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Mathematics Awareness Month  April 2013

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Logistic Equation:
The simplest model of population growth P(t) assumes that the rate of change of the population dP/dt is proportional to the size of the population:
This model leads to the prediction that population will grow without bound. While this may be true in the short term, over time a growing population will eventually be constrained by the finite resources (air, water, food, etc) of its environment. This aspect of population growth is captured by assuming the system has a maximum sustainable population N called the carrying capacity. Incorporating carrying capacity into the population models leads to the logistic equation:
In resource management, one studies the growth of a population where a certain amount h of the population is harvested each year. This harvesting could be catching fish for a fish population or chopping down trees for lumber. Including the harvesting term h gives the logistic model with harvesting:
Using this equation, one can show that there is a tipping point in the harvesting level. If one harvests at a level below this critical harvesting value, there is a long term, sustainable population level. But if one harvests at a level above this critical harvesting value, then over time the population will decrease to zero and die out.
While a greatly simplified model of a complex real world situation, the logistic equation with harvesting provides a stark warning of the dangers of placing short term profits, via unduly high harvesting levels, over long term sustainability of the resource.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area. 