Problem on cubic population model

Question

Assignment:

Consider the cubic population model
dN/dt=cN(N-κ)(1-N)
Where c and κ are given constants, such that c>0 and 0<κ<1. Note that the population N is measured in units of thousands.

Find and classify the steady states N₁23 of this equation.

If the initial population is N₀ describe, without proof, the future of the population, distinguishing various cases on the size of N₀ relative to the steady states N₁,N₂ and N₃.

Obtain the solution of the differential equation with initial condition N(0)  =2, assuming the parameters c=1 and κ=1/2.
Hints: If you encounter a quartic equation for N, consider the substitution K=N-1/2. For choosing the right signs in extracting roots, you should assume N≥1.

Provide complete and step by step solution for the question and show calculations and use formulas.