Nevertheless, it is often a good match for things that happen in the real world. We will see that this can, and did, apply to epidemics such as the covid pandemic. At this time, there was no community testing: this is the number of hospitalised confirmed cases of covid The number of new cases goes up with the number of existing cases. Our second way of understanding exponential growth is to think of doubling times.
Decide for yourself: is the number of cases roughly doubling between each highlighted row? Table 1. Reported cases of covid in the UK in the month before lockdown measures were announced. A simple way to look out for exponential growth is to try to spot a doubling time. A concerned newspaper reader in the Spring of might notice the apparent doubling between the 23 rd and 26 th of February, for example, and then keep watching the news to see if cases continue to double approximately every three days.
With each subsequent year, the amount of interest paid grows, creating rapidly accelerating, or exponential, growth. On a chart, this curve starts slowly, remains nearly flat for a time before increasing swiftly to appear almost vertical. It follows the formula:. The current value, V, of an initial starting point subject to exponential growth can be determined by multiplying the starting value, S, by the sum of one plus the rate of interest, R, raised to the power of T, or the number of periods that have elapsed.
While exponential growth is often used in financial modeling, the reality is often more complicated. The application of exponential growth works well in the example of a savings account because the rate of interest is guaranteed and does not change over time. In most investments, this is not the case. For instance, stock market returns do not smoothly follow long-term averages each year.
Other methods of predicting long-term returns—such as the Monte Carlo simulation, which uses probability distributions to determine the likelihood of different potential outcomes—have seen increasing popularity. Exponential growth models are more useful to predict investment returns when the rate of growth is steady. Portfolio Management. Risk Management. Investing Essentials. Fixed Income Essentials.
Interest Rates. Actively scan device characteristics for identification. Use precise geolocation data. Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population. In population ecology, density-dependent processes occur when population growth rates are regulated by the density of a population.
Most density-dependent factors, which are biological in nature biotic , include predation, inter- and intraspecific competition, accumulation of waste, and diseases such as those caused by parasites. Usually, the denser a population is, the greater its mortality rate. In addition, low prey density increases the mortality of its predator because it has more difficulty locating its food source. An example of density-dependent regulation is shown with results from a study focusing on the giant intestinal roundworm Ascaris lumbricoides , a parasite of humans and other mammals.
The data shows that denser populations of the parasite exhibit lower fecundity: they contained fewer eggs. One possible explanation for this phenomenon was that females would be smaller in more dense populations due to limited resources so they would have fewer eggs. This hypothesis was tested and disproved in a study which showed that female weight had no influence. The actual cause of the density-dependence of fecundity in this organism is still unclear and awaiting further investigation.
Effect of population density on fecundity : In this population of roundworms, fecundity number of eggs decreases with population density. Many factors, typically physical or chemical in nature abiotic , influence the mortality of a population regardless of its density.
They include weather, natural disasters, and pollution. An individual deer may be killed in a forest fire regardless of how many deer happen to be in that area. Its chances of survival are the same whether the population density is high or low. In real-life situations, population regulation is very complicated and density-dependent and independent factors can interact.
A dense population that is reduced in a density-independent manner by some environmental factor s will be able to recover differently than would a sparse population. For example, a population of deer affected by a harsh winter will recover faster if there are more deer remaining to reproduce. Privacy Policy. Skip to main content. Population and Community Ecology.
Search for:. Environmental Limits to Population Growth. Exponential Population Growth When resources are unlimited, a population can experience exponential growth, where its size increases at a greater and greater rate. Learning Objectives Describe exponential growth of a population size. This page has been archived and is no longer updated. The easiest way to capture the idea of a growing population is with a single celled organism, such as a bacterium or a cilliate. In Figure 1, a population of Paramecium in a small laboratory depression slide is pictured.
In this population the individuals divide once per day. So, starting with a single individual at day 0, we expect, in successive days, 2, 4, 8, 16, 32, and 64 individuals in the population. Thus we can see a relatively simple generalization, namely. Finally we note that this equation was derived from the specific situation shown in Figure 1, where one division per day was the hard and fast rule.
That is where the 2 comes from in Equation 1 — from each individual Paramecium we obtain two individuals the next day. Of course the division rate could be anything. So the division rate could be any number at all and the general equation becomes,. In Figure 2 we illustrate this equation for various values of R.
It is normally referred to as the exponential equation, and the form of the data in Figure 2 is the general form called exponential. Figure 2: Left: general form of exponential growth of a population equation 2. Right: actual numbers of Paramecium in a 1 cc sample of a laboratory culture.
Any value of R can be represented in an infinite number of ways e. The constant r is referred to as the intrinsic rate of natural increase Figure 2.
All sorts of microorganisms exhibit patterns that are very close to exponential population growth. For example, in the right hand graph of Figure 2 is a population of Paramecium growing in a laboratory culture.
The pattern of growth is very close to the pattern of the exponential equation. Another way of writing the exponential equation is as a differential equation, that is, representing the growth of the population in its dynamic form. Rather than asking what is the size of the population at time t , we ask, what is the rate at which the population is growing at time t. That constant rate of growth of the log of the population is the intrinsic rate of increase. The basic relationship between finite rate of increase and intrinsic rate is.
Note that Equation 6 and Equation 3 are just different forms of the same equation Equation 3 is the integrated form of Equation 6; Equation 6 is the differentiated form of Equation 3 , and both may be referred to simply as the exponential equation.
Figure 3: Hypothetical case of a pest population in an agroecosystem According to model 1 which has a relatively large estimate of R , the farmer needs to think about applying a control procedure about half way through the season. According to model 2 which has a relatively small estimate of R , the farmer need not worry about controlling the pest at all, since its population exceeds the economic threshold only after the harvest.
Clearly, it is important to know which model is correct. In this case, according to the available data blue data points , either model 1 or 2 appears to provide a good fit, leaving the farmer still in limbo. The exponential equation is a useful model of simple populations, at least for relatively short periods of time.
For example, if a laboratory technician needs to know when a bacterial culture reaches a certain population density, the exponential equation can be used to provide a prediction as to exactly when that population size will be reached.
Another example is in the case of agricultural pests. Herbivores are always potentially major problems for plants. When the plants subjected to such outbreaks are agricultural, which is to say crops, the loss can be very significant for both farmer and consumer.
Thus, there is always pressure to prevent such outbreaks. However, in recent years we have come to realize that these pesticides are extremely dangerous over the long run, both for the environment and for people. Consequently there has been a movement to limit the amount of pesticides that are sprayed to combat pests. The major way this is done is to establish an economic threshold, which is the population density of the potential pest below which the damage to the crop is insignificant i. When the pest population increases above that threshold, the farmer needs to take action and apply some sort of pesticide, or other means of controlling the pest.
Given the nature of this problem, it is sometimes of utmost importance to be able to predict when the pest will reach the economic threshold.
Knowing the R for the pest species enables the farmer to predict when it will be necessary to apply some sort of control procedure Figure 3. The exponential equation is also a useful model for developing intuitive ideas about populations.
The classic example is a pond with a population of lily pads. If each lily pad reproduces itself two pads take the place of where one pad had been each month, and it took, say, three years for the pond to become half filled with lily pads, how much longer will it take for the pond to be completely covered with lily pads?
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