Problem 11 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Some population scientists have argued that population density can get so low that reproduction will be less than natural attrition and the total population will be lost. Named the Allee effect. after W. C. Allee who wrote extensivelv about it \(^{6}\), this mav be a basis for arguing. for example that marine fishing of a certain species (Atlantic cod on Georges Bank, for example \({ }^{7}\) ee Paul Greenberg, "Four Fish, the Future of the Last Wild Food, The Penguin Press, 2010 . The model is a lot more complicated than we present here.) should be suspended, despite the presence of a small residual population. How should we modify the logistic differential equation, \(u^{\prime}=u(1-u),\) to incorporate such a threshold? Assume a fixed area and uniform density throughout the area and a threshold number, \(\epsilon .\) If the population number is less than \(\epsilon\) the population will decline; if the population number is more than \(\epsilon\) the population will increase. a. Modify the direction field for \(u^{\prime}=u(1-u)\) to account for the Allee effect. That is, a \(u, t\) plane, draw the line \(u=1\) and a threshold line \(u=\epsilon\) where \(\epsilon=0.1,\) say. Arrows below \(u=\epsilon\) should point downward; arrows between \(u=\epsilon\) and \(u=1\) should point upwards. Draw enough direction field arrows to indicate the paths of solutions for a threshold model. b. Draw the the logistic phase plane graph and the phase plane graphs for the following three candidates of a threshold logistic differential equation where \(\epsilon=0.1\). \(u^{\prime}=f(u)=u \times(1-u) \quad\) Logistic \(u^{\prime}=f_{1}(u)=u^{\frac{2}{3}} \times(u-\epsilon)^{\frac{1}{3}} \times(1-u) \quad\) Candidate 1 \(u^{\prime}=f_{2}(u)=u \times \frac{u-\epsilon}{u+\epsilon} \times(1-u)\) Candidate 2 \(u^{\prime}=f_{3}(u)=u \times(u-\epsilon) \times(1-u)\) Candidate 3

Step-by-Step Solution

Answer

Modify logistic equation for threshold impact; test phase planes for candidates.

Step 1: Understand the Logistic Differential Equation

The given logistic differential equation is \( u^{\prime}=u(1-u) \). This describes a population growth model where the growth rate is proportional to both the current population \( u \) and the remaining capacity \( (1-u) \). It assumes a simple scenario without any Allee effect.

Step 2: Understand the Allee Effect

The Allee effect describes a situation where population growth is negative when the population is too small. Below a threshold \( \epsilon \), population decline occurs because low density impacts reproduction negatively.

Step 3: Modify the Direction Field

Incorporate the threshold \( \epsilon = 0.1 \). Below \( u=0.1 \), draw arrows pointing downward indicating the population is declining. Between \( u=0.1 \) and \( u=1 \), draw arrows pointing upwards to show population growth. Ensure these arrows form a clear pattern across the \( u, t \) plane.

Step 4: Analyze Candidate 1

Candidate 1 is represented by \( u^{\prime}=f_{1}(u)=u^{\frac{2}{3}} \times (u-\epsilon)^{\frac{1}{3}} \times (1-u) \). This model increases growth response for small populations by modifying the exponent on \( u \) and the root around the \( u-\epsilon \) term.

Step 5: Analyze Candidate 2

Candidate 2 uses \( u^{\prime}=f_{2}(u)=u \times \frac{u-\epsilon}{u+\epsilon} \times (1-u) \). It provides a continuous function that smoothly transitions growth rates around the threshold \( \epsilon \).

Step 6: Analyze Candidate 3

Candidate 3 proposes \( u^{\prime}=f_{3}(u)=u \times (u-\epsilon) \times (1-u) \). Here, population growth linearly depends on \( u-\epsilon \), directly causing \( u^{\prime} \) to be negative if \( u < \epsilon \).

Step 7: Draw Phase Plane Graphs

For each candidate, depict the phase plane graphs. Illustrate stability and the equilibrium point behavior. Show how each function affects population growth or decline across the threshold \( u = 0.1 \). The arrows in the phase plane should reflect population dynamics below, at, and above this threshold in each candidate's context.

Key Concepts

Allee Effect

The Allee Effect is an important concept in ecology and population dynamics. It describes the phenomenon where a population's growth rate decreases at low population densities. This can result in negative growth, meaning the population declines instead of grows. Think of it as a biological challenge. With fewer individuals, the chances of finding mates decrease, or there might not be enough collective resources to protect themselves, leading to a decline.

In practical terms, imagine a species of fish in the ocean. If there's not enough of them in one area, they might not reproduce efficiently, leading to the population getting smaller over time. This critical decline happens below a certain threshold, denoted as \( \epsilon \).

Understanding this effect is crucial for managing species that are endangered or under threat due to low numbers. It gives scientists and conservationists insights into when a population is too low to sustainably recover, prompting actions like habitat conservation or fishing restrictions.

Population Density

Population density refers to the number of individuals living within a particular area. It’s a key factor in understanding the dynamics of populations. A very low population density can lead to the Allee effect, where reproduction rates decrease because individuals are spread too far apart. Conversely, too high a density can lead to competition for resources, which might not support everyone, thus slowing down growth.

Having a balance in population density is essential for the health and sustainability of a species. The right density allows for efficient reproduction and resource sharing while maintaining the ecosystem’s health. In mathematical models, like in this exercise, a density threshold \( \epsilon \) helps simulate and analyze these conditions.

By examining how population density affects individuals, models can predict potential outcomes, such as extinction or overpopulation. This makes population density a central concept in ecological studies and conservation efforts.

Mathematical Modeling

Mathematical modeling is a powerful tool in understanding population dynamics like those described by the logistic differential equation. This equation, \( u^{\prime}=u(1-u) \), models how a population grows. It highlights how growth is influenced by the current population size \( u \) and the remaining capacity to grow \( 1-u \).

By introducing the Allee effect into this model, we can modify the equation to include a threshold \( \epsilon \) below which the population starts to decline. The candidates in the exercise demonstrate different mathematical approaches to incorporate this threshold:

• Candidate 1: Considers power terms to reflect enhanced growth potential at different stages.
• Candidate 2: Uses a fraction to transition smoothly through the threshold \( \epsilon \).
• Candidate 3: Introduces linear adjustment around the threshold for simplicity.

Each approach provides insights into different scenarios, helping researchers choose the most appropriate model for studying specific populations.

Phase Plane Analysis

Phase plane analysis is a method used to study differential equations and visualize the behavior of dynamic systems such as population models. In a phase plane, each point represents a possible state of the system with the axes typically showing population size over time.

In this exercise, the phase plane analysis involves drawing graphs for each candidate model. These graphs visually illustrate how populations change over time, particularly around the threshold \( \epsilon \). Arrows are drawn to show whether the population is increasing or decreasing above and below this level.

By examining these phase plane graphs, we can:

• Identify equilibria where populations remain stable.
• Understand how quickly populations might recover from low densities.
• Visualize how populations respond to different conditions imposed by the logistic models.

This analysis is crucial for determining system stability and evaluating long-term population dynamics, providing a clear visual summary of the implications of each model considered.