Author: Peyton Caroll
Addressing world issues of food stability and bee population decrease through usage of objectbased computer simulations, Monte Carlo analysis, multifunction modeling and differential equations to determine the ideal crop organizational patterns to maximize pollination and therefore production.
Summary (Abstract)
Two major issues affecting the world moving into 2045 are food stability and bee populations. Currently, the world population is growing at a massive rate. Further, more land is becoming urbanized, leaving less space for farming. As such, if we continue the current farming methods, we will run out of food to feed the world. This means we need to innovate and discover new ways to efficiently produce food given the small amount of land farms occupy. To further compound the issue, the populations of bees have been decreasing throughout the world. Most staple food grains such as wheat and corn do not require bees, but foods which contain vital nutrients and vitamins such as nuts, fruits, herbs, and vegetables, require pollination, tasks done mostly by bees. To confront these issues, we simulated a farm using the java programming language and the intellij IDE. We then arranged the plants in different orientations, and measured the percentage of plants pollinated and the average number of pollinations. We then varied the number of bees, and collected data points for rectangular, triangular, and hexagonal patterns. As bees can move in a variety of directions, we used a Monte Carlo approach and conducted over 500 trials. We then modeled these functions using Logger pro, and differentiated the result. Using these differential equations, we determined which pattern would achieve the highest pollination as the bee population decreases. We concluded a crop pattern of hexagons yields more pollenations as the bee population decreases.
Introduction
Two huge issues facing the world as we approach 2045 is food security and the bee population. We aim to address both these issues through developing a revolutionary new approach to agriculture patterns. Currently, agriculture is farmed in rectangular patterns, but we propose using hexagonal planting patterns. These patterns have been proven to increase pollination in the simulations we have run.
The global issue of food insecurity occurs when a large section of the population does not have access to food, leading to starvation. Currently, 815 million people suffer from starvation across the globe(1). These people account for over 10% of the world population(1). Even further is the issue of malnutrition, where people do not have the necessary vitamins and minerals in their diet. This affects 1 in 3 people(2). Although these problems are apparent now, the problem will intensify as we move farther into the future. The population currently is 7.7 billion, and is estimated to reach 9.8 billion by 2050 and 11.2 billion in 2100(3)(4). As the population increases, agriculture production must also increase in parallel to continue to provide food for our growing population. As land is being developed at record rates to provide homes for people, we cannot simply make more farms, we must increase the production of our farms(5). Some processes such as fertilization and pesticides increase crop production, but at a price. These chemicals are not only unhealthy, but they pollute waterways and the air, destroying natural environments. To address this issue, we propose using hexagonal crop patterns to plant instead of rectangular ones. This solution is completely natural as it involves no chemicals.
The second issue we look to address is the declining bee population. From Oct. 1, 2018 to April 1, 2019, almost 40% of the managaned bee colonies died off in the United States(6). This issue, known as "colony collapse disorder,", has caused up to 90% of the bees to die throughout the United States over the past 15 years(6). Currently, 20 billion dollars of produce are generated from bees, half of those coming from commercial honey bees(6). As bees are vital to the production of food, when bees die off, the food production will decrease. This further compounds the issue of food insecurity, as we have shown we need to increase food production to provide for our growing population, while colony collapse disorder directly undermines this goal. We intend to circumvent this issue by planting crops in a hexagonal pattern. This pattern provides more directions for the bee to travel, something we will discuss in the discussion. Therefore we plan to solve the problem of decreasing bee populations not through growing more bees but to increase the efficiency of the bees we currently have.
Our research was aimed to determine the crop orientation which maximized pollination of crops as bee population decreased. We did this through a four step process. Firstly, we develop a simulation of a farm using java graphics and Intelij. Next, we ran the simulation for rectangular, triangular, and hexagonal crop orientation, and recorded the percentage of plants which were pollinated and the average number of pollenisations. We varied the bee population and reran the simulations to collect more data. Thirdly, we graphed this data to determine the line of best fit. Lastly, with these equations, we took the derivative to find how pollination rates change as the bee population decreases. Given these differential equations, we were able to determine which crop patterns pollinated the most food as the bee population declined.
Main
In this experiment, we attempted to determine which crop organizational pattern was most effective. We ran simulations on rectangular crops, as those are how crops are farmed currently and therefore what we compared our patterns to, triangular crops and hexagonal crops. We chose triangular shapes and hexagonal shapes because they are the only geometric shape to tessellate other than square.
We conducted our experiment through simulating a farm and recording the percentage of plants which were pollinated and the average number of pollenisations for those plants which were pollinated. We decided on these numbers because pollination is needed to bear a fruit, and therefore the more pollenisations and more fruit produced. Further, we needed to find how spread out the pollenisations were. We then conducted more trials by varying the bee populations. We conducted trials with 10, 20, 30, 40, 100, and 200 bees. With this data, we then plotted it in the application logger pro and modeled the best fit functions. In the table below are the modeled functions we obtained.

