+ Q| CD Page view | A Read aloud | T Add textDraw9. (a) Perform a qualitative analysis on the following competition model:dydtdxdtO Search= x(1- x - y)-= y(0.75-y-0.5x)dydtThat is: find the nullclines, the equilibrium values, and draw arrows in each ofthe regions describing how r and y are changing.(b) Based on your work, in the long-term, if both populations start out at non-zerovalue, will one population drive the other into extinction or will the populationsbe able to coexist?10. (a) Perform a qualitative analysis on the following slightly different competitionmodel:dxdtK= x(1 - x - y)-= y(0.5-0.25y - 0.75x)Highlight(b) Based on your work, in the long-term, if both populations start out at non-zerovalue, will one population drive the other into extinction or will the populationsbe able to coexist?11. (a) Use Euler's method to estimate x(0.3) for the differential equationdrdt2(1-x)WErase | 86 881REROA4) D3/8/2

For (a), We are given the following competition model. dxdt=x1-x-ydydt=y0.75-y-0.5x We…

Answered: (a) Perform a qualitative analysis on…

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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4. Monitoring oxygen consumption (y) each day during the 5-day BOD test shows the following results, which are expected to conform the equation: y = Lo(1-ekt). t (d) Y (mg/L) Perform regression analysis to fit the nonlinear equation and obtain Lo (mg/L) and k (d¹¹). 1 85 2 150 3 150 4 170 5 205
The Current (year 2020) population of the Earth is 7.5 billion people, and the yearly birthand death rates are β = 0.021 and δ = 0.009 respectively. Assuming the birth and deathrates remain constant, find the population of the Earth in the year 2120
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b) A metal component is subjected to heat-treatment called normalisation. The component is heated to a temperature T, (which is above the critical phase transformation temperatures) and cooled in air. The elementary change dT(t) of temperature T(t) of a heated component is determined from the heath energy balance equation cmdT(t) =-hA(T(t)– Tuir)dt where cmdT(t) is the energy which is lost due to cooling; hA(T(t)– Tuir )dt is the total heat flux from the component to the air during an infinitesimally small time interval dt. The constant c [J/(kg K)] is the specific heath capacity, m is the mass of the component in kg, h [W/(m? K)] is the heat transfer coefficient and A is the surface area in m?. Derive an equation which gives the temperature T(t) as a function of the cooling time t.
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Consider the following population linear regression model of individual food expenditure: Y = 50 + 0.5X + u, where Y is weekly food expenditure in dollars, X is the individual’s age, and 50+0.5X is the population regression line. Suppose we generate artificial data for 3 individuals using this model. This artificial sample, which consists of 3 observations, is shown in the following table: Answer the following questions. Show your working. (a) What are the values of V1 and V4? (b) Suppose we know that in this artificial sample, the sample covariance between X and Y is 150, and the sample variance of X is 100. Compute the OLS regression line of the regression of Y on X. (Hint: Assume these summary statistics and the OLS regression line continue to hold in parts (c)-(e).) (c) What are the values of V5 and V7?
Consider the regression model Y₁ = ẞX; +u₁ Where u; and X, satisfy the assumptions specified here. Let ẞ denote an estimator of ẞ that is constructed as ẞ= where Y and X are the sample means of Y, and X,, respectively. Show that ẞ is a linear function of Y₁, Y2.... Y Show that ẞ is conditionally unbiased. 1. E(XXX)= 2017-0 2. E(X₁. X2...X) = E Y -Yn) B X X2..... Xn)
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11. Consider the population model dy dt = 3y - y² - 2. This is a harvesting equation with a harvesting rate of 2. Find the steady states and their stability, and interpret your answer in terms of the population model. Is this overhar- vesting, underharvesting, or critical harvesting?
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…
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