Examples Lecture Three Plan Pros and cons of logistic model Checking the solution The method of integrating factors Differentiation from first principles (no longer required so will not cover)

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) PROS CONS

Solution of the logistic growth equation Population size Time (arbitrary units)

Stochasticity: one replicate Population size Time (arbitrary units)

Stochasticity: many replicates Population size Time (arbitrary units)

Problem with instantaneous responses One nice interpretation of logistic growth is as Note that dY/dt depends on population size Y at time t Easy to concoct examples where rate of births/deaths depends on popn some time ago, i.e. Y(t-  ), where  is a delay Gestation period Delay in effect of competition

Integrating Factors (Formula Sheet p7)

Differentiation from first principles What does it mean when we say that if y = x 2 then dy/dx = 2x? Really a statement about the slope of the tangent

The definition of the derivative of a function y(x) is in fact Gives a recipe for calculating the slope of the tangent in terms of a limit that can be calculated using the equation y(x) Differentiation from first principles INSERT SUITABLE PICTURE!!!

Differentiation from first principles:  x = 2

Differentiation from first principles:  x = 1

Differentiation from first principles:  x = 0.5

Differentiation from first principles:  x = 0.01