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## Life tables and K-factor analysis, Biology tutorial

Introduction:The population's growth potential has much to with how frequently individual members reproduce. Few species (e.g., most invertebrates) have only one reproductive event in the lifetime, whereas others (like, most birds and mammals) are capable of numerous events over extended portion of the lives. Former are known as semelparous and latter, iteroparous life cycles. There is large amount of variation, though, within these broad categories. The common form of semelparity in insects of temperate regions is the annual species. In this case, insect overwinters as egg or larval resting stage until spring, then grows during warm months and emerges in reproductive adult. Adults mate and lay eggs that, again, remain dormant all through winter.

Life Tables:Constructing the life table is frequently simple method for keeping track of births, deaths, and reproductive output in the population of interest. Essentially, there are 3 methods of constructing such a table:

1) Cohort life table follows the group of same-aged individuals from birth (or fertilized eggs) throughout their lives,

2) Static life table is made from data collected from all ages at one particular time it assumes age distribution is stable from generation to generation, and

3) Life table can be made from mortality data collected from the specified time period and also supposes stable age distribution.

Particular characteristics of stage-dependent life tables:There is no reference to calendar time. This is very suitable for analysis of poikilothermous organisms. Gypsy moth development depends on temperature but life table is comparatively independent from weather. Mortality procedures can be recorded individually and therefore, this type of life table has more biological information than age-dependent life tables.

1) Static (Vertical) Life Table Based on Living Individuals:

Most organisms have more complex life histories, and while it is possible to follow the single cohort from birth to death, it frequently too costly or time-consuming do so. Another, less correct, method is static, or vertical, life table. Rather than following the single cohort, static table compares population size from different cohorts, across complete range of ages, at single point in time. Static tables make two significant assumptions: i) population has stable age structure that is, proportion of individuals in each age class doesn't change from generation to generation, and ii) population size is, or nearly, stationary.

2) Static (Vertical) Life Table Based on Mortality Records:

Static life tables can also be made from knowing, or evaluating, age at death for individuals from population. This can be helpful method for secretive large mammals (like, moose) from temperate regions where it is hard to sample living members. As the highest mortality of large herbivores takes place during winter, an early spring survey of carcasses from starvation and predator kills can yield helpful information in constructing the life table. We can use data collected from cemetary grave markers to construct a static life table and reveal interesting features of human populations from past generations.

Population Features which can be computed from Life Tables:Besides R

_{0}, basic reproductive rate, many other population features can be determined from life tables. Some of the most common features are cohort generation time (T_{c}), life expectancy (e_{x}), and intrinsic growth rate (r). Cohort generation time is fairly easy to get from our first example, semelparous annual life cycle (T_{c}= 1 year), but generation time is less obvious for more complex life cycles. Generation time can be stated as average length of time between when the individual is born and birth of its offspring.Life expectancy is useful way of expressing probability of living (x) number of years beyond the given age. Though, this value is actually life expectancy at birth. One can also compute mean length of life beyond any given age for population. Life expectancy is somewhat complicated calculation. As l

_{x}is only proportion surviving to starting of particular age class, we must first compute average proportion alive at that age (Lx) :L

_{x}= (l_{x}+ l_{x}+ 1)/2Next, total number of living individuals at age x and beyond (T

_{x}) is:T

_{x}= L_{x}+ L_{x}+ 1 +¼+ L_{x}+ nLastly, the average amount of time yet to be lived by members surviving to particular age (e

_{x}) is:Ex = T

_{x}/l_{x}The following example illustrates life expectancy changes in the hypothetical population which experienced 50% mortality at each age:

Basic reproduction rate (R

_{0}) converts initial population size to new size one generation later as:N

_{T}= N_{0}x R_{0}If R

_{0}remains constant from generation to generation, then we can also utilize it to forecast population size numerous generations in future. To forecast population size at any future time, it is more suitable to use the parameter which already takes generation time in account. Intrinsic rate of natural increase and it can be computed (or estimated for complex life cycles) by the given equation:r = lnR

_{0}/TcFactor Analysis:Main factor analysis has been applied to the variety of animal species to evaluate role of natural enemies in population fluctuations. Generally, this method is not applicable to tropical insects as in most species generations overlap. Factor analysis originated in psychometrics, and is utilized in product management, operations research, behavioral sciences, marketing, social sciences, and other applied sciences which deal with large quantities of data. Factor analysis is associated to principal component analysis (PCA), but two are not identical. As PCA performs the variance-maximizing rotation of variable space, it takes in account all variability in variables.

