- Exercises 1-4, p. 307.
- Comment on Project 4.5 - ignore t!
- Questions from exercises? (Due Monday, April 16th)
- In general:
- Relationship to difference equations:
x(n+1)-x(n)=f(x(n)) We started out using difference equations because they were easy to understand, theoretically - making discrete steps is easier to follow than infinitesimal steps:
x(n+dt)-x(n)=f(x(n))dt - Autonomous versus non-autonomous differential equations:
dx=f(x)dt ordx=f(x,t)dt One solves for the function x(t). At each point in time a small change in time, dt, leads to a small change in x, dx, as described in the differential equation.
- D.E.s may be very hard (or impossible) to solve analytically - so
we approximate them numerically, which means turning them back into difference
equations!
- Compartmental diagrams ->
differential equations
- A linear model (p. 241)
- The logistic model (p. 241)
- has both inflow and outflow! rx inflow, rx^2/K outflow
- A linear system (p. 242)
- Note conservation law
- Asymptotic behavior: (0.233503,0.461929,0.304569)
- Why note solve as Markov chain? Asymptotic behavior?
- Relationship to difference equations:
- U.S. population model (p. 245-252)
- simple linear regression and residuals:
- multiple curvilinear regression and residuals: adjust
using square term in model
pop=a+bx+cx2:
- Try exponential model:
- models upon models: model rates, then use these in
exponential solution:
- multiple curvilinear regression: adjust using square term
in model ln(pop)=a+bx+cx2:
- different models, depending upon route
- simple linear regression and residuals:
- Models of epidemics - a
non-linear system
- Compartmental models
- The equations: SIR
- The
basic reproduction number and the implications for public health.
Sometimes qualitative information results even when the solution is not forthcoming!
- Compartmental models
- A non-linear D.E. model for the origin of sex in simple organisms
- Compartmental model
- Non-linear differential equations
- Qualitative effect of parameter changes