Seed dispersal of juniper and pinyon is a process in which frugivorous birds play an important role. Birds either consume and digest seeds or carry and cache them at some distance from the source tree. These transport...Seed dispersal of juniper and pinyon is a process in which frugivorous birds play an important role. Birds either consume and digest seeds or carry and cache them at some distance from the source tree. These transported and settled seeds can be described by a dispersal kernel, which captures the probability that the seed will move a certain distance by the end of the process. To model active seed dispersal of this nature, we introduce handling time probabilities into the dispersal model to generate a seed digestion kernel. In the limit of no variability in handling time the seed digestion kernel is Gaussian, whereas for uniform variability in handling time the kernel approaches a Laplace distribution. This allows us to standardize spatial movement (diffusion) and handling time (peak settling rate) parameters for all three distributions and compare. Analysis of the tails indicates that the seed digestion kernel decays at a rate intermediate between Gaussian and Laplace seed kernels. Using this seed digestion kernel, we create an invasion model to estimate the speed at which juniper and pinyon forest boundaries move. We find that the speed of seed invasion corresponding to the digestion kernel was faster than seeds resulting from Laplace and Gaussian kernels for more rapidly digested seeds. For longer handling times the speeds are bounded between the Laplace (faster) and Gaussian (slower) speeds. Using parameter values from the literature we evaluate the migration potential of pinyon and juniper, finding that pinyon may be able to migrate up to two orders of magnitude more rapidly, consistent with observations of pine migration during the Holocene.展开更多
The United States dollar was tied to a finite standard dating back to 1792. The dollar has survived multiple wars, armed conflicts short of war, economic recessions, and the Great Depression. At the end of World War I...The United States dollar was tied to a finite standard dating back to 1792. The dollar has survived multiple wars, armed conflicts short of war, economic recessions, and the Great Depression. At the end of World War II, the dollar, by international agreement, effectively became the world’s reserve currency. Other currencies were fixed to the dollar and the dollar was in turn fixed to the gold standard, that is, the value of a dollar was fixed at thirty-five dollars per ounce of gold. In 1971, the United States government severed the dollar from the gold standard leaving the size of the money supply free of any finite limit. Since that time, the United States money supply, national debt, and stock market indices have experienced exponential growth. The purpose of this paper is to investigate mathematically and model the relationship among these key economic indicators. Our work in this paper allows us to gain some insight into how each of these economic indicators can influence one another while giving us a better idea of how the economy functions. We were allowed to collect data for each of the indicators over a certain period which allowed us to demonstrate that a strong correlation exists among these three indicators. After collecting our data and making some logical assumptions, we modeled the money supply as a function of time. In turn, we modeled the Dow Jones stock market index as a function of the money supply. Finally, we produced a model of the National Debt as a function of the Dow Jones stock market index. From the data, we collected and our economic indicator models, we derived a series of differential equations that would help us observe the exponential growth trend of each graphically. By showing a comparison from our collection of raw data versus our results from our model, we conclude that, under prevailing circumstances, it is reasonable to expect growth in all three indicators and increasing market volatility for some time to come. We have answered the question of whether the monetary and fiscal policies of the past half-century have been of some benefit. However, we question whether this trend is good for all Americans and whether it is sustainable in perpetuity. This paper concludes with possible areas of interest warranting additional investigation to enable us to better understand the economy and develop effective financial strategies.展开更多
文摘Seed dispersal of juniper and pinyon is a process in which frugivorous birds play an important role. Birds either consume and digest seeds or carry and cache them at some distance from the source tree. These transported and settled seeds can be described by a dispersal kernel, which captures the probability that the seed will move a certain distance by the end of the process. To model active seed dispersal of this nature, we introduce handling time probabilities into the dispersal model to generate a seed digestion kernel. In the limit of no variability in handling time the seed digestion kernel is Gaussian, whereas for uniform variability in handling time the kernel approaches a Laplace distribution. This allows us to standardize spatial movement (diffusion) and handling time (peak settling rate) parameters for all three distributions and compare. Analysis of the tails indicates that the seed digestion kernel decays at a rate intermediate between Gaussian and Laplace seed kernels. Using this seed digestion kernel, we create an invasion model to estimate the speed at which juniper and pinyon forest boundaries move. We find that the speed of seed invasion corresponding to the digestion kernel was faster than seeds resulting from Laplace and Gaussian kernels for more rapidly digested seeds. For longer handling times the speeds are bounded between the Laplace (faster) and Gaussian (slower) speeds. Using parameter values from the literature we evaluate the migration potential of pinyon and juniper, finding that pinyon may be able to migrate up to two orders of magnitude more rapidly, consistent with observations of pine migration during the Holocene.
文摘The United States dollar was tied to a finite standard dating back to 1792. The dollar has survived multiple wars, armed conflicts short of war, economic recessions, and the Great Depression. At the end of World War II, the dollar, by international agreement, effectively became the world’s reserve currency. Other currencies were fixed to the dollar and the dollar was in turn fixed to the gold standard, that is, the value of a dollar was fixed at thirty-five dollars per ounce of gold. In 1971, the United States government severed the dollar from the gold standard leaving the size of the money supply free of any finite limit. Since that time, the United States money supply, national debt, and stock market indices have experienced exponential growth. The purpose of this paper is to investigate mathematically and model the relationship among these key economic indicators. Our work in this paper allows us to gain some insight into how each of these economic indicators can influence one another while giving us a better idea of how the economy functions. We were allowed to collect data for each of the indicators over a certain period which allowed us to demonstrate that a strong correlation exists among these three indicators. After collecting our data and making some logical assumptions, we modeled the money supply as a function of time. In turn, we modeled the Dow Jones stock market index as a function of the money supply. Finally, we produced a model of the National Debt as a function of the Dow Jones stock market index. From the data, we collected and our economic indicator models, we derived a series of differential equations that would help us observe the exponential growth trend of each graphically. By showing a comparison from our collection of raw data versus our results from our model, we conclude that, under prevailing circumstances, it is reasonable to expect growth in all three indicators and increasing market volatility for some time to come. We have answered the question of whether the monetary and fiscal policies of the past half-century have been of some benefit. However, we question whether this trend is good for all Americans and whether it is sustainable in perpetuity. This paper concludes with possible areas of interest warranting additional investigation to enable us to better understand the economy and develop effective financial strategies.
