What are partial differential equations?

Partial differential equations (PDEs) are mathematical equations that describe relationships between functions of multiple variables and their partial derivatives. In theoretical biology, PDEs are used to model dynamic processes that occur in space and time, such as population growth, disease spread, and tissue development.

What are the key features of PDEs?

• Multiple variables: PDEs involve functions of two or more variables, such as space and time.
• Partial derivatives: PDEs contain partial derivatives of the functions, which describe the change of the function with respect to one of the variables.
• Dynamic processes: PDEs are used to model dynamic processes, such as the change of a population over time or the spread of a substance through a tissue.

What is the importance of PDEs?

PDEs are important in theoretical biology for the following reasons:

• Realistic modeling: PDEs can realistically model complex processes in nature, including the effects of spatial and temporal variation.
• Analytical solutions: In some cases, PDEs can be solved using analytical methods, leading to exact and deep insights into the modeled process.
• Numerical simulations: In other cases, PDEs can be simulated numerically using computers, which allows for exploring a wide range of scenarios and parameters.

What are applications of PDEs in practice?

PDEs are used in a wide range of applications in theoretical biology, including:

• Population dynamics: PDEs can be used to model the growth and decline of populations, including the effects of predation, competition, and migration.
• Disease spread: PDEs can be used to model the spread of diseases, including the effects of contagion, quarantine, and vaccination.
• Tissue development: PDEs can be used to model the development of tissues, including the interactions between cells and the extracellular matrix.

Practical example

A PDE can be used to model the spread of a bacterial population in a Petri dish. The equation takes into account the growth of the bacteria, the diffusion of bacteria through the dish, and the death of bacteria due to antibiotics. The simulation can be used to study how the bacterial population evolves over time and how different factors, such as the concentration of antibiotics, affect the growth of the population.

Critical remarks

PDEs also have some drawbacks:

• Mathematical complexity: PDEs can be mathematically complex, making them difficult to solve and interpret.
• Data requirements: PDEs require detailed data on the parameters of the modeled process.
• Difficulties with validation: It can be difficult to validate whether the simulations of a PDE match reality.