% pollinated 
Average pollination 
Rectangular 

Triangular 

Hexagonal 
Given this table, we can then derive these functions to determine how % pollinated and average pollination change as the bee population decreases.

% pollinated derivative 
Average pollination derivative 
Rectangular 
y=0.000002x20.000404x+0.02311 
y=0.0000000603x20.000058x+0.05683

Triangular 
y=0.000002x20.000458x+0.02521

y=0.000002x20.000445+0.07573

Hexagonal 
y=0.000002x20.000427x+0.02385

y=0.000001x2+0.000143x+0.06633

From these equations, we can understand how the different crop organizational patterns respond to a decrease in bees.
In our research, we first developed a simulation about crops on a farm. We did this through using the development tool Intelij. We created the graphics with PApplet. In our program, we created a Point class, which held the x and y coordinates of the point, the speed, whether or not we could change its speed, and a boolean isPollinated. We created getters and setters for each of their variables. The point class resembles a point on our simulation. The point could either be a bee or a plant. We then created a main class called Pollinate, where we ran the simulation. In this simulation, we created specific methods to generate the rectangular, triangular, and hexagonal crop orientations. We then created a method which would spawn bees at random places. This allowed us to easily change the crop orientation and bee population. With this, we then simulated bee movement by moving them in random directions until they came into the vicinity of a plant. The bee then moved in a straight line to the plant, to which it would remain there for one frame and then leave. We then implement two counters, one to increase by one once the isPollinated boolean is true. Then the second counter was to increase by one every time the bee ran into the flower. We then printed into the consol the two data values. We recorded these values into a data sheet, where we then tan the simulation 27 times per orientation per bee population. Once this data was recorded, we averaged the trials and plugged the data into logger pro. We then modeled the data with cubic functions. We took the derivative of these functions to compare the effects of decreases in the bee population.
Conclusion
As can be seen in the results table in the results section, we have determined the differentials for the percent pollinated derivative and average pollination derivative. With these, we see the coefficient for the hexagonal average pollination derivative is 0.000001, half of the triangular and rectangular. This means the result of that function is lower. Because that function signifies the change in bee population over time, we see that the resulting decrease of a pollination as a result of lowering bee populations is lower in the hexagons. We hypothesize this is due to the higher number of directions a bee can go in the hexagonal pattern. As the hexagon has the most number of sides, the bees can move to six different plants around it, while the rectangles contain only four and the triangles only three.
To quantify the results, if we subtract the hexagonal differential equations from the rectangular ones, we get:
Therefore, the hexagonal formations will have
y = 0.000023x+0.00074
more pollination. This makes logical sense, as x signifies the change in bee population over time. As we discussed above, the bee population is decreasing, and therefore x will be negative. In this equation, if x is negative, the output y will be positive. As we obtained the equation from the difference between the hexagonal and rectangular patterns, we see that the difference will be positive. Therefore, the hexagonal pollination will be larger. This same trend is observed with the average number of pollenations. The difference here between the hexagonal differential equations and the rectangular ones are:
If x is negative here, the first term will not be affected as it is a square root and therefore will always be positive. The second term will become negative, but because the third term is 463.497 times larger than the coefficient of the second term, the bees will have to reach 0.215% of their current population for the rectangular method to surpass the hexagonal method. Therefore, we concluded the hexagonal pattern produces more average pollination.
From the calculations we have done, we have shown hexagonal patterns result in greater percentage of plants pollinated and average pollenisations as the bee population decreases. Therefore, we conclude the hexagonal pattern is more efficient as the bee populations decrease.
Conclusion
Given this data, we can conclude increase in production due to our hexagonal method. Here, we see that:
Represents the difference in production per acre. If we used the 40% increase number given in the introduction and know there are 2.78 million honey bees hives with 60,000 bees each, we can conclude there are 166800000000 bees on farms. If this continues the current pattern and decreases by 40%, the new number of bees will be 100080000000 bees with a decrease of 66720000000 bees. If we plug this into the equation above:
As we can see from this equation, a hexagonal crop pattern will produce 1534560.00074 more pollenations than rectangular patterns. This has massive global implications. If we apply this to a global scale, we can truly see the implications of this approach. If the United States can produce over 1.5 million crops by using this hexagonal method, we can help solve world hunger(8)(9). These numbers apply only to the United States as well. As such, the benefits will be amplified on a world scale. Not only can these crops help end world hunger, but they can also help end malnutrition. As bees pollinate crops with vital nutrients such as antioxidants, vitamin C, potassium, and vitamin K(10).
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 Shapes That Tessellate, www.gwydir.demon.co.uk/jo/tess/grids.htm.
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 Shahbandeh, M. “Honey: Number of Bee Colonies in the U.S. 2019.” Statista, 24 Apr. 2020, www.statista.com/statistics/755263/beecoloniesus/#:~:text=This statistic shows the number,2.78 in the previous year.
“Honey Bees Are Pollinators.” Pollination Facts  American Beekeeping Federation, www.abfnet.org/page/PollinatorFacts#:~:text=As honey bees gather pollen,dependent on honey bee pollination.