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 eigenvalues. (Weisstein, n.d.).
Eigenvalue is a number that is derived from a square matrix where a square matrix is an assembly of n rows and n numbers. A Greek letter lambda (λ) is used to symbolize eigenvalue. When there exist an n ×n matrix P, then a non-zero vector x in R^n is known as eigenvector of P. Ax is a scalar multiple of x that is Ax=λx, for some scalar of λ. P represents the square matrix, while x is the non-zero vector and λ is a non-zero value. In other words, eigenvector is a vector that will sustain its course after going through linear transformation whereas the scalar value that was multiplied with the eigenvector during linear conversion is known as eigenvalue (Sharma, n.d.).
Predation is a positive or negative interaction which includes pre-predator, plants and herbivores and interactions in between parasite host. They are very vital in the ecosystem as they are the primary motivators of energy throughout the food chain (“Trophic links”, 2005). A creature that feeds on another creature is known as predator whereas prey is the organism which are consume by the predator. As an illustration, the pair of prey and predator are deer and lion, rabbit and fox, flies and frogs, fish and bears and etc. Prey...