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d N d T  =  b N     d N  =  ( b   -   d ) N

Notice that the “ d ” associated with the first term refers to the derivative (as the term is used in calculus) and is different from the death rate, also called “ d .” The difference between birth and death rates is further simplified by substituting the term “ r ” (intrinsic rate of increase) for the relationship between birth and death rates:

d N d T  =  r N

The value “ r” can be positive, meaning the population is increasing in size; or negative, meaning the population is decreasing in size; or zero, where the population’s size is unchanging, a condition known as zero population growth    . A further refinement of the formula recognizes that different species have inherent differences in their intrinsic rate of increase (often thought of as the potential for reproduction), even under ideal conditions. Obviously, a bacterium can reproduce more rapidly and have a higher intrinsic rate of growth than a human. The maximal growth rate for a species is its biotic potential, or r max , thus changing the equation to:

d N d T = r max N
Both graphs (a) and (b) plot population size versus time. In graph (a), exponential growth results in a curve that gets increasingly steep, resulting in a J-shape. In graph (b), logistic growth results in a curve that gets increasingly steep, then levels off when the carrying capacity is reached, resulting in an S-shape
When resources are unlimited, populations exhibit exponential growth, resulting in a J-shaped curve. When resources are limited, populations exhibit logistic growth. In logistic growth, population expansion decreases as resources become scarce, and it levels off when the carrying capacity of the environment is reached, resulting in an S-shaped curve.

Logistic growth

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 will survive to pass on their own characteristics and traits (which we know now are transferred by genes) 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, slowing the growth rate. Eventually, the growth rate will plateau or level off ( [link] ). This population size, which represents the maximum population size that a particular environment can support, is called the carrying capacity, or K .

The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression “ K N ” is indicative of how many individuals may be added to a population at a given stage, 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:

d N d T = r max d N d T = r max N ( K   -   N ) K

Questions & Answers

what is abiotic and biotic factors?
Hira Reply
which of the following shows the correct sequence of the cell cycle
Kameishia Reply
who is name virus
Shivam Reply
centromere consist of
Shivam
meeting point of two chromatids
Cffrrcvccgg
Explain the function of nematocysts in cnidarians?
Israel Reply
The nemotocyst is used by Cnidarians (hydra, jellyfish, sea anemones) to sting their prey and any threatening enemy.
Lee
photosynthesis in plants is an example of what ? (a) excretion (b) irritability (c) nutrition (d) reproduction
Lee Reply
If a Hox 13 gene in a mouse was replaced with a Hox 1 gene, how might this alter animal development?
Israel Reply
Which of the following organisms is most likely to be a diploblast?
Israel
what are reactions of photosynthesis?
Maria Reply
what are the probabilities of blood genotypes for the offspring from a cross between a mother lAlA blood and a father with lBi blood?
dayana Reply
what is matter
Emmanuel Reply
matter is anything that has mass and can occupied space
Alice
weight
Alice
example of matter
Oyekemi
You serves as an example of matter Because matter is anything that has mass and occupy space e.g man and every other things that exist on earth.. So think of every other things around you ...
Biola
and you too
Oyekemi
We generally
Biola
What is ecological management
Oyekemi
how the kidney functions as osmoregulatory organ
Sam Reply
That true
Banda
what is the major connection for sugars in glycolysis?
Ibrahim Reply
Simple term of science
Palesa Reply
what does it mean
gopal
it's means what do u know about biology?
Phathu
what is immunisation
Melysa
the action of making a person immune to infections ,for immunisation
Kalia
what is the biology? what do you know about biology
Phathu Reply
biology is the study of living organisms, divided into many specialized fields that cover their morphology, physiology, anatomy, behavior, origin, and distribution.
Julia
The study of all aspects of life. The study of all living organisms (such as animal cells and plant cells) in greater detail (their structure and how they function). It's a very broad science.
juanita
what is prokaryotic
Bhaskar Reply
what is pathogens
Bhaskar
pathogens are a bacterium, virus, or other microorganism that can cause disease.
Lee
transistion metals....
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Source:  OpenStax, Biology. OpenStax CNX. Feb 29, 2016 Download for free at http://cnx.org/content/col11448/1.10
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