Asymmetry Demystified: Strict CLFs and Feedbacks for Predator-Prey Interconnections

Abstract

The difficulty with control of population dynamics, besides the states being positive and the control having to also be positive, is the extreme difference in the dynamics near extinction and at overpopulated states. As hard as global stabilization is, even harder is finding CLFs that are strict, don't require LaSalle arguments, and permit quantification of convergence. Among the three canonical types of two-population dynamics (mutualism, which borders on trivial, predator-prey, and competition, which makes global stabilization with positive harvesting impossible), predator-prey is the ``sweet spot'' for the study of stabilization. Even when the predator-prey interaction is neutrally stable, global asymptotic stabilization with strict CLFs has proven very difficult, except by conservative, hard-to-gain-insight-from Matrosov-like techniques. In this little note we show directions for the design of clean, elegant, insight-bearing, majorization-free strict CLFs. They generalize the classical Volterra-style Lyapunov functions for population dynamics to non-separable Volterra-style constructions. As a bonus to strictification as an analysis activity, we provide examples of concurrent designs of feedback and CLFs, using customized versions of forwarding and backstepping (note that, in suitable coordinates, predator-prey is both strict-feedforward and strict-feedback), where the striking deviations from these methods' conventional forms is necessitated by the predator-prey's states and inputs needing to be kept positive.

Figures (7)

• Figure 1: $V(X,Y)$ in \ref{['clf1']}. Open-loop trajectories for the systems \ref{['pp-sf']} and \ref{['pp-both']} for $U=1$.
• Figure 2: $V(X,Y)$ in \ref{['clf1']}. Positive definite and radially unbounded. Blue dot: equilibrium at $X=Y=1$.
• Figure 3: The barrier Lyapunov functions $\Psi(S) = S-1-\ln(S)$ (blue) and $\Psi(1/S) = 1/S - 1+ \ln(S)$ (red). Both are highly asymmetric but the red dominates on the interval $(0,1)$ at the rate ($1/S$, whereas the blue dominates on $(1,\infty)$ at the rate $S$. The logarithmic growth is always weaker---both near $S=0$ and for large $S$.
• Figure 4: $V(X,Y)$ in \ref{['clf2']} for $\varepsilon=0.5$.
• Figure 5: Level sets of $V(X,Y)$ in \ref{['clf2']} for $\varepsilon=0.9$ and trajectories of the closed-loop system \ref{['pp-both-closed']}. The level sets of $V$ may only appear to be like the open-loop trajectories in Figure \ref{['fig:open-loop']} but a closer inspection reveals that they are definitely not.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof