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# during exponential growth a population always

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Growth curve

A growth curve is an empirical model of the evolution of a quantity over time. Growth curves are widely used in biology for quantities such as population size, body height or biomass. Values for the measured property can be plotted on a graph as a function of time; see Figure 1 for an example.

Growth curves are employed in many disciplines besides biology, particularly in statistics, which has an extensive literature on growth curves. In mathematical statistics, growth curves are often modeled as being continuousstochastic processes, e.g. as being sample paths that almost surely solve stochastic differential equations.

## Bacterial growth

In this example (Figure 1, see Lac operon for details) the number of bacteria present in a nutrient-containing broth was measured during the course of an 8 hour cell growthexperiment. The observed pattern of bacterial growth is bi-phasic because two different sugars were present, glucose and lactose. The bacteria prefer to consume glucose (Phase I) and only use the lactose (Phase II) after the glucose has been depleted. Analysis of the molecular basis for this bi-phasic growth curve led to the discovery of the basic mechanisms that control gene expression.

## Cancer cell growth

Cancer research is an area of biology where growth curve analysis [http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Search&db=books&doptcmdl=GenBookHL&term=%22growth+curve%22%5BAll+Fields%5D+AND+cmed6%5Bbook%5D+AND+351845%5Buid%5D&rid=cmed6.section.10278] plays an important role. In many types of cancer, the rate at which tumors shrink following chemotherapy is related to the rate of tumor growth before treatment. Tumors that grow rapidly are generally more sensitive to the toxic effects that conventional anticancer drugs have on the cancer cells. Many conventional anticancer drugs (for example, 5-Fluoro Uracil) interfere with DNA replication and can cause the death of cells that attempt to replicate their DNA and divide. A rapidly growing tumor will have more actively dividing cells and more cell death upon exposure to such anticancer drugs.

In the example shown in Figure 2, a tumor is found after the cell growth rate has slowed. Most of the cancer cells are removed by surgery. The remaining cancer cells begin to proliferate rapidly and cancer chemotherapy is started. Many tumor cells are killed by the chemotherapy, but eventually some cancer cells that are resistant to the chemotherapy drug begin to grow rapidly. The chemotherapy is no longer useful and is discontinued.

## The growth of children

Children who fall significantly below the normal range of growth curves for body height [http://www.cdc.gov/nchs/data/nhanes/growthcharts/set1clinical/cj41c021.pdf] can be tested for growth hormone deficiency and might be treatable with hormone injections [http://www.nlm.nih.gov/medlineplus/ency/article/001176.htm].

## Exponential growth

Some growth curves for certain biological systems display periods of exponential growth. Typically, periods of exponential growth are of limited duration due to depletion of some rate-limiting resource.

Population decline

Population decline can refer to the decline in population of any organism, but this article refers to population decline in humans. It is a term usually used to describe any great reduction in a human population. It can be used to refer to longterm demographic trends, as in urban decay or rural flight, but it is also commonly employed to describe large reductions in population due to violence, disease, or other catastrophes.

Sometimes known as depopulation, population decline is the reduction over time in a region's census. It can be caused for several reasons; notable ones include sub-replacement fertility (along with limited immigration), heavy emigration, disease, famine, and war. History is replete with examples of large scale depopulations. Many wars, for example, have been accompanied by significant depopulations. Prior to the 20th century, population decline was mostly observed due to disease, starvation and/or emigration. The Black Death in Europe, the arrival of Old World diseases to the Americas, the tsetse fly invasion of the Waterberg Massif in South Africa, and the Great Irish Famine have all caused sizable population declines. In modern times, the AIDSepidemic has caused declines in the population of some African countries. Less frequently, population declines are caused by genocide or mass execution; for example, in the 1970s, the population of Cambodia underwent a period of decline due to wide-scale executions by the Khmer Rouge.

During the Age of Imperialism, Europeans migrating to new continents brought with them not only devastating new means of waging warfare but also, often inadvertently, infectious diseases such as smallpox to which indigenous peoples had no resistance. These factors, particularly the latter, sometimes had a devastating impact on the indigenous inhabitants.

Some notable historical examples of large depopulation of entire continents include:

Some examples of depopulation of large regions brought about mainly by warfare include:

Famine has also frequently played a role in depopulation, whether as a result of war, climatic conditions, human incompetence and so on.

