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B. How do populations change; "the motion picture" (population dynamics)

  2. Age-specific population dynamics and why do we care

a. What will the future bring (population projections)

Given two populations with the same b and d:

• one in which chance of death was equal for all organisms in the population
• one in which chance of death was much greater for young organisms in the population

would you predict that the changes in population sizes would be the same? 

Would the equation from the previous lecture predict this?

N_t+T=  (e^(b-d)T) * N_t

If you were in charge of setting prices of insurance policies for various people or planning a maternity hospital, what kind of information would be useful?

 
 
 
 

How are age specific rates measured?

Mortality (q_a) and survivorship (l_a)

Life Tables for Red Deer

a	La	la	Da	qa	ea
age	number surviving to that age	probability of surviving to that age	number dying during that age interval	probability of dying during that age interval	life expectancy (probable years of life from that age onward)
1	1000	1.0	0	0	4.35
2	1000	1.0	61	0.06	3.35
3	939	0.939	185	0.20	2.53
4	754	0.754	249	0.33	2.03
5	505	0.505	200	0.37	1.79
6	305	0.305	119	0.39	1.63
7	186	0.186	54	0.29	1.35
8	132	0.132	107	0.81	0.70
9	25	.025	25	1.0	0.50

Note on notation: age is denoted in most ecology text as 'x' (e.g.  q_x instead of q_a), but here I use 'a' to denote the age interval

Where:

Either L_a orD_a is measured directly, and the other is back-calculated from it according to:
 

D_a  =  L_a - L_a+1    or   L_a+1 = L_a - D_a        *
In plain english, the number dying over a certain age interval is simply the difference between the number that initially reach that age and the number that are alive at the end of the age interval.  For example :

 
(click on figure to enlarge)

And l_a and q_a are calculated as:

l_a  = L_a / L₀    *
In plain english, survivorship at age a is the number surviving to that age class relative to the proportion of all individuals alive at age 0

 

q_a   =  D_a / L_a    *
In plain english, mortality at age a is the number dying as individuals pass through that age class relative to the proportion of all individuals that entered that age class
 
 

As is often the case in ecological studies, which of the above parameters are actually measured and which are estimated depends on the type of information that can be obtained from the population being study.
 

For example:

• Following a single cohort over their lifetime and tabulating number of survivors (L_a) as the cohort enters each age class (a).

   

 
 
• Following all cohort over a single age (time) interval and measuring the number of death occurring in each age class  (D_a) either directly or indirectly from remains


 
 
• If  birth and death rates have been constant resulting in a stable age distribution, than the proportion of individuals in each age class (i.e. the age distribution) itself (N_a)  reflects number surviving  (L_a).

 
 

Examples of patterns in mortality:
 
 

Survivorship curves plot  survivorship against age.

Can survivorship curves have a positive slope?
In these examples, why do survivorship curves look like they do?

 

     
 

Possible curves:

Type I-high survival of young
Type II-constant  rate of death - independent of age
Type III - low survival of young

In reality not always smooth curves or straight

(where x is age)

Often these curves are plotted as the log of survivorship because the logarithmic function transforms processes undergoing a proportional change into a linear relationship (e.g. in type II survival, death rate is constant, meaning a certain constant proportion is lost in each age class and when log transformed this survivorship 'curve' is a straight line).

 
Is death rate the inverse of suvivorship?
(remember q_a is a proportion relative to the number of survivors in that age class, where as l_a is the relative to the number of survivors in  Age Class 0)


 

Why might male and female patterns differ?
 
 
An example of determining patterns of mortality
 

 
 
 

Life expectancy  (e_a)

Life expectancy is the average probable years of survival for a given age class from that point on in life

Calculated for each age class by considering the survivorship in all future (older) age classes relative to survivorship in that age class.  (I won't bore you with the equation here).

Life expectancy can actually increase in older age classes. How?

   

Natality
 

gross reproductive rate  (m_a)  = number of females born in each female age group (age-specific fecundity or 'average fecundity enjoyed by an individual at age a').
(expressed as females because population increases are directly a function of the females)
 

 

Does gross reproductive rate consider mortality?

 

net reproductive rate (R) (or replacement rate) is the probable number of females offspring produced during a lifetime by an average female.

 
Considers that mortality has an effect of the number of offspring a population will produce because not all live to their full reproductive potential.

R₀ =  summation (l_a* m_a)             *

In plain english, number of offspring produced during a lifetime is the sum for all age classes of the mean number of females born in each group adjusted for survivorship

If:
R₀=  < 1, then population will decline
R₀=  1, then  population size constant, i.e. replacing itself
R₀= > 1, then  population expanding (growing)

Examples of R₀ calculation:

          

 
 
 
 

reproductive value (v_a) (age-specific expectation of future growth) is the probable number of offspring produced over the rest of life from that age on.
 

i.e. the reproductive equivalent of life expectancy.

Calculated for each age class by considering reproductive rates of future age classes and the survivorship in future (older) age classes  (again, I won't bore you with the equation here).
 
 

Note that v_a can be lower early in life due to changes in mortality (v₀ = R₀ ).

 

 

Life Tables for female White-Crowned Sparrows 

a	la	ma	lama	va
age	probability of surviving to that age	number of females born in each female group	number of offspring produced per original individual during each stage or class	probable number of offspring over the rest of life from that age on
1	1.0	0	0	1.02
2	0.167	3.142	0.525	6.13
3	0.083	3.333	0.277	6.01
4	0.048	3.556	0.171	4.63
5	0.012	3.750	0.045	4.28
6	0.0016	4.000	0.006	4.00
			R0 =  sum = 1.023

To play with this life table, click here to download the Excel spreadsheet
 
 
 
  Why are life tables important?

1) Infer factors influencing mortality and natality
 

2) Predict changes in population size based on age-specific natality and mortality:

• A crude estimate of the direction of change is to calculate R₀  and compare to 1.0
 
 
• A somewhat more precise approach is to consider broad age groups.  For example for Spotted Owl can be divided into three functional age class (From A.R. Ives in Ecology by S. Dodson)

• No. of next year's adults are a function of the proportion of adults and subadults surviving (designated as s_X): 
  

(s_adults * N_adults ) + (s_subadults * N_subadults )

• No. of next year's subadults are a function of the proportion of juveniles surviving: 

(s_juveniles * N_juveniles )

• No. of next year's juveniles are a function of  the birth rate of surviving  adults and subadults:
  

(b * s_adults * N_adults ) + (b * s_subadults * N_subadults )

Click here to download a nifty Excel version population projection for this example

• For more precise prediction, use a projection matrix (involving linear algebra-matrices and vectors) to determine (predict) future age-structure given a set of m_a and l_a
 
Why consider age-structure when considering population growth?

Such projection can be used to explain why the population of China continues to grow despite the last few decades in which the number of offspring per couple has been less than 2 (i.e. r<0).  ie. Population Momentum
 

Why? Because there more people are presently entering reproductive ages.  Given two populations of the same size and same m_a  values, with Population A having a greater proportion of individuals in reproductive ages classes relative to Population B, which population will produce more offspring over the next time period?
 

Click here for animated age distributions over time or China
For example, India has a greater proportion of young than China and will become the world's most populated nation in a few decades (click on image to enlarge)


 
 
 
 
 
 
 

3) Used to test explanations of  'life history patterns', e.g. why populations reproduce when, how often, how large...(the subject of the next lecture)

 
 
 
 

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