Population ModelingBy Ryan JahansoozAssisted by Zac Foster, Hunter Wolfbear Drobenaire, and Tyler VossBIO 001 Lab Section 16 10/30/2013Abstract:This lab was a quick experiment involving a computer program known as Excel which allowed us to efficiently create a simulation of a population in an environment. The equations that were put into excel made it so that the numbers for each aspect of a populations growth could be easily changed to create any outcome. Using this, it was simple to see how populations of prey and predators grow in relation to one another, as well as how a single population might grow depending on the resources and land available to it. It was found that the populations of prey and predators are so dependent on one another that over time they come to a stable population that does not change. It was also found that any population will eventually reach its limit.Introduction:Exponential growth is when a population is allowed to grow forever with no limitations. As the population grows, the number of new births at the end of each cycle also increases. This allows for a population to spiral out of control quite quickly and end up with improbably large numbers of people. The equation to find the population of a group with exponential growth is simply the growth rate multiplied by the population in this equation: ΔN = r x NWhere N is the original population and r is the growth rate.Logistic growth is a much more realistic system of population expansion because it adds in the limiting factors that come from a population’s environment. Things such as space, food, disease, and other factors all add together to form a maximum population for an area. As the size of a population increases, it will begin to taper off into being stable rather that shoot upwards infinity. The maximum population of the area is commonly called the areas “carrying capacity”. To find the population of a group with logistic growth, use the following equation:ΔN = r x N x (1-N/K)Where N is the original population, r is growth rate, and K is the carrying capacity.The predator/prey simulation is one that incorporates two separate groups to see how their relations determine their final populations. Neither group can grow out of control because as one group gets bigger, the other keeps it in check. It ends with both populations stabilizing after several cycles of inverted growth. The population of either group can be found using the following equation:ΔN = r x N x (1-N/K) – β x N x PWhere N is the original population of the prey, r is the growth rate, K is the areascarrying capacity, β is the predation rate, and P is the predator’s population.Once familiar with the equations and the different forms of population growth, it was easy to use Excel to determine the populations of a group at any time in its history. A series of graphs and tables made the information easy to collect as well as making it easy to see the changes over time on a scale that made sense. Methods and Materials:The experimental procedures for this lab were adapted from a previously supplied protocol .There were no deviations from protocol.Results:While inputting data and checking the results for the exponential growth model, it became clear which parts of a group have the greatest impact on its population. Changing anything just slightly resulted in regularly different outcomes. It did seem that the initial population had a larger impact on more immediate cycles while growth rate had a larger impact after a few dozen. One of the groups that were used as a model was the human population, which was found to reach ten billion people by the year 2025 if allowed to grow exponentially.In the logistic growth model, however, the simulation output that it would take mankind almost one hundred and twelve years to reach the carrying capacity of the earth (eleven billion). This Is due to the fact that the earth does have a carrying capacity for humans. There was also a simulation for a smaller group with only a few thousand inhabitants. As you can see in Fig 1, the rate of change in a population is determined by the carrying capacity. All populations with logistic growth tend to adapt the “S” shape in their graphs. The predator/prey interaction simulation was the most interesting because it dealt with two populations instead of just one. As one population increased, the other would increase or decrease respectively. In the end, though, both would even out to stable and constant population. They both achieve steady- state at the same time and both underwent a period of oscillation beforehand. Also, when the initial populations and growth rates of each group were modified, it was found that the steady-state was achieved at the same time regardless of rate. If both initial populations were equal to the steady state, then there was oscillatory period at all. When modifying some of the more delicate details in the predator/prey simulation, it was found that the initial population is the most volatile. A high initial population of one group will cause the population growth curve to spike and oscillate violently before stabilizing. The second most influencing factor was the death rate of the predator. A high death rate resulted in extinction and a high prey population, while a low death rate changed the steady -state.Fig 1: Effect of Carrying Capacity on Curve Shape and PopulationCarrying Capacity (K) Overall Shape of Curve Population at year 10010000 Exponential 244.71000 S 202100 Flat S 73.1 The most shocking discovery was that the prey population was not as dependent on the predator’s population as the predators population was on the preys population. The prey had a constant steady- state of .3 regardless of the population of the predators, while the predator’s steady- state changed a lot depending on its hunting efficiency. This can be seen on Fig 2, where N is constant and P is increasing. This is attributed to the oscillating, inverted relationship between the two populations. Fig 2: Steady State for Prey and Predator with Changing Predation RateSteady StatePredation Rate N P0.8 .3 .870.85 .3 .820.95 .3 .820.6 .3 1.170.4 .3 1.740.1 .3 7Discussion:While surprising, it is still clear that the phenomenon in Fig 2 does take place in real life. A perfect example is the population of foxes and rabbits. As the number of foxes increases, the need for food increases as well. The population of rabbits lowers as they are hunted