Discrete-Time Optimal Control of Species Augmentation for Predator-Prey Model

TL;DR

This work develops two discrete-time optimal control models for species augmentation in predator–prey systems with strong Allee effects, comparing two distinct orders of events: growth–predation–augmentation (Model A) and augmentation–growth–predation (Model B). Model A is treated with a discrete Pontryagin framework yielding an explicit, bounded control law, while Model B requires sequential quadratic programming due to the nonconvex, cubic Hamiltonian. Numerical results show that the order of events materially affects final populations and objective values, with different patterns of augmentation timing yielding higher gains under varying parameter regimes. The findings underscore the importance of event sequencing in discrete-time augmentation strategies and provide a foundation for exploring additional orders and species scenarios in conservation planning.

Abstract

Species augmentation is one of the methods used to promote biodiversity and prevent endangered species loss and extinction. The current work applies discrete-time optimal control theory to two models of species augmentation for predator-prey relationships. In discrete-time models, the order in which events occur can give different qualitative results. Two models representing different orders of events of optimal augmentation timing are considered. In one model, the population grows and predator-prey action occurs before the translocation of reserve species for augmentation. In the second model, the augmentation happens first and is followed by growth and then predator-prey action. The reserve and target populations are subjected to strong Allee effects. The optimal augmentation models employed in this work aim to maximize the prey (target population) and reserve population at the final time and minimize the associated cost at each time step. Numerical simulations in the two models are conducted using the discrete version of the forward-backward sweep method and the sequential quadratic programming iterative method, respectively. The simulation results show different population levels in the two models under varying parameter scenarios. Objective functional values showing percentage increases with optimal controls are calculated for each simulation. Different optimal augmentation strategies for the two orders of events are discussed. This work represents the first optimal augmentation results for models incorporating the predator-prey relationship with discrete events.

Figures (10)

• Figure 1: Plots of the states with no control and plot with optimal control of the discrete augmentation Model A where the population is allowed to grow and then predator-prey action happens before augmentation at each time step using the baseline parameter values (\ref{['baselineeqn']}). The red dash-dot lines indicate the plots of the populations without augmentation, and the gray dotted lines indicate the Allee thresholds ($m = n = 0.25$) for the prey and the reserve populations, respectively. The corresponding optimal control and the objective functional values are: ${\bf h}^\ast = [0,0,0,0,0.04,0.30]$, $J(0) = 0.4413$ (with no control), and $J({\bf h}^{\ast}) = 0.4896$ (with optimal control).
• Figure 2: Plots of the states with no control and plot with optimal control of the discrete augmentation Model B where the population is augmented and then grows and then predator-prey action at each time step using the baseline parameter values (\ref{['baselineeqn']}). The red dash-dot lines indicate the plots of the populations without augmentation, and the gray dotted lines indicate the Allee thresholds ($m = n = 0.25$) for the prey and the reserve populations, respectively. The corresponding optimal control and the objective functional values are: ${\bf h}^\ast = [0,0, 0,0.05,0.08,0.15]$, $J(0) = 0.4413$ (with no control), $J({\bf h}^{\ast}) = 0.4825$ (with optimal control).
• Figure 3: Plots of the states and plot of the optimal control of the discrete augmentation Model A where the population is allowed to grow and then predator-prey action happens before augmentation at each time step using the baseline parameter values (\ref{['baselineeqn']}) except for $M_2 = 0$. The corresponding optimal control and the objective functional values are: ${\bf h}^\ast = [0,0,0.01,0.06,0.05,0.48]$, $J(0) = 0.4413$ (with no control), $J({\bf h}^{\ast}) = 0.5794$ (with optimal control).
• Figure 4: Plots of the states and plot of the optimal control of the discrete augmentation Model B where the population is augmented and then grows and then predator-prey action at each time step using the baseline parameter values (\ref{['baselineeqn']}) except for $M_2 = 0$. The corresponding optimal control and the objective functional values are: ${\bf h}^\ast = [0,0,0.05,0.08,0.11,0.21]$, $J(0) = 0.4413$ (with no control), $J({\bf h}^{\ast})= 0.5379$ (with optimal control).
• Figure 5: Plots of the states and plot of the optimal control of the discrete augmentation Model A where the population is allowed to grow and then predator-prey action happens before augmentation at each time step using the baseline parameter values (\ref{['baselineeqn']}) except for $M_2 = 0, N= 0.10$. The corresponding optimal control and the objective functional values are: ${\bf h}^\ast = [0, 0, 0.01, 0.06, 0, 0.9]$, $J(0) = 0.1215$ (with no control), $J({\bf h}^{\ast}) = 0.4662$ (with optimal control).

Theorems & Definitions (3)

• Theorem 3.1
• proof
• Remark 3.2