Constructing stage-structured matrix population models from life tables: comparison of methods

Early maturation types

For the early maturation types, two types of survivorship and three types of fecundity schedules were incorporated. For the first type of survivorship, individuals experience a constant risk of mortality. This leads to an exponentially declining survivorship curve ($l\left(x\right)$) with age: (1)${\displaystyle l\left(x\right)={\mathrm{e}}^{-{\alpha }_{1}x},}$ where α₁ is the constant risk (i.e., hazard rate). For the second type of survivorship, individuals experience an exponentially increasing risk of mortality with age: (2)${\displaystyle l\left(x\right)={\mathrm{e}}^{\frac{{\alpha }_2}{{\beta }_2}\left(1-e^{{\beta }_{2}x}\right)},}$ where α₂ and β₂ are the parameters for the increasing risk. Under this model, the hazard function is given by $h\left(x\right)={\alpha }_2{e}^{{\beta }_{2}x}$ (Siler, 1979). This type of risk may be due to aging. These two types of survivorship curves are shown in Fig. 1A.

Figure 1B shows the three types of fecundity schedules. The first type assumes the fecundity $b\left(x\right)$ is proportional to the cube of the body length of the individuals, which in turn increases according to von Bertalanffy growth equation: (3)${\displaystyle b\left(x\right)=R_1{\left[L_{\infty }\left(1-e^{-\kappa x}\right)\right]}^3,}$ where L_∞ and κ are the parameters in the growth model (Von Bertalanffy, 1968). This assumes reproduction is approximately proportional to the mass of individuals. The second type assumes a constant fecundity with age (4)${\displaystyle b\left(x\right)=R_2.}$ Finally, the third type assumes exponentially declining fecundity: (5)${\displaystyle b\left(x\right)=R_3{e}^{-\gamma x},}$ where γ is the parameter determining how fast fecundity declines with age. Parameter R_m (where m = 1, 2, or 3) are constants with different units depending on the type of fecundity, and these constants were used for adjusting λ’s in this study (see ‘Analytical Procedure’). For all of the early maturation types in this study, individuals are assumed to mature at age 1.5 (i.e., x₁ = x₂ = 1.5).

Delayed maturation types

For the delayed maturation types, two kinds of survivorship curves are incorporated into a juvenile stage. One incorporates a constant risk of mortality (Eq. (1)), which results in exponentially declining survivorship (dashed curve in Fig. 1C). The other assumes an exponentially declining risk of mortality in addition to a constant risk: (6)${\displaystyle l\left(x\right)={\mathrm{e}}^{-{\alpha }_{1}x}{e}^{\left[-\frac{{\alpha }_3}{{\beta }_3}\left(1-e^{-{\beta }_{3}x}\right)\right]},}$ where α₃ and β₃ are the parameters for the declining risk. In this model, the exponentially declining hazard is given by $h\left(x\right)={\alpha }_3{e}^{-{\beta }_{3}x}$ (Siler, 1979). This causes initial fast descent of survivorship (solid curve in Fig. 1C). This type of survivorship can be observed when organisms grow out of a mortality risk such as predation as they age. Once individuals mature, they reach a constant risk of mortality (Eq. (1)) and constant fecundity (Eq. (4)) under the delayed maturation types.

In addition, three types of maturation schedule are incorporated. The first type assumes a fixed age of maturation at age 10.5 years (x₁ = x₂ = 10.5). The second type assumes a constant rate of maturation, and the third type assumes an exponentially increasing rate of maturation with age. In the latter two cases, maturation begins at age 8.5 years (x₁ = 8.5), and the parameters are chosen so that the mean age of maturity is approximately 10.5 years (x₂ = 10.5); however, the mean age varies slightly depending on the level of mortality rate during the maturation period. Under the latter two types of maturation schedules, the densities of individuals in immature (all stages prior to maturation) and adult stages are described conveniently with a system of ordinary differential equations (ODEs): (7)${\displaystyle \begin{array}{c}\frac{d{n}_1\left(x\right)}{dx}=-m\left(x\right)n_1\left(x\right)-h\left(x\right)n_1\left(x\right)\hfill \\ \frac{d{n}_2\left(x\right)}{dx}=m\left(x\right)n_1\left(x\right)-a{n}_2\left(x\right)\hfill \end{array},}$ where $n_1\left(x\right)$ and $n_2\left(x\right)$ are the densities of immature individuals and adults at age x, respectively, $m\left(x\right)$ is a per-capita instantaneous maturation rate at age x, and $h\left(x\right)$ is the age-dependent hazard function of mortality. For the second type of maturation schedule, $m\left(x\right)$ is set at m₁ = 0.5 for x ≥ x₁ and 0 otherwise. For the third type, $m\left(x\right)=m_1{e}^{m_2\left(x-x_1\right)}$ for x ≥ x₁, and $m\left(x\right)=0$ otherwise. The proportion of individuals that are mature at a given age for the three types of maturation schedule is shown in Fig. 1D. When the system of ODEs is solved with the initial condition $\left[\begin{array}{cc}{n}_1\left(0\right)\hfill & n_2\left(0\right)\hfill \end{array}\right]=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]$ using a numerical ODE solver, the survivorship is given by $l\left(x\right)=n_1\left(x\right)+n_2\left(x\right)$. Note Eq. (7) does not include reproduction, and the initial condition is 1 in the immature stage; therefore, $l\left(x\right)\le 1$ for all x ≥ 0.