According to 2002 reports by the United Nations Population Division and the US Census Bureau, population decline is occurring today in some regions. According to the UN, below-replacement fertility is expected in 75% of the developed world by the year 2050. The US Census Bureau notes that the 74 million people added to the world's population in 2002 were fewer than the high of 87 million people added in 1989â€“1990. The annual growth rate was 1.2 percent, down from the high of 2.2 percent in 1963-64.

"Census Bureau projections show this slowdown in population growth continuing into the foreseeable future," stated the Bureau's brief on the findings. "Census Bureau projections suggest that the level of fertility in many countries will drop below replacement level before 2050... In 1990 the world's women, on average, were giving birth to 3.3 children over their lifetimes. By 2002 the average was 2.6, and by 2009, 2.5. This is marginally above the global replacement fertility of 2.33. This fall has been accompanied by a decline in the world's population growth rate and in the actual annual population increase.

Sometimes the term underpopulation is applied in the context of a specific economic system. It does not relate to carrying capacity, and is not a term in opposition to overpopulation, which deals with the total possible population that can be sustained by available food, water, sanitation and other infrastructure. "Underpopulation" is usually defined as a state in which a country's population has declined too much to support its current economic system. Thus the term has nothing to do with the biological aspects of carrying capacity, bu

Population genetics

Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of population subdivision and population structure. It attempts to explain such phenomena as adaptation and speciation.

Population genetics was a vital ingredient in the emergence of the modern evolutionary synthesis. Its primary founders were Sewall Wright, J. B. S. Haldane and R. A. Fisher, who also laid the foundations for the related discipline of quantitative genetics.

## Fundamentals

Population genetics concerns the genetic constitution of populations and how this constitution changes with time. A population is a set of organisms in which any pair of members can breed together. This implies that all members belong to the same species and live near each other.

For example, all of the moths of the same species living in an isolated forest are a population. A gene in this population may have several alternate forms, which account for variations between the phenotypes of the organisms. An example might be a gene for coloration in moths that has two alleles: black and white. A gene pool is the complete set of alleles for a gene in a single population; the allele frequency for an allele is the fraction of the genes in the pool that is composed of that allele (for example, what fraction of moth coloration genes are the black allele). Evolution occurs when there are changes in the frequencies of alleles within a population of interbreeding organisms; for example, the allele for black color in a population of moths becoming more common.

To understand the mechanisms that cause a population to evolve, it is useful to consider what conditions are required for a population not to evolve. The Hardy-Weinberg principlestates that the frequencies of alleles (variations in a gene) in a sufficiently large population will remain constant if the only forces acting on that population are therandom reshuffling of alleles during the formation of the sperm or egg, and the random combination of the alleles in these sex cells during fertilization. Such a population is said to be in Hardy-Weinberg equilibrium as it is not evolving.

### Hardyâ€“Weinberg principle

The Hardyâ€“Weinberg principle states that both allele and genotype frequencies in a population remain constantâ€”that is, they are in equilibriumâ€”from generation to generation unless specific disturbing influences are introduced. Outside the lab, one or more of these "disturbing influences" are always in effect. Hardy Weinberg equilibrium is impossible in nature. Genetic equilibrium is an ideal state that provides a baseline to measure genetic change against.

Allele frequencies in a population remain static across generations, provided the following conditions are at hand: random mating, no mutation (the alleles don't change), no migration or emigration (no exchange of alleles between populations), infinitely large population size, and no selective pressure for or against any traits.

In the simplest case of a single locus with two alleles: the dominant allele is denoted A and the recessivea and their frequencies are denoted by p and q; freq(A)&nbsp;=&nbsp;p; freq(a)&nbsp;=&nbsp;q; p&nbsp;+&nbsp;q&nbsp;=&nbsp;1. If the population is in equilibrium, then we will have freq(AA)&nbsp;=&nbsp;p2 for the AAhomozygotes in the population, freq(aa)&nbsp;=&nbsp;q2 for the aa homozygotes, and freq(Aa)&nbsp;=&nbsp;2pq for the heterozygotes.

Based on these equations, useful but difficult-to-measure facts about a population can be determined. For example, a patient's child is a carrier of a recessive mutation that causes cystic fibrosis in homozygous recessive children. The parent wants to know the probability of her grandchildren inheriting the disease. In order to answer this question, the genetic counselor must know the chance that the child will reproduce with a carrier of the recessive mutation. This fact may not be known, but disease frequency is known. We know that the disease is caused by the homozygous recessive genotype; we can use the Hardyâ€“Weinberg principle to work backward from disease occurrence to the frequency of heterozygous recessive individuals.

