The exponential model describes how a population grows when resources are unlimited and conditions are ideal. It is represented by the differential equation , where N is the population size and k is the growth rate. If k is positive, the population increases rapidly over time, producing a curve that becomes steeper as the population grows, as shown by the solid line on the graph. If k is negative (-), the population decreases and eventually approaches zero (0), shown by the dotted line. The rabbit population example illustrates positive (+) exponential growth: starting from small numbers, the population rises sharply each year from 50, 80, 128, 205, and so on. This is because the number of new rabbits produced each year is proportional to the number already present. This pattern reflects the key idea of exponential growth: the larger the population becomes, the faster it grows.
![The exponential model describes how a population grows when resources are unlimited and conditions are ideal. It is represented by the differential equation [math]\frac{dN}{dt}=kN[/math], where[i] N [/i]is the population size and[i] k[/i] is the growth rate. If [i]k[/i] is positive, the population increases rapidly over time, producing a curve that becomes steeper as the population grows, as shown by the solid line on the graph. If [i]k[/i] is negative (-), the population decreases and eventually approaches zero (0), shown by the dotted line. The rabbit population example illustrates positive (+) exponential growth: starting from small numbers, the population rises sharply each year from 50, 80, 128, 205, and so on. This is because the number of new rabbits produced each year is proportional to the number already present. This pattern reflects the key idea of exponential growth: the larger the population becomes, the faster it grows.](https://www.geogebra.org/resource/bspwgjav/0vcEWntYVswRwdEC/material-bspwgjav.png)