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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 photosynthesis
Victor Reply
The process plants use to convert sunlight into food (energy).
juanita
Some other organisms use
juanita
types of photosynthesis
Victor
ps1 and ps2
smritilekha
what is used to determine phylogeny?
Israel Reply
which condition is the basis for a species to be reproductively isolated from other members?
Israel Reply
Why do scientists consider vestigial structures evidence for evolution?
Israel
8.Which statement about analogies is correct?
Israel
What is true about organisms that are a part of the same clade?
Israel
Why is it so important for scientists to distinguish between homologous and analogous characteristics before building phylogenetic trees?
Israel
(CH2O)n is the stoichiometric formula of
Marcellus Reply
what are nucleotide
Anastijjaninaiya Reply
Methane,ammonia,water and sugar are dissolved to form nuceotide
Me
Introduction To Biology
Tanveer Reply
can ringworm be caused by bacterium
fred Reply
Nope
ejikeme
Branches of biology
Tanveer
no it does not occurs by bacterium
gopal
what is a brick?
Istifanus Reply
what is gene in biology?
yousaf Reply
it is a heredity unit
Me
what is DNA
yousaf Reply
carrier of genetic information
missy
deoxyribonucleic acid
gopal
it contains genetic information and brings it to one generation to other
gopal
it is of two Types circular DNA and linear DNA
gopal
plasmids are the type of small circular DNA which lies outside the genomic DNA
gopal
And what makes a virus to be difficult to destroy
Mosongo
what observation is made when dry seeds and soaked seeds are put in a vacuum flask
Robin Reply
there is respiration from the soak seeds which shows on the walls of the vacuum flask
Israel
what's mammals ?
Istifanus Reply
mammals are vertebrates ,any member group of vertebrates animals in which the young are nourished with milk from special mammary glands of the mother.
yousaf
what are actin and myosin
Praveen Reply
they are muscle filaments
Israel
they make up the microfibrils of the muscle ,relaxing and contracting to cause movement
Israel
Please did anybody know the questions that will come out in the coming practical?
Oladimeji
Why it is importantthat there are different types of protein in plasma membraine for the transport materials into and out of a cell?
Louellie Reply
the nerve cell
Mustapha Reply

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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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