Optimizing Impulsive Releases: A Species Competition Model

This study focuses on optimizing species release $S_2$ to control species population $S_1$ through impulsive release strategies. We investigate the conditions required to remove species $S_1$, which is equivalent to the establishment of $S_2$. The research includes a theoretical analysis that examines the positivity, existence, and uniqueness of solutions, the conditions ensuring global stability, and a sufficient condition for controlling the $S_1$-free solution. In addition, we formulate an optimal control problem to maximize the effectiveness of $S_2$ releases, manage the population of $S_1$, and minimize the costs associated with this intervention strategy. Numerical simulations are conducted to validate the proposed theories and allow visualization of population dynamics under various release scenarios.

💡 Research Summary

This paper addresses the problem of eradicating a target species S₁ by periodically releasing a competing species S₂, using an impulsive differential‑equation framework and optimal control theory. The authors begin by adapting a continuous competition model originally proposed for wild and Wolbachia‑infected Aedes aegypti mosquitoes. The adapted model (equations 1a–1b) incorporates a frequency‑dependent Allee effect acting only on S₁, intrinsic growth rates r₁ = ψ₁ − δ₁ and r₂ = ψ₂ − δ₂, and carrying capacities K₁, K₂. The impulsive component adds a discrete jump to the S₂ population at times t = kτ, with release magnitude uₖ constrained by 0 ≤ uₖ ≤ u_max. Parameter assumptions ψ₁ > δ₁, ψ₂ > δ₂ together with ψ₂ < ψ₁, δ₂ > δ₁, r₂ < r₁ ensure that S₁ is intrinsically more robust than S₂, reflecting realistic biological scenarios where the wild population out‑competes the introduced strain.

Mathematical analysis proceeds in three stages. First, the authors establish well‑posedness: using the class V₀ of functions and the upper right‑hand derivative D⁺, they prove existence, uniqueness, positivity, and boundedness of solutions for any non‑negative initial condition and any admissible control sequence {uₖ}. A comparison theorem (Theorem 2) guarantees that trajectories cannot become negative and remain confined to a biologically meaningful region.

Second, equilibrium analysis reveals four biologically relevant steady states: a repelling node (K_b, 0) representing the Allee threshold for S₁, a saddle point (S₁*, S₂*) corresponding to unstable coexistence, and two attracting nodes (K_*, 0) and (0, K₂) representing dominance of S₁ or S₂ respectively. The authors delineate basins of attraction based on the initial S₁ level relative to K_b. The central theoretical contribution is a sufficient condition for global asymptotic stability of the S₁‑free solution (0, S₂). By constructing a Lyapunov function and applying the comparison principle, they derive an explicit inequality linking the release period τ, the minimum release amount u_min, and model parameters (K₀, K₁, K₂, r_i, δ_i). If each impulse satisfies uₖ ≥ u_min, the S₁ population is driven below the Allee threshold and inevitably collapses, while S₂ converges to its carrying capacity K₂.

Third, the paper formulates an optimal control problem aimed at minimizing the total number of released individuals while guaranteeing that S₁ falls below the Allee threshold by a prescribed final time T. The objective functional J = ∑_{k=0}^{N‑1} uₖ penalizes release effort; the terminal constraint S₁(T) ≤ K₀ enforces eradication. Existence of an optimal control is proved using standard variational arguments and the compactness of the admissible control set. Necessary optimality conditions are derived via Pontryagin’s Minimum Principle: the Hamiltonian incorporates the state dynamics, the co‑state (adjoint) equations evolve continuously between impulses, and jump conditions adjust the co‑states at each release time. These conditions lead to a bang‑bang‑type structure where optimal releases are either zero or at the maximal feasible level u_max, depending on the sign of the switching function.

Numerical simulations illustrate the theory. Parameter values are calibrated to the wild‑female/Wolbachia‑infected‑female mosquito system. In the first set of experiments, the authors choose τ = 7 days and uₖ = 4000 individuals, satisfying the derived sufficient condition. Simulations confirm that S₁ declines rapidly to extinction while S₂ stabilizes at K₂, validating the global stability result. In a second set, the optimal control algorithm is applied with a horizon T = 60 days. The computed optimal release schedule achieves S₁ ≤ K₀ well before T, while reducing the cumulative release effort by roughly 30 % compared with a constant‑rate strategy. Trajectory plots, control profiles, and cost evolution graphs are presented, demonstrating both the practical feasibility and economic advantage of the optimal impulsive strategy.

The paper concludes that integrating impulsive modeling, rigorous stability analysis, and optimal control yields a powerful framework for biological control programs based on competitive release. The sufficient condition provides a clear, implementable guideline for the minimal release effort required to guarantee eradication, while the optimal control formulation offers a systematic way to further reduce costs. Future work is suggested on spatial extensions, stochastic parameter variations, and multi‑species interactions, which would broaden the applicability of the proposed methodology to more complex ecological and epidemiological settings.