# Mathematics Awareness Month - April 2013 Mathematics of Sustainability

## Logistic Equation 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.