(b) Suppose that we want to model the evolution of the population of a cer-tain type of organisms. Observations indicate that if the population dropsbelow a survival level of 10° individuals, it goes extinct. Moreover, thepopulation growth is limited: the available resources of space and foodcan sustain at most 106 individuals. We treat the population size P(t) asa continuous function of time.(i) Explain briefly how the following model incorporates the above ob-servations:dP— К(А- Р)(Р — В), k>0,dtwhere P(t) denotes the population size at time t and B < A. Usingthe informations given in the text of the question, determine the val-ues of the constants A and B. Can you also determine the value of[2](ii) Suppose that A < P < B. At which value of the population is the[2]k?growth fastest?(c) Perform a linear stability analysis for the model below:dPP(1 – P)e-P*,P > 0.dtFind the equilibrium values and determine their stability.[6]

Given Information:Population drops below the survival level of 10^2 individuals.The resources…

Answered: (b) Suppose that we want to model the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3) When modeling population growth of a new species (or virus), we often use the very important logistic curve P(t) =- , where A is some constant. [Note: You may have seen populations modeled exponentially as P(t) = Poe"t, or something similar. This model grows without bound, and so is not a very accurate in the long run. We are fancier now, and can use better models!] a. Graph P(t) on your calculator and sketch it here. Why might this model a population's growth well? b. We want to describe how fast it is growing at a given time, so we need to take the derivative [You may first want to find the derivative of e-t]. Sketch the derivative. What is lim P'(t), and what does it mean?
The table shows the total assets (in trillions) held in a foreign bank for the given years. Complete parts (a) through (c) below. 2000 | 2005 | 2007 | 2009 | 2011 | 2013 | 2015 117 150 199 Year Assets 17 42 52 78 (a) Find an exponential model of the form f(t) = y,b' for these data, where t= 0 corresponds to the year 2000. If you do not have suitable technology, use the first and last data points to find a function. If you have a graphing calculator or other suitable technology, use exponential regression to find a function. The exponential model for the data is f(t) = O
Can you show an example similar to this problem part A - but not this problem - of how to calculate an exact percentage change?
6. A home improvement store carries two models of power drills. Lifetime (i.e., time to failure) of Model 1 drills are exponentially distributed with a rate of of 0.15 failures per year, whereas Model 2 drills have a lifetime that is log-normally distributed with parameters u = 0.75 and o = 1.2. (a) Which model of power drill is more likely to fail within a year? (b) Which brand produces cables that are more likely to last at least 2 years? (c) Which model of power drill has longer expected lifetime?
a) Logistically, the growth of Mrs. Fields cookies in different states is closely modeled after this: 19440 P(t) = 1 + 485 e -0.4444t P(t) is number of U.S. cities with a Mrs. Fields cookie storet years after the year 2000, and t is time in years after the year 2000. Use this logistic model to find the number of cities that had a Mrs. Fields cookie store in the year 2000, when they first opened. U.S. cities in the year 2000. b) How many U.S. cities had a Mrs. Fields cookie store in the year 2015. (round to the nearest whole number) U.S. cities in the year 2015.
This exercise uses the population growth model. It is observed that a certain bacteria culture has a relative growth rate of 11% per hour, but in the presence of an antibiotic the relative growth rate is reduced to 5% per hour. The initial number of bacteria in the culture is 24. Find the projected population after 24 hours for the following conditions. (Round your answers to the nearest whole number.) (a) No antibiotic is present, so the relative growth rate is 11%.bacteria (b) An antibiotic is present in the culture, so the relative growth rate is reduced to 5%.bacteria
Question 3 (a) Consider a model of market equilibrium in which the current supply of firms is a function of the price that is expected to prevail when the product is sold. Assume that the market supply equation is q³(t) = F + Gp and pe is the expected price and F and G are constant parameters of the supply equation. Assume further that suppliers use information about the actual current price and its first and second derivatives with respect to time to form their prediction of the price that will prevail when their product reaches the market. In particular, assume that dp pe=p+b+c- dt d²p dt² tdpd²p=0, then suppliers and b> c> 0. If the current price is constant, so that dt expect the prevailing price to equal the current price. If the current price is rising, so that > 0, then suppliers expect the prevailing price to be higher than the current dt price. How much higher depends on whether the current price is rising at an increasing rate, > 0; or at a decreasing rate, p 0 is a constant…
An ecologist is studying the impacts of air pollution on giant panda populations, specifically on panda birth rates. Due to the historical interest in giant pandas, the ecologist has access to data from the last 30 years that allows her to fit two models to the evolution of giant panda populations over time (one for 30 years and one for the last 5 years after a new chemical plant was built close to the panda habitat). She will use these models to determine the long-term impacts of this pollution on panda populations. She has the following information to help her model the two sets of population dynamics data on adult and baby pandas : i. 1) Without pollution, for every 1 adult female panda a year earlier, 0.75 baby pandas are born each year (reproduction rate = 0.75). 2) With pollution, for every 1 adult female panda a year earlier, 0.7 baby pandas are born each year (reproduction rate = 0.7). 3) 3 in 4 of the adult female pandas survive to the next year. 1 in 3 female baby pandas…
Question 4: Suppose that in a certain chemical process the reaction time y (hr) is related to the temperature (°C) in the chamber in which the reaction takes place according to the simple linear regression model with equation y = 5.00 -0.01x. (a) What is the expected change in reaction time for a 1°C increase in temperature? For a 10°C increase in (b) (c) temperature? What is the expected reaction time when temperature is 98°C? When temperature is 120°C? Suppose o = 0.08. Consider the temperature 98°C. What is the probability that the reaction time will exceed 4 hours? (d) What is the probability that two independently observed reaction times for temperatures 1° apart are such that the time at the higher temperature exceeds the time at the lower temperature?

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