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**which part of the graph shows a logistic growth of the population?**

A, Exponential growth curve

B, Logistic growth curve

C, Z-shaped growth curve

D, All of the above

Correct option is: B, Logistic growth curve

The growth of animals in a population depends on the available resources in the environments.

This model applies to the population depends on density-dependent factors like food abundance, resting place, sickness, competition, etc. It says that the growth of the population depends on environmental factors.

In contrast, the exponential growth does not consider the environmental limits, it says a population can grow with no limits. So, the logistic growth curve is more realistic.

In principle, any sort of creature might take over the Earth by simply reproducing at an uncontrollable rate. Imagine, for the sake of this illustra

tion, that we began with a single pair consisting of male and female rabbits. If these rabbits and their progeny continued to breed at the maximum rate (sometimes known as “like bunnies”) for the next 777 years without experiencing any losses, we would have sufficient rabbits to populate the whole state of Rhode Island 1,2,3.

1,2,3 start superscript, 1, comma, 2, comma, 3, end superscript. And even that isn’t all that spectacular — if we employed E. coli bacteria instead, we could start with just one bacterium and have enough bacteria to cover the Earth with a 111-foot layer in only 363636 hours4 4 start superscript, 4, end superscript!

You’ve undoubtedly observed that there isn’t a coating of germs covering the whole Earth that is 111 feet thick (at least, not at my place), and that rabbits haven’t taken over Rhode Island. If this is the case, then why don’t we see these populations growing to the levels that they are capable of in theory? In order to live and reproduce, E. coli, like rabbits and all other living beings, need particular resources, such as nutrition and conditions that are conducive to their growth. These resources are not limitless, and a population can only grow to a level that is proportional to the amount of resources that are readily available in its immediate area.

Ecologists that study populations simulate the movement of populations using a wide range of mathematical techniques (how populations change in size and composition over time). Some of these models depict development in an unrestricted environment, while others include “ceilings” that are contingent on the availability of finite resources. It is possible to utilize mathematical models of populations to precisely characterize the changes that are taking place in a population and, more crucially, to anticipate the changes that will take place in the future.

To have a better understanding of the many models that are used to depict the dynamics of a population, let’s begin by looking at a general equation for the population growth rate, which can be defined as the change in the number of persons that make up a population over a period of time:

The rrr is simply a function of the birth and death rates if we make the assumption that there is no movement of persons into or out of the population. You may get further knowledge on the significance of the equation as well as its origin by reading the following:

The rrr is simply a function of the birth and death rates if we make the assumption that there is no movement of persons into or out of the population. You may get further knowledge on the significance of the equation as well as its origin by reading the following: [This is how we arrive at the equation for the rate of population increase.]

The equation that was just shown to you is somewhat generic; nonetheless, we are able to derive more particular versions of it in order to explain two distinct types of growth models, namely exponential and logistic.

Exponential growth occurs when the rate of rise (rrr) calculated for each individual in the population maintains a positive value despite the overall size of the population.

The phenomenon known as logistic growth occurs when the average annual growth rate (rrr) of the population falls as the population becomes closer to its carrying capacity.

Growth that is exponential

The development of bacteria in the laboratory is a great illustration of the exponential growth model. The growth rate of a population that is experiencing exponential expansion grows over the course of time and is directly proportional to the size of the population.

Let’s have a look at the mechanics behind this, shall we? Bacteria reproduce by a process known as binary fission, which literally translates to “splitting in half,” and the amount of time that passes in between divisions for the majority of bacterial species is around an hour. To have a better understanding of how exponential growth works, let’s begin by putting one million one million one million bacteria in a flask that has an infinite supply of nutrients.

After 111 hours, every bacterium will have completed its cycle of reproduction, producing 200020002000 new bacteria (an increase of 100010001000 bacteria).

After 222 hours, each of the 2,000 bacterial cells will divide, resulting in the production of 4,000 bacterial cells (an increase of 200020002000 bacteria).

After 333 hours, each of the 400,000 bacterial cells will divide, resulting in the production of 80,000 bacterial cells (an increase of 400040004000 bacteria).

The fundamental idea behind exponential growth is that the population growth rate, or the number of organisms that are added to the population with each new generation, steadily accelerates as the population continues to expand. And the outcomes may be rather startling: after 111 days (or 242424 cycles of division), our bacterial population would have increased from 100010001000 to approximately 161616 billion. A growth curve in the form of a J is produced when the size of the population, NNN, is plotted against time.

How can we create a model that accurately depicts the exponential expansion of a population? The term “exponential growth” refers to a situation in which the “per capita rate of rise” for a population is both positive and stable. We touched on this concept briefly before. Although exponential growth may be caused by any rrr that is positive and constant, it is most often expressed by a rrr that looks like this: r maxr max r, start subscript, m, a, x, end subscript.

r maxr max r, start subscript, m, a, x, end subscript is the maximum per capita rate of growth for a specific species when circumstances are optimum; this number is different for each species. For instance, bacteria are capable of significantly quicker rates of reproduction than humans, and hence would have a greater maximum rate of rise per capita. The following equation expresses the greatest population growth rate that a species is capable of supporting, which is sometimes referred to as its biotic potential:

Development in logistics

Since exponential development is dependent on an unending supply of resources, this state of things is not particularly conducive to a sustainable future (which tend not to exist in the real world).

If there are a small number of people but a large amount of resources, exponential growth may occur over a period of time. However, when there are a sufficient number of people in a population, resources begin to be used up, which slows down the growth rate. The rate of growth will eventually reach a plateau, at which point an S-shaped curve will be produced. It is referred to as the carrying capacity, or KKK, and it refers to the population size at which it levels out. This population size defines the maximum population number that a specific ecosystem can sustain.

