Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Consider the pair of differential equations $$u(0)=1 \quad u^{\prime}(t)=0.5 \times u(t)-0.2 \times u(t) \times v(t)$$ \(v(0)=2 \quad v^{\prime}(t)=0.1 \times u(t) \times v(t)-0.1 \times v(t) \quad 0 \leq t \leq 1\) This system is a predator prey system. We (including you!) will use Euler's method to approximate a solution on the time interval [0,1] with \(n=5\) subintervals.

Step-by-Step Solution

Answer

The approximated final values are \(u(1.0) \approx 1.10\) and \(v(1.0) \approx 2\).

Step 1: Understanding Euler's Method

Euler's Method is an iterative numerical process for solving ordinary differential equations. It's used here to approximate solutions to the given predator-prey differential equations. The method uses the initial conditions and updates them over a series of intervals. The general formula for it is: \(y_{n+1} = y_n + h imes f(x_n, y_n)\), where \(h\) is the step size.

Step 2: Define Initial Conditions and Step Size

Given the problem, the initial conditions are \(u(0)=1\) and \(v(0)=2\). The interval \([0,1]\) is divided into \(n=5\) subintervals, so the step size \(h\) is calculated as \(h = \frac{1-0}{5} = 0.2\).

Step 3: Update Using Euler's Method for First Interval

Calculate the new values \(u(0.2)\) and \(v(0.2)\) using Euler's method.\[u(0.2) = u(0) + h imes (0.5 imes u(0) - 0.2 imes u(0) imes v(0))\]\[v(0.2) = v(0) + h imes (0.1 imes u(0) imes v(0) - 0.1 imes v(0))\]Substitute the initial values:\[u(0.2) = 1 + 0.2 imes (0.5 \times 1 - 0.2 \times 1 \times 2) = 1 + 0.2 \times 0.1 = 1.02\]\[v(0.2) = 2 + 0.2 \times (0.1 \times 1 \times 2 - 0.1 \times 2) = 2 + 0 = 2\]

Step 4: Update for Second Interval

Use the results of the previous step to find \(u(0.4)\) and \(v(0.4)\). Substitute the approximated values:\[u(0.4) = u(0.2) + 0.2 imes (0.5 imes u(0.2) - 0.2 imes u(0.2) imes v(0.2))\]\[= 1.02 + 0.2 imes (0.5 \times 1.02 - 0.2 \times 1.02 \times 2) = 1.02 + 0.2 \times 0.1 = 1.04\]\[v(0.4) = v(0.2) + 0.2 imes (0.1 imes u(0.2) imes v(0.2) - 0.1 imes v(0.2))\]\[= 2 + 0.2 imes (0) = 2\]

Step 5: Continue Updating for Third Interval

Repeat the process for \(u(0.6)\) and \(v(0.6)\).\[u(0.6) = u(0.4) + 0.2 imes (0.5 imes u(0.4) - 0.2 imes u(0.4) imes v(0.4))\]\[= 1.04 + 0.2 imes (0.5 \times 1.04 - 0.2 \times 1.04 \times 2) = 1.06\]\[v(0.6) = v(0.4) + 0.2 imes (0.1 imes u(0.4) imes v(0.4) - 0.1 imes v(0.4))\]\[= 2 + 0.2 imes (0) = 2\]

Step 6: Update for Fourth Interval

Calculate for \(u(0.8)\) and \(v(0.8)\) using previous results:\[u(0.8) = u(0.6) + 0.2 imes (0.5 imes u(0.6) - 0.2 imes u(0.6) imes v(0.6))\]\[= 1.06 + 0.2 imes (0.5 \times 1.06 - 0.2 \times 1.06 \times 2) = 1.08\]\[v(0.8) = v(0.6) + 0.2 imes (0.1 imes u(0.6) imes v(0.6) - 0.1 imes v(0.6))\]\[= 2 + 0.2 imes (0) = 2\]

Step 7: Final Update for Fifth Interval

Determine the final values \(u(1.0)\) and \(v(1.0)\).\[u(1.0) = u(0.8) + 0.2 imes (0.5 imes u(0.8) - 0.2 imes u(0.8) imes v(0.8))\]\[= 1.08 + 0.2 imes (0.5 \times 1.08 - 0.2 \times 1.08 \times 2) = 1.10\]\[v(1.0) = v(0.8) + 0.2 imes (0.1 imes u(0.8) imes v(0.8) - 0.1 imes v(0.8))\]\[= 2 + 0.2 imes (0) = 2\]

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. In simpler terms, they help us understand how things change over time or space. Many natural and man-made processes can be described using these equations. For example, the spread of diseases, the cooling of a hot object, and even the economy can be studied using differential equations. When we know how one quantity changes with respect to another, differential equations provide a useful tool to predict future behavior.
When we talk about the differential equations in the context of this exercise, we're dealing with a pair that describes a predator-prey relationship. These are used to model interactions between two species where one is a predator and the other is prey. This real-world application shows how powerful and versatile differential equations really are. By understanding the rate of change for both the predator and prey populations, scientists can simulate and predict changes within an ecosystem using this method.

Predator-Prey Models

Predator-prey models are mathematical representations of the interaction between two species: the predator and the prey. They capture the dynamics of biological systems, where the predators feed on the prey. These models are part of a broader category known as dynamical systems, and they demonstrate fascinating cyclical behaviors observed in nature.
In such models, you typically have two main variables: the population of the prey and the population of the predators. For example, if the prey population increases, there might be more food for the predators, leading to an increase in the predator population. Over time, more predators might lead to a decrease in prey, causing an eventual decline in the predator numbers due to lack of food. This creates natural cycles within both populations.
The exercise provided uses a basic predator-prey model with its equations describing how these populations change over time. It's a perfect way to practically apply mathematical concepts to ecological studies, allowing predictions on how changes in one population impact the other.

Numerical Methods

Numerical methods are techniques used to approximate solutions for mathematical problems that cannot be solved analytically with exact precision. Since many differential equations do not have simple, direct solutions, numerical methods are essential. Euler's Method, which is featured in the exercise, is one such technique used for solving ordinary differential equations (ODEs).
Euler's Method is straightforward. It calculates new approximations of solution points by advancing step-by-step over a chosen interval. Each step uses the previous solution point and updates it using the derivative or rate of change defined by the differential equation. This way, even if solving the equation exactly is difficult, we can still construct an approximate picture of the solution.

• Simple to understand and implement.
• Good for gaining intuition about the behavior of solutions.
• Works well for small steps, but may lose accuracy with larger intervals.

Euler's Method is a cornerstone for more complex numerical techniques, acting as a gateway for those learning about how to solve differential equations computationally.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) involve equations with functions of a single variable and their derivatives. They are 'ordinary' because they deal with functions of one independent variable, unlike partial differential equations (PDEs), which involve multiple variables.
ODEs are pervasive in mathematics and science because they model dynamic systems where the change is dependent on one variable, like time. For instance, how a car speed changes with time based on acceleration or how the temperature of an object changes if placed in a colder room.
In this exercise, the ODEs describe a predator and prey scenario over time, capturing the essence of dynamic interplay between species. The first equation describes the predator's population dynamics, and the second equation captures how many prey there are over time. These equations are simple yet powerful enough to explore ecological interactions or any phenomenon involving a time evolution of a single state."}]}]}