Problem 10 - Advanced Engineering Mathematics

Question

Effect For an initial population \(P_{0}\), where \(P_{0}>K\) the logistic population model (3) predicts that population cannot sustain itself over time so it decreases but yet never falls below the carrying capacity \(K\) of the ecosystem. Moreover, for \(0

Step-by-Step Solution

Answer

Modify to \( \frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) \left( \frac{P}{A} - 1 \right) \).

Step 1: Understand the logistic equation

The logistic population model is given by the differential equation \( \frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) \), where \( P \) is the population size, \( r \) is the intrinsic growth rate, and \( K \) is the carrying capacity of the environment.

Step 2: Identify the desired characteristics

We want to model a population with the logistic characteristics but that also becomes extinct when below a critical threshold \( A \). The population should decrease to extinction if \( P < A \), increase if \( A < P < K \), and saturate at \( K \) if \( P > K \).

Step 3: Modify the logistic equation

To include the threshold \( A \), modify the logistic equation to \( \frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) \left( \frac{P}{A} - 1 \right) \). This equation includes a new term \( \left( \frac{P}{A} - 1 \right) \) which contributes negatively when \( P < A \), thus driving the population to extinction.

Step 4: Construct phase portrait analysis

The new equation \( \frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) \left( \frac{P}{A} - 1 \right) \) has three equilibrium points: \( P = 0 \), \( P = A \), and \( P = K \). Analyze stability: \( P = 0 \) is unstable (population extinction), \( P = A \) is semi-stable (if \( P \) is perturbed below \( A \), it goes to 0), and \( P = K \) is stable (sustained population).

Key Concepts

Differential Equations

Differential equations are essential tools in modeling how populations change over time. In general, they describe the relationship between a function and its derivatives. For population dynamics, the rate of change of the population size is expressed as a derivative, and this is influenced by factors like current population size and environmental limitations.

The logistic population model is a common example of a differential equation used in ecology. It models populations that experience a rapid growth phase followed by a slowdown as they approach a maximum sustainable size, known as the carrying capacity. The logistic differential equation is given as:
\[ \frac{dP}{dt} = rP \left( 1 - \frac{P}{K} \right) \]
Here, \( P \) is the population size, \( r \) is the intrinsic growth rate, and \( K \) is the carrying capacity. This equation highlights that population growth not only depends on its current size but also on how far it is from the environment's threshold for supporting the population. Understanding differential equations helps in predicting long-term population trends and managing resources effectively.

Carrying Capacity

Carrying capacity is a crucial concept when discussing population dynamics. It refers to the maximum number of individuals that an environment can sustainably support without degrading.

In the context of the logistic model, carrying capacity \( K \) influences how the population grows and stabilizes. Initially, when the population is small, resources are abundant, and the growth rate is high. However, as the population size \( P \) approaches \( K \), resources become limited, and the growth rate slows down until it halts at \( K \). This is mathematically captured by the term \( 1 - \frac{P}{K} \) in the logistic equation.

Understanding carrying capacity is essential for ecological management and conservation efforts. For Example:

• It helps predict how ecosystems react to changes.
• Ensures that exploitation of resources remains at sustainable levels.

By maintaining populations near their carrying capacity, ecosystems can provide resources and services continuously without depletion.

Allee Effect

The Allee effect describes a scenario where populations at low densities have difficulty surviving and reproducing. Warder Clyde Allee highlighted this phenomenon, showing that certain populations could face extinction if reduced below a particular threshold.

The modified logistic equation includes this Allee effect by incorporating a threshold \( A \), demonstrating that populations below this level \( A < P \) are driven to extinction. This is accounted for using an additional term \( \left( \frac{P}{A} - 1 \right) \), which negatively affects population growth when \( P < A \).

This reflects real-world situations like:

• Inadequate group protection in smaller populations.
• Difficulties finding mates or cooperating for survival.

Recognizing the Allee effect in population models helps in understanding and mitigating risks to populations, particularly in conservation biology where maintaining healthy population sizes is critical. Addressing these factors ensures that even small populations can be managed for long-term sustainability.