Stage-specific survival rate

All the methods for calculating a stage-specific survival rate from age-specific survival rates investigated in this study assume that the beginning and ending ages of a stage are given by the mean ages of transition into and from the stage, respectively, and that these ages are known. The former assumption can potentially produce bias when calculating λ and generation time because stage transitions often do not occur at a fixed age (i.e., individuals gradually transition from one stage to another over a range of ages). Therefore, the performance of conversion methods was also investigated when the age of stage-transitions was not fixed (Eq. (7)). It is also possible that stage-transitions are completely age-independent (e.g., as in some plant species). This situation is beyond the scope of the current analysis, but it will be briefly discussed in the ‘Discussion’ section.

An age-specific survival rate from age x to x + 1 is given by (8)${\displaystyle s_x=\frac{l\left(x+1\right)}{l\left(x\right)}.}$ Hereafter, lower case s is used for denoting an age-specific survival rate whereas the upper case S is used for denoting a stage-specific survival rate. Suppose the stage begins at age $x^{\left(i\right)}$ and ends at $x^{\left(j\right)}-1$ (i.e., age $x^{\left(j\right)}$ is the next stage), then the survival rate of individuals in the stage is given by the mean survival rate over age classes within the stage. Note that an index in the superscript within parentheses denote a stage. Here, nine different ways of calculating the mean are compared. First, the mean is given by geometric mean, arithmetic mean, or harmonic mean (Table 2). These are three common methods for obtaining a stage-specific survival rate. However, the number of individuals in age classes are different, and it is more appropriate to put weight based on the proportion of individuals in a given age class, which is given by a survivorship curve. This leads to a weighted geometric mean, weighted arithmetic mean, and weighted harmonic mean (Table 2). Although weighting with survivorship is fine when a population is neither growing nor declining (i.e., λ = 1), when a population is growing (or declining), there are more (or less) individuals in younger age classes than predicted by a survivorship curve. Therefore, it is necessary to discount the weight using λ. This leads to weighted means where the weight is given by both survivorship and λ (Table 2). Hereafter, the latter weight is termed “a discounted weight.”

Conditional transition rate

Transition rate from stage i to stage j conditional on their survival (p_j,i) is calculated in three ways (Table 3). The first method (T1) matches the expected duration in the stage assuming exponentially declining time to transition into the following stage. The second method (T2) matches the expected proportion of individuals making the transition into the following stage assuming the survivorship curve gives the distribution of individuals among age-classes. The third method (T3) is the same as the second method except that the distribution is discounted by a population growth rate. These methods are described in more detail in Caswell (2001).

Fertility rate

Fertility rates in matrix population models are different from fecundity in a life table. A fertility rate gives the per capita rate of contribution from one stage to the next over one year so that adults will have to survive to reproduce, and/or offspring must survive to appear in the next stage. On the other hand, fecundity is the number of offspring produced. The latter does not include any survival of adult or offspring. Consequently, there are two steps in converting life table data into a stage-structured fertility rate: calculating age-specific fertility rates and converting them into a stage-specific fertility rate. There are many approaches for the first step depending on the reproductive schedules of organisms within a year (see Caswell, 2001) or among years (e.g., Crowder et al., 1994). Evaluating them is beyond the scope of this study. Here, reproduction was assumed to occur at any time of year (i.e., a birth flow model), and an age-specific fertility rate (f_x) was estimated as follows: (9)${\displaystyle f_x=l\left(0.5\right){\int }_{z=x-0.5}^{x+0.5}b\left(z\right)\frac{l\left(z\right)}{l\left(x-0.5\right)}dz,}$ where $b\left(z\right)$ is the instantaneous per-capita fecundity rate at age x. In the equation, $l\left(z\right)∕l\left(x-0.5\right)$ gives the survival rate of adults from age x − 0.5 to age z (where z > x − 0.5), and the integral calculates the total number of offspring produced between ages x − 0.5 and x + 0.5 per adult that was alive at age x − 0.5. Then, all births are attributed to the half way point (i.e., age x) so that offspring will have to survive half a year on average to appear in the first stage; this survival rate is given by $l\left(0.5\right)$, which is the survival rate from age 0 to 0.5.

