# On stochastic Lotka-Volterra predator-prey model

I am currently working (as a mathematician) on some estimations involving an stochastic predator-prey type model in which some of the coefficients have been perturbed by a Brownian Motion yielding to a system of stochastic differential equations.

The basic formulation of this problem can be found for instance in this article.

We have the following model

$$\begin{equation}\label{0} \begin{cases} dX(t)=X(t)\left(a-bX(t)-\frac{sY(t)}{\beta+Y(t)}\right)dt\\ X(0)=x>0,\\ dY(t)=Y(t)\left(\frac{h X(t)}{\beta+Y(t)}-c-fY(t)\right)dt\\ Y(0)=y>0. \end{cases} \end{equation}$$

where we use Holling II response function.

In the literature (mostly mathematics literature) the authors propose to perturb the birth rate of the preys $$a\mapsto a+\dot{B}_1(t)$$ and the mortality rate of predators $$c\mapsto c+\dot{B}_2(t)$$.

This yields to the following (stochastic) system

$$\begin{equation}\label{1} \begin{cases} dX(t)=X(t)\left(a-bX(t)-\frac{sY(t)}{\beta+Y(t)}\right)dt+\sigma_1 X(t)dB_1(t),\\ X(0)=x>0,\\ dY(t)=Y(t)\left(\frac{h X(t)}{\beta+Y(t)}-c-fY(t)\right)dt+\sigma_2 Y(t)dB_2(t),\\ Y(0)=y>0 \end{cases} \end{equation}$$

I know nothing about biology but I am concerned about a couple of things regarding this particular formulation:

1. If we assume that the Brownian motions are perturbing the birth/mortality rates, how can I interpret the fact that this new "perturbed" rates can become negative (due to the effect of the BM). Does it have any sense (in the framework of this particular model) to talk about "negative birth rate" or "negative mortality rate"?
2. What if the "perturbed" mortality rate goes beyond $$100 \%$$? Mathematically nothing "wrong" will happen, but what about the interpretation?

Notice that the "noise" and the model as a whole can be interpreted differently (ignoring the fact that we are actually modelling the dynamics of two species) but my main issue is that if I(together with many authors) am stating that we perturb a certain parameter I believe that we must respect the basic assumptions of the model!

I hope everything's clear and I thank you all in advance, any opinion or suggestion will be welcome!

• Have you considered to change how the noise scales with the population sizes? If done properly, this avoids negative population sizes and other unrealistic results and may be more realistic anyway. More specifically, if the noise is bounded by the square root of the abundance (for small abundances), everything should be fine. A typical example for a model that does this is the minimal market model. – Wrzlprmft Oct 9 '20 at 8:54

## 1 Answer

1. To interpret $$a$$ and $$c$$ as a birth and mortality rates is somewhat inaccurate, as $$a$$ and $$c$$, as written are density-independent rates that grow or shrink populations exponentially. If $$a$$ were negative, the prey population would shrink and most people would find a problem like that unexciting unless it was, say, a sink population maintained by immigration from another population. This is true of $$c$$ being negative, where the predator population would grow independent of the prey population. More accurately, $$a$$ can be interpreted as the difference between birth, $$b$$, and deaths, $$\delta$$; i.e., $$a = b - \delta$$. In this case, depending on the kinds of stochasticity being added to the model, negative $$a$$ is not just normal, but expected. The same is true for $$c$$. Ultimately, a perturbations like the white noise in this equation is simultaneously perturbing the difference between birth and death rates, which explains the negative difference.
2. The value of the rates seem to have the same interpretation if they are constant or are perturbed to an arbitrary magnitude.

I hope this helps!

• thank you very much, now it's clear that my understanding about the true meaning of the parameters was actually wrong. In case you care I leave you the link to the preprint of the article I mentioned in the question arxiv.org/abs/2009.14516 – Chaos Oct 8 '20 at 13:00
• You're welcome. It seems to be inconsequential to your work and tends to be prevalent in the mathematical literature, so it's not a big deal. The work looks interesting and I will study it later. Since you are working in 2D, I also wanted to pass along a method that is of growing interest to find the quasi-potential of the system to study stability and first-passage time: Nolting and Abbott 2015. (The supplemental material is where all the math is hidden.) – Chris Moore Oct 9 '20 at 15:27
• Ey Chris thanks for the reference, the approach seems very interesting and uses some techniques that I've never seen! I'll be reading it in detail since my idea was that of further studying the two dimensional case! – Chaos Oct 10 '20 at 10:21