Population Growth Model

Homework Statement

I'm trying to model how a population would grow wrt the following constraints:

1. Number of people within the population
2. How many people each person of the population informs
3. Probability of each person of the population informing a member outside the population.
4. There is an upper limit to the population growth, but this is not the most important constraint as of now.

It's almost like a virus growth model and I have some idea of where to begin, but it's been a while since I've solved these problems.

Homework Equations

Because of the first constraint, this looks like an exponential growth model to begin with. Thus,

$$\frac{dN}{dT}=kN$$

is the basic equation to be used.

The Attempt at a Solution

From the second constraint onwards, I'm having a problem forming the DE.

The growth is dependent on the probability of each person within the population telling others about it and the number of people each person will tell.

Thus, if $$\alpha (t)$$ is the probability of a person within the population telling others
and $$m(t)$$ are the number of people each person tells, then I think the model would look like this:

$$\frac{dN}{dT}=\alpha (t)m(t)N$$

Is this right? Again, it's been a while since I've done this stuff, so I'm not sure, but I think I'm on the right track.

For the final constraint, if I have to put an upper limit on the model, I think I can do a (k-N) growth model where k is the upper limit.

I believe that the model I'm trying to derive is akin to the Logistic Growth Model. However, I cannot understand the derivation of the same. Could somebody guide me through the process?

From what I understand, as there is an upper limit on the population here (k), the rate of growth is directly proportional to (k-N). However, the model is also proportional to the current population, N.

I do not understand why we multiply the two and not add them. Thus, why is the growth proportional to N(k-N) ?

Furthermore, the logistic growth model gives the following equation:

$$\frac{dN}{dt}=\frac{r}{k} N(k-N)$$

Why do we divide throughout by k? Also, what is r?

Can someone help me out with this? This is what I have so far:

Since the rate of growth is directly proportional to the current population, N and there is a cap on the total possible population K.

As each person reaches out to m other persons, the probability of telling someone who is not a part of the current population is given by $$\alpha k C m(1-\frac{N}{K})$$

Thus, if the acceptance rate is R, then the rate of growth should be:

$$\frac{dN}{dt}=R\alpha kCm(1-\frac{N}{k})N$$

Is this right?