May 21, 2018, Monday

# MBW:Life Table Analysis of Age Structured Populations in Seasonal Environments

Like any mathematical model, a population projection depends on the assumptions of the system. The first successful age structured model for populations was developed by Leslie (1945) and assumes that the population has a stable age distribution. In populations that have seasonal births/deaths, a stable age distribution may not be the most accurate model. Life Table Analysis of Age Structured Populations in Seasonal Environments by J. Wild Manage[1] discusses the differences and relationships among popular seasonal environments. It also includes a comprehensive discussion of the ease of estimation and parameters.

## Executive Summary

Estimating population dynamics in systems with seasonal birth/death can be easily and accurately modeled with discrete mathematics. There are several popular models that are used to estimate age structured populations in seasonal environments. The basic age structured model is the Leslie matrix which assumes uniform distribution in each age. Relaxing the constraints on the time of census allows the recruitment rates to be more easily measured. Life Table Analysis of Age Structured Populations in Seasonal Environments compares and contrasts several methods of taking censuses and estimating populations.

## Project Categorization

Mathematics Used

1. Leslie Matrix. All of the following models are based on the Leslie Matrix (model)
2. Michod and Anderson Model
3. Pielou Model
4. Combination Model

Type of Model

The models considered in this paper are stage based population models, where each stage represents a different age bracket.

Biological System Studied

No specific biological systems were studied. Instead, the Leslie Model and the Combination Model were compared for their value to field researchers.

## Context/History

Modeling populations and projecting how they will grow and decline has long been of interest for mathematical biologists. There are many different types of models that use a variety of techniques. The first person to begin using an age structured population in an attempt to model and project a population was Alfred Lotka in his 1907 paper. He later extended his work in 1911 with F.R. Sharpe and again in 1925 with Louis Dublin. Lotka focused on modeling populations with continuous models as opposed to discreet models. Continuous models require following the population being modeled continuously to obtain the parameters used in the model. Most censuses are taken on periodic time intervals therefore leading to discrete models being better than continuous. Discrete models are much better suited for populations in seasonal environment and pulse birth populations. In 1945, Leslie published a very successful discrete, age-structured model. This model uses a matrix filled with physical parameters of the population at each stage and is multiplied by a vector representing the age distribution of the population at time t. This gives the age distribution of the population at time t+1. Although ground breaking, the parameters used in the Leslie model are very difficult to find or estimate. Field studies, to determine these parameters are very expensive and take large amounts of time to complete. In the discrete life table model published by Cole (1954) and Pielou (1974), they tweaked the Leslie model to make the parameters easier to measure. M. Taylor and J. S. Carley (1988) compared the Leslie Matrix and the life table.

## Derivation and Discussion of Models

### Basic Similarities of all Models

Figure 1. Leslie Model Example

Each model is an alteration of the Leslie Model[2] which describes evenly distributed age structures. An example of a Leslie Matrix can be seen in Figure 1. In all three approaches analyzed in this paper, the population is described by the number of viable females $(N_{{x,t}})$ in each age class and the time interval of each age class is the same. The primary differences in the approaches result from the timing of the census, which occurs at the beginning of the time interval in the Leslie approach and at the end of the time interval in the life table approach.

The growth rate, λ, of the Leslie Matrix model of each of these models is described in equation 1. $(1)\lambda =N_{{x,t-1}}/N_{{x,t}}$

The Leslie model uses asurvival rate ($p_{x}$) giving the following equation:

$(2)N_{{x+1,t+1}}={p_{x}{{\cdot }}N_{{x,t}}}$

$p_{x}$ = fraction of females of age x at census in time interval t who are alive at age x + 1 at census in time interval t + 1.

For more examples of models that employ Leslie Matrices, see Pod Specific Demography of Killer Whales and Stage-Based Population Model.

