Was sind die Vorteile eines mathematischen Modells im Vergleich zu einem verbalen Modell?
Was sind die 3 Hauptbestandteile eines mathematischen Modells?
Malthusian model:
- aim of the model?
- differential equation and its parameters
- solution to the differential equation
- Graph: N(t) over time for r = 0, r>0, r<0? (also b and d paramters?)
- flaws of the malthusian model?
AIm: to find a model describing population growth
Diff eq.: \(N'(t) = rN(t)\), where r = population's intrinsic rate of increase and r = b-d, where b: instantaneous birth rate and d: inst. death rate
Solution to diff eq.: \(N(t) = N(0)e^{rt}\)
Flaws:
- all individuals are considered equal (no difference between age, gender etc)
- r: is considered constant
- r > 0, N(infinite)
What is the growth rate per capita according to Malthus?
\(N'(t) = rN(t)\) ⇔ \(r = {N'(t)/N(t)}\) = growth rate per capita (per individual)
Logistic model:
- aim
- differential equation
- solution to differential equation
- Graphs:
-> per capita growth rate over population size
-> population over time
-> growth rate over population size
Definition eines equilibrium punktes?
Ein equiblibriumspunkt der Gleichung u' = f(u) ist eine konstante Lösung der Gleichung, i.e. eine, für die u' = 0 gilt. Man muss also die Nullpunkte von f(u) finden.
Levin's paradox und die Lösung
GInzburg paradox
Wenn man einen mortaliätsterm substrahiert, aber N > K, dann wird der Mortalitätsterm positiv -> i.e. zusätzliche Mortalität führt zu neuen individuen