# Is Insulin-Glucose dynamic Lotka-Volterra?

From Wikipedia:

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

\begin{align} \frac{dx}{dt} = \alpha x - \beta x y \\ \frac{dy}{dt} = \delta x y - \gamma y \end{align} This looks very similar to how Insulin and Glucose interact with each other in the body. Glucose uptake release insulin and glucagon offsets the effect of insulin through glycogenesis.

Can the Glucose-Insulin dynamic be described as Lotka Volterra?

• Is there a model for glucose-insulin relationship. I guess there may be several. Lotka-Volterra is basically a negative feedback model and there are several examples of negative feedbacks (they are not referred to as Lotka-Volterra because it is model for a specific case and not a modelling strategy). You should clarify your question. Without looking at the model no conclusions can be made. – WYSIWYG Jul 22 '15 at 8:57

Is the standard Lotka-Volterra (LV) model an exact fit for insulin-glucose (IG) dynamics? No. Can a similar model built on the same principles capture most of the essential features of the IG dynamics? Absolutely.

# How to capture most of the insulin-glucose dynamics using a slightly modified Lotka-Volterra model

We can figure out how to change the LV equations to fit the IG dynamics by figuring out how our assumptions have changed. Like Daan mentioned, neither insulin nor glucose undergo self-reproduction. So we'll drop those terms from the equation, and we'll represent an influx of glucose (as from, say, a meal) as a simple time-dependent linear spike. Your rate equations will now look like:

$\frac{dx}{dt} = \alpha[t_{gi} < t< t_{gf}]-\beta x y$

$\frac{dy}{dt} = \delta x-\gamma x y$

where $x$ is glucose concentration, $y$ is insulin concentration, and $[t_{gi} < t< t_{gf}]$ is equal to 1 if the current time $t$ is greater than the time when the glucose spike starts $t_{gi}$ and less than the time when the glucose spike ends $t_{gf}$, and is equal to zero if the time $t$ does not meet this condition.

## Simulation of insulin-glucose dynamics using the above LV model

I ran a simulation of the above model in Matlab, and here's what it looks like: The basic features of the insulin-glucose dynamics from the example graph the OP posted are present. Just like in the real system, first the glucose spikes, then the insulin spikes after a short lag, then they both return to baseline.

# With some small improvements, the LV model can capture the dynamics of the infamous "sugar crash" phenomenon

Now for the really cool part.

As can be seen on the OP's graph, glucose levels tend to fall somewhat below their baseline following a large glucose spike. This is the physiological basis for the sugar crash that people sometimes get a little while after eating something really sugary like a candy bar. Regardless of whether the psychological symptoms actually occur, the dip in glucose levels is a real thing.

In order for a model to be considered "good", it has to be able to capture these kinds of complex but important phenomena in their underlying systems. The glucose dip did not show up in my original LV model, so the question is, can the model be modified in a simple way so that it will capture the dip? Let's take another crack at it!

Part of the problem with trying to study phenomena related to baseline levels using the previous model is that the baseline levels of both insulin and glucose are effectively fixed at 0 in that model. We can set the baseline levels by adding what's called a zeroth-order production term (essentially a fixed constant that isn't multiplied by the concentration of anything) to each of our rate equations. The rate equations now look like:

$\frac{dx}{dt} = \alpha[t_{gi} < t< t_{gf}]+\epsilon-\beta x y$

$\frac{dy}{dt} = \delta x+\eta-\gamma x y$

where $\epsilon$ is the 0th order glucose production term, $\eta$ is the 0th order insulin production term, and everything else is exactly the same.

## Simulation of insulin-glucose dynamics using the "dip-enabled" LV model

Again using Matlab, I ran a simulation of the "dip-enabled" LV model: and this time the dip shows up.

### Notes:

-The term with the times in it is called an Iverson Bracket.

-You could make this model more physiologically relevant by using a more complex function for the glucose spike, but for simplicity's sake we'll stick with a linear function. A smoother function, like for example a Hill function, would probably help to reproduce your original graph of the IG dynamics.

### Matlab code

function dxdt = insulin_glucose_birth_death_dxdt(t, x, kip, bip, kid, kgp, bgp, kgd)
dxdt = [kip*x(2) + bip - kid*x(1),
kgp*(t>1 && t<2) + bgp - kgd*x(1)*x(2)];
end

function [t,x] = insulin_glucose_birth_death_odesol(tspan, x0, kip, bip, kid, kgp, bgp, kgd)
opts=odeset();
[t,x]=ode45(@(t,x)insulin_glucose_birth_death_dxdt(t, x, kip, bip, kid, kgp, bgp, kgd), tspan, x0, opts);
end

function [t,x] = insulin_glucose_birth_death_plot(tspan, x0)
%run with:
%   tspan = [-10, 5];
%   x0 = [0,0];
%   [t,x] = insulin_glucose_birth_death_plot(tspan,x0);

kip = 2;
bip = 0;  % set to .01 for the "dip" simulation
kid = 1;
kgp = .1;
bgp = 0;   % set to .1 for the "dip" simulation
kgd = 50;

