Examples Lecture Three Plan 1)Finish sketching logistic growth 2)Talk about the pros and cons of logistic model 3)Introduce the method of integrating factors 4)(First principles in next Examples lecture, since the practical where we will need it is now later)

Effects of varying parameters Note that in the following slides (and in Fig 6.3 in your handout) we concentrate on a slightly more general version of logistic growth with allowance for a lower asymptote greater than zero

 varying Amount of disease Time (arbitrary units)  =.250

 varying Amount of disease Time (arbitrary units)  =.250  =.125

 varying Amount of disease Time (arbitrary units)  =.250  =.125  =.0625

 varying Amount of disease Time (arbitrary units)  = 30

 varying Amount of disease Time (arbitrary units)  = 30  = 20

 varying Amount of disease Time (arbitrary units)  = 30  = 20  = 10

 u varying Amount of disease Time (arbitrary units)  u = 85

 u varying Amount of disease Time (arbitrary units)  u = 85  u = 80

 u varying Amount of disease Time (arbitrary units)  u = 85  u = 80  u = 60

 l varying Amount of disease Time (arbitrary units)  l = 15

 l varying Amount of disease Time (arbitrary units)  l = 15  l = 10

 l varying Amount of disease Time (arbitrary units)  l = 15  l = 10  l = 5

Advantages and disadvantages of Logistic Growth Advantages and disadvantages of Logistic Growth 1) Simple 2) Includes density dependence 3) Can solve analytically to find Y(t) 4) Can be derived from plausible biological assumptions 5) Parameters mean something 5) Reflects sigmoidal pattern often seen in nature 6) Good starting point for more complex models 1) Arguably over-symmetric about t = , Y = K/2 2) Only linear density dependence 3) Constant parameters in time 4) No population structure (eg. age, sex,…) 5) No stochasticity 6) No spatial effects 7) Continuous reproduction 8) Instantaneous responses (i.e. no delay terms) PROSCONS

Deterministic skeleton (“logistic growth” Population size Time (arbitrary units)

Demographic stochasticity: one replicate Population size Time (arbitrary units)

Demographic stochasticity: many replicates Population size Time (arbitrary units)