I am looking at an application from population dynamics in biology and trying to understand the dynamics of system of delay differential equations below. The model is for a single species with a multi-generational evolution. That is, the member of this group will go through a maturation process over time, so there are transitions of individuals between the compartments of the delay differential equation system below.
The first step in understanding systems like this is often to non-dimensionalize the constants in the model, to reduce the number of constants to track. But I was not sure if I can actually non-dimensionalize the system below. Hence I would appreciate it if someone with more experience could tell me if there is a way to non-dimensionalize the system, or if there is a different method by which to understand the dynamics. I think that without some way to reduce the number of parameters, it is too difficult to understand the dynamics or behaviors in the system.
As I said, there is one species but each member of that species passes through a maturation process. The model below traces the evolution of the population over time. There are 3 population categories $A, B, C$, corresponding to each stage in maturation. Each population group has a birth, death, and migration process.
$$ \frac{dN_A}{dt} = \beta N_C - \gamma_A N_A - \nu_A N_A(t - \tau_A) \\ \frac{dN_B}{dt} = \nu_A N_A(t - \tau_A) - \gamma_B N_B - \nu_B N_B(t - \tau_B) \\ \frac{dN_C}{dt} = \nu_B N_B - \gamma_C N_C \\ $$
So constants $\beta, \gamma, \nu, \tau$ represent constants for the birth rate, death rate, migration-maturation rate, and time lag. In my actual model I have 2 species that compete, but this model above gives a flavor of the dynamics. I would like to include some carrying capacity constraints too, but that can come later.
I was looking in the book by Murray on Mathematical Biology volume 1, which describes delay differential equations, but not how to non-dimensionalize a system of delay differential equations.
I would appreciate it if anyone can suggest either how to non-dimensionalize the system, or how to understand the dynamics by somehow limiting the number of parameters, etc.
Thanks.