We know that with ample space and food resources, the rate of change of the population...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
We know that with ample space and food resources, the rate of change of the population of a species may, at least for a while, be proportional to the population. Namely, = ky. Here the k>0 term represents the proportionality constant. Now, assume that to make sure that our population doesn't grow too much we harvest/remove some of the population at a certain time-dependent rate. Suppose that our new model for the population is dy dt = ky - et y(0)=yo, (1) (2) where g(t) e' is the rate at which we are harvesting. 2. We see that this equation is neither autonomous nor separable. However, we can still solve it. One way of solving this problem is to exploit the properties of the equation. We can do it as follows: (a) Assume first that the solution is of the form y(t) = eu(t). Then plug this into the equation and write a new differential equation in terms of the function u. (b) Solve the equation terms of u and find the answer to your original model by writing y(t) = e'u(t). 3. We could also solve this equation without assuming that the solution is of the form y(t) = e*u(t). This new method will be much more useful when we deal with first order linear differential equations where can't clearly see a way to assume a certain form for the solution that simplifies the problem. We can do that as follows: (a) First, collect all the y terms to the left hand side (LHS). Then, find a function , such that when we multiply the equation by this (t) the left hand side becomes a full derivative of the function (t)y(t). That is, d(ay) = (t)e. dt (3) (b) Once you figure out this function, you can solve the differential equation (3) to get an answer for our model with given initial population. As we can see, the solutions we got from questions 1 and 2 are the same. We know that with ample space and food resources, the rate of change of the population of a species may, at least for a while, be proportional to the population. Namely, = ky. Here the k>0 term represents the proportionality constant. Now, assume that to make sure that our population doesn't grow too much we harvest/remove some of the population at a certain time-dependent rate. Suppose that our new model for the population is dy dt = ky - et y(0)=yo, (1) (2) where g(t) e' is the rate at which we are harvesting. 2. We see that this equation is neither autonomous nor separable. However, we can still solve it. One way of solving this problem is to exploit the properties of the equation. We can do it as follows: (a) Assume first that the solution is of the form y(t) = eu(t). Then plug this into the equation and write a new differential equation in terms of the function u. (b) Solve the equation terms of u and find the answer to your original model by writing y(t) = e'u(t). 3. We could also solve this equation without assuming that the solution is of the form y(t) = e*u(t). This new method will be much more useful when we deal with first order linear differential equations where can't clearly see a way to assume a certain form for the solution that simplifies the problem. We can do that as follows: (a) First, collect all the y terms to the left hand side (LHS). Then, find a function , such that when we multiply the equation by this (t) the left hand side becomes a full derivative of the function (t)y(t). That is, d(ay) = (t)e. dt (3) (b) Once you figure out this function, you can solve the differential equation (3) to get an answer for our model with given initial population. As we can see, the solutions we got from questions 1 and 2 are the same.
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
You have been provided with the description of a programming language, J, intended for scripting applications. Its syntax is similar to a cut-down version of Java in that it consists of function...
-
Planning is one of the most important management functions in any business. A front office managers first step in planning should involve determine the departments goals. Planning also includes...
-
A particle of mass m moves in a certain plane P due to a force F whose magnitude is constant and whose vector rotates in that plane with a constant angular velocity . Assuming the particle to be...
-
Refer to Google's recent balance sheet in Appendix A. What property, plant and equipment assets does Google list on its balance sheet? What is the book value of its total net property, plant and...
-
McQueen Motor Company manufactures automobiles. During September 2022, the company purchased 5,000 head lamps at a cost of $15 per lamp. Fifty of these lamps were used to replace the head lamps in...
-
On December 14, 2011, appellant Aaron Olson contracted to receive telephone service from respondent CenturyLink and also applied for reduced-rate service that CenturyLink provides through Minnesotas...
-
Dr. Roger Jones is a successful dentist but he is experiencing recurring financial difficulties. For example, Dr. Jones owns his office building, which he leased to the professional corporation that...
-
Find the range of the function defined by the equation and the given domain. f(x) = 4x3; domain= {0, 1, 2, 3}
-
Adaptive selling requires salespeople to adjust their sales strategy based on the: hard data about the customer's previous purchasing patterns relationship and product needs of each customer sales...
-
In March 2012, the French bank, RCI Banque, issued an 18-month bond with a face value of 10,000, and an annual coupon rate of 2 per cent, paid every quarter. The issue price was 9,984.50. What was...
-
A 30-year gilt is issued with face value of 100, paying interest of 20 per year. If market yields increase shortly after the bond is issued, what happens to the: (a) Coupon rate (b) Price (c) Yield...
-
If Severn Trent plc were to distribute all its earnings, it could maintain a level dividend stream of 0.67 per share. How much is the market actually paying per share for growth opportunities?
-
The market consensus is that Pinto plc has an ROE of 9 per cent, a beta of 1.25, and plans to maintain its plowback ratio of 0.67. This years earnings per share was 3. The consensus view of the...
-
Regenboog NV earned 68 million for the fiscal year ending yesterday. The firm also paid out 25 per cent of its earnings as dividends yesterday. The firm will continue to pay out 25 per cent of its...
-
In 2007, the weighted average combined fuel deficiency of cars and light trucks in the United States, combined, was 8.67 km/L. The average vehicle traveled 18,861 km in 2007. The amount of CO2...
-
Find the center of mass of a thin triangular plate bounded by the y-axis and the lines y = x and y = 2 - x if (x, y) = 6x + 3y + 3.
-
The soft landing of a lunar module descending on the moon can be modeled as shown in Figure P3.8. Define the state variables as x1 = y, x2 = dy/dt, x3 = m and the control as u = dm/dt. Assume that g...
-
A unity feedback system has the loop transfer function Sketch the Bode plot of the loop transfer function and indicate how the magnitude 20 log |L(jÏ)| plot varies as K varies. Develop a table...
-
The Radisson Diamond uses pontoons and stabilizers to damp out the effect of waves hitting the ship. The block diagram of the ship's roll control system is shown. Determine the feedback gains K2 and...
-
Define human communication.
-
Perceiving your communication as effective while those around you perceive it as ineffective is known as ______________.
-
Visit any of the websites for major social media companies (Facebook, Twitter, LinkedIn, etc.). Under the About Us section, read the company mission statement and goals for the organization. Do you...
Study smarter with the SolutionInn App