Stochastic hybrid delay population dynamics: well-posed models and extinction.

Area of Science:

• Ecology
• Mathematical Biology
• Dynamical Systems

Background:

• Nonlinear differential equations are foundational in population dynamics, modeling species fluctuations and interactions.
• Traditional models have evolved to incorporate delays, stochasticity, and discrete dynamics for greater realism.
• Stochastic hybrid delay population dynamics (SHDPD) represent a comprehensive framework integrating these advanced features.

Purpose of the Study:

• To investigate the mathematical properties of stochastic hybrid delay population dynamics (SHDPD).
• To establish conditions ensuring well-posedness and realistic behavior of SHDPD models.
• To determine criteria for predicting ecosystem extinction within SHDPD frameworks.

Main Methods:

• Analysis of nonlinear differential equations with stochastic and delay terms.
• Development of sufficient conditions for global positive, ultimately bounded solutions.
• Investigation of extinction criteria for SHDPD models.

Main Results:

• Sufficient conditions were derived for SHDPD to exhibit global positive, ultimately bounded solutions.
• These conditions are crucial for ensuring the biological relevance and mathematical soundness of population models.
• Specific criteria were established to predict the extinction of species within SHDPD-modeled ecosystems.

Conclusions:

• SHDPD offers a robust framework for modeling complex ecological dynamics.
• The derived conditions ensure the validity and interpretability of population models.
• The study provides critical insights into the factors driving ecosystem extinction.