Analysing a dynamical system

Question

I am developing a model and my model equations are

${dA\over dt}=r_1A(1-{A+B\over k})-a_1 AE-m_1{AN\over (xN+A)}-m_4A$

${dB\over dt}=r_2B(1-{A+B\over k})+a_1 AE-m_2{BN\over (xN+B)}$

${dC\over dt}=a_2AE-m_eC$

${dE\over dt}=\gamma P -a_2AE-\gamma E$

${dN\over dt}= \alpha (R-N)(A+B)-m_3N$

Where P, R are constant.

I want to see what parameters are related with the initial exponential increase seen in the model. For this I thought of using a method similar to the question in where $r_0$ is the initial exponential growth rate and I am trying to come up with a relation similar to what was obtained in part 2 in that question. For this I linearized the above system. But, my question is if I set $f= r_1A(1-{A+B\over k})-a_1 AE-m_1{AN\over (xN+A)}-m_4A $ when linearizing I get terms such as $\frac{\partial f}{\partial A}=…..-d_1{xN^2\over {(xN+A)}^2}$

Therefore, can I linearize the above system and can I use a method similar to that used in the the other question in order to obtain a relationship between the parameters and the exponential growth rate.

2. In this system I want to analyse whether there are linked parameters, that is if I increase one parameter there may be another parameter that I have to decrease in order to maintain the same behaviour. This I want to do because for some parameters in the long run the model starts oscillating and for some it decreases to zero. I want to analyse which parameter values or ranges give these different behaviours.
   The different behaviours of the model are
   
   

3. Also, I tried to find the equilibrium solutions to the system, but Mathematica couldn’t come up with a solution for a long time. Why was it? Shouldn’t it be able to at least come up with the equilibrium point (0,0,0, P,0)

Solve [ { r1 * a [ t ] * ( 1 - ( ( ( a [ t ] + b [ t ] ) ) / k ) ) - a1 * a [ t ] * E1 [ t ] - m1 * a [ t ] * n [ t ] / ( ( x * n [ t ] + a [ t ] ) ) - m4 * a [ t ] ⩵ 0 , r2 * b [ t ] * ( 1 - ( ( ( a [ t ] + b [ t ] ) ) / k ) ) + a1 * a [ t ] * E1 [ t ] - m2 * b [ t ] * n [ t ] / ( ( x * n [ t ] + b [ t ] ) ) ⩵ 0 , a2 * a [ t ] * E1 [ t ] - de * C [ t ] ⩵ 0 , gamma * P - a2 * a [ t ] * E1 [ t ] - gamma * E1 [ t ] ⩵ 0 , alpha * ( R - n [ t ] ) * ( a [ t ] + b [ t ] ) - m3 * n [ t ] ⩵ 0 } , { a [ t ] , b [ t ] , C [ t ] , E1 [ t ] , n [ t ] } ]