## Scope and theoretical considerations

The mathematics of population genetics were originally developed as part of the modern evolutionary synthesis. According to Beatty (1986), it defines the core of the modern synthesis.

According to Lewontin (1974), the theoretical task for population genetics is a process in two spaces: a "genotypic space" and a "phenotypic space". The challenge of a complete theory of population genetics is to provide a set of laws that predictably map a population of genotypes (G1) to a phenotype space (P1), where selection takes place, and another set of laws that map the resultin

Question:A) if it is limited only by density-dependent factors B) until it reaches carrying capacity C) if there are no limiting factors D) if it is a population with an equilibrial life history E) if it shows logistic growth

Answers:C) if there are no limiting factors

Question:I have noted the comments in response to my question so far, and agree that the example of the disappearance of certain types of marine life can be caused by pollution and that CO2 emissions are not the sole cause of increasing sea acidity. However, that was not the question. That is why I chose the best answer in the way I did. By describing the problem of global over-population as " media-hype " or the effects of the population on pollution which is leading to climate change - gradual or otherwise, in the same way, is to deny that anything is really happening at all. In fact even to the extent that the whole " conspiracy " is purely to get us to pay more money in "green" taxation. That is what I call cynisism. The example experiment of taking maggots and placing them in an enclosed glass container with enough food and air is the most graphic method of predicting our future. More and more maggots are added to the container to simulate human population growthat present levels . Soon all the food is consumed, the faeces pollute their environment and they run out of breathable air, producing an eventual mass extinction. However we humans manage the environment, if the population continues to grow exponentially, then inevitably the pollution that we produce will kill us, if not all, most. All the arguements, bickering, money, wars, religions etc will not produce a change of direction unless there is a realistic - and doubtless to say, very harsh, global policy on limiting the population voluntarily. This is a horriblr concept and morally unacceptable to many, but I believe, inevitable - or......... The other alternative, of course, is to allow things to drift on as they are, always aiming for " sustained growth ",( in the words of the British Chancellor, Alistair Darling). 9 billion....11 billion.....15 billion...20 billion...................? If we are having trouble feeding 6.4 billion, where are we going ? And what of the waste products in sustaining such a population.? When someone suggests " getting real " and that this is all " media-hype ", I suggest they start doing some basic maths. You don't have to be a rocket scientist to come to a logical conclusion.

Answers:Neither climate change nor overpopulation are "media hype". You won't change the minds of people who think they aren't happening, though. They have what is called a "fixed delusion" in psychiatry. Fixed delusions don't rely on facts or logic. No matter how many facts are thrown at them, they will still cling to their ignorant positions for whatever reason, be it fear or a religious or political agenda.

Question:The population in year 0 was 6.079 billion. In year 3, it was 6.302 billion. Write a differential equation that can represent the population growth from year 0 to year 3. I need help getting the answer, not just the answer. Thanks :)

Answers:You can write any number of differential equation to model the population growth during this time. It all depends on what model you use for how the population grows with time. The simplest model would be to assume that the population grows linearly with time over this time interval (i.e., the population grows at a constant rate, regardless of the value of the population). In this case, the differential equation that represents the population growth is: dP/dt = k where k is a constant = ((6.302 - 6.079) billion people)/(3 years) k = 74.33 million people/year The equation that describes the number of people is then simply: P(t) = (6079 + 74.33/year * t) million people where t is measured in years. Now, for small changes in population (or, equivalently, over small time periods), this linear approximation is actually pretty good, but the population doesn't really change linearly. Rather, it grows almost exponentially. In this case, the relevant differential equation is: dP/dt = K*P where K is a different constant from the one in the linear case, above. The solution to this separable differential equation is given by: dP/p = K dt ln(P/Po) = K*t P(t) = Po * exp(K*t) where Po is the population at t = 0. We are told what Po is, and we can use the information given in the problem to solve for K: ln(6.302/6.079) = (3 yr)*K 3.603*10^-2 = (3 yr)*K 1.201*10^-2 yr^-1 = K The population as a function of time is then given by: P(t) = (6.079 billion)*exp(t*(1.201*10^-2 yr^-1)) You can go on to try other models for population growth, for instance: dP/dt = B*P - D*P^2 where D is the death rate and B is the birth rate.

Question:2 examples of exponential growth? Thanks in advance.

Answers:population growth. and stock market growth.