Differential Equation Project
FAST School of Computing

Project Differential Equations (MT-224)

Due Date: 14th, June 2021. Max Marks: 70

A Brief Literature Review:

We have studied the population growth model i.e., if P represents population. Since the population varies over time, it is understood to be a function of time. Therefore we use the notation P (t) for the population as a function of time. If P (t) is a differentiable function,

then the first derivative dP

dt represents the instantaneous rate of change of the population

as a function of time, which is proportional to present population in case of the exponential growth and decay of populations and radioactive substances. Mathematically

dP

dt ? P.

We can verify that the function P (t) = P0e rt satisfies the initial-value problem

dP

dt = rP, P (0) = P0.

This differential equation has an interesting interpretation. The left-hand side represents the rate at which the population increases (or decreases). The right-hand side is equal to a positive constant multiplied by the current population. Therefore the differential equation states that the rate at which the population increases is proportional to the population at that point in time. Furthermore, it states that the constant of proportionality never changes.

One problem with this function is its prediction that as time goes on, the population grows without bound. This is unrealistic in a real-world setting. Various factors limit the rate of growth of a particular population, including birth rate, death rate, food supply, predators, diseases and so on. The growth constant r usually takes into consideration the birth and death rates but none of the other factors, and it can be interpreted as a net (birth minus death) percent growth rate per unit time. A natural question to ask is whether the population growth rate stays constant, or whether it changes over time. Biologists have found that in many biological systems, the population grows until a certain steady-state population is reached. This possibility is not taken into account with exponential growth. However, the concept of carrying capacity allows for the possibility that in a given area, only a certain number of a given organism or animal can thrive without running into resource issues.

• The carrying capacity of an organism in a given environment is defined to be the maxi- mum population of that organism that the environment can sustain indefinitely.

• We use the variable K to denote the carrying capacity. The growth rate is represented by the variable r. Using these variables, we can define the logistic differential equation.

dP

dt = rP

( 1 ?

P

K

) .

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• An improvement to the logistic model includes a threshold population. The threshold population is defined to be the minimum population that is necessary for the species to survive. We use the variable T to represent the threshold population. A differential equation that incorporates both the threshold population T and carrying capacity K is

dP

dt = rP

( 1 ?

P

K

) ( 1 ?

P

T

) , (1)

where r represents the growth rate, as before, which is known as threshold logistic differential equation.

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Project Statement:

In the 2018 Marvel Studios blockbuster, “Avengers: Infinity War,” the villain Thanos snaps his fingers and turns half of all living creatures in the universe to dust. He was concerned that overpopulation on a planet would eventually lead to the suffering and extinction of the entire population.

This is evident in the following quote from Thanos. Avengers: Infinity War. Marvel, 2018.

“ Little one, it’s a simple calculus. This universe is finite, its res-

-ources finite. If life is left unchecked, life will cease to exist.

It needs correction.”

In this activity, we will investigate the validity of Thanos’ claims using mathematical models for population dynamics.

(a) There is a bit to unpack in Thanos’ quote. What are some of the assumptions that Thanos is making?

(b) Model#1

(i): Model the situation with the help of differential equations that Thanos is describing? and Solve the initial value problem (by taking P (0) = P0) and determine what would happen to a population in the long run. Explain why its solution reflects the Thanos Claim.

(ii): Thanos’ plan is to eliminate half of all living creatures in the universe. What would happen if the population size was suddenly cut in half? How could that be represented with this model? What parameters would change?

(c) Model# 2 Use the concept of carrying capacity, to formulate the another initial value problem.

(i) How does each parameter affect the growth of the population?

(ii) For what value(s) of P if any, would the population stay constant? This value will be called an equilibrium solution.

(iii) Note that an equilibrium solution is considered stable if all solutions close to the equilibrium value approach the equilibrium. Otherwise, the equilibrium value is unsta- ble. For each equilibrium value, determine the stability.

(iv) Solve the initial value problem and determine what would happen to a population in the long run. Explain why your answer makes sense in terms of the differential equation.

(v) Thanos’ plan is to eliminate half of all living creatures in the universe. How would halving the initial population impact the overall dynamics of the system?

(vi) Do these assumptions seem more or less reasonable than the first model for describ- ing Thanos’s version of reality?

