Problem 14 - Calculus for Biology and Medicine

Question

Use a graphing calculator to sketch solution curves of the given Lotka- Volterra predator-prey model in the \(N-\) P plane. Also graph \(N(t)\) and \(P(t)\) as functions of \(t\) \(\frac{d N}{d t}=3 N-2 P N\) \(\frac{d P}{d t}=P N-P\) with initial conditions (a) \((N(0), P(0))=(1,3 / 2)\) (b) \((N(0), P(0))=(2,2)\) (c) \((N(0), P(0))=(3,1)\) In Problems 15 and 16 , we investigate the Lotka-Volterra predatorprey model.

Step-by-Step Solution

Answer

Use a graphing calculator to input the given equations with initial conditions, sketch the phase plane plot, and plot \(N(t)\) and \(P(t)\). Observe cyclic oscillations in all scenarios.

Step 1: Understand the Lotka-Volterra Model

The Lotka-Volterra model consists of two differential equations that describe the dynamics of predator (P) and prey (N) populations. The model assumes an unlimited prey population growth rate, a constant kill rate for predators, and no external factors such as disease or migration.

Step 2: Understand the Differential Equations

The first equation, \( \frac{dN}{dt} = 3N - 2PN \), represents the rate of change of prey population. The term \(3N\) suggests the prey grows exponentially in the absence of predators, whereas \(-2PN\) accounts for the loss due to predation. The second equation, \( \frac{dP}{dt} = PN - P \), describes the rate of change of predator population, where \(PN\) represents the growth due to feeding on prey and \(-P\) accounts for the natural decay of the predator population.

Step 3: Input Initial Conditions into a Graphing Calculator

Using a graphing calculator capable of solving differential equations, input the equations \( \frac{dN}{dt} = 3N - 2PN \) and \( \frac{dP}{dt} = PN - P \). Enter each set of initial conditions: (a) \((N(0), P(0))=(1, 3/2)\), (b) \((N(0), P(0))=(2, 2)\), and (c) \((N(0), P(0))=(3, 1)\).

Step 4: Sketch the Phase Plane Plot

On the graphing calculator, observe the plot generated in the \(N-P\) (prey vs. predator) plane. Notice the closed, cyclic trajectories typical of a Lotka-Volterra system, indicating stable oscillations between predator and prey populations.

Step 5: Graph Time-Dependent Functions

On a separate display, plot \(N(t)\) and \(P(t)\) against time \(t\) using the calculator, checking for each initial condition set. Observe the oscillating nature of each population over time, showcasing their interdependent dynamics.

Step 6: Analyze and Compare the Results

Compare the different trajectory shapes and time-based plots for each initial condition set. Note how varying initial conditions affect the frequency and amplitude of population oscillations.

Key Concepts

Predator-Prey Dynamics

In the natural world, predator-prey dynamics describe the interactions between two species: predators, which hunt, and prey, which are hunted. This is a key concept in ecology and is captured mathematically by the Lotka-Volterra model. This model analyzes how the populations of these two groups affect each other over time. Think of it like this: if the prey population increases, predators have more to eat and may also increase in number. However, if predators become too numerous, they can significantly reduce the prey population, leading eventually to a reduction in predators. This causes natural oscillations, or cycles, in both populations.

In the Lotka-Volterra model:

• The prey population grows rapidly in the absence of predators.
• Predators rely entirely on the prey population for their survival and growth.
• Neither population has limits, assuming no external constraints like environment changes.

These simple rules create intriguing and complex dynamics, which can be explored through mathematical modeling.

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes over time. In our case, these equations help us track the populations of predators and prey. Let's peek closer at the Lotka-Volterra model's differential equations:

- The first equation, \( \frac{dN}{dt} = 3N - 2PN \), explains how the prey population changes. Here:

• \(3N\) represents the natural exponential growth of the prey without any predators. It's as if the prey have unlimited resources and space to grow.
• \(-2PN\) subtracts from this growth, as it factors in the loss of prey due to being hunted by predators.

- The second equation, \( \frac{dP}{dt} = PN - P \), looks at predators:

• \(PN\) represents predators growing in number when they have sufficient prey to feed on. More food means more predator offspring.
• \(-P\) shows the natural death or decline in predator population when they aren't feeding on enough prey.

With these equations, we can predict population changes over time, illustrating the delicate balance in ecosystems.

Phase Plane Analysis

Phase plane analysis is a method used to visualize the behavior of dynamic systems. It takes the predator-prey model equations and translates them into a two-dimensional plot. This plot is known as the phase plane, with one axis typically representing the prey population and the other, the predator population.

Once plotted, we often notice cyclic or circular patterns. These are trajectories that show us how the predator and prey populations evolve together over time. Such plots usually reveal:

• Stable cycles where populations return to starting levels, highlighting natural rebalancing tendencies.
• The relationship and effect of one species population on the other.
• Dependency of population cycles on initial conditions, a crucial insight for understanding real-world ecosystems.

Using phase plane analysis, students can visualize and better understand oscillations in populations and how mathematical models predict natural behavior.

Initial Conditions

Initial conditions in mathematical models specify the exact state from which the system starts. For the Lotka-Volterra predator-prey model, this means defining the initial numbers of predators and prey.

These initial conditions are central because they directly influence the system's dynamic outcomes. In our exercise:

• For condition (a), the initial values are \((N(0), P(0))=(1, 3/2)\).
• Condition (b) has \((N(0), P(0))=(2, 2)\) as starting values.
• Condition (c) begins with \((N(0), P(0))=(3, 1)\).

By entering different initial conditions into a calculator or computational tool, you see how trajectories in a phase plane and population graphs shift. This demonstrates:

• The sensitivity of populations to starting values.
• How initial imbalances might lead to more dramatic oscillations in predator and prey numbers.
• The diversity of possible outcomes despite a fixed model and unchanging ecological rules.

Emphasizing initial conditions teaches students about the real-world complexity and unpredictability inherent in natural systems.