Problem 32 - Calculus: Early Transcendental Functions

Question

Many species of trees are plagued by sudden infestations of worms. Let \(x(t)\) be the population of a species of worm on a particular tree. For some species, a model for population change is \(x^{\prime}=0.1 x(1-x / k)-x^{2} /\left(1+x^{2}\right)\) for some positive constant \(k .\) If \(k=10,\) show that there is only one positive equilibrium solution. If \(k=50,\) show that there are three positive equilibrium solutions. Sketch the direction field for \(k=50 .\) Explain why the middle equilibrium value is called a threshold. An outbreak of worms corresponds to crossing the threshold for a large value of \(k\) ( \(k\) is determined by the resources available to the worms).

Step-by-Step Solution

Answer

For \(k = 10\), there is only one positive equilibrium at \(x = 0\). But for \(k = 50\), we find three positive equilibrium solutions, with the middle one acting as a threshold that, when surpassed, leads to an outbreak of the population.

Step 1: Set Up the Equation

The differential equation given in the problem is \(x^{\prime} = 0.1x(1 - x/k) - x^{2}/(1+x^{2})\). We want to find values of \(x\) at equilibrium. At equilibrium, the rate of change \(x^{\prime}\) is 0, so set the equation equal to 0.

Step 2: Find the Equilibrium for \(k=10\)

Plug \(k = 10\) into the differential equation and solve for \(x\). We get \(0 = 0.1x(1 - x/10) - x^{2}/(1+x^{2})\). After simplifying, the only positive equilibrium we obtain is \(x = 0\).

Step 3: Find the Equilibrium for \(k=50\)

Using the same process, set \(k = 50\) in the differential equation and solve for \(x\). We get \(0 = 0.1x(1 - x/50) - x^{2}/(1+x^{2})\). After solving this equation, it yields three positive equilibrium solutions.

Step 4: Establish the Threshold

The middle equilibrium value corresponds to a 'threshold'. When the population surpasses this threshold, it causes an outbreak, meaning the population begins to increase rapidly. This threshold is known as the bifurcation point where behavior shifts between stable and unstable.

Key Concepts

Differential Equations

Differential equations play a crucial role in modeling how populations change over time. These equations describe the relationship between a function and its derivatives. In population dynamics, like in the case of worms infesting trees, the differential equation \[x^{\prime} = 0.1x(1 - x/k) - x^{2}/(1+x^{2})\]represents the rate of change of the worm population, \(x(t)\), at any given time \(t\).

To find equilibrium solutions, we set the equation to zero, because at equilibrium, the population does not change, i.e., \(x' = 0\). Thus, the task is to find values of \(x\) such that the right-hand side of the equation becomes zero. This modeling helps scientists and ecologists predict future population sizes and understand under which conditions populations grow or decline.

Population Dynamics

Population dynamics is a field of study that examines how populations change over time. With worms, trees, and other ecological systems, it involves analyzing birth rates, death rates, and interactions with the environment.

The equation for population dynamics often includes terms that represent growth rates and limiting factors. In the worm model, the term \(0.1x(1 - x/k)\) describes how resources limit growth as the population reaches the carrying capacity \(k\). Meanwhile, the term \(x^2/(1+x^2)\) represents other biological factors affecting the population, such as disease or predation.

• Initial Population: Starting value of \(x\) at \(t = 0\).
• Carrying Capacity: The maximum population size the environment can sustain.
• Surrounding Resources: How availability of resources influences \(x\).

Understanding these dynamics helps manage populations and prevent overgrowth or underpopulation, which is critical for ecological balance.

Bifurcation Point

Bifurcations are points in the behavior of a dynamical system where a small change in a parameter value, such as \(k\), causes a sudden qualitative change in its long-term behavior.

In our worm population model, a bifurcation point is reached when the parameter \(k\) changes, altering the number of equilibrium points. For example, at \(k=10\), there is only one equilibrium point, but at \(k=50\), three appear. This implies a shift in the system dynamics.

• Stable Equilibrium: Small deviations diminish; population returns to equilibrium.
• Unstable Equilibrium: Small deviations grow; population moves away from equilibrium.
• Threshold or Bifurcation Point: A neutral point where the system's stability pivots.

Understanding bifurcation points is key to predicting dramatic changes in population behavior, such as outbreaks, and helps in creating strategies for managing and controlling populations.