Consider a population dynamics with n subpopulations:
dXi(t)/dt=rixi, i=1,2,...,n.
in which xi(t) is the population size of the ith subpopulation, constant ri is its per capita growth rate. let xi (1 ≤i ≤n) present the fraction of the subpopulation xi among the total population:
(a) Derive the corresponding system of differential equations for x(t)
(b) Show that
n
Σ (dx,(t)/dt)=0
i=1
and
n
d/dt(Σ rixi) ≥ 0
i=1
