Problem 17 - Calculus for Biology and Medicine

Question

Assume that \(N(t)\) denotes the density of an insect species at time \(t\) and \(P(t)\) denotes the density of its predator at time \(t\). The insect species is an agricultural pest, and its predator is used as a biological control agent. Their dynamics are given by the system of differential equations $$ \begin{array}{l} \frac{d N}{d t}=5 N-3 P N \\ \frac{d P}{d t}=2 P N-P \end{array} $$ (a) Explain why $$ \frac{d N}{d t}=5 N $$ describes the dynamics of the insect in the absence of the predator. Solve (11.85). Describe what happens to the insect population in the absence of the predator. (b) Explain why introducing the insect predator into the system can help to control the density of the insect. (c) Assume that at the beginning of the growing season the insect density is \(0.5\) and the predator density is \(2 .\) You decide to control the insects by using an insecticide in addition to the predator. You are careful and choose an insecticide that does not harm the predator. After you spray, the insect density drops to \(0.01\) and the predator density remains at \(2 .\) Use a graphing calculator to investigate the long-term implications of your decision to spray the field. In particular, investigate what would have happened to the insect densities if you had decided not to spray the field, and compare your results with the insect density over time that results from your application of the insecticide.

Step-by-Step Solution

Answer

Without predators, insects grow exponentially. Predators control insects by preying on them. Spraying decreases insects quickly, helping predators stabilize the population faster.

Step 1: Understanding Insect Dynamics Without Predators

The equation \(\frac{dN}{dt} = 5N\) models the insect population change when there are no predators present. This is a form of the exponential growth model where the population grows at a constant rate proportional to its size. Insects reproduce and the population increases by a factor of 5 times its current population.

Step 2: Solving the Insect Growth Equation

To solve the differential equation \(\frac{dN}{dt} = 5N\), we separate variables and integrate:\[ \int \frac{1}{N} \, dN = \int 5 \, dt \]which gives:\[ \ln |N| = 5t + C \]Taking the exponential of both sides gives us:\[ N(t) = N_0 e^{5t} \]where \(N_0\) is the initial insect density. This shows the insect population grows exponentially without predators.

Step 3: Effect of Predators on Insect Population

The term \(-3PN\) in \(\frac{dN}{dt} = 5N - 3PN\) reduces the insect growth rate because predators consume insects. The presence of predators represents a controlling factor, decreasing the insect population growth as it interacts with predators.

Step 4: Introducing Predator Control

When predators are introduced, the equation for the predator becomes \(\frac{dP}{dt} = 2PN - P\). This shows that predator density increases when insects are available since insects are food for predators. Thus, predators can naturally control insect densities in a biological control sense.

Step 5: Initial Conditions and System Response

With initial conditions of \(N(0) = 0.5\) and \(P(0) = 2\), the predator exerts pressure on the insect population. If we had not sprayed, the predator population would naturally control the insect density over time due to this predator-prey interaction.

Step 6: Effect of Insecticide on Population Dynamics

Spraying reduces the insect density to \(N=0.01\), shifting the balance. This sharp drop gives predators time to suppress insect growth even further as their density remains at \(P=2\). This intervention initially disrupts but eventually stabilizes under predator control.

Step 7: Long-term Implications

Using a graphing calculator or simulation, the long-term behavior with spraying shows a faster approach to low insect densities due to predators maintaining control over fewer initial insects. Without spraying, the decline is slower but still trends to lower levels as predator dynamics naturally control the insect population.

Key Concepts

Predator-Prey Model

In the fascinating world of mathematical biology, the predator-prey model is a popular way to express the dynamics between two species: one as a predator and the other as its prey. This model shows how these two populations influence each other over time.
In this specific case, the interaction between the insect (as the prey) and its predator is captured in the system of differential equations:

• \( \frac{d N}{d t} = 5 N - 3 P N \)
• \( \frac{d P}{d t} = 2 P N - P \)

These equations describe how the rate of change for each population depends on the current number of insects \((N)\) and predators \((P)\). The equations visually highlight two important aspects:

• Growth of Prey: The term \(5N\) indicates the natural growth rate of insects without predation. Without any predators, the insects would grow exponentially.
• Impact of Predators: The term \(-3PN\) in the first equation shows how the presence of predators lessens the insect population due to predation.
• Predator Diet: The term \(2PN\) in the second equation represents how the predator population grows when there are enough insects to feed on.

The predator-prey model forms a cycle of interaction, often leading to oscillations in population sizes over time.

Exponential Growth

Exponential growth is a phenomenon where a quantity, such as a population, increases at a constant proportional rate over time. For insects in our model, the equation \( \frac{dN}{dt} = 5N \) illustrates this concept clearly in the absence of predators.
In this scenario, the population of insects grows by a factor proportional to their current numbers. Mathematically, when solved, it yields the formula:
\[N(t) = N_0 e^{5t} \]where \(N_0\) is the initial population density. This equation highlights:

• Rapid Increase: The population grows very quickly, doubling and increasing exponentially.
• No Limitations: Such growth assumes no limits on resources or space. It would continue indefinitely.

However, in nature, uncontrolled exponential growth typically doesn't sustain due to factors such as limited food, space, and the presence of predators, which naturally slow down and regulate growth rates.

Biological Control

Biological control refers to the method of managing pest populations by introducing natural predators or competitors. In our context, the predator acts as a biological control agent against the agricultural pest, the insect.
In the given equations, the term \(-3PN\) represents the control exerted by predators on the insect population, while the equation \( \frac{dP}{dt} = 2PN - P \) reflects predators' growth when insects are available. This method of control offers several advantages:

• Eco-Friendly: Unlike chemical controls like insecticides, biological control is a natural approach and leaves no chemical residues.
• Sustainable: Predators continuously control pest populations as part of a balanced ecosystem.
• Targeted: Predators specifically reduce the numbers of the pest without affecting non-target species.

However, the introduction of predators alone may not be sufficient in severe cases of pest outbreak. In such scenarios, a combination of methods, including safe chemical interventions, might be necessary to achieve desired control levels.