Problem 4 - Calculus: Early Transcendentals

Question

Lynx eat snowshoe hares and snowshoe hares eat woody plants like willows. Suppose that, in the absence of hares, the willow population will grow exponentially and the lynx population will decay exponentially. In the absence of lynx and willow, the hare population will decay exponentially. If \( L(t), H(t), \) and \( W(t) \) represent the populations of these three species at time \( t, \) write a system of differential equations as a model for their dynamics. If the constants in your equation are all positive, explain why you have used plus or minus signs.

Step-by-Step Solution

Answer

System is:\(\frac{dW}{dt} = rW - aWH, \frac{dH}{dt} = -mH + bWH - dHL, \frac{dL}{dt} = -pL + cHL.\)

Step 1: Understanding Population Dynamics

To model the interactions in this ecosystem, we need to understand the relationships between the populations. Willow populations grow exponentially in the absence of hares. Hares feed on willows, thus affecting the growth rate of both populations. Lynx predate on hares, influencing each other's population. Therefore, the interactions are: willow-hare and hare-lynx.

Step 2: Exponential Growth of Willow Without Hares

In the absence of hares, the willow population grows exponentially. We represent this by the equation \( \frac{dW}{dt} = rW \), where \( r \) is the growth rate constant for willows.

Step 3: Hares Consume Willows

Hares consume willows, which affects the growth of willows. Thus, the willow population decreases due to consumption by hares. This is represented by \( -aWH \) in the willow equation, where \( a \) is a positive constant representing the rate of consumption.

Step 4: Exponential Decay of Hares Without Lynx and Willow

Without willows or lynx, the hare population decays exponentially. This is captured by the equation \( \frac{dH}{dt} = -mH \), where \( m \) is a positive constant representing the natural decay rate of hares.

Step 5: Hares Feeding on Willows

Hares grow by consuming willows, so the hare population increases with the availability of willows. This is represented by \( +bWH \) in the hare equation, where \( b \) is a positive constant showing increase due to consumption of willows.

Step 6: Lynx Prey on Hares

Lynx feed on hares, so the lynx population growth depends positively on hare population, represented by \( +cHL \) where \( c \) is a positive constant for predation.

Step 7: Exponential Decay of Lynx Without Hares

Without hares, the lynx population decays exponentially, given by \( \frac{dL}{dt} = -pL \), where \( p \) is the natural decay rate of lynx.

Step 8: Assembling the Differential Equations

Combining these interactions, we have the system: \[\begin{align*}\frac{dW}{dt} &= rW - aWH, \\frac{dH}{dt} &= -mH + bWH - dHL, \\frac{dL}{dt} &= -pL + cHL.\end{align*}\]All constants are positive and signs are chosen to reflect interactions: growth (+) and decline (-) due to predation or consumption.

Key Concepts

Population Dynamics

Population dynamics explores how the sizes of populations change over time and the factors driving these changes. In an ecological setup, this may involve several species interacting with each other and the environment. For example, in our exercise, we have a system of willows, hares, and lynx. Each species' population changes based on interactions with the others.

• Willows grow by consuming resources like sunlight and nutrients.
• Hares eat willows, influencing willow growth by their consumption rate.
• Lynx, predators of hares, have their survival and birth rates tied to the availability of hares.

This interplay results in complex dynamics captured by equations that model these relationships. Understanding these interactions is key to predicting how changes in population size can impact an ecosystem over time.

Exponential Growth

Exponential growth occurs when the growth rate of a population is proportional to its current size. This means that as the population grows, the increase over successive time periods becomes larger and larger. In the absence of limiting factors, such as predators, populations can grow rapidly. For instance, if willows are left unchecked by hares, their population will increase exponentially, represented mathematically by the equation: \[\frac{dW}{dt} = rW\]Here, \( W \) is the willow population, \( r \) is a positive growth rate constant, and \( \frac{dW}{dt} \) signifies the change in willow population over time. This equation highlights the basic principle that more willows lead to even more willows in the next time step, provided resources are unlimited.

Predator-Prey Model

The predator-prey model is an essential aspect of population dynamics that explains the interaction between two species: a predator and its prey. In our example, hares are preyed upon by lynx, and this relationship impacts their population sizes over time. The equations used to express these dynamics are derived from the basic Lotka-Volterra equations, capturing both exponential decay and growth:

• The hare population, \( H \), increases as it feeds on willows (\( +bWH \)), and decreases naturally and due to lynx predation (\(-mH\) and \(-dHL\)).
• The lynx population, \( L \), grows by feeding on hares (\( +cHL \)), but diminishes naturally without enough prey (\(-pL\)).

This interplay creates a cyclical pattern where lynx populations rise following increases in hare populations, only to fall once the hares become scarce. This steady back-and-forth is a hallmark of predator-prey dynamics, providing fascinating insights into the balance of ecosystems.