Chapter 7.2, Problem 19E

Interpretation Introduction

Interpretation:

To interpret the three terms on the right hand side of the equations ${\dot{\text{R}}\text{ = -R + A}}_{\text{S}}{\text{+ kSe}}^{\text{-S}}$, and ${\dot{\text{S}}\text{ = -S + A}}_{\text{R}}{\text{+ kRe}}^{\text{-R}}$. To find what the functional form of the third terms ${\text{kSe}}^{\text{-S}}$ and ${\text{kRe}}^{\text{-R}}$ signifies about how Rhett and Scarlett react to each other’s endearments. To show that all trajectories that begin in the first quadrant $\text{R,S}\ge \text{0}$ stay in the first quadrant forever, and to interpret that result psychologically. To prove that the model has no periodic solutions. To plot the phase portrait for the system, assuming parameter values ${\text{A}}_{\text{S}}\text{ = 1}\text{.2}$, ${\text{A}}_{\text{R}}\text{ = 1}$, and $\text{k = 15}$. To plot the predicted trajectory for what happens in the first stage of their relationship, assuming that $\text{R(0) = S(0) = 0}$.

Concept Introduction:

Dulac’s Criterion states that if $\dot{\text{x}}$ be a continuously differentiable vector field in $\text{R}$, and there exists a continuously differentiable, real-valued function $\text{g(x)}$ such that $\nabla \text{.(g}\dot{\text{x}}\text{)}$ has one sign throughout $\text{R}$, then there are no closed orbits lying entirely in $\text{R}$.