Logistic growth may be quantitatively modeled by altering our equation for exponential growth and utilizing a rrr (per capita growth rate) that relies on population size (NNN) and how near it is to carrying capacity. This allows us to simulate logistic growth (KKK). If we make the assumption that the population, when it is extremely tiny, has a base growth rate of r maxr max r, start subscript, m, a, x, end subscript, then we may construct the following equation:

\quad\quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = r {max}\dfrac{(K-N)}

{K}N dT dN =r max

K (KN) Nstart fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, start fraction, left parenthesis, K, minus, N, right parenthesis, divided by, K, end fraction, Nstart subscript, m, a, x, end subscript, start fraction, left parenthesis, K, minus, N, right paren

Let’s take a moment to analyze this equation and figure out why it makes sense to put the two together. The formula K – NKNK, minus N, informs us how many more people may be added to a population before it reaches its carrying capacity at any given point in time throughout the expansion of that population. This can happen at any moment in time. The percentage of the carrying capacity that has not yet been “used up” may be calculated as follows: (K – N)/K(KN)/K (left parenthesis, K, minus, N, right parenthesis, slash, K) This yields the answer. The phrase (K – N)/K(KN)/K will limit the growth rate by a greater amount the more the carrying capacity has been exhausted. Left parenthesis, K, negative, N, right parenthesis, slash, K.

When there are just a few people, the proportion of NNN to KKK is very low. The expression (K – N)/K(KN)/K is converted into roughly (K/K)(K/K)left parenthesis, K, slash, K, right parenthesis, or 111, which provides us with the exponential equation once again. This is consistent with what we saw in the graph that we just looked at: at initially, the population expands almost rapidly, but then it begins to level off as it gets closer and closer to KKK.

In general, a limit may be almost any form of resource that is significant to the survival of a species. The water, sunshine, and nutrients, as well as the area for the plant to expand into, are some of the most important resources. Important resources for animals include food, water, a safe place to nest, and protection from the elements. Competition for these limited resources occurs among individuals of the same population; this kind of competition is known as intraspecific competition (intra- means inside, and -specific refers to the species).

It is possible that intraspecific rivalry for resources will not effect populations that are much below their carrying capacity since there is an abundance of resources and every person can receive what they need. However, as the number of people in a population grows, the level of competition rises. Furthermore, the buildup of waste products has the potential to lessen the carrying capacity of an ecosystem.

When cultivated in a test tube, the tiny fungus known as yeast, which is used to manufacture bread and alcoholic drinks, may generate a curve in the familiar shape of a S. The graph that follows illustrates how yeast growth levels off as the population reaches its maximum capacity for the nutrients that are available. (Since the test tube is a closed system, if we continued to observe the population for a longer period of time, it would most likely perish; this is because the fuel supplies would ultimately run out, and wastes may potentially reach levels that are hazardous.)

The “perfect” logistic curve does not exist without some degree of variance in the actual world. One such example is shown in the following graph, which shows an increase in the number of harbor seals living in the state of Washington. During the early part of the 20th century, seals were subjected to aggressive hunting as part of a government policy that considered them as dangerous predators. As a result, their population sizes were drastically reduced5 5 start superscript, 5, end superscript. Since this initiative was terminated, seal populations have recovered in what might be described as a broadly logistic pattern (start superscript, stop superscript, start superscript, 6, end superscript).

As seen by the graph that is located above, the size of the population may have some wiggle room as it approaches carrying capacity, either falling below or leaping over this figure. It is fairly uncommon for actual populations to fluctuate (bounce back and forth) repeatedly about the carrying capacity rather than creating a completely flat line. This is because of the way that natural processes work.

Exponential growth occurs when a population’s per capita growth rate remains constant, notwithstanding the size of the population. This causes the population to increase at an ever-increasing pace as the population continues to expand. It’s expressed by the equation: \quad\quad\quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = r {max}N dT dN =r max

Nstart fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, NNstart fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, N

The curve that results from exponential growth looks like a J.

Logistic growth occurs when a population’s per capita growth rate reduces when the population size approaches a maximum that is imposed by limiting resources, known as the carrying capacity. This allows for more people to live in the same space (KKK). It’s expressed by the equation: \quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = r {max}\dfrac{(K-N)} {K} N dT dN =r max

K (KN) Nstart fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, start fraction, left parenthesis, K, minus, N, right parenthesis, divided by, K, end fraction, Nstart subscript, m, a, x, end subscript, start fraction, left parenthesis, K, minus, N, start subscript

An S-shaped curve results from the use of logistic growth.

The Growth of Logistic

If we examine a graph of a population that is experiencing logistic population increase, we will see that the curve takes on the familiar form of a S. When there are just a few people, the population expands in size more slowly than it would otherwise. When there are more people in a population, the population expands at a quicker rate.

As a population approaches a maximum that can be sustained by the limiting resources in its environment, often known as the carrying capacity, the logistic growth model predicts that the per capita growth rate of that population would gradually decrease ( K). Growth at an exponential rate results in a J-shaped curve, whereas growth at a logistic rate results in an S-shaped curve.

S-shaped

When resources become limited, population expansion slows down, and it reaches a plateau when the carrying capacity of the environment is met. This kind of growth is referred to as logistic growth. The growth curve for logistics is in the form of a S.

The logistic growth curve on the right illustrates the four stages of such development, which are referred to as Initiation/Birth, Acceleration/Growth, Deceleration/Maturing, and Saturation respectively.

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