Kind of factor analysis:Exploratory factor analysis (EFA): is utilized to uncover underlying structure of the relatively large set of variables. Researcher's a priori assumption is that any indicator may be related with any factor. There is no prior theory and one uses factor loadings to intuit factor structure of data.

Confirmatory factor analysis(CFA): looks to find out if number of factors and loadings of measured (indicator) variables on them confirm to what is estimated on basis of pre-established theory. Indicator variables are selected on the basis of prior theory and factor analysis is utilized to see if they load as predicted on expected number of factors.

Types of factoring:Principal component analysis (PCA): The most common form of factor analysis, PCA seeks linear combination of variables such that maximum variance is extracted from variables. It then removes this variance and seeks second linear combination that describes maximum proportion of remaining variance, and so on. This is known as principal axis method and results in orthogonal (uncorrelated) factors.

Canonical factor analysis: It is also known as Rao's canonical factoring, is a different method of calculating same model as PCA that uses principal axis method.

Common factor analysis: It is also known as principal factor analysis (PFA) or principal axis factoring (PAF), seeks least number of factors that can account for common variance (correlation) of set of variables.

Image factoring: based on correlation matrix of predicted variables rather than actual variables, where each variable is predicted from others using multiple regressions.

K-values:K-value is simply another measure of mortality. Major benefit of k-values as compared to percentages of died organisms is that k-values are additive: k-value of the combination of independent mortality procedures is equal to sum of k-values for individual processes. Mortality percentages aren't additive. For instance, if predators alone can kill 50% of population, and diseases alone can kill 50% of population, then combined effect of these process won't result in 50+50 = 100% mortality. If two mortality processes are present, then organism survives if it survives from each individual process. For instance, the organism survives if it was concurrently not infected by disease and not captured by the predator.

Suppose that survival from one mortality source is s

_{1}and survival from second mortality source is s_{2}. Then survival from both procedures, s_{12}, (if they are independent) is equal to product of s_{1}and s_{2}:S

_{12}= S_{1}S_{2}This is survival multiplication rule. If survival is replaced by 1 minus mortality [s=(1-d)], then this equation becomes:

d

_{12}= 1-(1-d_{1})(1-d_{2})Varley and Gradwell (1960) recommended calculating mortality in k-value which is negative logarithms of survival:

k = -ln(s)

We utilize natural logarithms (with base e=2.718) instead of logarithms with base 10 utilized by Varley and Gradwell. Benefit of using natural logarithms will be given below.

It is easy to illustrate that k-values are additive:

k

_{12}=ln(S_{12})=-ln(S_{1}S_{2})=[-ln(S_{1})]+[-ln(S_{2})]= k_{1}+k_{2}k-values for entire life cycle (K) can be evaluated as sum of k-values for all mortality processes:

K= Σk

_{i}K-value = instantaneous mortality rate multiplied by time. Population which experience constant mortality during specific stage (like, larval stage of insects) change in numbers according to exponential model with the negative rate r. We can't call r intrinsic rate of natural increase as this term is utilized for entire life cycle. According to exponential model:

N

_{t}=N_{o}.exp(r.t)Population numbers decrease and therefore, N

_{t}< N_{0}. Survival is: s = N_{t}/N_{0}. Now we can evaluate the k-value:k = -r t.

Instantaneous mortality rate, m, is equal to the negative exponential coefficient because mortality is the only ecological process considered (there is no reproduction):

m = -r,

k = m t

Exponential coefficient r is negative (as population declines), and mortality rate, m, is positive. We proved that if mortality rate is constant, then k-value is equal to instantaneous mortality rate multiplied by time. If instantaneous mortality rate changes with time, then k-value is equal to integral over time. In the similar way, in physics, distance is essential of instantaneous speed over time. Annual mortality rates of oak trees due to animal-caused bark damage are 0.08 in first 10 years and 0.02 in age interval of 10-20 years. We require evaluating total k-value (k) and total mortality (d) for first 20 years of oak growth.

k = 0.08 × 10 + 0.02 × 10 = 1.0

d = 1 - exp(-k) = 0.63

Therefore, total mortality during 20 years is 63%.

Limitation of k-value concept: All organisms are supposed to have equal dying probabilities. In nature, dying probabilities may differ because of spatial heterogeneity and individual variation (both inherited and non-inherited).

Estimation of k-values in natural populations:Estimation of k-values for individual death processes is hard as these processes frequently go simultaneously. Problem is to forecast what mortality could be expected if there was only one death process. To separate death processes it is significant to know biology of species and its interactions with natural enemies.

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