Stochastic parameters in population growth equations

Question

I'm more of a mathematician than a biologist, so I had a question about an application of the population growth equation to real-life models, and thus would like to ask biologists for their insights. The equation I'm looking at is the following (in reality, I have n-species but it's just an extension of this two dimensional problem):

$$ \frac{dx_1}{dt} =x_1\Big(b_1-a_{11}x_1-a_{12}x_2 \Big) $$ $$ \frac{dx_2}{dt} =x_2\Big(b_2-a_{21}x_1-a_{22}x_2 \Big) $$

where the $b$ are growth parameters, and the $a$'s are all parameters describing interactions among the species (and carrying capacity). The papers I've looked at for stochastic perturbations with these equations have primarily dealt with stochastically perturbing either the $b$ parameters while holding the others constant, or perturbing the $a$ parameters while holding $b$'s constant. However, these papers are written by mathematicians with no biological experience, so the decision on which sets of parameters to perturb are somewhat arbitrarily decided without any biological basis.

For the biologists, can someone suggest some cases for where it would be more biologically reasonable to perturb the growth parameters and some cases were it would make more biological sense to perturb the interacting parameters? I ask because I'm a part of some interdisciplinary neuroscience project which is using equations of this form to study interacting neuron clusters, but since I'm a math person and not a biologist, I am trying to understand reasonable explanations for what stochastic perturbations mean biologically for these parameters.

Answer 1

Since you are asking for the biological interpretations about these parameters, it is important to realize that the model you are presenting is a non-dimensionalized version of this model:

$$ \frac{dx_1}{dt} =b_1x_1\Big(1- \Big(\frac{x_1 + \beta_{12}x_2}{K_1} \Big)\Big) $$ $$ \frac{dx_2}{dt} =b_2x_2\Big(1- \Big(\frac{x_2 + \beta_{21}x_1}{K_2} \Big)\Big), $$

which can also be written as:

$$ \frac{dx_1}{dt} =b_1x_1\Big(1- \frac{x_1}{K_1} - \beta_{12}\frac{x_2}{K_1}\Big) $$ $$ \frac{dx_2}{dt} =b_2x_2\Big(1- \frac{x_2}{K_2} - \beta_{21}\frac{x_1}{K_2}\Big). $$

This clearly shows how this competition model is merely the logistic Verhulst model, with an extra term attached. This extra term is simply describing to what extent each individual in "species" 2 is using up the resources of "species" 1 ($K_1$), where $\beta_{12}$ describes this relationship (~proportion of species 2 that is counted as species 1). The parameters in the model formulations above are also more easily interpreted in ecological/biological terms than the $a_{ij}$s in your model.

To go from this latter formulation to yours you substitute: $$a_{11} = b_1/K_1$$ $$a_{12} = \frac{b_1\beta_{12}}{K_1}$$ $$a_{22} = b_2/K_2$$ $$a_{21} = \frac{b_2\beta_{21}}{K_2}$$

As you can see, this means that the $a_{ij}$s are composites of other parameters, and perturbing $a_{ij}$s could correspond to either $b_i$, $K_i$ or $\beta_{ij}$ being perturbed. It also means that perturbing e.g. $b_1$ in your model while holding $a_{11}$ fixed means that you are actually also perturbing $K_1$ to offset the change in $b_1$ (since $a_{11} = b_1/K_1$), which will introduce a positive temporal correlation between $K_1$ and $b_1$. This is a strong assumption to make. This also means that $a_{ij}$s are not strictly "interaction parameters" as you call them, but they include other aspects as well.

Since realistic cases and scenarios for perturbations may often be expressed in terms of the parametrization I give above (e.g. random fluctuations in population growth rate or carrying capacity), these relationships are essential to keep in mind.

Answer 2

You just have 6 parameters in this model and you can easily perturb each of them individually while holding the others constant. This will let you know the individual effect of different parameters.

To assess the global sensitivity of the system to "environmental" perturbations, what you can do is to a random multivariate sampling of parameters. You will get different combinations of parameter values. Then you can plot the distribution of any metric that you intend to study.

However, if you intend to perturb a single parameter or a few parameters in combination, the choice of which ones to perturb depends on what you intend to study. System can, in general be insensitive to certain parameters. Local sensitivity analysis can be done to study the effect of parameter on the steady state. This you can do by taking a derivative of the expression for steady state with respect to the parameter. For a simple model such as:

$$\frac{dx}{dt}=x(b-ax)$$

local sensitivity of the steady state $x^*=b/a\ $ to $a$ would be $$\frac{dx^*}{da}=-\frac{b}{a^2}$$.

From here you can see that the effect of perturbation of $a$ on $x^*$ would be higher than that of $b$.

However local sensitivity analysis does not tell anything about dynamics and for complicated systems it may be difficult to guess which parameter would have the strongest effect.

For such cases you may do a random sampling and then run a classification/regression algorithm to determine which parameter had the strongest effect on your metric.

Another approach is to guess which parameters are likely to get perturbed in the natural course of events. This, you would know from prior knowledge of the system. If we assume steady environment, then the carrying capacity is not likely to change much. However, for a population of great phenotypic diversity, the growth rate may vary a lot.