Mathematical Methods For Engineers 2 Leslie Matrix M

Mathematical Methods For Engineers 2 Math1064leslie Matrix Matlab Gr

Mathematical Methods for Engineers 2 (MATH1064) Leslie matrix Matlab group project Due no later than 2 pm on Friday 10th October, 2014 Graduate Qualities: This project is designed to help the student achieve course objective 4: solve simple applied problems using software such as Matlab, and to develop Graduate Qualities 1 & 3, namely operating effectively with and upon a body of knowledge, and effective problem solving. Assessment: The assessment will take into account all of your documentation of the mathematical analysis of the problem, your Matlab m-file(s), your Matlab output, the correctness of the final solutions and the presentation of your whole report. Groups should contain two or three people.

It will be assumed that each member of the team contributed equally and will be awarded individually the mark allocated to the report. If this is not the case, then a lesser percentage for one or more members must be agreed by the team and clearly indicated. This especially will apply to absences from the practical class or non-attendance at agreed team meetings. The University policy on plagiarism will apply between different groups. Students who wish to can submit a peer assessment form which can be found on the course webpage.

How to divide the work: Each team member must participate in all aspects of the project: mathematical calculations, Matlab work and report writing. Only one copy of your project report is required for each group. Summary: In this project you will: • Investigate the Leslie matrix model for a population • Explain how a Leslie matrix can be used to calculate the population in each age class from time to time • Use Matlab to draw plots of age class populations evolving over time • Use Matlab to study the long term behaviour of population numbers Your report must be typed, and submitted through LearnOnline by one member of your group. It should include: • Written worked answers to all questions where this is required. • Appropriately labeled figures where required. • A listing of your Matlab script file should be included at the end of your report in an appendix. • A coversheet is not needed but your report must have a title page that lists the names and student identification numbers of all members of the group. • The group’s .m file must be submitted as a separate file via LearnOnline. Be sure to list all group members at the top of the file; only one copy per group is required. There will be marks awarded for submitting this file, so don’t forget. Your .m file may be run and checked during the marking process.

1 Leslie Matrix Model Invented by Patrick H. Leslie in the 1940s, the Leslie Matrix is a mathematical model of population growth for a species.

Time is divided into discrete periods, with individual members of the population progressing through discrete age classes at given survival rates. Here is a simplified example: The Central Australian Budgericoot (CAB) cannot live beyond five years of age. We shall discretise time into years, and we shall count the number of CABs in each of the 5 possible age classes at the start of each year. Only 80% survive the first year after birth. Of those that survive the first year, only 50% live for another year and enter their third year of life.

Of those, 40% survive the next year, and of those, 25% make it into their fifth year, but none ever reach their fifth birthday at the end of that year. (This means that only 0.80 × 0.50 × 0.40 × 0.25 = 0.04 = 4%, on average, live beyond four years.) This gives a survival rates array p = [0.80 0.50 0.40 0.25 0.00] = [p1 p2 p3 p4 p5]. The sustainability of a species depends not only upon survival rates, but also on the birthrate of females of the species. Suppose that female CABs cannot give birth during their first two years or during their fifth year, should they live that long. On average, a female of this species in her third year can produce 2 female offspring, while in her fourth year on average a female produces 1.5 female offspring.

This information may be assembled into the Leslie matrix for this species: A = [0 f1 f2 f3 f4 f5; p1 0 0 0 0 0; 0 p2 0 0 0 0; 0 0 p3 0 0 0; 0 0 0 p4 0 0; 0 0 0 0 p5 0]. The matrix is all zeros except for the fertility vector across the top row, and the survival rates situated below the leading diagonal. Suppose that an investigator starts monitoring the female CABs in a particular area in a given year. At that time, there are 100 females between birth and their first birthday, another 100 females in their second year of life, another 40 females in their third year, 40 females in their fourth year, and 20 females that have survived until their last possible year.

Thus the total female population is 300. The female population can be represented by the initial age population vector y(1) = [100; 100; 40; 40; 20]. In general, the population in age class j in the time period k + 1 is given by yj(k + 1) = pj−1 yj−1(k), with j = 2, 3, ..., n, since the population in age class j comes from survivors of the population in age class j−1 in the previous time period k. The population in age class 1 (newborns) in time period k + 1 should be considered separately, since it arises only by birth from the population of other age classes in the previous time period.

In the case of n age classes, y1(k + 1) = f1 y1(k) + f2 y2(k) + ... + fn yn(k), where f is the average fertility array as in the example above. The population vector y(k) = [y1(k); y2(k); ...; yn(k)] then satisfies the matrix equation y(k + 1) = A y(k), where A is the Leslie matrix. Typically, in a Leslie model, the numbers in the various age classes, and the total population itself, exhibit fluctuations until transient effects disappear. After a sufficiently long time, the population changes at a rate r. If r > 1, the population eventually increases; if r

Paper For Above instruction

The objective of this report is to investigate and analyze the Leslie matrix model for a population, using MATLAB to simulate the evolution of age class populations over time, and to understand the long-term behavior of such populations through eigen analysis. The Leslie matrix offers a structured method to project future populations based on survival and fertility rates among discrete age classes, providing insights into population growth or decline, stability, and extinction risk. This report explains the mathematical foundation of the Leslie matrix, demonstrates how to construct and apply it, and utilizes MATLAB to visualize and analyze population trends over a specified timeframe. We focus specifically on the example of the Central Australian Budgericoot (CAB), a species with a maximum age of five years, and explore how initial population data evolve through successive time periods. The analysis includes calculating population vectors, plotting population dynamics, and studying the dominant eigenvalues to infer growth rates and stability. Finally, the impact of modifying fertility rates is examined to illustrate how changes influence long-term population behavior, providing practical insights for ecological modeling and conservation strategies.

The first part of the report details the mathematical formulation of the Leslie matrix model for the specified species. It involves calculating the population for subsequent years based on initial data, interpreting the biological significance of the matrix operations, and understanding the long-term implications of the eigenvalues. The second part implements the model in MATLAB, generating population projections over 30 and 100 years, and analyzing eigenvalues for population stability. Furthermore, the report investigates how altering fertility rates affects the growth rate and population stability, as reflected in the dominant eigenvalue. The analysis highlights the utility of the Leslie matrix in ecological modeling, offering valuable predictions for species persistence or extinction risk, and demonstrating the effectiveness of MATLAB in handling complex numerical simulations. Overall, this comprehensive study serves as an educational example of mathematical and computational population modeling within an engineering context, with implications for biological conservation and resource management.