Question

The following problems add in a minimal threshold value for the species to survive, \(T\), which changes the differential equation to \(P^{\prime}(t)=r P\left(1-\frac{P}{K}\right)\left(1-\frac{T}{P}\right)\)Draw the directional field of the threshold logistic equation, assuming \(K=10, r=0.1, T=2 .\) When does the population survive? When does it go extinct?

Short answer

The population survives if it exceeds the threshold \(T = 2\). It goes extinct if \(P < 2\).

Step-by-step solution

Understand the Directional Field

A directional field is a visual representation of a differential equation showing the slope of solutions passing through different points in the plane. In this case, it's generated using the logistic equation with a threshold factor.

Identify Parameters

We are given the parameters: the carrying capacity, \(K = 10\), the growth rate, \(r = 0.1\), and the threshold value \(T = 2\). These parameters will be used to determine the dynamics of the population \(P\).

Set Up the Differential Equation

The differential equation is: \[ P'(t) = r P \left(1 - \frac{P}{K}\right)\left(1 - \frac{T}{P}\right) \]Substituting the given values, we have:\[ P'(t) = 0.1 P \left(1 - \frac{P}{10}\right)\left(1 - \frac{2}{P}\right) \]

Analyze Fixed Points

To find the fixed points, set \(P'(t) = 0\). Solve the equation:\[ 0.1 P \left(1 - \frac{P}{10}\right)\left(1 - \frac{2}{P}\right) = 0 \]The solutions are \(P = 0\), \(P = 2\), and \(P = 10\).

Determine Stability

Analyze the stability of these fixed points:- At \(P = 0\), the population goes extinct. This point is unstable since any small amount of population will grow if it exceeds \(T\).- At \(P = 2\), this is a stable threshold point, meaning if the population reaches this point, it will survive.- At \(P = 10\), this is the stable equilibrium, representing the carrying capacity when the population fully stabilizes without external factors influencing it.

Sketch the Directional Field

Create a sketch or use software to illustrate the directional field:- Draw arrows pointing right (increasing) where \(P > 2\) and \(P < 10\).- Draw arrows pointing left (decreasing) where \(P < 2\).- No change at \(P = 2\) and \(P = 10\) as these are equilibrium points.

Key concepts

Directional Field

Understanding a directional field is fundamental in the study of differential equations. It is essentially a graphical tool used to visualize the behavior of solutions to these equations. When you plot a directional field, you plot small line segments or arrows. Each of these indicates the slope of the solution at a particular point in the plane. This gives you an overview of how solutions behave without solving the equation fully.

For the threshold logistic equation provided, the directional field will showcase how the population changes over time, depending on its initial size. This allows us to see where the population would increase or decrease, which is incredibly useful for understanding the overall dynamics of the system.

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. These equations are important in modeling situations where change occurs continuously. For example, they are widely used in physics, engineering, and economics.

The specific differential equation in question is a threshold logistic equation, which is an extension of the standard logistic equation. It is written as: \[ P'(t) = r P \left(1 - \frac{P}{K}\right)\left(1 - \frac{T}{P}\right) \] This equation considers not only the carrying capacity but also a threshold value, altering how population growth is represented.

Population Dynamics

Population dynamics is the study of how populations change over time. This can be influenced by factors like birth rates, death rates, immigration, and emigration. In the context of the threshold logistic equation, population dynamics is modeled mathematically to understand how a population will grow or decline under certain conditions.

In this scenario, given the threshold value, populations below this threshold might shrink, risking extinction. On the other hand, populations above this threshold can grow towards the carrying capacity, stabilizing over time once they reach it.

Threshold Value

The threshold value is a specific parameter within the threshold logistic equation, represented by \(T\). It signifies a critical population size below which the population cannot sustain itself.

If the population size falls below this threshold, the rate of growth becomes negative, leading to a decline.

• A population size \(P < T\) could mean extinction, as seen from the test exercise.
• Conversely, when \(P > T\), the population growth may become positive, allowing the population to increase towards the carrying capacity.

The threshold value thus acts as a crucial tipping point in population dynamics.

Fixed Points

Fixed points in a differential equation are the solutions where the derivative, or rate of change, is zero. This means the population size does not change at these points; it remains constant.

For the equation \[ P'(t) = 0.1 P \left(1 - \frac{P}{10}\right)\left(1 - \frac{2}{P}\right) \], the fixed points are found by solving \( P'(t) = 0 \). By doing so, we identify \( P = 0 \), \( P = 2 \), and \( P = 10 \) as fixed points. Each represents a different scenario:

• \(P = 0\): The population is extinct.
• \(P = 2\): Represents the threshold for survival.
• \(P = 10\): Represents the carrying capacity of the environment.

Stability Analysis

Stability analysis involves determining whether a fixed point is stable or unstable. A stable fixed point will attract nearby trajectories, while an unstable one will repel them. In the threshold logistic equation context, this means determining whether slight changes in population will result in a return to the fixed point or a divergence from it.

- At \(P = 0\), it's unstable; the population will grow if it starts above \(T\).
- At \(P = 2\), it's a stable threshold. If population reaches this point, small deviations will not cause collapse.
- At \(P = 10\), it is stable, reflecting the natural limit due to environmental resources.

This understanding helps in predicting the long-term viability of a population subject to various pressures.