Up868093 In this text I describe the exponential growth and its constraints in terms of initial population, time interval, and growth. Furthermore, I provide with some examples to clairfy the meaning of the paraphrased text requested. In many studies scientists have shown that the population of a species in an area tends to increase rapidly, over a measured period of time. In the book introduction to population ecology numerous examples show how the number of a species increase exponentially taking into consideration the local food supply, and the lack of other predator species (Rockwood, 2006). Various mathematical analyses performed show that the increase in the numbers of a species does not follow a linear progression, but rather an exponential increase, depending on several significant factors. Population growth is an example of an exponential increase that is met in demographics. Murray has studied the demographics in most states in the Middle East, and showed that it follows an exponential rate (Murray, 1999). Specifically, a population growth of 3.2%, 18.7 million was recorded in 1995, and suggests strong connections with large oil exportations that allowed the import of goods, such as food, to be ample. Therefore, the problem of exponential growth is a factor of initial numbers, period of time, and several constraints that I will try to analyse further.An exponential increase of a factor is usually reflected in the number of the quantity measured. For such a measurement to be valid there are two main conditions that have to be concomitantly satisfied; namely, the time period considered, and the amount of the quantity that accumulated in such a time interval. The first condition defines the amount of time lapsed in every repeating measurement. As an example, I consider the water lily growth in a pond (Washington & Cook, 2011; Sweeney & Meadows, 2010). According to this riddle the size of the water lily doubles every day. The unit time considered here is a single day. If the time period was two days instead, then there would be a slower increase of the water lily’s size. This means that the time interval assumed must be selected logically based on the measured quantity. The second condition relates to the amount of a factor being accumulated. To understand this condition an example will be used from the finance sector; that of an exponential growth of interest in a bank account (Moles, Parrino, & Kidwell, 2011). Let’s assume that a person has 100£ in his account. The interest rate that the bank offers is 1 % per year. This means that at the end of the first year the amount of money that will be earned is 100+100*0.01=101£, at the end of the second year 101+101*0.01=102.01£, and so on. In this example, we see that the amount of money earned from interest at the end of each year depends on the amount of money accumulated in the same way the year before.One of the consequences of the exponential growth is overshooting (Cattor, 1980). Overshoot manifests when a specific change in the numbers of a factor increases extremely fast reaching to a point at which the population cannot be sustained any longer. Overshoot is therefore linked to unprecedented growth and environmental constraints. As an example of uncontrolled growth is the study of a rabbit population over time in an isolated island (Feldman, 2012). It is only reasonable to assume that the number of the rabbits in the island will increase rapidly. If we assume that there are 4 rabbits initially in the island, and that the population doubles in every generation, then 4 rabbits will become 8, 8 will become 16, 16 will become 32, and so on. In the beginning the growth looks insignificant. Only 4, 8, or 16 rabbits are considered. Thereafter, the growth becomes faster, and this is a feature of the exponential growth. When the growth is graphed, what is observed is that it starts slowly, but then it grows big in numbers. The island soon will be full of rabbits, leading to overshoot. However, the food supplies are limited, which means that the growth cannot continue for ever. To control exponential growth and balance the ecosystems, nature has introduced predators and finite supplies. These are two constraints that limit the infinite growth, which in turn prevents from overshoot.In conclusion, I discussed the meaning of the exponential growth providing with some examples. Since this is a cause and effect, explanation to examples approach, I focused on the analysis of these examples, especially the main issues related to the exponential growth. These are the time interval, the accumulated numbers, and the existence of constraints. Constraints such as finite supplies are normally imposed to such systems in order to avoid the infinite growth and overshoot. Further discussion will require an in-depth analysis of each system separately. Bibliography Cattor, W. R. (1980). OVERSHOOT, The Ecological Basis of Revolutionary Change. University of Illinois Press.Feldman, D. P. (2012). Chaos and Fractals. Oxford University Press.Moles, P., Parrino, R., & Kidwell, D. (2011). Corporate Finance. John Wiley & Sons Ltd.Murray, W. (1999). The Emerging Strategic Environment: Challenges of the Twenty-first Century. Greenwood Publishing Group.Rockwood, L. L. (2006). Introduction to Population Ecology. Blackwell Publishing.Sweeney, L. B., & Meadows, D. (2010). The Systems Thinking Playbook. Chelsea Green Publishing.Washington, H., & Cook, J. (2011). Climate Change Denial, Heads in the Sand. Earthscan.