# Population numbers in the presence of restricted resources

I wonder if the following simple system is considered in population dynamics, under which name, and in which textbooks.

Consider a population $$X$$ (also indicating its number of individuals) with a default birth rate $$\beta_0$$ and a default death rate $$\delta_0$$. Let the effective birth rate be given by

$$\beta = \beta_0\,\frac{\alpha}{\rho}$$

where $$\alpha$$ is the amount of resources available to each individual, and $$\rho$$ is the amount of resources needed by each indiv idual. We have $$\beta > \beta_0$$ when there are more resources available than needed, and $$\beta < \beta_0$$ when there are less resources available than needed.

For the sake of simplicity let's assume that the effective death rate is given by

$$\delta = \delta_0\,\frac{\rho}{\alpha},$$

i.e. the effective death rate decreases with increasing $$\alpha/\rho$$ and increases otherwise.

The equilibrium condition $$\dot{X} = 0$$ of the system $$\dot{X} = \beta\,X - \delta\,X$$ is given by

$$\frac{\beta_0}{\delta_0} = \frac{\rho^2}{\alpha^2}$$

With the final assumption of restricted but fully exploited resources, specified by

$$X\,\alpha = R = \text{const.}$$

we get the equilibrium condition

$$\frac{\beta_0}{\delta_0} = \frac{\rho^2}{R^2}X^2$$

Resolved with respect to $$X$$ we have the equilibrium population

$$$$X_0 = \frac{R}{\rho} \, \sqrt{\frac{\beta_0}{\delta_0}}$$$$

The next step would be to consider systems of populations $$X_i$$ competing for the same resources $$R$$, especially with different values of $$\alpha_i$$ (indicating different fitnesses in the struggle for resources), but also with different values of $$\rho_i$$ (indicating different efficiencies in utilizing the resources). But I'd like to take the first step first. Is it a sensible approach?

• Birth rate cannot increase exponentially with the number of avaliable resourses. Each birth requires some amount of resources, so it increases at best linearly. In reality, there is some limit on how much resources each specie can process, so the birth rate is limited. Commented Jan 1, 2022 at 14:55
• There are still some aspects of this that are unclear to me. Is $\alpha$ meant to be a constant valuable rather than a dynamical variable? That wouldn't make sense because then we'd just have exponential growth (and your constraint wouldn't make sense). If $\alpha$ is meant to be a dynamic variable then we have $\alpha \propto 1/X$ and we get something of the form (I think) $\dot X = c_1 - c_2 X^2$, or $d(\log X)/dt = c_1/X - c_2 X$. As @user31264 points out, the per capita birth rate diverges at $X=0$, which is weird. Commented Feb 21, 2022 at 20:00
• This is somewhat similar to a single-species version of ratio-dependent predator-prey dynamics (e.g. pubmed.ncbi.nlm.nih.gov/19083063 ), which has been argued about a lot "Since the ratio-dependent model always has difficult dynamics in the vicinity of the origin ..." Commented Feb 21, 2022 at 20:01
• @BenBolker: I consider the total amount of resources $R$ to be a constant, the amount of resources available to a single individual (of which there are $X(t)$ at time $t$) is $\alpha(t) = R/X(t)$. Commented Feb 22, 2022 at 8:05

If we express your equations in their simplest form (by substituting $$R/X(t)$$ for $$\alpha(t)$$) we get

$$\dot X = (\beta_0 R/\rho) - (\delta_0 \rho/R) X^2$$

or

$$(\dot X)/X = \dot{(\log X)} = \frac{(\beta_0 R/\rho)}{X} - (\delta_0 \rho/R) X$$

(assuming I didn't screw up the algebra — however, the important point here is the form of the dependence on $$X$$).

This means that the per capita reproductive rate diverges as $$X \to 0$$, which is weird/unbiological. That's not necessarily a deal-breaker — there are plenty of models that are pedagogically or practically useful (e.g. see Lehman et al 2020 for models where the population density diverges in finite time) — but it does ring an alarm bell.

An initially linear growth rate (i.e. constant absolute growth rate near 0) is more typical of abiotic resources that are being considered to be added to a system at a constant rate, as in a chemostat model.

This model reminds me of ratio-dependent predator-prey models, which have been extensively analyzed in the math-bio world (see this Google Scholar search), but much less used by ecologists. Abrams and Ginzburg (2000), whose co-authors are respectively a strong opponent and a strong proponent of ratio-dependent models, might be a good starting point in the ecological side of the literature.

Abrams, Peter A., and Lev R. Ginzburg. “The Nature of Predation: Prey Dependent, Ratio Dependent or Neither?” Trends in Ecology & Evolution 15, no. 8 (August 1, 2000): 337–41. https://doi.org/10.1016/S0169-5347(00)01908-X.

Lehman, Clarence, Shelby Loberg, Adam T. Clark, and Daniel Schmitter. “Unifying the Basic Models of Ecology to Be More Complete and Easier to Teach.” BioScience 70, no. 5 (May 1, 2020): 415–26. https://doi.org/10.1093/biosci/biaa013.