Eigenvalues and eigenvectors is one of the important topics in linear algebra. The purpose of this assignment is to study the application of eigenvalues and eigenvectors in our daily life. They are widely applicable in physical sciences and hence play a prominent role in the study of ordinary differential equations. Therefore, this assignment will provide explanations on how eigenvalues and eigenvectors will be functional in a prey-predator system. This will include background, history of the concept and explanation on what is meant by eigenvalues, eigenvectors and prey-predator system. Other than that, models and application of the eigenvalues and eigenvectors in prey-predator system will also be included in this assignment. Necessary appendix such as graphs will be attached with the assignment.
BACKGROUND AND CONCEPT
Linear algebra is the study of linear transformations of linear equations which are represented in a matrix form by matrices acting on vectors. Eigenvalues, eigenvectors and Eigen space are properties of a matrix (Sharma, n.d.). The prefix “Eigen” which means “proper” or “characteristics” was originally developed in German and invented by a German mathematician. Latent roots, characteristic roots, proper values or characteristics value are few common terms of eigenvalues. They are a special set of scalars allied with a linear system of equations for instance a matrix equation. In engineering and physics field, knowledge about eigenvalues and eigenvectors are very crucial where it is corresponding to diagonalization of matrix. They are practice in vibrating system with small oscillations, concepts of rotating bodies, as well as stability analysis. Corresponding eigenvectors will be paired with their eigenv...
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Trophic Links: Predation and parasitism. (2005, February). Retrieved March 14, 2014, from http://www.globalchange.umich.edu/globalchange1/current/lectures/predation/predation.html
Weisstein, E.W. (n.d.). Eigenvalue. Retrieved March 14, 2014, from http://mathworld.wolfram.com/Eigenvalue.html
APPENDICES
The graph shows that matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.
The above picture illustrates the prey-predator relationship.
This pictures demonstration the level of food chain in the ecosystem. The bottom level is the producer and the higher level will be the predator.
This indicates changes in numbers of the hare and throughout the years.
This diagram shows the Lotka-Volterra model of solving prey-predator problem.
The Bald Eagle is at the top of it’s food chain as a tertiary consumer. It
1) Inspiration: Grey wolves are considered as apex predators, meaning that they are at the top of the food chain. Grey
A food chain begins with the producers. Since plants get their energy from sunlight, they are producers; one of the common producers in the Sonoran Desert is the prickly pear cactus. Many different animals eat the fruit of the prickly pear cactus, including Harris's antelope squirrel. The squirrel is a consumer because it gets its energy from other organisms. In this case, the squirrel gets its energy from the fruit of the prickly pear cactus. The food chain starts with a producer, the prickly pear cactus, which obtains its energy from sunlight. The prickly pear is eaten by Harris's antelope squirrel, which, because it is the first consumer in the food chain, is called the primary consumer. The squirrel is eaten by the diamondback rattlesnake,
This implies that the prey population does not return to some particular equilibrium after deviation. The predation limiting hypothesis involves a density independent mechanism. The mechanism might apply to one prey - one predator systems (Boutin 1992). This hypothesis predicts that losses of prey due to predation will be large enough to halt prey population
they are at the top of the food chain of the grassland plants and animals. Grizzly bears are powerful, top-of-the-food-chain predators, yet much of their diet consists of nuts, berries, fruit, leaves, and roots. Bears also
Purpose: The purpose of this lab is to investigate the various components of different ecosystems in a smaller representation and study the conditions required for the ecosystem’s sustainability as well as the connections between
Ecosystems can be as small as a tide pool or a rotting log or as large as a body of water, forest or desert. Each system can consist of a community of plants and animals and this community is sustained by raw materials, chemical elements and water. Each community is surrounded by soil, water, climate and other conditions of the environment. An example of an ecosystem is a pond it supports a community of fish, frogs, insects and plants. Basic food is small organisms and plankton for the fish to consume. If the ecosystem does not maintain a balance it can weaken and the species will suffer. The systems are affected by fire, disease and severe climate changes not including predators and man such as building, pollution and mining.
On the second level, the introduced species starts to reproduce faster than other native species around it. It also out-competes native species for resources
The plural of focus is foci. The midpoint of the segment joining the foci is called the center of the ellipse. An ellipse has two axes of symmetry. The longer one is called the major axis, and the shorter one is called the minor axis. The two axes intersect at the center of the ellipseThe center of the ellipse is at (h, k). The radius of ellipses are not a constant distance from the center. To find the distance to the curve from the center you have to find the distance from the center to the curve for the x and y separately, these points are called vertices. The vertices are on the major axis and minor axis. The major axis is the longer axis and the minor axis is the shorter axis through the center of the ellipse. To find the distance from the center in the x direction you take the square root of a2. To find the distance from the center in the y direction you take the square root of b2. You then will have two points on the x direction and two points in the y direction and you use these four points to draw your ellipse. Ellipses are symmetrical across both of there
asteroid was on a line with Earth, the computer would show us and enable us
In 1891, a German zoologist named Karl Semper introduced the concept of a food chain, a process that is requisite for all living creatures. The chain consists of different levels. On the bottom are plants, then herbivores, the animals that eat plants. Next are carnivores, animals that eat other animal species, and the last are the animals that eat carnivores. The chain tends to overlap due to animals that eat more than one kind of food. Some people choose to be carnivores, while others choose to be herbivores due to the feeling that it is wrong to eat another living being. Humans are usually thought of as the superior animals on the Earth and living in modern society many nutritious foods are provided, especially meat. Some people choose to live herbivorous lifestyles due to moral and ethic reasoning, which can easily result in malnutrition as well as health risks that could have easily been avoided had they eaten carnivorously.
...ke the vanishing determinant, a fixed value of λ = λk, is chosen accordingly. Therefore, at λ = λk, the coefficients of the unknown amplitude Aj in equation (2.12) will become fixed and then it will be possible in obtaining the solution Ajk (the additional subscript k will be used to indicate the correspondence with the particular values of λk). Such a system of equations does not determine the Ajk uniquely but gives their ratios. A convenient mathematical solution designated by the quantities mjk are defined in terms of an arbitrary solution A_jk^' by the formula
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.
Computers have changed the way that the world works in many different ways. Some of these changes are positive and some of these changes have had negative effects on our lives. From an industrial standpoint most of these changes have been helpful to businesses and the economy. In the medical field computers have had an impact in many different areas, ranging from the way appointments are made to the carrying out of everyday tasks.
The late 20th century has successfully witnessed the boom of sophisticated technology, which gave birth to a wide range of technological outcomes, including computer technology. Computer technology, that is a skill to manage information, communicate and a kind of entertainment media, has become an important part in modern life at work, in recreation and social networking. Whether computer technology is better or not is a complicated issue. There are different arguments that need to be examined. Most people support that computer technology is better for individual lifestyles. It improves quality of working and study, provides a wide range of entertainment and is a wonderful tool of communication. On the other hand, others believe that computer