Introduction
Let us consider the classic Lotka-Volterra predator-prey model with two variables: $x(t)$, the prey population size over time, and $y(t)$, the predator population size over time.
Population changes over time depend on:
- the prey reproduction rate, $\alpha$
- the prey mortality rate due to predators, $\beta$
- the predator production rate based on prey consumed, $\delta$
- the predator mortality rate, $\gamma$.
Lotka and Volterra independently proposed the following equations:
$$\left\{\begin{array}{l}x'=x(\alpha-\beta y)\\ y'=y(\delta x-\gamma)\end{array}\right.$$
Modeling a viability problem for the prey
In their article
Térence Bayen, Alain Rapaport. Minimal time crisis versus minimum time to reach a viability kernel: a case study in the prey-predator model. Optimal Control Applications and Methods, 2019, 40 (2), pp.330-350. ⟨10.1002/oca.2484⟩. ⟨hal-01943636v2⟩
the authors addressed the problem of preserving the prey population in the face of predators by keeping their numbers above a given threshold $\underline{x} > 0$ as much as possible. This amounts to ensuring that the state remains within the set
$$K(\underline{x}):=\{(x,y)\,|\, x>=\underline{x}\},$$
based on the premise that low prey density exposes the population to a risk of extinction that should be avoided as much as possible.
Using the original Lotka-Volterra equations, the authors assume constant values for $\alpha=r$, $\beta=1$, and $\delta=1$. However, a control can influence predator mortality; thus, the authors set $\gamma = m + u$, where $m$ is a constant and $u$ is an excess mortality rate that can vary, with $u(t) \in [0, \bar{u}]$.
The viability problem can be summarized as follows:
\begin{equation}
\left\{
\begin{array}{l}
x'=x(r- y)\\
y'=y( x-m-u)\\
u(t) \in [0;\bar{u}]\\
\left(x(t),y(t)\right) \in K(\underline{x}):=\{(x,y)\,|\, x>=\underline{x}\}
\end{array}
\right.
\end{equation}
Solving this problem means finding the set of states $(x,y)$ from which it is possible to maintain a prey population level above $\underline{x}$ over time by manipulating prey excess mortality—in other words, the viability kernel, using the terminology of viability theory.
| Dynamics |
Controls |
Uncertainties |
Constraints |
Target |
Viability concept |
|
Continuous time
Continuous in 2D space
\begin{equation}
\left\{
\begin{array}{l}
x'=x(r- y)\\
y'=y( x-m-u)\\
\end{array}
\right.
\end{equation}
Parameters : $r, m$
|
$u\in U=\left[ 0,\bar{u}\right]$
Parameters : $\bar{u}$
|
None |
$\{(x,y)\,|\, x>=\underline{x}\}$
Parameters : $\underline{x}$
|
None |
Viability kernel |
Results
The authors provide an analytical description of this viability kernel. When $\bar{u}\geq \underline{x}-m$, by considering the invariant of this Lotka-Volterra system (which depends on $u$), $W_u(x,y):=x-(m+u)\ln x+y -r\ln y$, they construct the boundary of the viability kernel using a closed curve composed of three subsets:
- $\{(x,y) | y\geq r \text{ and }W_{\bar{u}}(x,y)=W_{\bar{u}}(\underline{x}, r) \}$.
- Let $x^+>\underline{x}$ such that $W_{\bar{u}}(x^+,r)=W_{\bar{u}}(\underline{x}, r)$,
$\{(x,y) | y\leq r \text{ and }W_{0}(x,y)=W_{0}(x^+, r) \}$
- Let $r^-$ such that $W_{0}(\underline{x},r^-)=W_{0}(x^+, r)$,
$\{(x,y) | x=\underline{x} \text{ and } y\in [r^-,r]\}$
The following figure shows the result for the parameters: $r=1$, $m=1$, $\bar{u}=0.5$, $\underline{x}=0.8$.
Because an analytical description of its boundary exists, this viability problem allows for testing the precision of approximations generated by generic software.
Shown below is a comparison—for $r=1$, $m=1$, $\bar{u}=0.5$, and $\underline{x}=0.8$—between the viability kernel boundary derived from its analytical description (in red) and the approximate result obtained using the ViabLab software on a regular grid of 4001×4001 points (light blue set with a dark blue boundary):
It can be seen that the kernel approximation is well performed from the outside.
We provide below the files necessary to compute the above approximations thans to ViabLab software.
Please refer to training material :