Population growth models are mathematical models that seek to represent the rate of growth in a population over a given period of time. Because it’s difficult to incorporate all the factors that might influence the growth or decline of a population, mathematicians begin with basic models that assess growth and death rates and then build on those by inserting other factors as needed.
Exponential Growth Model

The exponential growth model describes populations that are constantly increasing, getting exponentially bigger. For example, if two cats produce four cats, and each of those cats produces four cats of its own, if growth is consistent and none of them die, over time, they could produce a small army of cats. However, populations typically only grow exponentially for short periods, because as they grow, the amount of food to which they have access declines. The limitation on resources leads to a dropoff in population rates. Sustained exponential growth is rare and typically only occurs when strict controls are placed on the external forces that enable it to happen.
Malthusian Growth Model

The Malthusian growth model is a variation on the exponential growth model positing that populations tend to grow exponentially while the supply of available food grows linearly. Inevitably, this leads to a crisis in which a large population does not have enough food to sustain itself, resulting in a reduction in population. According to Malthus, nature has a builtin systems of checks and balances that limit the amount of damage that can be caused by a large group by reducing the population of that group once it reaches a certain point.
Logistics Growth Model

The logistics growth model is a population growth model that describes populations that are able to grow substantially, but not indefinitely. After a period of significant development, the rate of growth slows or plateaus, because the availability of living space and resources, along with other factors, limits its growing ability. The logistics growth model is a more reliable measure of population growth than the exponential model because it accounts for the realworld factors that inhibit population growth.
Mechanistic Population Model

According to the recently developed mechanistic growth model, a correlation exists between food availability, agricultural production, fertility rates and the size of a population. This model suggests that an increase in the amount of farming increases the amount of food available, while leaving the rate of growth unaffected. Conversely, a decrease in fertility lowers the growth rate while leaving food production unaffected. Based on these projections, some scientists theorize that the amount of calories available to a population in a given year increases fertility while enabling survival, thereby lowering the death rate.
References
 Austin Community College: Unit 3  Population Growth and Regulation
 Math Is Fun: Exponential Growth and Decay
 Mathematical Association of America: An Introduction to Population Ecology—The Logistic Growth Equation
 Duke University: Education: Population Growth Models
 Richland University: Exponential and Logarithmic Models
 MathWords: Logistic Growth
 Stanford University: Models of Population Growth
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