Math Exploration

For my SL math exploration, I have chosen to model the carrying capacity of earth. Carrying capacity is a variable which denotes the number of people, other living organisms, or crops that a region can support without environmental degradation. In this exploration, I will be using different mathematical models to explore the growth rate of a population and at what point the population reaches the carrying capacity of earth. Reproduction is proportional to the number of individuals. In exponential growth, the population grows faster and faster, continuing to double in size at regular intervals. Exponential growth may be a good model for early stage populations such as bacteria.

However, exponential cannot continue for long for a population to reach an enormous value. Exponential growth cannot continue for long because the environment in which the population lives cannot support an infinite large population of species. Based on the amount of available food, space, water, and other essentials, an environment will have a carrying capacity. As a result, as the populations reach the environmental limits, the growth rates decrease, and in other cases, even if the population reaches environmental limits, the growth rate will not change.

In any given environment, there is always going to be a constrain on its growth of the species that live in that environment. As mentioned before, these environmental limits may be lack of resources such as food, water, space, etc. These limitations act as a ‘ceiling’ to the growth of the population. Populations may grow and gradually reach a maximum size, or they will continue to grow without a change in the growth rate. To be more concrete, there are 2 different types of population growth: logistic and exponential. These two functions have been evidently and accurately verified as the only models for population growth of species.

Logistic model

In nature, populations start by growing rapidly and exponentially however, due to environmental restrictions, they are limited. Although the following reasons are only fatal and don’t have association with the carrying capacity, a population’s growth rate may also slow down due to drop in fertility and birth rates, increase in mortality as a result of lack of health care, violence, disease, and other natural catastrophes. In logistic growth, the growth rate decreases as the population approaches a maximum size determined by the environmental setting, known as the carrying capacity. The variable ‘K’ represents carrying capacity.