Stabilization of Predator-Prey Age-Structured Hyperbolic PDE when Harvesting both Species is Inevitable

TL;DR

The paper tackles stabilization of age-structured predator–prey dynamics modeled as nonlinear IPDEs with a common harvesting input $u(t)$, facing positivity and saturation constraints. It converts the IPDE into a two-dimensional ODE driven by two IDEs via a state transformation, and develops two control designs: (i) a simple CLF-based feedback $u=u^*+\beta(\phi_1(\eta_1)+(1+\varepsilon)\phi_2(\eta_2))$ that globally stabilizes the ODE; (ii) a saturated positive-dilution controller $u=u^*+\varepsilon\phi_2(\eta_2)+\beta\frac{\varphi(\eta)}{\sqrt{\delta^2+(\min(0,\varphi(\eta)))^2}}$ ensuring $u(t)>0$. The analysis yields global ODE stabilization with potential regional stability for the IPDE under IDE disturbances, and explicit regions of attraction are derived for the IPDE. Simulations with biologically inspired kernels illustrate the two controllers: the unrestricted design can yield negative harvesting, while the positive-dilution design maintains positivity and yields robust stabilization under initial predator–prey imbalances. The work extends control of infinite-dimensional positive systems to multi-species age-structured dynamics and provides a framework for positive-input feedback in complex biological settings.

Abstract

Populations do not only interact over time but also age over time. It is therefore common to model them as age-structured PDEs, where age is the space variable. Since the models also involve integrals over age, both in the birth process and in the interaction among species, they are in fact integro-partial differential equations (IPDEs) with positive states. To regulate the population densities to desired profiles, harvesting is used as input. But non-discriminating harvesting, where wanting to repress one species will inevitably repress the other species as well, the positivity restriction on the input (no insertion of population), and the multiplicative nature of harvesting, makes control challenging even for ODE versions of such dynamics, let alone for their IPDE versions on an infinite-dimensional nonnegative state space. We introduce a design for a benchmark version of such a problem: a two-population predator-prey setup. The model is equivalent to two coupled ordinary differential equations (ODEs), actuated by harvesting which must not drop below zero, and strongly disturbed by two autonomous but exponentially stable integral delay equations (IDEs). We develop two control designs. With a modified Volterra-like control Lyapunov function, we design a simple feedback which employs possibly negative harvesting for global stabilization of the ODE model, while guaranteeing regional regulation with positive harvesting. With a more sophisticated, restrained controller we achieve regulation for the ODE model globally, with positive harvesting. For the full IPDE model, with the IDE dynamics acting as large disturbances, for both the simple and saturated feedback laws we provide explicit estimates of the regions of attraction. The paper charts a new pathway for control designs for infinite-dimensional multi-species dynamics and for nonlinear positive systems with positive controls.

Figures (7)

• Figure 1: The prey $x_1$ and predator $x_2$ interact via the terms $g_i$. Each species is affected by mortality $\mu_i$ and new population can only enter the system through birth $k_i$. The dilution input $u$ has a repressive effect on both species: harvesting both species is inevitable and represents a challenge for stabilization.
• Figure 2: Periodic behavior of the population densities $x_i$ when $u=u^*$, along with the states of the controllable ODE system $\eta_1$ (), $\eta_2$ () and the autonomous but stable IDEs $\psi_1$ (), $\psi_2$ (). The parameter set (\ref{['eq:parameters']}) and ICs (\ref{['eq:ic-1']}) used are specified in Section \ref{['sec:simulations']}.
• Figure 3: Level sets of $V_1$ () for two different different choices of $\varepsilon$, $\beta$. with bounds $H_1$, $H_2$ () for $\eta_1$, $\eta_2$. The gray area is the set $\mathcal{D}$, namely, the set in which $\eta_1>-H_1$, $\eta_2<H_2$, and $u>0$. The largest level set of $V_1$ () is contained within the actual region of attraction of $\eta=0$, for the case $\psi=0$. The left plot indicates the results with the control parameters $\varepsilon$, $\beta$ as used in the simulation shown in Section \ref{['sec:simulations']}.
• Figure 4: Control A with initially underpopulated predator and overpopulated prey: Population densities $x_i$ from system (\ref{['eq:karafyllis_system']}) with steady-state $x_i^*(a)$ (), and transformed state variables $\eta_1$ () and $\eta_2$ () from representation (\ref{['eq:transformed-system']}) under control law (\ref{['eq:control_law']}), and with parameter set (\ref{['eq:parameters']}), ICs (\ref{['eq:ic-1']}), $\varepsilon=0.2$, $\beta=0.6$.
• Figure 5: Control B with initially underpopulated predator and overpopulated prey: Population densities $x_i$ from system (\ref{['eq:karafyllis_system']}) with steady-state $x_i^*(a)$ (), and transformed state variables $\eta_1$ () and $\eta_2$ () from representation (\ref{['eq:transformed-system']}) under control law (\ref{['eq:global-stabilization-controller']}), and with parameter set (\ref{['eq:parameters']}), ICs (\ref{['eq:ic-1']}), $\delta=0.2$, $\beta=0.13$, $\varepsilon=0.01$.

Theorems & Definitions (16)

• Lemma 1: Lotka-Sharpe condition sharpe1911problem
• Proposition 1: Equilibrium
• Remark 1
• proof : Proof of Proposition \ref{['prop:steady']}.
• Lemma 2
• proof
• Proposition 2: System Transformation
• proof : Proof of Proposition \ref{['prop:trafo']}.
• Theorem 1
• Theorem 2