Once age-specific fertility rates are obtained, a stage-specific fertility rate F_i is calculated in three different ways. First, an arithmetic mean is taken with equal weights on all age classes, (10)${\displaystyle F_i=\frac{1}{x_3-x_2}\sum _{x=x_2+0.5}^{x_3-0.5}{f}_x.}$ The second approach is to take a weighted arithmetic mean, (11)${\displaystyle F_i=\frac{1}{\sum _{x=x_2+0.5}^{x_3-0.5}{\omega }_{1,x}}\sum _{x=x_2+0.5}^{x_3-0.5}{\omega }_{1,x}{f}_x,}$ where ${\omega }_{1,x}=l\left(x\right)$. The third method is to use a discounted weight by replacing ω_1,x in Eq. (11) with ${\omega }_{2,x}=\frac{1}{{\lambda }^{x-x_0}}l\left(x\right)$. In all calculations, the same weight (or no weight) is used for calculating a stage-specific survival rate and a stage-specific fertility rate.

Euler-Lotka equation

The Euler-Lotka equation (Kot, 2001) is given by (12)${\displaystyle 1={\int }_0^{x_3}{\lambda }^{-x}b\left(x\right)l\left(x\right)dx.}$ The fecundity $b\left(x\right)$ and survivorship $l\left(x\right)$ are defined for a specific life history type as described in the previous section (Fig. 1). Provided R_m is fixed, the only unknown is λ, which can be found numerically by searching the value that satisfies Eq. (12) using a root-finding algorithm.

Generation time in this study is defined as the mean age of parents where the mean is calculated over all offspring born at a given time. More specifically, generation time G₁ is given by (13)${\displaystyle G_1={\int }_0^{x_3}x{\lambda }^{-x}b\left(x\right)l\left(x\right)dx.}$ (Keyfitz & Caswell, 2005). In the case where individuals mature over a range of ages, some individuals do not reproduce in certain age ranges. Therefore, generation time G₂ is instead given by (14)${\displaystyle G_2={\int }_0^{x_3}x{\lambda }^{-x}b\left(x\right)q\left(x\right)l\left(x\right)dx,}$ where $q\left(x\right)$ is the proportion of individuals that are mature at age x.

Age-structured matrix population models (Leslie matrix)

An age-structured population matrix is given as (15)${\displaystyle \mathbf{A}=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill f_2\hfill & \hfill \dots \hfill & \hfill f_{x_3-1.5}\hfill & \hfill f_{x_3-0.5}\hfill \\ \hfill s_{0.5}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill s_{1.5}\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill s_{x_3-2}\hfill & \hfill 0\hfill \end{array}\right].}$ In this study, the first age class is assumed to begin at age 0.5, and subsequent age class is incremented by one year. It should be noted that survival rate s_x gives the proportion of individuals that survive from time x to x + 1 and fertility f_x gives the mean number of offspring of age 0.5 produced by a parent alive at age x − 0.5 by reproducing between age x − 0.5 and x + 0.5 (see Eq. (9)). It is assumed that the reproduction begins at the mean age of maturation x₂.

In the case where individuals mature over a range of age, the population matrix is given instead as (16)${\displaystyle \mathbf{A}=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill q_2{f}_2\hfill & \hfill \dots \hfill & \hfill q_{x_3-1.5}{f}_{x_3-1.5}\hfill & \hfill q_{x_3-0.5}{f}_{x_3-0.5}\hfill \\ \hfill s_{0.5}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill s_{1.5}\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill s_{x_3-2}\hfill & \hfill 0\hfill \end{array}\right].}$

Once the population matrix is constructed, λ is obtained by taking its dominant eigenvalue. Furthermore, generation time is given by (17)${\displaystyle G_3=\frac{\lambda \mathbf{vw}}{\mathbf{vFw}}}$ where v and w are the left and right eigenvectors (row and column vectors), respectively, of the population matrix, and F is a modified population matrix only consisting of fertility terms (Bienvenu & Legendre, 2015).

Stage-structured matrix population models (Lefkovitch matrix)

Stage-structured matrix population models are given in two different forms in this study. For early maturation types, a two-stage population matrix is used (18)${\displaystyle \mathbf{A}=\left[\begin{array}{cc}0\hfill & F_2\hfill \\ S_1\hfill & S_2\hfill \end{array}\right]}$ where the duration of stage 1 is from age x₀ = 0.5 to age x₁ = x₂ = 1.5 and the duration of stage 2 is from age 1.5 to age x₃ = 40.5.

For organisms with delayed maturation, three-stage matrix population models are used (19)${\displaystyle \mathbf{A}=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill F_3\hfill \\ \hfill S_1\hfill & \hfill S_2\left(1-P_{3,2}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill S_2{P}_{3,2}\hfill & \hfill S_3\hfill \end{array}\right].}$ The duration in stages 1, 2, and 3 are assumed to be from age x₀ = 0.5 to 1.5, from age 1.5 to x₁ = x₂ = 10.5, and from age x₂ = 10.5 to x₃ = 40.5. Using the stage-structured matrix, λ and generation time can be calculated in the same way as with the age-structured matrices.