Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the growth rate function for a population model is positive, then the population is increasing. b. The solution of a stirred tank initial value problem always approaches a constant as \(t \rightarrow \infty\) c. In the predator-prey models discussed in this section, if the initial predator population is zero and the initial prey population is positive, then the prey population increases without bound.

Short answer

Answer: True.

Step-by-step solution

Statement a: Understanding population growth rate function

The population growth rate function is usually represented as a derivative with respect to time, denoted as \(p'(t)\). When this function is positive, it implies that the rate of change of the population with time is positive, meaning the population is increasing. If this function is negative, it means the population is decreasing.

Statement a: Conclusion

Since the growth rate function is positive, the population is, in fact, increasing in this case. Therefore, statement a is true.

Statement b: Stirred tank initial value problem

A stirred tank initial value problem involves a liquid in a tank being mixed (or stirred) in which the concentration of a substance undergoes changes over time. The behavior of this system is governed by a differential equation that describes the rate of change of the concentration. Considering an ordinary stirred tank problem (assuming perfect mixing and no accumulation), the solution is generally an exponential decay of the form \(c(t) = c_0 e^{-kt}\), where \(c(t)\) represents the concentration at time \(t\), \(c_0\) is the initial concentration, \(k\) is a constant and \(e\) is Euler's number.

Statement b: Conclusion

As time (\(t\)) goes to infinity, the exponential term \(e^{-kt}\) approaches zero. Therefore, the concentration \(c(t)\) approaches zero as well. For this typical stirred tank problem, the solution always approaches a constant (in this case, zero) as the time goes to infinity. Thus, statement b is true. However, keep in mind that this doesn't have to hold for any stirred tank problem, there might be specific cases where this is not true.

Statement c: Predator-prey models

In predator-prey models, we use a system of differential equations to model the interaction between predator and prey populations in an ecosystem. A famous example of such a model is the Lotka-Volterra model, consisting of the following two equations: \(dP/dt = \alpha P - \beta P Q\) (Change in prey population) \(dQ/dt = \delta P Q - \gamma Q\) (Change in predator population) where \(P\) represents the prey population, \(Q\) represents the predator population, \(dP/dt\) and \(dQ/dt\) are the rates of change of prey and predator populations respectively, and \(\alpha\), \(\beta\), \(\delta\), and \(\gamma\) represent some constants. Now, let's analyze the given scenario: Predator population is initially zero, \(Q(0) = 0\), and prey population is initially positive, \(P(0) > 0\).

Statement c: Evaluating the prey population when the predator population is initially zero

Since \(Q(0) = 0\), the predator population equation simplifies to: \(dQ/dt = -\gamma Q\). Since \(Q = 0\) initially, there will be no growth in the predator population throughout the time evolution. Now let's examine the effect on the prey population equation. Since \(Q=0\), we have: \(dP/dt = \alpha P\). This simplifies to a first-order linear equation whose solution can be written as \(P(t) = P(0) e^{\alpha t}\), where \(P(0)\) is the initial prey population and \(\alpha > 0\) is a constant.

Statement c: Conclusion

As time (\(t\)) goes to infinity, the term \(e^{\alpha t}\) increases without bound (as \(\alpha > 0\)). Hence, in the given case, with no predator population to control the prey population, the prey population will increase without bound, as described by the solution we derived. Thus, statement c is true.

Key concepts

Population Growth Rate

Understanding the population growth rate is critical in assessing how a population, be it of organisms, cells, or even people, changes over time. The growth rate function, typically denoted as \(p'(t)\), signifies the rate of change in population size with respect to time. Positive values of the growth rate function indicate an increase in population size, which implies an environment conducive to growth, whether it be ample resources or low predation.
A key point to remember is that a positive growth rate may not always equate to exponential growth; it simply means the population is increasing at the time of measurement. Various factors, such as carrying capacity and environmental resistance, can affect long-term trends, possibly leading to a population plateau or even a decline if conditions change. Hence, it's important not to view population growth in a vacuum but as a dynamic process influenced by a multitude of factors.

• The population growth rate function reflects the rate at which a population size changes over time.
• A positive value indicates population increase while a negative implies a decrease.
• Long-term population trends are influenced by additional ecological and environmental factors.

Stirred Tank Initial Value Problem

The stirred tank initial value problem is a fascinating example of how differential equations can model real-world systems. Here, we delve into the context of a substance being mixed in a stirred tank, leading to concentration changes over time. This problem typifies scenarios in fluid dynamics which are crucial in engineering and various scientific fields.
In an ideal scenario, where the tank is perfectly mixed and there's no accumulation, we often find an exponential decay in concentration, expressed with the equation \(c(t) = c_0 e^{-kt}\). However, if considering more complex scenarios involving factors like an influx of substances, nonlinear relationships, or imperfect mixing conditions, we face differing behaviors that defy the simple tendency towards a constant concentration.

• The stirred tank model showcases how concentrations change in a liquid substance over time as it's mixed.
• In simple scenarios, the model predicts that concentration levels will stabilize, or approach a constant value as time progresses.
• More complex situations may deviate from this pattern, stressing the diversity of real-world systems.

Lotka-Volterra Model

The Lotka-Volterra model stands as a seminal concept in population dynamics, showcasing the interaction between predators and their prey through a set of equations. It outlines the sustenance of predator populations by the consumption of prey and, conversely, how the burgeoning of prey populations is tempered by predation.
The model includes equations \(dP/dt = \alpha P - \beta P Q\) for the prey and \(dQ/dt = \delta P Q - \gamma Q\) for the predators. Here, a key feature is the cyclical interaction: as prey increases, predators also increase, leading to a higher predation rate that eventually decreases the prey, which in time also reduces the predator population, and so forth.
This model not only provides insight into the interdependence of species within an ecosystem but also illustrates how mathematical models can mirror biological processes.

• The Lotka-Volterra model mathematically formulates the biological interactions between predator and prey species.
• It highlights cyclic dynamics driven by the mutual influence of both populations on each other.
• The model lays the groundwork for understanding complex ecosystem interactions through a mathematical lens.

Predator-Prey Dynamics

Predator-prey dynamics form a cornerstone of ecological studies, explaining how the existence and interaction of predator and prey species shape population sizes and ecosystem stability. These dynamics adhere to a fundamental principle: predators survive by consuming prey, while prey must evade predation to survive and reproduce.
Understanding these relationships helps in recognizing the delicate balance within ecosystems, where too many predators can lead to the depletion of prey, and an overabundance of prey without natural predators can cause environmental degradation due to overgrazing or overpopulation.
In a scenario where there are no predators, as seen with \(Q(0) = 0\), the model predicts an unbounded growth of the prey population given by \(P(t) = P(0) e^{\alpha t}\), emphasizing the critical role predators play in regulating prey numbers and maintaining ecological equilibrium.

• Predator-prey dynamics are pivotal in sustaining the ecological balance.
• The absence of predators can lead to unrestricted prey population growth.
• These dynamics are essential in maintaining the integrity and health of ecosystems.