Life science math exam notes

>Malthusian
dP/dt = kP

>Logistics
dP/dt = kP(1-P/M)
k= intrinsic growth rate;
M= carrying capacity of the environment

>Competition
dx/dt = k1x(1-x-ay)
dy/dt = k2y(1-y-bx)

>Predation
Predation efficiency: ε
if ε>1, the predator population survives in the long run
dx/dt= k1x(1-x/ε-y)
dy/dt=-k2y(1-x)

>Volterra eqnt.
dx/dt = (-a+by)x
dy/dt = (c-kx)y
eq. @ (0,0) or (c/k, a/b)
Laws of Volterra:
1. Flutuations are periodic, the period increases with the amplitude of perturbation.
2. The average predator & prey over a period are, respectively, c/k, a/b

>Harvest
dx/dt = (-a+by)x - H1x
dy/dt = (c-kx)y - H2y
*Harvest is bad for the predator, better for the pray

>Markov process (genetics of inbreeding)
*discrete-time dynamics based on probability, see page 500 in the textbook.