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Consider the following model of predator-prey dynamics :

$\dot x = x(\lambda − x − y),\ \\ \dot y = y(−1 + x − y)$

The number and type of equilibria of the system depend on the parameter $\lambda$, and there are essentially three different cases corresponding to three different ranges of $\lambda$.

Find these three ranges for $\lambda$ and the number, type and stability of the equilibria in each case. There are two values of $\lambda$ that give rise to borderline cases; one of these corresponds to a zero eigenvalue and may be ignored, but for the other borderline case you should make a careful analysis of the stability type.

I Seem to be quite a dunce at dynamics, I would appreciate some help on how to go about this problem.

erfink
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Max Denner
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1 Answers1

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  1. Find the nullclines, i.e., set $\dot x = 0$ and $\dot y = 0$ and solve independently. For example, $x=0$ is always an $x$-nullcline.
  2. Equilibrium points always occur when the $x$- and $y$-nullclines intersect. For example, $(0,0)$ will be an equilibrium point for any value of $\lambda$. The location of other equilibrium points will depend on $\lambda$.
  3. Linearize the system at each equilibrium point by computing the Jacobian matrix; classify the equilibrium point. For example, the Jacobian matrix at $(0,0)$ is $$J_{(0,0)} = \begin{bmatrix} \lambda & 0 \\ 0 & -1 \end{bmatrix}.$$ How does the stability of this point change as $\lambda$ varies?
erfink
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  • How would you go about describing the change in stability, lets say at the point (0,0) for this matrix? – Max Denner Mar 22 '17 at 13:53
  • Since $J_{(0,0)}$ is diagonal, we can directly read off the eigenvalues. If $\lambda >0$, then $(0,0)$ is a saddle; If $\lambda < 0$, then $(0,0)$ is a sink; If $\lambda =0$, then more sophisticated analysis is required (can no longer ignore higher order terms). – erfink Mar 22 '17 at 22:33