birth-death models, equilibria and stability For the following two differential equations which model birth rate in relation to population density


*

*$\dfrac {dN}{dt} = bN^2 - aN$,

*$\dfrac {dN}{dt} = bN^2 \left(1 - \dfrac N K \right) - aN$
where $a$ and $b$ are positive constants
I need to locate the equilibria of $N$, determine stability, and sketch solution curves for "various starting values" $N_0$.
I have no idea how to go about this. 
 A: To find the equilibria, just find the values of $N$ for which the right hand side equals zero.  Note that, to be biologically meaningful, $N$ must be a non-negative real number.  For any population model without influx terms, $N = 0$ will be a trivial equilibrium; to find the non-trivial equilibria, dividing by $N$ is usually a good start.

To determine stability, differentiate the right hand side of the equation with respect to $N$ and look at the value of the resulting mized derivative $\frac{d^2N}{dNdt}$ at the equilibrium points: if it is positive, small increases in $N$ will increase the population growth rate above $0$, while small decreases in $N$ will reduce it, and so the equilibrium will be unstable. Conversely, if $\frac{d^2N}{dNdt}$ is negative, small perturbations of $N$ will lead to opposite changes in the growth rate, and so the equilibrium will be stable.
(Edit: For unstructured population models described by a single variable $N$, you don't actually need to differentiate anything: since $\frac{dN}{dt}$ can only change sign at the equilibria, once you've found all of them it's enough to look at the sign of $\frac{dN}{dt}$ at some arbitrary point within each interval bounded by adjacent equilibria.  However, calculating $\frac{d^2N}{dNdt}$ is still a useful consistency check, as well as good practice for multidimensional models where you need to look at the eigenvalues of the Jacobian to determine stability.  Then again, looking at the sign of $\frac{dN}{dt}$ also generalizes usefully to the multidimensional case, in the form of phase plane analysis.)

To sketch the solution curves, the easy solution is to use a computer program that can solve ordinary differential equations numerically.  (I generally use Maple, but there are plenty of other tools that can do it.  Even a good graphing calculator should be enough.)
Alternatively, you can make pretty good sketches by hand just by remembering that near stable equilibria the orbits will generally decay exponentially closer to the equilibrium, while near unstable equilibria they will grow exponentially away from it.  When you have an orbit starting near an unstable equilibrium and ending near a stable one, the ends will connect to form a sigmoid shape.  The details will depend on the exact form of the growth rate function, but they don't affect the general rule that orbits move away from unstable equilibria and towards stable ones.
Ps. I deliberately avoided showing any of the actual calculations in my answer.  Let me know if you need more help with something — or, better yet, show us what you've managed to do so far, so that we can help if you get stuck on a particular detail.
