 to n2 and from n2 to n3 are P1 and P2, respectively.
Ramets classifiable by size
Characteristic equation:
𝐺1𝐺2𝐹3𝜆−α+1
𝐺1𝐺2𝐹4𝜆−α+1
𝐺2𝑉3
1=
−
−
(𝜆 − 𝑃1)(𝜆 − 𝑃2)(𝜆 − 𝑃3) (𝜆 − 𝑃1) … (𝜆 − 𝑃4) (𝜆 − 𝑃2)(𝜆 − 𝑃3)
𝐺2𝐺3𝑉4
−
(𝜆 − 𝑃2) … (𝜆 − 𝑃4)
Eigenvectors:
𝑊1 = 1 −
𝐺2𝑉3
𝐺2𝐺3𝑉4
−
, 𝑉1 = 𝑊1
(𝜆 − 𝑃2)(𝜆 − 𝑃3) (𝜆 − 𝑃2) … (𝜆 − 𝑃4)
𝐺1𝜆−α+1
𝐺2𝑉3
𝐺2𝐺3𝑉4
=
[
−
]
(𝜆 − 𝑃1) (𝜆 − 𝑃2)(𝜆 − 𝑃3) (𝜆 − 𝑃2) … (𝜆 − 𝑃4)
𝑊2 =
𝐺1𝜆−α+1
(𝜆−𝑃2)
𝐺1𝐺2𝜆−α+1
𝑊3 = (𝜆−𝑃2)(𝜆−𝑃3)
𝐺2𝐹3
𝐺2𝐺3𝐹4
𝑉2 = (𝜆−𝑃2)(𝜆−𝑃3) + (𝜆−𝑃2)…(𝜆−𝑃4)
𝐹
𝐺3𝐹4
𝑉3 = (𝜆−𝑃3) + (𝜆−𝑃3)(𝜆−𝑃4)
𝐺1𝐺2𝐺3𝜆−α+1
𝐹4
𝑊4 = (𝜆−𝑃2)…(𝜆−𝑃4)
𝑉4 = (𝜆−𝑃4)
The stages in this life cycle represent size classes and not age classes. An individual of size i may
grow to size i + 1 with probability Gi or may retain in size class i with probability Pi. A good
example is the soft coral Alcynium sp., which reproduces asexually by binary fission, typically
producing two (occasionally three) daughter colonies of approximately equal size (fission occurs
throughout the year), and sexually with internal fertilization on a yearly cycle (McFadden,
1991). Therefore, during each year, a colony could undergo different types of transitions: it
could 1) grow into a larger size class; 2) shrink to a smaller size class; 3) remain in the same size
class; 4) undergo fission or 5) die. The author observed that sexual reproduction occurred at
very low frequencies and the population dynamics were dominated by the births and deaths of
asexual ramets.
Genets
Characteristic equation:
𝐺1𝐹2
𝐺1𝐺2𝐹3
1=
−
−⋯
(𝜆 − 𝑃1)(𝜆 − 𝑃2) (𝜆 − 𝑃1)(𝜆 − 𝑃2)(𝜆 − 𝑃3)
Eigenvectors:
𝑊1 = 1
𝑉1 = 1
𝐺1
𝑊2 = (𝜆−𝑃2)
𝐹2
𝐺2𝐹3
𝐹3
𝐺3𝐹4
𝑉2 = (𝜆−𝑃2) + (𝜆−𝑃2)(𝜆−𝑃3) +
𝐺1𝐺2
𝑊3 = (𝜆−𝑃2)(𝜆−𝑃3)
𝐺𝑥
𝑊𝑥 + 1 = ((𝜆−𝑃𝑥+1)) 𝑊𝑥
𝐺2𝐺3𝐹4
(𝜆−𝑃2)(𝜆−𝑃3)(𝜆−𝑃4)
+⋯
𝑉3 = (𝜆−𝑃3) + (𝜆−𝑃3)(𝜆−𝑃4) + ⋯
𝐹𝑥
𝐺𝑥
𝑉𝑥 = (𝜆−𝑃𝑥) + (𝜆−𝑃𝑥) 𝑉𝑥 + 1
Genets are classified by their size (i.e. the number of ramets they have accumulated) and what
is termed as ‘clonal reproduction’ is not actually reproduction, but growth. The transition of
individual from one size to another is made by the addition of new ramets. Gi and Pi are the
size-specific probabilities of growth and of remaining in the same size class, respectively.
Regeneration in the columnar cactus Stenocereus eruca seems to occur mainly through clonal
propagation. The life cycle diagram and the projection matrix model of this cactus species is
shown below (extracted from Clark-Tapia et al., 2005).
Size classes are: Seedling, non-reproductive, reproductive I-V. The elements inside the matrix A have the
following regions: fecundity (F, in the first row); stasis (S, in the main diagonal); retrogression in size or
clonal propagation (R and C, respectively in the upper diagonal); and finally growth to later stages (G, in
the sub diagonals).
Literature Cited
Caswell, H. 1985. The evolutionary demography of clonal reproduction. pp. 187-224. In: J. B. C.
Jackson, L. W. Buss and R. E. Cook (eds.) Population Biology and Evolution of Clonal Organisms.
Yale Univ. Press.
Clark-Tapia, R., Mandujano, M.C., Valverde, T., Mendonza, A. & Molina-Freaner, F. 2005. How
importante is clonal recruitment for population maintenance in rare plant species?: The case of
the narrow endemic cactus, Stenocereus eruca, in Baja California, Mexico. Biological
Conservation, 124, 123-132.
Leslie, P.H. 1945. On the Use of Matrices in Certain Population Mathematics. Biometrika, 33,
183-212.
Magalhães, W.F., Bailey-Brock, J.H. & Davenport, J.S. 2011. On the genus Raphidrilus Monticelli,
1910 (Polychaeta: Ctenodrilidae) with description of two new species. Zootaxa, 2804, 1-14.
McFadden, C.S. 1991. A comparative demographic analysis of clonal reproduction in a
temperate soft coral. Ecology, 72(5), 1849-1866.