Systems of Differential Equations

Suppose that $\mathbf{A}_{n\times n}$ is diagonalizable, and let $\mathbf{P} = \begin{pmatrix} \overrightarrow {x}_1 \mid \overrightarrow {x}_2 \mid \cdots \mid \overrightarrow {x}_n \end{pmatrix}$ be a matrix whose columns are a complete set of linearly independent eigenvectors corresponding to eigenvalues $\lambda_i$. Show that the solution to $\overrightarrow {u}' = \mathbf{A}\overrightarrow {u}$, $\overrightarrow {u}(0) = \overrightarrow {c}$, can be written as

in which the coefficients $\xi_i$ satisfy the algebraic system $\mathbf{P}\overrightarrow {\xi } = \overrightarrow {c}$.

Using only the eigenvalues, determine the long-run behavior of the solution to $\overrightarrow {u}' = \mathbf{A}\overrightarrow {u}$, $\overrightarrow {u}(0) = \overrightarrow {c}$ for each of the following matrices.

Consider two species that coexist in the same environment but compete for the same resources. Suppose that the population of each species increases proportionally to the number of its own kind but decreases proportionally to the number in the competing species---say that the population of each species increases at a rate equal to twice its existing number but decreases at a rate equal to the number in the other population. Suppose that there are initially 100 of species I and 200 of species II.

Consider two species that survive in a symbiotic relationship in the sense that the population of each species decreases at a rate equal to its existing number but increases at a rate equal to the existing number in the other population.