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Population growth  =   r N

The value of r can be positive, meaning the population is increasing in size (the rate of change is positive); or negative, meaning the population is decreasing in size; or zero, in which case the population size is unchanging, a condition known as zero population growth    .

Logistic growth

Extended exponential growth is possible only when infinite natural resources are available; this is not the case in the real world. Charles Darwin recognized this fact in his description of the “struggle for existence,” which states that individuals will compete (with members of their own or other species) for limited resources. The successful ones are more likely to survive and pass on the traits that made them successful to the next generation at a greater rate (natural selection). To model the reality of limited resources, population ecologists developed the logistic growth    model.

Carrying capacity and the logistic model

In the real world, with its limited resources, exponential growth cannot continue indefinitely. Exponential growth may occur in environments where there are few individuals and plentiful resources, but when the number of individuals gets large enough, resources will be depleted and the growth rate will slow down. Eventually, the growth rate will plateau or level off ( [link] b ). This population size, which is determined by the maximum population size that a particular environment can sustain, is called the carrying capacity    , or K . In real populations, a growing population often overshoots its carrying capacity, and the death rate increases beyond the birth rate causing the population size to decline back to the carrying capacity or below it. Most populations usually fluctuate around the carrying capacity in an undulating fashion rather than existing right at it.

The formula used to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression “ K N ” is equal to the number of individuals that may be added to a population at a given time, and “ K N ” divided by “ K ” is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation:

Population growth  =   r N   [ K N K ]

Notice that when N is almost zero the quantity in brackets is almost equal to 1 (or K / K ) and growth is close to exponential. When the population size is equal to the carrying capacity, or N = K , the quantity in brackets is equal to zero and growth is equal to zero. A graph of this equation (logistic growth) yields the S-shaped curve ( [link] b ). It is a more realistic model of population growth than exponential growth. There are three different sections to an S-shaped curve. Initially, growth is exponential because there are few individuals and ample resources available. Then, as resources begin to become limited, the growth rate decreases. Finally, the growth rate levels off at the carrying capacity of the environment, with little change in population number over time.

Questions & Answers

How many bones are in the human skeleton
Treasure Reply
203
Oyeleke
procce of digestion of proteins a long human alimentarycanal
Carson Reply
what are the properties of lipids?
Isiah Reply
They are: Fatty acids, fats, oils, waxes, phospholipid, glycolipids, steroids and some vitamins
Rachel
explain why a fresh water fish excrete ammonia
Leonard Reply
plz answer my question
Leonard
sorry i meant it has a nucleous unlike plant cells lol
Lailah
Ammonia is the end product of protein catabolism and is stored in the body of the fish in high concentrations relative to basal excretion rates. Ammonia, if allowed to accumulate, is toxic and is converted to less toxic compounds or excreted
Rachel
What are eukaryotic cells?
Nwosueke Reply
cell with no nucleous so not a plant cell
Lailah
eukaryotic cells are membrane bound organelles that have a membrane bound nucleus
ojeen
where does the cell get energy for active transport processes?
A'Kaysion Reply
IDK maybe glucose
Lailah
what is synapsis
Adepoju Reply
how many turns are required to make a molecule of sucrose in Calvin cycle
Amina Reply
why Calvin cycle occurs in stroma
Amina
why do humans enhale oxygen and exhale carbondioxide?
Maryam Reply
why do humans enhale oxygen and exhale carbondioxide? For the purpose of breaking down the food
dil
what is allele
uzoka Reply
process of protein synthesis
SANTOSH Reply
what is cell
Zulf Reply
a cell is a smallest basic, structural and functional unit of life that is capable of self replication
Lucas
why does a fresh water fish excrete ammonia
Leonard
plz answer my question
Leonard
Ammonia is a toxic colorless gas and when its inside the fish biological system is converted to a less toxic compound then excreted in the form of urea. However too much ammonia will kill the fish " Ammonia Poisoning " which is a very common disease among fish.
This
what is cytoplasm
uzoka Reply
cytoplasm is fluid of cell.
Deepak
how many major types of Cloning
Saeed Reply
two
amir
two
Zulf
comparative anatomy of gymnosperms?
Meenakshi Reply
anatomy of gymnosperms
Meenakshi

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Source:  OpenStax, Concepts of biology. OpenStax CNX. Feb 29, 2016 Download for free at http://cnx.org/content/col11487/1.9
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