Mathematical models of colony population dynamics and hive placement

Animal pollinators have been supporting the lives of human beings on Earth. Bee pollinators are the biggest contributors to the pollination of crops, providing humans with food. Given such circumstances, this paper investigates the population of honeybee colonies and the processes of bee pollination. We constructed Honeybee Colony Population Model (BCPM) to predict the population of a honeybee colony over time. We first outlined the life cycle of a honeybee, including eggs, larval stage, pupal stage, and adult bee stage. Within the adult bee stage, bees transition back and forth between foragers and hive bees depending on the number of available resources and the workload of nursing tasks. By listing out factors that affect the population in each stage, we established equations representing the rate of change in each of the stages of a honeybee's life cycle, as well as an equation describing the change in resource storage. We also evaluated the death rate and the resources in each month of the year and calculated each group's typical maximum, minimum, and mean population in a honeybee colony: 3 years after the establishment of the colony, the total adult population follows a seasonal change with recurring patterns each year, giving a maximum of 100862 bees and a minimum of 35676 bees. The annual average population is found to be 64877 bees. We then conducted a sensitivity analysis on BCPM and found that the initial number of bee hives and the initial amount of available resources have the most significant impact on the population of the colony. We also observed an unusual pattern in the cross-analysis of the two factors and constructed Simplified Colony Collapse Disorder Model (SCCDM) to predict whether a colony will collapse using only one equation. In response to estimate the number of hives needed to support the pollination of a specific land area, we constructed Hive Deployment Model (HDM). We first divided the land into 20 nodes and then found the most appropriate locations to place the hives. After establishing the equations for movements between nodes per day per forager group, we developed an iterating algorithm to find the number of hives needed to pollinate crops on 20 acres of land. We collected data for 9 typical bee-pollinated plants and found the number of hives needed for each type of plant based on the algorithm, with blueberries being the most demanding, requiring 83 hives, whilst apples and roses only required 2 hives at the other end of the spectrum. Then, we established a sensitivity analysis to ensure the stability of the model by changing two arbitrary parameters. Finally, we discussed the potential advantages and disadvantages of our model. We have also created a non-technical blog that summarizes our investigation, presenting our results in a simplified way