# Analysis of the Influence of Broad Deaths on Honeybee Population
###### Abstract  
Many mathematical models using ordinary differential equations (ODEs) have been used to investigate what type of stressors cause honeybee colonies collapse. We propose a simple model of a delayed differential equation system (DDE) to describe the effect of insecticides over brood death rate and its influence over honeybee population dynamics. First, we remember some basic facts for the model with no delay. To analyze our model, we study the equilibria and perform stability and sensitivity analysis of the DDE system. Next, by using the delay time \(\tau\) as a bifurcation parameter, we find that no Hopf bifurcation could arise as the time lag \(\tau\) varies within biologically plausible ranges. Numerical simulations with real data are studied for the biological significance of the model. honeybee population; brood death; time delay; colony collapse disorder; extinction; bifurcation 20242020241141202411412024114  
Queens are central to the colony's reproduction and genetic diversity. A newly emerged virgin queen embarks on a series of orientation flights, followed by mating flights where she mates with multiple drones. This genetic diversity is crucial for the colony's resilience to diseases and environmental changes. After mating, the queen returns to the hive to lay eggs, ensuring the colony's continuity. Drones, while having a singular focus on reproduction, contribute significantly to the genetic health of the bee population. Their presence during the mating season ensures that queens mate with drones from different colonies, thus promoting genetic variation. The colony's dynamic lifecycle and division of labor are critical for its success. Worker bees' ability to adapt to different roles based on the colony's needs demonstrates their remarkable flexibility and the complex social structure of the hive. This adaptability is key to the colony's ability to thrive in varying environmental conditions. Understanding the sophisticated dynamics of honeybee colonies, including the roles and lifecycle of different castes, is fundamental for developing accurate models that predict their behavior and response to stressors. These models are essential for devising strategies to protect and sustain honeybee populations, which are vital for biodiversity and agriculture through their pollination activities. Honeybee populations typically follow a seasonal cycle influenced by the availability of resources [1]. In spring (April through June), the lengthening days and increased pollen supply lead to a birth rate that exceeds the death rate, causing the population to expand. The population peaks toward the end of June and stays elevated until mid-summer, after which it starts to decline. This downward trend continues into autumn and winter, reaching its lowest point in February. Regarding brood dynamics, the queen's egg-laying activity is minimal during late winter but rises sharply in spring to take advantage of the flowering season. In summer, the egg-laying rate remains elevated but slightly lower than spring levels. In autumn, certain plant species like ironweed bloom, triggering a second, smaller peak in egg-laying. After this brief increase, the queen's egg-laying rate drops to zero by December [2;3]. The seasonal fluctuations in honeybee populations are primarily driven by environmental factors and the availability of resources. In the spring, longer days and abundant pollen from blooming flowers provide optimal conditions for the queen to increase her egg-laying rate. This results in a surge in the number of brood and, subsequently, adult bees. The peak population during late June supports the hive's activities, including foraging and storing food for the coming months. As summer progresses, the initial abundance of resources begins to wane. The reduction in pollen and nectar sources causes the population growth to slow and eventually decline. By mid-summer, the hive experiences a gradual decrease in population as the death rate begins to exceed the birth rate.