$1.$ Competition $-$ colonization models

A metapopulaion model for two species can be given by the system of nonlinear differential equations

dp$1$

dp$2$

dt$=$ c$1$p$1(1$ p$1)$ m$1$p$1$

dt$=$ c$2$p$2(1$ p$1$ p$2)$ m$2$p$2$ c$1$p$1$p$2$

where pi is the fraction of patches occupied by species i$,$ and ci and mi are the species specific rates

of colonization and extinction of patches, respectively. These equations assume that any patch has at

most one species, and species $2$ patches can be taken over by species $1,$ but not vice versa.

$($a$)$ Suppose that m$1=$ m$2=3,$ c$1=5,$ and c$2=30.$ Find all equilibria.