In Lecture 2, we non-dimensionalized the logistic growth model to…

In Lecture 2, we non-dimensionalized the logistic growth model to obtain,
dn

= n(1 − n)
where n = N/K, the ratio of the population size N and carrying capacity K. And where τ = r0t,
the product of the basic reproduction number r0 and time t.
(a) Show that the solution of this differential equation is,
n(τ ) =
n0
n0 + (1 − n0)e−τ ,
(1)
where n0 is the initial population value. [7 marks]
(b) Sketch equation (1) when n0 = 0.5 and when n0 = 2. Interpret what your sketch tells us
about the long term behaviour of this model. [7 marks]

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