General framework
Let
- Oft, R is a function of
. - Often,
- non-overlapping generations.
Constant R cases
- Density independent population growth.
- So,
. Exponential growth. and as .- For R = 1, b=d and
as .
R linearly reducing with population
- Model:
for , leading to . Taking (aka rescaled population to get “population density”), this can be simplified to for - Logistic population growth. Population converges to some finite value.
- Equilibria: \(x_{t+1} = x_t = x^\). So \(x^= R_0(1-x^*)\)
for all for all- So, 1 is a bifurcation point, where stability of the population changes.
shows oscillations about the equilibrium - Entirely because of internal dynamics of the population.
R exponentially decays
Model: