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

  1. Simple population growth models
 

Population size is a function of what processes?  (Hint: What do you need to know in order to predict changes in water level?)

 

  Basic population (demographic) parameters: 

Population growth rate is change in the total number of individuals in a population over a given time period.

Per capita population growth rate is the mean contribution by each individual in the population (=population growth rate/population size).  

What does per capita population growth rate depend on?

 

equals

minus

 

If we can estimate per capita reproductive rate (B) and per capita death rate (D), then
 

population size at next time interval is size of population at previous time plus the number born minus the number that die

or

N_t+1 =   N_t + B N_t  - D N_t  =    (1 + B_t  - D_t ) * N_t    =   R * N_t

where:
R is the per capita population growth rate, which is a function of B - D  (R is expressed as λ in the Molles textbook)
N_tis population size at time t and  N_t+1is population size at time t+1

If B and D remain constant (i.e. keeping things simple):
iterating the above equation  N_t+1 = R * N_t   yields:

 

Nt+2	=  R * Nt+1	for the second time period, or
	=  R * R * Nt	or
	=  R2 * Nt

from this it follows that

N_t+3 =  R³ * N_t

and

N_t+T =  R^T * N_t

where T is number of time units in the future

 

Example:

For bacteria a reasonable R is 1.035 min-1.  If we start with a petri dish of 100 individuals (N0=100) at 1 min: N1 = 100 * 1.035 = 103.5 at 2 min: N2 = 103.5 * 1.035  = 107.2 at 3 min: N3 = 107.2 * 1.035  = 111.0 Note that the constant is being multiplied by a larger and larger number. The calculation would become tedious if N20 to be calculated, so we use the more concise formula derived above: Nt+T =   RT * Nt at 20 min: N20 = 100 * 1.035 20  = 200 at 40 min: N40 = 100 * 1.035 40  = 400 at 60 min: N60 = 100 * 1.035 60  = 800 at 80 min: N80 = 100 * 1.035 80  = 1600

 

 
An enormously important consequence:  populations can grow exponentially even when the per capita rates are constant.
 

 

In other words, if birth rates are greater than death rates and even if these remain constant over time:
 

The time it takes the population to increase by "x" amount becomes less and less.

or

The population increases by a greater absolute amount over each succeeding time period even if the per capita growth rate remains constant over time  (note that the population increases by a constant proportion in each unit of time).

 

Fictional "Tribbles" from Star Trek: One of the more peculiar species encountered, the defining characteristic of the Tribbles is their extreme reproductive rate. Over half of a Tribbles metabolism is devoted to reproduction, allowing them to bear a litter of young every twelve hours.  With an average litter of ten, a single Tribble can therefore create a population of 1,771,561 within three days, and an amazing 304,481,639,541,400,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 in thirty days!

Why is growth exponential?
Analogous to investing an initial sum of money at a fixed return rate
where:

amount of money is analogous to population size

interest (return) rate is analogous to per capita rate of growth

 

Under what conditions could exponential growth occur?

                  

 
 

The rate of increase is usually expressed as r, where:
 

 r = lnR or   e^r = R

where r is the "intrinsic rate of increase"

so that:

N_t+T =  R^T * N_t

becomes
 

N_t+T =  (e^rT) * N_t      *    (sometimes expressed as N_t =  N₀e^rt )
 In plain english, the population size at a future time can be determined from the population size at any previous time and the per capita rate of population change (the intrinsic rate of increase, r, or r_max in some textbooks).  

 

The reason why r  is used (rather than R) is that it is a conceptually easier term.  For example, if r > 0 then population size increase and if r < 0 population size decreases.  In general, changes that are 'proportional' are easier to see and to work with when converted using  log functions.

To 'play' with this equation, download this Excel spreadsheet
 

What does the above equation assume about r?

 

We can slightly modify this equation by defining

 
r = b - d

where b and d are the per capita birth and per capita death rates

so:  

N_t+T=  (e^(b-d)T) * N_t

(If b = d, then e⁰ = 1,so N_t+T =  N_t  which is to say population size is stable over this time period).

 

What this equation is doing is calculating population size at the next time period on an instantanous basis.
 

The miracle of calculus lets us derive this equation from:

  dN/dt = bN - dN

or

dN/dt= rN  *

In plain english, the rate a which the population grows is a function of the rate at which individuals can replace themselves (per capita rate of growth) and the number of individuals in the population.

dN/dt for the graph below (or dx/dt in some textbooks) is the slope at any given point along the curved line:

Bottom line: How fast a population grows is not only a function of the average mortality and natality of individuals, but also of how many individuals there are in the population.   Larger populations can grow faster because there are more individuals reproducing.  Simple, but enormously important.

 

Implications for Populations in Nature:

Which population is growing faster,  'Population A' of 1000 individuals where intrinsic rate of increase (r) is 0.1 years^-1 or  'Population B' of 100 individuals where intrinsic rate of increase (r) is 0.2 years^-1 ?     If r remains constant, will the answer always remain the same as these populations increase?

Under what conditions would such exponential growth occur?
 
 

Or, one might wish to know which of two populations has the greatest potential to increase rapidly by
solving the equation N_t+T =  (e^rT) * N_t for  r given measurements of population size over a time interval for each population.

 
 
 
 

Practical applications of this apply not only to comparing the potenial grow of organisms in the environment, but also to understanding immune cells and disease-causing organisms in the body.
Click here for a demonstration of a simple model
(for more complex models see http://cancerres.aacrjournals.org/cgi/content/full/65/17/7950 ).

 
 
 
 
 

 The rate of population has slowed, but is the global population  still increasing rapidly?

http://www.iiasa.ac.at/Research/ LUC/Papers/gkh1/chap1.htm

 

Globally, 'fertility rates' (~ b) have declined over the last couple of decades.  Why?

Four major periods in human population growth: 1. Hunter-gathers  2. Agriculture - land supports more humans 3. Industrial revolution - both improved health and agriculture  4. Full industrialization (developed nations) - decreased birth rates

James Watt and his steam pump

 

"Demographic transition" due to industrialization - Lower death rates is followed later by lower birth rate results in growth then later stablization

 
 
 However the global population continues to increase rapidly.  Why?  Hint: Where is most of the world along the curves in this demographic transition?

Are these population growing?

  

World population breaks down into 3 groups:

• 20% affluent - high resource consumption
• 60%self-sufficient- sufficient for basic health and nutrition
• 20% destitute- insufficient water, food, energy
  

 

80% of worlds population lives in developing nations. At the beginning of the last century (1900), 70% did so. By 2050, the share of the world population living in the currently less developed regions will have risen to 90%.   Can the Earth sustain the current global population size if all the world lived the lifestyle that Americans do?

  
http://www.undp.org/popin/wdtrends/p98/p98cht1.htm

"90% of growth takes place in developing nations"

  So, is the cause of current global population growth a result of population size of developing nations (N) or of the per capita growth rate of developing nations (r), or both?  

 
 
   
Questions to consider for the next lectures:   

• In China, birth rates are lower than death rates, but populations continue to grow?  Why?
• Given that organisms can only invest so much energy into survival and reproduction, which is the better investment, genes that improve survival or genes that improve reproduction?  Or, another way to ask this, is why do some species invest more in reproductive effort, while others in survival?

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