Problem 67 - Calculus for Biology and Medicine

Question

The two-dimensional diffusion equation $$ \frac{\partial n(\mathbf{r}, t)}{\partial t}=D\left(\frac{\partial^{2} n(\mathbf{r}, t)}{\partial x^{2}}+\frac{\partial^{2} n(\mathbf{r}, t)}{\partial y^{2}}\right) $$ where \(n(\mathbf{r}, t), \mathbf{r}=(x, y)\), denotes the population density at the point \(\mathbf{r}=(x, y)\) in the plane at time \(t\), can be used to describe the spread of organisms. Assume that a large number of insects are released at time 0 at the point \((0,0)\). Furthermore, assume that, at later times, the density of these insects can be described by the diffusion equation (10.41). Show that $$ n(x, y, t)=\frac{n_{0}}{4 \pi D t} \exp \left[-\frac{x^{2}+y^{2}}{4 D t}\right] $$ satisfies (10.41).

Step-by-Step Solution

Answer

The proposed solution \(n(x, y, t) = \frac{n_{0}}{4 \pi D t} \exp \left[-\frac{x^{2}+y^{2}}{4 D t}\right]\) satisfies the diffusion equation.

Step 1: Understand the Diffusion Equation

The two-dimensional diffusion equation describes how a quantity, such as population density \(n(\mathbf{r}, t)\), evolves over time. The partial differential equation is given by \(\frac{\partial n(\mathbf{r}, t)}{\partial t} = D\left(\frac{\partial^{2} n(\mathbf{r}, t)}{\partial x^{2}} + \frac{\partial^{2} n(\mathbf{r}, t)}{\partial y^{2}}\right)\), where \(D\) is the diffusion coefficient.

Step 2: Write Down the Proposed Solution

The problem proposes a solution of the form \(n(x, y, t) = \frac{n_{0}}{4 \pi D t} \exp \left[-\frac{x^{2} + y^{2}}{4 D t}\right]\). Our task is to verify if this solution satisfies the diffusion equation.

Step 3: Compute the Time Derivative

Calculate the time derivative of the proposed solution: \(\frac{\partial n}{\partial t}\). Applying the chain rule, we have:\[\frac{\partial n}{\partial t} = \left(-\frac{x^{2} + y^{2}}{4Dt^2} + \frac{1}{t}\right) \cdot \frac{n_{0}}{4 \pi D t} \cdot \exp\left(-\frac{x^{2} + y^{2}}{4Dt}\right).\]

Step 4: Compute the Second Spatial Derivatives

For \(\frac{\partial^{2} n}{\partial x^{2}}\) and \(\frac{\partial^{2} n}{\partial y^{2}}\), calculate:\[\frac{\partial^{2} n}{\partial x^{2}} = \frac{n_{0}}{4\pi Dt} \cdot \left(\frac{1}{2Dt} - \frac{x^2}{4D^2t^2}\right) \cdot \exp\left(-\frac{x^{2} + y^{2}}{4Dt}\right),\]\[\frac{\partial^{2} n}{\partial y^{2}} = \frac{n_{0}}{4\pi Dt} \cdot \left(\frac{1}{2Dt} - \frac{y^2}{4D^2t^2}\right) \cdot \exp\left(-\frac{x^{2} + y^{2}}{4Dt}\right).\]

Step 5: Sum the Second Spatial Derivatives

Add the expressions for \(\frac{\partial^{2} n}{\partial x^{2}}\) and \(\frac{\partial^{2} n}{\partial y^{2}}\):\[\frac{\partial^{2} n}{\partial x^{2}} + \frac{\partial^{2} n}{\partial y^{2}} = \frac{n_{0}}{4\pi Dt} \cdot \left(\frac{1}{Dt} - \frac{x^2 + y^2}{4D^2t^2}\right) \cdot \exp\left(-\frac{x^{2} + y^{2}}{4Dt}\right).\]

Step 6: Verify the Diffusion Equation

Plug all computed derivatives into the diffusion equation. Multiply the result of Step 5 by \(D\) to get:\[D\left(\frac{\partial^{2} n}{\partial x^{2}} + \frac{\partial^{2} n}{\partial y^{2}}\right) = \left(-\frac{x^{2} + y^{2}}{4Dt^2} + \frac{1}{t}\right) \cdot \frac{n_{0}}{4 \pi D t} \cdot \exp\left(-\frac{x^{2} + y^{2}}{4Dt}\right).\]This expression is exactly equal to the time derivative from Step 3, confirming the proposed solution satisfies the diffusion equation.

Key Concepts

Two-Dimensional Diffusion

Diffusion is a process where particles spread out from areas of high concentration to low concentration over time. In a two-dimensional space, this process allows us to observe how something like population density changes over a plane. Although diffusion may seem complex, the concept boils down to understanding how particles move and spread across a two-axis system, such as x and y.

When dealing with two dimensions, it's important to consider how substances diffuse laterally. In the case of organisms or particles released at a point, such as at \(0,0\), the spread over time can be modeled mathematically. The two-dimensional diffusion equation is an essential tool here.

The mathematical form of this equation involves the Laplacian operator, which captures the essence of particles diffusing equally in all directions from a point source.

• The diffusion coefficient \(D\) is a measure of how fast the diffusion process occurs.
• The equation examines how the concentration of particles changes over time, incorporating derivatives concerning both x and y dimensions.

Through this equation, we can visualize the spread of a population or substance across a plane as a function of time.

Population Density

Population density refers to the measurement of organisms in a given area at a particular time. When applied to our problem, it's about understanding how densely packed a population is sitting on a plane, like how insects might be distributed over a field.

Understanding population density in diffusion scenarios helps in predicting how organisms are likely to spread out over time. With the initial release of insects at a central location, population density will undergo significant changes. This change can be modeled by the two-dimensional diffusion equation.

The equation tells us that as time progresses, the population density decreases at the point of release as the insects spread out. However, while it seems they are less dense at the center, the density at other locations increases as the insects reach those places.

• This results in a gradually smoothing distribution of insects across the plane.
• The distribution pattern becomes a known shape described by a Gaussian function.

This behavior is essential in fields like ecology, where understanding and predicting population behavior under natural movement is critical.

Partial Differential Equation

A partial differential equation (PDE) is used to describe various phenomena involving continuous change, like diffusion. Such equations involve rates of change with respect to multiple variables. In our case, the PDE in focus deals with time and two spatial variables (x, y).

The key structure of a PDE allows us to solve complex problems involving dynamic systems, such as finding out how population density evolves over time.

• The specific form of the PDE given in the problem includes spatial second derivatives across both the x and y axes.
• The temporal derivative reflects how the state of the system changes over time.

By solving this equation, we find expression for \(n(x, y, t)\), indicating how population density is distributed across the plane at any time \(t\). This is not just a mathematical exercise; understanding and using PDEs are important in physics, biology, and engineering.

Through this, we learn how to translate the behavior of particles or populations in space and time into a comprehensible framework using mathematical tools.