Consider the following deterministic models for the mean (average) development of the size of populations:

(1) Let \(m(t)\) be the mean number of individuals of a population at time \(t\). It is reasonable to assume that a change of the population size, namely \(d m(t) / d t\), is proportional to \(m(t), t \geq 0\), i.e., for a constant \(h\) the mean number \(m(t)\) satisfies the differential equation

\[\frac{d m(t)}{d t}=h m(t)\]

a) Solve this differential equation assuming \(m(0)=1\).

b) Is there a birth and death process the trend function of which has the functional structure of \(m(t)\) ?

(2) The mean population size \(m(t)\) satisfies the differential equation \[\frac{d m(t)}{d t}=\lambda-\mu m(t)\]

a) With a positive integer \(N\), solve this equation under the initial condition \[m(0)=N\]

b) Is there a birth and death process the trend function of which has the functional structure of \(m(t)\) ?