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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 hominids
Dauba Reply
what is genetics
Musa Reply
what is meiosis?
list three vertebrate animal and three invertebrate animal
Katie Reply
lizard, goat, frog ,earthworm, housefly, tapeworm
what is meant by radial symmetry?
what is meaning of polysaccharide
mark Reply
water as a function in cell for inorganic substances
chisom Reply
hi john....wat about for mineral salts, its function in a cell and structure
hi chisom which topic is this
what is sepsis
this biology is for which grade?
Janet Reply
state any one specialized animal cell and it's function
Pascal Reply
sperm cell: Fertilization
Can DNA and RNA be praticalized
Oyewale Reply
you can make models of them and show how they replicate
what is dentition
Tisa Reply
Study of teeths and their functions
what are the function
it's not teeths. teeth is plural as well as singular noun. the function of the teeth is to cut and grind food for easier swallowing and for increasing the surface area if the fish as the substrate in which digestive enzymes work
dentition is not the study of teeth but the arrangement of teeth in the mouth of species or individuals
what are the two types of teeth
upper lower jaw and upper jaw teeth
Function of probosis
Adeola Reply
why do we study biology?
where biology began
What are chromesome
Talkmore Reply
condensed DNA wrapped with histone
chromosomes are gins that we take from our parents.when mom get pregnant the baby inside is made up of 23 chromosomes from his dad and 23 chromosomes from his mom.so a baby is made up of totally 46 chromosomes
chromosome are rod or thread shape body located in the nucleus of a cell.
what are chromosomes
what is chromosome?
Royd Reply
How can a person who has been in the vascular disease repairs his or her DNA
Rabson Reply
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Of wat
explanation for wat
Of war na so dat I can provide
Is d process which food is converted into substance dat can be utilize by d body

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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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