This study develops and analyzes an eco-epidemiological predator–prey model describing the interaction between copepods and Atlantic horse mackerel (Trachurus trachurus) along the Moroccan coast. The predator population is stratified into susceptible and infected subclasses, with nonlinear prey refuge mechanisms and selective harvesting incorporated for each group. Predation follows a Holling type I functional response for each class, while disease transmission occurs through contact between susceptible and infected predators. Rigorous mathematical analysis establishes the existence, uniqueness, positivity, and boundedness of solutions, guaranteeing the biological well-posedness of the system. Stability analysis demonstrates that the disease-free equilibrium is globally asymptotically stable whenever <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R_0 < 1\)</EquationSource> </InlineEquation>, while a transcritical bifurcation at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R_0=1\)</EquationSource> </InlineEquation> gives rise to a stable endemic equilibrium. Disease persistence or elimination is shown to be critically governed by the transmission coefficient, prey refuge intensity, and harvesting pressures applied to each predator class. Hopf bifurcation analysis further reveals the emergence of sustained oscillatory dynamics under certain parameter regimes. To account for environmental stochasticity, the deterministic framework is extended to a stochastic differential equation system, for which the existence of a unique global positive solution and sufficient conditions for disease extinction are rigorously derived. Numerical simulations corroborate the theoretical results, demonstrating that harvesting strategies, prey refuge mechanisms, and stochastic perturbations collectively govern the long-term dynamics and resilience of marine predator–prey systems. The proposed harvesting and refuge strategies contribute to reducing infection prevalence while maintaining ecological balance, providing insights for sustainable and climate-resilient fisheries management under environmental variability. Graphic Abstract This graphical abstract summarizes the structure, analytical framework, and key findings of a deterministic–stochastic eco-epidemiological predator–prey model incorporating harvesting and refuge strategies, applied to marine populations along the Moroccan Mediterranean coast. The deterministic component analyzes the existence and stability of equilibria, including the disease-free equilibrium <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(P_6\)</EquationSource> </InlineEquation>, and characterizes infection dynamics through the basic reproduction number <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(R_0\)</EquationSource> </InlineEquation> and bifurcation analysis. Sensitivity and contour plots illustrate the influence of biological parameters and harvesting efforts on predator and prey densities. The stochastic extension introduces environmental noise via Itô stochastic differential equations, ensuring the existence and positivity of solutions while identifying noise-induced extinction and persistence conditions. Numerical simulations highlight the impact of environmental fluctuations on infected and susceptible predator populations. The graphical abstract emphasizes that optimized harvesting and refuge strategies contribute to ecosystem resilience and long-term population persistence, even under stochastic environmental perturbations. These results provide useful insights for sustainable marine resource management and eco-epidemiological modeling.

A Deterministic and Stochastic Eco-epidemiological Model for the Copepod–Atlantic Horse Mackerel System Along the Moroccan Coast: Implications for Sustainable Fisheries Management

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发表日期：

2026年5月13日

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Nossaiba Baba,
Mohamed Hafdane,
Yamna Achik,
Kuldeep Singh Rautela,
Rizwan Niaz,
Gamal El Afandi

摘要

This study develops and analyzes an eco-epidemiological predator–prey model describing the interaction between copepods and Atlantic horse mackerel (Trachurus trachurus) along the Moroccan coast. The predator population is stratified into susceptible and infected subclasses, with nonlinear prey refuge mechanisms and selective harvesting incorporated for each group. Predation follows a Holling type I functional response for each class, while disease transmission occurs through contact between susceptible and infected predators. Rigorous mathematical analysis establishes the existence, uniqueness, positivity, and boundedness of solutions, guaranteeing the biological well-posedness of the system. Stability analysis demonstrates that the disease-free equilibrium is globally asymptotically stable whenever \(R_0 < 1\) , while a transcritical bifurcation at \(R_0=1\) gives rise to a stable endemic equilibrium. Disease persistence or elimination is shown to be critically governed by the transmission coefficient, prey refuge intensity, and harvesting pressures applied to each predator class. Hopf bifurcation analysis further reveals the emergence of sustained oscillatory dynamics under certain parameter regimes. To account for environmental stochasticity, the deterministic framework is extended to a stochastic differential equation system, for which the existence of a unique global positive solution and sufficient conditions for disease extinction are rigorously derived. Numerical simulations corroborate the theoretical results, demonstrating that harvesting strategies, prey refuge mechanisms, and stochastic perturbations collectively govern the long-term dynamics and resilience of marine predator–prey systems. The proposed harvesting and refuge strategies contribute to reducing infection prevalence while maintaining ecological balance, providing insights for sustainable and climate-resilient fisheries management under environmental variability.

Graphic Abstract

This graphical abstract summarizes the structure, analytical framework, and key findings of a deterministic–stochastic eco-epidemiological predator–prey model incorporating harvesting and refuge strategies, applied to marine populations along the Moroccan Mediterranean coast. The deterministic component analyzes the existence and stability of equilibria, including the disease-free equilibrium \(P_6\) , and characterizes infection dynamics through the basic reproduction number \(R_0\) and bifurcation analysis. Sensitivity and contour plots illustrate the influence of biological parameters and harvesting efforts on predator and prey densities. The stochastic extension introduces environmental noise via Itô stochastic differential equations, ensuring the existence and positivity of solutions while identifying noise-induced extinction and persistence conditions. Numerical simulations highlight the impact of environmental fluctuations on infected and susceptible predator populations. The graphical abstract emphasizes that optimized harvesting and refuge strategies contribute to ecosystem resilience and long-term population persistence, even under stochastic environmental perturbations. These results provide useful insights for sustainable marine resource management and eco-epidemiological modeling.