In a population showing exponential growth the individuals are not limited by food or disease.

However, in most real populations both food and disease become important as conditions become crowded. There is an upper limit to the number of individuals the environment can support. Ecologists refer to this as the "carrying capacity" of the environment. Populations in this kind of environment show what is known as logistic growth. If a population has a constant birth rate through time and is never limited by food or disease, it has what is known as exponential growth. With exponential growth the birth rate alone controls how fast (or slow) the population grows.

Logistic function
From Wikipedia, the free encyclopedia

For the recurrence relation, see Logistic map.

Standard logistic sigmoid function

A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre Franois Verhulst who studied it in relation to population growth. A generalized logistic curve can model the "S-shaped" behaviour (abbreviated S-curve) of growth of some population P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops. A simple logistic function may be defined by the formula

where the variable P might be considered to denote a population, where eis Euler's number and the variable t might be thought of as time.[1] For values of t in the range of real numbers from to +, the S-curve shown is obtained. In practice, due to the nature of the exponential function et, it is sufficient to compute t over a small range of real numbers such as [6, +6]. The logistic function finds applications in a range of fields, including artificial neural networks, biology, biomathematics, demography,economics, chemistry, mathematical psychology, probability, sociology, political science, and statistics. It has an easily calculated derivative:

Definition What is Logistic growth? A logistic growth curve is an S-shaped (sigmoidal) curve that can be used to model functions that increase gradually at first, more rapidly in the middle growth period, and slowly at the end, leveling off at a maximum value after some period of time.

Derivation In 1838, the Belgian Pierre-Francois Verhulst suggested a revised model which would eliminate the undesirable effect of unlimited growth from the exponential model.. which is the same as
r birthrate of the population, under optimal conditions carrying capacity-the level of the population that can be K sustained by the given environment. m mortality constant

Sometimes it can be helpful to think of the equation in this form:

Graph

Value of y

0 0

0<y<K +

K 0

y>K -

We notice that y decreases and thus the slope of the tangent lines to be drawn is negative whenever y>K but that whenever 0<y<K, the slopes are positive, and y increases. Analysis of the logistic growth curve: The initial part of the curve is exponential; the rate of growth accelerates as it approaches the midpoint of the curve. At the midpoint (K/2), the growth rate begins to decelerate but continues to grow until it reaches

an asymptote, K which is called the "Carrying Capacity" for the environment. Application Population growth, spreading of a rumor or virus,newtons law of cooling

The logistic growth model is a more accurate representation of population growth becauseit takes into account that there are outside factors that may effect growth. It is frequently used to model biological growth patterns where there is an initial exponential growth period followed by a leveling off as more of the population is infected or as the food supply or some other factor limits further growth.

r and K selection

Organisms that live in stable environments tend to make few, "expensive" offspring. Organisms that live in unstable environments tend to make many, "cheap" offspring. Imagine that you are one of the many invertebrate organisms which existed during the Cambrian or one of their descendents living today. Maybe you live in a tide pool which is washed by waves. A storm appears on the horizon. The waves increase in height. You feel yourself being dashed upon the rocks or into the mouth of a much larger and predatory animal. Finally, you begin to see your brothers and sisters die, one by one, as the forces of nature change your unpredictable environment.
If you could design a "strategy" to overcome the problems created by an unpredictable environment, you would have two choices - go with the flow or cut and run to a more stable environment.

Suppose you stayed. Then, one thing you could do would be to increase the number of offspring. Make lots of cheap (requiring little energy investment) offspring instead of a few expensive, complicated ones (requiring a lot of energy investment). If you lose a lot of offspring to the unpredictable forces of nature, you still have some left to live to reproductive age and pass on your genes to future generations. Many invertebrates follow this strategy - lots of eggs are produced and larvae are formed but only a few survive to produce mature, reproductive adults. Many insects and spiders also follow this strategy. Alternatively, you could adapt to a more stable environment. If you could do that, you would find that it would be worthwhile to make fewer, more expensive offspring. These offspring would have all the bells and whistles necessary to ensure a comfortable, maximally productive life. Since the environment is relatively stable, your risk of losing offspring to random environmental factors is small. Large animals, such as ourselves, follow this strategy. Plants are also subject to the same sorts of forces as animals. Some live in unstable environments such as a floodplain near a river or a gap in the forest caused by falling trees. Others live in a quite stable environment, such as a climax forest.

The two evolutionary "strategies" are termed r-selection, for those species that produce many "cheap" offspring and live in unstable environments and K-selection for those species that produce few "expensive" offspring and live in stable environments. Of course, the animal or plant is not thinking: "How do I change my characteristics?" Natural selection is the force for change, not the individual's conscious decision. But, natural selection has produced a gradation of strategies, with extreme r-selection at one end of the spectrum and extreme Kselection at the other end.
The following table compares some characteristics of organisms which are extreme r or K strategists:

Unstable environment, density independent small size of organism energy used to make each individual is low many offspring are produced early maturity short life expectancy each individual reproduces only once type III survivorship pattern in which most of the individuals die within a short time but a few live much longer

Stable environment, density dependent interactions large size of organism energy used to make each individual is high few offspring are produced late maturity, often after a prolonged period of parental care long life expectancy individuals can reproduce more than once in their lifetime type I or II survivorship pattern in which most individuals live to near the maximum life span

The terms "r-selected" and "K-selected" come from a description of the population growth regimes of the two types of organisms. If you are in an unstable environment, you are unlikely to ever have population growth to the point where density dependent factors come into play. The population is still at low values relative to the carrying capacity of the environment and thus is growing exponentially with intrinsic reproductive rate r (when it is not subject to environmental perturbations.), hence the name rstrategist.
An extreme K-strategist lives in a stable environment which is not seriously affected by sudden, unpredictable effects. Thus the population of a K-strategist is near the carrying capacity K.

Surviorship curves give us additional insight into r and Kselected strategies. Notice that the vertical axis of the survivorship plots is on a log scale and that horizontal axis is scaled to the maximum lifetime for each species. One of the interesting differences between r and K strategists is in the shape of the survivorship curve. We can generate a survivorship curve by ploting the log of the fraction of organisms surviving vs. the age of the organism. To compare different species, we normalize the age axis by stretching or shrinking the curve in the horizontal direction so that all curves end at the same point, the maximum life span for individuals of that species. Notice that the vertical axis is on a log scale, dropping from 1.0 (100%) to 0.1 (10%) to 0.01 (1%) to 0.001 (0.1%) in equally spaced intervals.
Extreme r-strategists, such as the oyster, lose most of the individuals very quickly, relative to the maximum life span for the species. But, a very few individuals do survive much longer than the rest. But, for extreme K-strategists, such as man, most individuals live to old age (again relative to the maximum life span for the species). These survivorship data are very valuable when studying the ecology of various organisms. Two components are involved in reproduction: 1) How many females survive to each age and 2) the average number of female offspring produced by females at each age. By using these data, we can compute the intrinsic rate of reproduction, r, a key parameter in models of population growth.

- Selected Species They are populations of a roughly constant size whose members have low reproductive rates. The offspring produced require extensive postnatal care until they have sufficiently matured. They are very limited in resourses therefore they are a very competitive species. Humans are examples of a kselected species.

R-Selected Species They are populations that experience rapid growth of the J-curve variety. The offspring produced are numerous, mature quite rapidly, and require very little postnatal care. Consequently, this population grows fast, reproduces quickly, and dies quickly. Bacteria are examples of r- selected species.