I am working on a problem from economics, to understand how populations evolve within business organizations. I have found that some of the population dynamics literature is very relevant to my own work, but I am not very familiar with the vast biology and ecology literatures on this topic. I have been reading Murray's Mathematical Biology volume 1 bible, and it is really great--but it only gives a taste of the models out there.

I was hoping someone could help point me towards articles or models that deal with multi-generational evolution of a population with time delays. I have having a little trouble deciphering some of the biology or ecology terminology, so I am not sure if my search terms are correct--I have not been able to find anything like this, though I am certain it exists.

I am looking at a very simple model, something on the order of a system of differential equations with delays. So a simplified examples is below with just one species that matures over time. So for the same species, there are 3 populations, A, B, C, which represent stages in its lifecycle. I can refer to these in the model as $N_A, N_B, N_C$. Then there is a birth and death rate for each population, as well as a migration rate between the different populations--as individuals mature. I am thinking of the populations in continuous time, though of course suggestions for articles about discrete dynamics for such systems would also be valuable.

So the continuous system would be as below:

$$ \frac{dN_A}{dt} = \beta N_C - \gamma_A N_A - \nu_A N_A(t - \tau_A) \\ \frac{dN_B}{dt} = \nu_A N_A(t - \tau_A) - \gamma_B N_B - \nu_B N_B(t - \tau_B) \\ \frac{dN_C}{dt} = \nu_B N_B(t - \tau_B) - \gamma_C N_C \\ $$

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.

Like I said, it is easy enough to express this model in a discrete form as well. But I was hoping someone might be able to reference articles or something that shows someone analyzing discete multi-generational models--so that I can see a flavor of how they deal with the nondimensionalization, bifurcations, chaotic dynamics, etc.

I was hoping to see if other biologists or ecologists had studied this kind of continuous system, and how they had analyzed it. It seems like the first thing to do is nondimensionalize it, as I have way too many parameters. But I am not sure of a good scheme or equilibrium condition to use to do that.

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    $\begingroup$ "evolutionary game theory" plus "delay differential equation" (or "discrete lag") would be a good search terms to start with $\endgroup$
    – Ben Bolker
    Jul 2, 2022 at 22:23
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    $\begingroup$ e.g. scholar.google.ca/… . Can you say more specifically what kind of qualitative conclusions you want to draw (invasibility, evolutionary stable states, etc.) ? $\endgroup$
    – Ben Bolker
    Jul 2, 2022 at 22:31
  • $\begingroup$ @BenBolker Okay, I can start to search under these headings. I am seeing a lot of gene expression articles come up--can you tell me if gene expression models would apply here? Thanks for suggestions, at least I know I am starting to look in the correct places. $\endgroup$
    – krishnab
    Jul 3, 2022 at 3:38
  • $\begingroup$ @BenBolker good question about what I am looking for. So I would like to understand if there are changes in the qualitative behavior of this system as I vary the parameters or move in and out of different parameter regimes. But right now there are like 9 parameters in the model. So I don't believe that I can explore the dynamics or find bifurcations without non-dimensionalizing the system or such to reduce the number of parameters. But yes, I would like to see if there are fixed points in the model, find bifurcations, any potential for chaos, etc. $\endgroup$
    – krishnab
    Jul 3, 2022 at 3:57
  • $\begingroup$ Did you leave out a delay term in the final equation, i.e. should the first term on the RHS be $\nu_B N_B(t-\tau_B)$ ? $\endgroup$
    – Ben Bolker
    Jul 5, 2022 at 22:47

1 Answer 1


You use the word "evolution" here to mean "change in populations over time", as is typical in (say) astrophysics (or, I guess, economics?). My initial comments and suggestions were based on a misreading of "evolution" having its biological meaning, i.e. change of genetic make-up of a population over time.

With that out of the way: delay differential equations are slightly less common than ordinary DEs in population biology, but still very common. I wouldn't say there are any special mathematical tricks that aren't also used by mathematical analysts or people studying dynamics in other fields ...

  • What you have appears to be a linear DDE, which is considerably simpler than the typical biological DDE, which will almost always have at least one bilinear term representing a species interaction (and often a stronger nonlinearity representing something like a saturating predator-prey functional response, see e.g. Chattopadhyay et al ref below). To be honest, I don't think that this system is going to admit any very bizarre behaviour, because of that linearity. The model is simple enough that it's possible that someone has actually analyzed this case ...
  • Fixed point analyses for DDEs are actually relatively simple, because when the system is at the fixed point the time delays are irrelevant (e.g. $N_A^* = N_A(t) = N_A(t-\tau_A)$) so your equilibrium equations simplify to (stars suppressed for laziness)

$$ \begin{split} 0 & = \beta N_C - (\gamma_A + \nu_A) N_A \\ 0 & = \nu_A N_A - (\gamma_B + \nu_B) N_B \\ 0 & = \nu_B N_B - \gamma_C N_C \end{split} $$

which is a pretty simple linear system, general solution shouldn't be bad at all. Stability of DDEs is harder, you'd have to look in a book ...

  • For bifurcation analyses, I think that standard numerical bifurcation analysis methods (continuation methods) can in principle handle DDEs as well, although to be honest I haven't checked them out -- some very basic notes here

  • You're right that a 9-dimensional parameter space is going to be hard to handle; in general you will only be able to get rid of two parameters by non-dimensionalizing (one for space/density and one for time). For the rest, you'll have to use the standard kinds of simplifications, e.g.

  • plausible symmetries/special cases

  • search the whole space over ranges that are physically/economically/biologically plausible

Here are a couple of fairly haphazardly chosen biological dynamics papers with DDEs:

Bonsall, M. B., S. M. Sait, and R. S. Hails. “Invasion and Dynamics of Covert Infection Strategies in Structured Insect-Pathogen Populations.” Journal of Animal Ecology 74, no. 3 (2005): 464–74.

Chattopadhyay, J., R. R. Sarkar, and A. el Abdllaoui. “A Delay Differential Equation Model on Harmful Algal Blooms in the Presence of Toxic Substances.” Mathematical Medicine and Biology: A Journal of the IMA 19, no. 2 (June 2002): 137–61. https://doi.org/10.1093/imammb/19.2.137.

  • $\begingroup$ Yes, thank you so much for the information it was very helpful. You certainly validated some of my intuitions, and also directed me towards the right places to look. I will read through the references you suggested. I will also think a bit more about the bilinear terms you mentioned. That is something that is still an open question and not yet resolved--but I understand what you mean, because this bilinear term occurs in the Lokta-Volterra equation or SIR models. $\endgroup$
    – krishnab
    Jul 11, 2022 at 15:41
  • $\begingroup$ Also, the nondimensionalization information was really helpful. Many books emphasize those methods, but they seem to only work for small systems with a handful of parameters. I have had trouble finding any good references or treatments of how to reduce the dimension of the parameters. Like you said, I have seen cases where people use some conservation law or equilibrium condition for this, but have never found a good resource explaining the intuitions and ideas. I will check with a dynamicist friend, but if you have a suggestion, please pass it along. $\endgroup$
    – krishnab
    Jul 11, 2022 at 15:46

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