(d) If last four digits of roll number of one of your group members are abcd then for P0 = abc millions, k = a% L = abcd millions.

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(i) Draw the phase line plot, direction fields (using MATLAB) and also classify solution as stable or unstable.

(ii) Use MATLAB the results for Model#1 and Model#2

(iii) Discuss in detail whether your plot in part (ii) supports your discussions done in (b) and (c).

(iv) When does population increase is the fastest in the logistic equation. Connect it with the concept of maximum value of the function in Calculus.

(e): Make your own threshold logistic equation model as in ??. Explain your model that includes your assumptions and description of parameters.

(f ): Final Report:

A Report for Thanos. Thanos claims to be a logical person. In the sequel, “Avengers: End Game,” time travel is used to undo Thanos’ work. Suppose you go back in time and work your way through to become a part of Thanos’ inner circle. Prepare a report to Thanos to encourage him to rethink his plan. Your report should discuss the assumption and results from both of the mathematical models discussed.

Summarize your work such a way that the reader can rapidly become acquainted with the material. It should contain a brief description of the problem, important background information, a discussion of pertinent assumptions, a short description of your method- ology, concise analysis, and your main conclusions. Assume the reader is familiar with the basics of calculus and differential equations, so there is no need to walk through every step of your solution process or include equations. However, you should still de- scribe the processes and mathematical techniques you used to reach your conclusions and explain why you used them. Refer the reader to the appendices as needed.

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Report Requirements: Students are required to submit a complete report of the project prepared in MS Word in

their own words, including:

Sr. No. Deliverable Marks

1. Objectives and introduction of the problem. 5 2. A step-by-step analytical solution. Clearly state the assumptions and

values that you use for the solution. 20 3. A step-by-step example demonstrating the MATLAB solution. Also

provide an instruction’s manual to run the MATLAB program to obtain the MATLAB solution demonstrated in the example. 10

4. Line by line explanation of MATLAB program. The part must include the explanation of the commands, functions, and toolboxes used. 5

5. Flowchart of the solution methodology. 5 6. Detailed results section. Present results and graphs of your analytical

and MATLAB solution in this section, compare and discuss your results including their physical interpretation. 10

7. Conclusions. In this section include conclusions related to this project, the difficulties that you faced during this project and how

you overcame those difficulties. 5 8. Complete and well commented MATLAB code. 10

Each report element should be documented under a separate heading. Report must not exceed 12 A4 size pages including table of contents as well as a single title page with project title, student names, ids, section, and name of the course. 3 marks will be deducted for every extra page. Each page should be numbered. The report should be written in Calibri or Times New Roman typeface only. The size of the font should be 12. The size of first and second level of headings should be 14 bold, and 12 bold, respectively. The alignment of the report should be justified, while pictures and tables should be center aligned with relevant captions. The option to align the text left, right, center, and justify can be found under paragraph options on Home tab. Line and paragraph spacing should be set as 1.5.

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Project Submission Guidelines: This project is an open-ended problem designed to demonstrate the applications of differential equations in real life. The open-ended nature of the problem means that this problem can be solved in more than one way using various techniques and methodologies, some of these techniques have been covered in this course. You are free to adopt any technique and solution methodology to solve this problem. Solution techniques and methodologies that are not part of the course outline can also be used to solve the problem. You will have to do extensive research to completely solve the problem. Project guidelines are summarized below:

• This is a group project and carries 70 marks.

• A group can have maximum of 3 students. One of the aims of this project is to en- able students to work effectively in a team. Therefore, this project cannot be done individually.

• Plagiarized work (from internet or fellow students) will result in zero marks.

• Deadline for complete project submission on google classroom (one MS Word file and one pdf of the same Word file including all the codes and by-hand solutions) is Wednesday 14 June 2021 latest by 04:30pm. Do not submit your project in a .zip or .rar format. You can submit additional files such as .m files, however, the single PDF and MS Word file must also include all these files.

• Name of your project report file must be as per following format: ID1 ?ID2 ?ID3 ? MT 224 ?Project?Section.(e.g., 123456 ? 654321 ? 987654 ?MT 224 ?Project?I)

• Do not submit your project via email, it will not be considered.

• Late submissions will not be considered.