### Michod and Anderson Model

In the Michod and Anderson (1980)[3] model a survivorship $(l_{x})$ term is introduced. Survivorship is the survival rate from age 0 to age x. It is used by the life table:

$(3)N_{{x+1,t+1}}=(\prod {p_{x}}_{{i=0}}^{x}){\cdot }N_{{0,t-x}}=l_{{x+1}}{\cdot }N_{{o,t-x}}$

where $l_{{x+1}}=$ fraction of females of age zero at census in time interval t - x who are alive at age x + 1 at census in time interval t + 1. Using equations (1-3) we find the following relationships among the numbers of females in relative age classes at census t (i.e., the stable standing age distribution $[S_{x}]$), the survivorship schedule $(l_{x})$, and the population growth rate:

$(4)S_{x}=N_{{x,t}}/{N_{{0,t}}}=l_{x}/{\lambda ^{x}}=(\prod _{{i=0}}^{x}{p_{x}}){\cdot }p_{x}^{{-1}}{\cdot }\lambda ^{{-x}}$

Figure 2. Michod and Anderson Model

The approaches differ principally in their estimation of $N_{{0,t}}$. In the Leslie matrix approach we consider both the births of daughters and their survival (i.e., recruitment) in the initial (x = 0) age class. Age specific recruitment rates are defined by the Leslie model as (the number of daughters who will enter age class zero in time interval t + 1)/(females alive in age class x censused at the beginning of time interval t). To determine all the daughters being recruited into age class zero during time interval t + 1, we need to sum the contributions from the females of all age classes:

$(5)N_{{0,t+1}}=\sum _{{x=0}}^{w}(F_{x}{\cdot }N_{{x,t}}).$

By recognizing that $N_{{0,t+1}}=\lambda {\cdot }N_{{0,t}}$ and substituting for $N_{{0,t+1}}$ in eq (5), λ can be defined in terms of the Leslie age specific recruitment term $F_{x}$ and the stable age frequency data determined at time t $(N_{{x,t}}/N_{{0,t}})$ as:

$(6)\lambda =\sum _{{x=0}}^{w}(F_{x}{\cdot }N_{{x,t}}/N_{{0,t}}).$

### Pielou Model

The Pielou model (1974)[4] is estimated with reference to a different group of females than in the Leslie matrix approach. Rather than considering the number of females found in the census at the beginning of the time interval, the life table approach considers those that survive to the end of the time interval t. Life table recruitment is defined as the number of daughters born in time interval (t + 1)/females alive in age class x (censused at the end of time interval t + 1), $(m_{x}^{{LT}})$. The total recruitment to age class zero is:

$(7)N_{{0,t+1}}=\sum _{{x=0}}^{{w-1}}(m_{{x+1}}^{{LT}}{\cdot }p_{x}{\cdot }N_{{x,t}})=\sum _{{x=1}}^{w}(m_{x}^{{LT}}{\cdot }N_{{x,t+1}})$

We have one fewer age class included in the discrete version of Lotka's (1907a,b) equation because the age class zero can receive recruits but cannot produce them (Fig. 1). We can rewrite equation (7) as:

$(7.5)1=\sum _{{x=1}}^{w}m_{x}^{{LT}}{\cdot }(N_{{x,t+1}}/N_{{0,t+1}})$

or from equation (4) as:

$(8)1=\sum _{{x=1}}^{w}(m_{x}^{{LT}}{\cdot }l_{x}{\cdot }\lambda ^{{-x}})$

Equation (8) is the usual discrete analog to the Lotka identity (Cole 1954). We can also write an expression for x from equations (1) and (7):

$(9)\lambda =\sum _{{x=1}}^{w}[m_{x}^{{LT}}{\cdot }(N_{{x,t+1}}/N_{{0,t}})]$

### Combination Model

The Michod and Anderson (1980)[3] approach is based on the number of female births occurring in the interval t (i.e., $B_{t}$) and then on the number of those animals that survive to enter age class zero during census t + 1 (using the survivorship $l_{o}$):