[t,x] = insulin_glucose_birth_death_odesol(tspan, x0, kip, bip, kid, kgp, bgp, kgd);

figure(gcf)
hold on
plot(t, x(:,1), 'b', t, x(:,2), 'r')
legend('insulin','glucose')
xlabel('time')
ylabel('concentration')
axis([0 inf -inf inf])
hold off
end

• You need not include the code. That is not really necessary. Instead you can explain the mathematical model better. – WYSIWYG Jul 25 '15 at 5:32

Interpreting your question as "would the Lotka-Volterra predator-prey model be a good model for the glucose-insulin system?" my answer is "no". The predator-prey equations capture assumptions about how prey and predator interact with each other, and how they would fare on their own. These assumptions are not equivalent to any reasonable assumptions about the glucose-insulin system. The fact that both prey and predator population sizes as well as glucose and insulin concentrations oscillate is not enough to say the two systems may be modelled by the same equations. In fact, the oscillations in the figures you show are very different form each other.

Let us go through some of the assumptions of the Lotka-Volterra predator-prey model to see why most of them do not resemble anything related to the glucose-insulin system.

1) In the absence of predators, the prey population grows exponentially without limits. This follows from self-reproduction. If glucose were likened to prey, glucose would increase exponentially in the absence of insulin. If prey were likened to insulin, insulin would increase exponentially in the absence of glucose. Is that reasonable?

2) The population growth rate of the prey is reduced by a quantity proportional to the population sizes of both prey and predator. The underlying assumption is that prey and predator meet a rate proportional to their population densities and that a fraction of these encounters leads to the death of the prey. The assumption about the encounter rate is called the Law of Mass-Action, and applies well to chemical compounds in a liquid, like glucose and insulin. However, glucose and insulin do not interact directly, and it is not clear to me how realistic assumptions abut their interaction would work out.

3) New predators are born at a rate proportional to the encounter rate in 2). In the glucose-insulin system this would mean that more insulin is produced if more insulin is present. Is that realistic?

The answer to these questions is, according to me, "no".

(On a side note: the Lotka-Volterra model is structurally unstable, meaning that slightest change in the assumptions would fundamentally alter the outcome of the model. One of these outcomes are neutrally stable predator-prey cycles, that is, cycles of which the amplitude and period depend on the initial population sizes. Changing the assumption that the prey population grows exponentially by including a term - $m \times x^2$ to the first equation changes the oscillations into damped oscillations so that the populations settle at a stable equilibrium.)

No, the Lotka-Volterra model is a description of predator-prey dynamics. Although in some respects over-simplifying it is well suited for educating population dynamics and basic research.

The physiology of carbohydrate homeostasis is different. Although, similar to population count in the Lotka-Volterra model, the elimination of both glucose and insulin depend on their respective concentrations (which represents a very basic form of autoregulation), their interdependence is different. Here, we have to consider distribution processes, receptor binding, enzyme kinetics etc.

Numerous mathematical and cybernetic models have attempted to describe glucose-insulin homeostasis. The following review articles provide a good overview of the topic:

1. Ajmera I, Swat M, Laibe C, Novère NL, Chelliah V. The impact of mathematical modeling on the understanding of diabetes and related complications. CPT Pharmacometrics Syst Pharmacol. 2013 Jul 10;2:e54. doi 10.1038/psp.2013.30. PMID 23842097; PMCID PMC3731829.

2. Palumbo P, Ditlevsen S, Bertuzzi A, De Gaetano A. Mathematical modeling of the glucose-insulin system: a review. Math Biosci. 2013 Aug;244(2):69-81. doi 10.1016/j.mbs.2013.05.006. Epub 2013 Jun 1. Review. PMID 23733079.

There is an important difference in the dynamics of the 2 situations.

The Lotka-Volterra model undergoes repeated oscillations in predator and prey levels. Changes in one population affect the changes in the other population, and vice versa.

In the glucose-control model, what you have plotted are the changes in glucose and insulin levels following a meal or infusion of glucose. Note that there is no subsequent peak of glucose after the insulin levels have dropped, unlike the peaks in prey levels after predator levels have dropped in the Lotka-Volterra model.

• Negative feedbacks can lose oscillations if certain parameters are changed. This is called the Hopf bifurcation. Just because a system ceases to oscillate doesn't mean that the model structure has changed. Anyways, since the OP has not really shown the mathematical equations underlying the glucose-insulin plot, nothing can be said. I am surprised why this question is not closed yet. – WYSIWYG Jul 25 '15 at 5:36