Back to Table of Contents

Back to  Ecology Home

Dr. D.'s Home Page

Department of Biological  and Physical Sciences

School of Science  and Mathematics

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

  3. Regulation of populations
 

From a previous lecture
 

Nt+T =  (erT) * Nt  dN/dt= rN

What does  this model of population growth assume?
 

Hint:

Bacteria innoculated in a broth  (N₀=100)
Bacteria double every 20 min (r=1.035 min^-1)
population size after 36 hours = e^1.035(36*60)* 100  =  2 X 10³⁴
or enough bacteria to cover the entire earth 1 ft deep!

What really happens to natality and mortality as a population grows beyond a certain size?  Why?
 

Resource = "environmental component used by an organism" e.g. food, shelter, light, space.
 
 

The population size at which population growth rate is zero (no slope,  dN/dt=0) is the 'carry capacity' or 'equilibrium population density' (K).  It is the maximum number the environment can support indefinitely.  At some point, something has to limit population growth.
 
 
 

There are numerous ways that limitations to population growth can be modelled and the most appropriate model for a given population would be a function of the type of factors that most affect that population.  Two general types of models:
 

Deterministic models - The population density at a future time can be predicted based on present density.  These models should work best for populations in which intrinsic or density-dependent factors dominate.  That is, the density of the population directly determines the rate of births and/or deaths.   For example:

• competition for a limiting resource affects the level of that resource, resulting in higher rates of starvation when population density is high.
• higher population density attracts more predators.

• weather events that occassionally kill off some of the population so that population size never reaches the point that resources become limiting.
• the level of the resource determining carrying capacity is not constant but determined by factors other than the population being modelled.

Here are two examples of these models:

 

A simple deterministic model of population growth

To model this, previous equations must be altered:
dN/dt= rN * (1 - ^N/_K)    *


where K is the 'carrying capacity' or 'equilibrium population density'

In plain english: population growth is not just a function of the intrinsic rate of growth and population size; it is limited to K number of individuals (i.e. is a function of intrinsic or density-dependent factors).


The term (1 - ^N/_K) esssentially converts r from r_max to r_actual which is expected to be a function of current population size relative to carrying capacity.



Look at what this equation does mathmatically:

• When N is very small relative K, growth is essentially exponential as modelled by dN/dt= rN
• As  N approaches K, ^N/_K appoaches 1 so 1 minus this is near 0 resulting in essentially no population growth (no slope).
• What happens to population growth rate if N is greater than K?


What happens graphically if carry capacity is increased?
What happens graphically if the per capita intrinsic rate of increase is increased?
Click here to run a simulation using the above equation
 
 

This model is 'artifical'(as are all models). 


• The term r in these models is constant and is actually r_max , the maximum per capita rate of increase under optimal environmental conditions.   In reality, the per capita rate of growth is a function of birth and death rates that change of populations size (N).

• K is a theoretical term that forces predicted population growth to not exceed that particular population size, and does not directly model resource levels that affect birth and death rates.

If artifical, what good is the model?

The model predicts the general shape population growth for some species.

Now can see where the terms r and K selection come from in the previous lecture.  An r-strategist reproduces rapidly because it experiences an environment w/ abundant resources and hence per capita growth is near r_max , whereas  a K-strategist experiences an environment where resources are scarce so population size ~K.
 
 
 

The above model can be used to project change in population size over time by iterating the following equation:

N_t+1 =  (e^{r(1 - N/K)}) * N_t
 

At higher values of r, there are some surprizing changes in population size over the long-term (click here to run a simulation).
 

• where r is near 0 (e.g.  r = 0.5) population size becomes stable at size K.

 
• where r higher (e.g.  r = 2.3) the population becomes cyclic ("stable 2-point").  This model under such conditions might simulate that, at higher per capita growth rates, population size 'overshoots' available resources leading to high mortality and subsequent 'crash' of the population below carrying capacity.

Note that such time delays can be added into this model by substracting a time delay term from the time N is evaluated relative to K (see text) (click on here to run a simulation).

 
• where r is very high (e.g. r = 2.9), the population becomes chaotic
Is a complex system always the result of many components interacting?
No, not always.
Traditional view of complexity:
A complex system is the result of a large number of variables acting together.
For example, the U.S. economy.

A more recent idea, Complexity/Chaos Theory:
Simple relationships among a few variables can also generate complex patterns.
For example, fractals.

http://www2.capcollege.bc.ca/~mfreeman/mand.html

Consequence of Complexity Theory: Larger complex processes are highly contingent on initial, small-scale conditions (the "butterfly effect").

 
 
 


So what?  While other factors may contribute to erratic fluctuations, even simple intrinsic factors influencing population change can produce complex patterns.
 

Are there other factors that might cause population size to fluctuate erratically over time?
 
 
 
 
 


A simple stochastic model of population growth
To model this, must alter the previous equations:
N_t+1 =  (e ^{€t + r(1 - N/K)}) * N_t


where €_t  is a 'random' variable that represent changes in the intrinsic rate of increase due to unpredictable extrinsic factors.

In plain english: population growth is a function of both intrinsic and extrinsic factors
Click here to run a simulation using the above equation

 What factors might be extrinsic and tend to be erratic?





 

More complex models might include components that model resource levels as a function of intrinsic and extrinsic factors.
 
 
 
 
 
 

 
A major debate in ecology in the 1950's was on whether most populations were stable and density-dependent factors dominated or fluctuate and density-independent factors dominated.
 
 

1957 Symposium at Cold Spring Harbor: A. J. Nicholson advocated population regulation by  density-dependent influences.  In support Elton wrote, "it is becoming increasingly understood by population ecologists that the control of populations, i.e., ultimate upper and lower limits set to increase, is brought about by  density-dependent factors"  Andrewartha and Birch disagreed: density-dependent factors "are not a general theory because . . . they do not describe any substantial body of empirical facts."