Figure 3. Life Table

$(10)N_{{0,t+1}}=l_{0}{\cdot }B_{t}$

$(11)B_{t}=\sum _{{x=0}}^{w}(m_{x}^{{MA}}{\cdot }N_{{x,t}})$

where $m_{x}^{{MA}}$ is the number of daughters born in the time interval t/female alive in age class x (censused at the beginning of time interval t), and $l_{0}$ is the fraction of female offspring born in time interval t that survive to enter age class zero at the time interval t + 1. From these definitions (eqs [10, 11]) we have the following relationship:

$(12)F_{x}=l_{0}{\cdot }m_{x}^{{MA}}$

Hence the expression for λ:

$(13)\lambda =l_{0}{\cdot }\sum _{{x=0}}^{w}[m_{x}^{{MA}}{\cdot }(N_{{x,t}}/N_{{0,t}})]$

is identical to that for the Leslie matrix approach (eq [6]). This formulation of Michod and Anderson's (1980) central result (their eq [3], our eq [13]), avoids the potentially confusing mixture of continuous and discrete time notation in their paper.

### Results of Derivations

Michod and Anderson (1980) argue that the discrete formulation of Lotka's identity is technically incorrect except for ideal birth pulse populations. Pielou (1982)[5] has commented that this conclusion is not valid because Michod and Anderson (1980) confused the meaning of $m_{x}^{{MA}}$ and $m_{x}^{{LT}}$. The life table and Michod and Anderson's (1980) results are reconciled by noting that, for x > 0:

$(14)m_{x}^{{MA}}=(\lambda /l_{0}){\cdot }m_{x}^{{LT}}$

This relationship (eq [14]) can be derived from the identity given by Pielou (1982):

$(15)N_{{x,t+1}}{\cdot }m_{x}^{{LT}}=F_{x}{\cdot }N_{{x,t}}$

Example of Census

and equations (1) and (12) above. The x > 0 condition is necessary because the zero age class of the life table model can only receive recruits, whereas the zero age class of the Leslie model can receive and produce recruits. This fundamental difference also explains why the life table equation is summed from age 1 when the initial age class is zero. Michod and Anderson (1980) did not specify a time of census. They cite Leslie (1945) as their source, so we assume their time of census is the beginning of the time interval. The relationship (eq [15]) between recruitment terms was derived for the particular classical definitions of $F_{x}$ and $m_{x}$, (Leslie 1945; Cole 1954; Pielou 1974, 1982).

All 3 approaches lead to equivalent descriptions of the dynamics of age structured populations. In principle, any of these methods can be used to estimate the population growth rate when estimates of survivorship (or survival rate) and the appropriate measures of recruitment (eqs [6, 9, or 13]) are available.

To study another type of mathematical model for Age-Size structures of a population view MBW:A New Model for Age-Size Structure of a Population.

## Example/Analysis

This table is an example of the differences between the Leslie matrix and the life table methods as described above. The table gives data from three different censuses all having growth rate λ = 1.3186 and the population age distribution is stable. The differences in the parameters and where the census of the population can be seen by looking at the table. In the life table, the young born that year are counted as age class 0, and in the Leslie model, the age class 0 are the yearlings. This table, along with the diagram of the models help illustrate the models and their differences.

The usefulness of these models is based on how easy the parameters, used in the model, are to find or estimate. The recruitment values for the life table are very easy to find due to the fact that they are unaffected by the age distribution of the population. However, in the Leslie matrix, to find this value, a field researcher much know the population growth rate (λ) or follow a cohort over the course of the censuses, both of which are very difficult and time consuming. The source of this difference is where the different models take the census. In the Leslie matrix, the census is taken at the beginning of the time period an individual is assumed to survive and reproduce at the same time were as in the life table, an individual survives then reproduces and the census is taken at the end of the time period. These assumptions in the life table allow field researchers to assume that all new births occur simultaneously, right before the census thus giving an $l_{0}$ value of 1. The recruitment term $(F_{x})$ in the Leslie matrix is based upon the number of females in age group x at the time of the census and therefore involves the survivability of the age class x, the reproductive rate of age class x, and the survivability of the newborns. Therefore, population projections using a Leslie matrix are only useful when working theoretically because the parameters in the life table are much easier to obtain in the field.