Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.

Bacterial populations exhibit exponential growth when conditions such as nutrient availability and temperature are favorable. In this phase, cells reproduce through binary fission, where each cell divides into two identical daughter cells. This process causes the population to double at regular intervals, resulting in a growth rate that is directly proportional to the current number of cells. As the population increases, the number of new cells formed during each generation also grows, creating...

Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is...

Plants grow throughout their lives; this is called indeterminate growth, and it distinguishes plants from most animals. Although certain parts of plants stop growing (e.g., leaves and flowers), others grow continuously—like roots and stems.

Vascular plants, which account for over 90% of the Earth’s vegetation, all undergo primary growth—which lengthens roots and shoots. Many land plants, notably woody plants, also undergo secondary growth—which thickens roots and shoots.

Chondrocytes form a temporary cartilaginous model by dividing and secreting a thick gel-like extracellular matrix. Once the chondrocytes undergo programmed cell death, osteoblasts enter the site of the cartilaginous model. The process of replacing the temporary cartilaginous model with bone in an ordered manner is called endochondral ossification. In endochondral ossification, not all of the cartilage is replaced by bone tissue. Some cartilage that performs a protective and supportive function...