0
$\begingroup$

Let $V:[0,\infty]$x$[0,\infty] \rightarrow \mathbb{R}^2$ be the vector field given by $[0,\infty]\in(x,y)\rightarrow V(x,y):=(.4x-.4xy,-.1y+.2xy)$ so that the associated differential equation describing the dynamical system is $$\dot {x}=.4x-.4xy$$ $$\dot y=-.1y+.2xy$$

The dynamical system describes a simple model for predator and prey.$x$ is the prey and $y$ is the predator.

$a)$calculate the critical points of $V$

$b)$ in physical form, describe (don't calculate) the integral curves of $V$ in the vicinity of the critical points. What happens when you move away from the critical points but still remain within the domain $[0,\infty]$x$[0,\infty]$? Also describe the integral curve for the initial conditions where $x(0)=0,y(0)=0$.

My attempt: $a)$ $V(x,y)=0$ when $x=0,y=0$ and $x=1/2,y=1$

$b)$ Not sure. Don't understand the concept of an integral curve

  • 1
    I think you mean $x' = \dots, y' = \dots$. Also, integral curves are covered [here](https://en.wikipedia.org/wiki/Integral_curve). – 2017-02-14
  • 0
    @Moo still stuck – 2017-02-14
  • 0
    does the integral curve start to have a straighten out its slope as you move away from the critical points? – 2017-02-14

0 Answers 0