This thesis expands the classical framework by examining an equation with variable parameters and a time-varying carrying capacity, providing a more realistic representation of biological systems. This study investigates the oscillatory behavior and asymptotic properties of all solutions of Nicholson’s Blowflies equation for color flies with variable parameters and a variable carrying capacity. Unlike previous research that focused on constant parameters, this work addresses a more general and complex case where the parameters evolve over time. Since the inclusion of variable coefficients and carrying capacity in the Nicholson’s Blowflies equation leads to multiple equilibrium, it is necessary to determine an optimal equilibrium for calculations and data analysis. To find such an equilibrium, we assumed a constant value for the coefficient function and the carrying capacity function that lies within their corresponding ranges, so that the equilibrium is the desired equilibrium.

The work in this study consists of two types of Nicholson’s Blowflies equations: the differential delay Nicholson equation and the difference delay Nicholson equation. Conditions were found to ensure that all solutions oscillate around equilibrium and other results to ensure every solution converge to equilibrium for both types of Nicholson equation. These conditions are determined through a set of auxiliary problems, which prove effective in supporting the main results. In addition, illustrative examples are provided to illustrate how to choose fixed optimal values ​​for the functions to calculate the optimal equilibrium. The results in this thesis contribute to a broader understanding of delay equations (differential or difference) in insect community dynamics and provide a theoretical framework applicable to other time-delay systems in biology, ecology, and applied sciences.

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