 
Is this population stable?  Does it fluctuate around "a particular mean"?
 


http://www.kangaroo-industry.asn.au/morinfo/BACKGR1.HTM

Four species of moths in a German pine forest.

Phytoplankton in Lake Erie
Theoretically, can density-dependent factors result in erratic fluctuations?

So how can these processes be distinguished?
 

Erratic fluctuations in population size are likely due to DENSITY-INDEPENDENT processes where the affects of a particular influence do not change proportionately with population size.  

Constant or cyclic fluctuations are likely due to DENSITY-DEPENDENT proceses,  whose influence is a proportional to it population size (But, theoretically, can density-dependent factors result in erratic fluctuations?)

A hypothetical numerical example:
Which mortality patterns are density-dependent and density-independent?
 

 

population size (N)	?	?	?
100	50 deaths (50%)	50 deaths (50%)	20 deaths (20%)
1000	50 deaths (5%)	500 deaths (50%)	500 deaths (50%)
10,000	50 deaths (0.5%)	5000 deaths (50%)	8000 deaths (80%)


 
 
 

Do the data in the table below support density-dependent or density-independent influences?

Mortality in the commercial catch of fish after severe cold weather on the Texas Gulf Coast in Winter of 1940:
Locality	N before cold weather	N after cold weather	%Decline
Matagorda Bay	16,919	1,089	93.6
Aransas Bay	55,224	2,552	95.4
Laguna Madre	2,016	149	92.6

Are real populations likely to be either soley density-dependent or density-independent?

What would happen if over the long run resources could not be renewed as fast as they were consumed?
 
 
 
   
 
 

Causes of density-dependent effects

 
The study of such effects is sometimes referred to as "Population Regulation"

Do individuals purposely decrease births (or commit suicide) as numbers increase?  Are there adaptations that arise for "the good of the species"?

   
For example  Wynne-Edwards proposed that some bird populations regulate their clutch size in hard times to benefit the population as a whole"  Can this reduction in clutch size be explained more simply? 

A. Ives writes  that  population regulation  wrongly "implies that something is doing the control, and that the actions of the controller have some purpose.  In fact, population regulation is used by ecologists in a much more neutral way simply to describe patterns of population dynamics...there is no reason to suppose that a purpose exists.  For example, a popular concept of "the balance of nature" implies that natural processes act in a purposeful way to create greater stability and harmony in the natural world.  In fact, no scientific evidence supports this."

i.e. What we see in ecological systems can most easily be explained by simple Darwinian selection on individuals.
 

From your 'quick review of evolution',  a community is a collection of individuals evolving to take advantage of available resources resulting in a dynamic 'balance of power'.


 

Given this, which of the statements below is more accurate way of describing a population competing for a limiting resource?

"As population density increases toward a point at which resources are insufficient, individuals in competition reduce their intake of food."

"As population density increases toward a point at which resources are insufficient, the intake per individual is reduced due to shortages"


 

The negative effect of density on population growth can result from  decreases in births or increases in deaths or dispersal.  This may appear to "be for the good of the population" but can be explained much more easily either through direct causes or Darwinian selection of individuals:

1. Increased mortality
• directly due limited energy/nutrients necessary for survival as a result of intraspecific competition
• due to less direct physiological effects such as weaken immunity, increased transmission of disease, increased predation...


 
 


2. Decreased reproducton

• directly due limited energy/nutrients necessary for reproduction.


 
• due to a physiological mechanism that decrease reproductive rates as density increases.  Can this be explained by natural selection or is this an example of an adaptation for the good of the species?



 

3. Increased dispersal

• due to need to increase search area for resource.
• due to change in territorrial behavior in which some individuals "move out".  Can this be explained by natural selection or is this an example of an adaptation for the good of the species?



 


 
 
 
 

One more time at applying ecological thinking to human populations: How will resources affect the future of human population growth?
 

Will  human populations be limited by insufficient resources? Are there sufficient resources (agricultural land, water, energy) left to accommodate the world's population growth ?

 
 

Do we need to think about resources needed by humans differently than we think about resources needed by nonhuman populations?
 

• dependence on non-renewable resources

 
 
• degradation of the environment that produces renewable resources



               Soil loss
 
 
 
 

• differences in resource consumption per individual

Global distribution of wealth:

20% affluent

60% self-sufficient

20% destitute



How affluent are you relative to the rest of the world?  
 
 
 

Is consumption of world resources unequal?
 

The affluent fifth of the world consumes 2/3 of the metals and 3/4 of the world's energy, and produces 3/4 of the world's pollution.
On a per capita basis, individuals in developing nations use 1.0 kw/person whereas developed nation use 7.5 kw.


http://www.bp.com/worldenergy/oil/main.htm
 

 

And the rate of consumption is increasing.  In the U.S. since 1950's, the average American consumes twice as much steel, meat, wood, and energy and 21 times as much plastic.




 

What factors have led to the disparity in consumption rates?

 
  


 

Should goals of consumption be questioned?

 
Relationships between resource consumption and happiness are not obvious.

New York Times October 4, 2005	From Science, 1995, Vol 267	From the film "The Gods Must be Crazy


                                                     
 

What drives consumption?

Is consumption ingrained in our system of values?

 
 
 
 

Does extreme consumption affect others?


 
 
 



 
 

---


* These blue astericks indicate the equations you should know.  You do not have to memorize them, but instead be able to explain what they mean and be able to work problems using these equations.


---

Back to Table of Contents

Back to  Ecology Home

Dr. D.'s Home Page

Department of Biological  and Physical Sciences

School of Science  and Mathematics