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

Provide an explanation or a 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 approaches a constant as time goes to infinity. c. In a predator-prey model, if the initial predator population is zero, and the initial prey population is positive, then the prey population will increase without bound. Answer: a. True. If the growth rate function is positive, the change in population over time is also positive, which indicates an increasing population. b. False. A counterexample is provided by the stirred tank system governed by the equation \(\frac{dx}{dt} = \sin(t)\), where the solution oscillates and does not approach a constant as \(t \rightarrow \infty\). c. True. When the predator population is initially zero, the prey population will not have any predators to limit their growth, and their population will increase exponentially without bound.

Step-by-step solution

Statement A: Growth Rate Function and Population Increase

If the growth rate function for a population model is positive, it means that the population is growing at some rate. In other words, the change in the population over time (dP/dt) is positive. Mathematically, if \(g(P) > 0\), then \(\frac{dP}{dt} > 0\). Since the rate of change in population is positive, it implies that the population is indeed increasing. So, statement A is true. The explanation is that if the growth rate function is positive, the change in population over time is also positive, which indicates an increasing population.

Statement B: Stirred Tank Initial Value Problem

The statement is not necessarily true. In order to assess whether a stirred tank initial value problem approaches a constant as \(t \rightarrow \infty\), we need to analyze the differential equation that governs the stirred tank system. The behavior of the solution depends on the parameters and the functional form of the governing equations. There can be cases where the solution does not approach a constant or may oscillate. As a counterexample, consider a stirred tank problem with the governing equation \(\frac{dx}{dt} = \sin(t)\). In this case, the solution does not approach a constant value as \(t \rightarrow \infty\) but continues to oscillate between -1 and 1. So, statement B is false. A counterexample is provided by the stirred tank system governed by the equation \(\frac{dx}{dt} = \sin(t)\), where the solution oscillates and does not approach a constant as \(t \rightarrow \infty\).

Statement C: Predator-Prey Model

In predator-prey models like the Lotka-Volterra equations, if the predator population is initially zero (\(P(0) = 0\)) and the prey population is initially positive (\(V(0) > 0\)), then the systems of equations become: 1. \(\frac{dP}{dt} = -aP + bPV\), where \(P\) is the predator population, 2. \(\frac{dV}{dt} = rV - cPV\), where \(V\) is the prey population. Since the initial predator population is zero, the equations simplify to: 1. \(\frac{dP}{dt} = 0\), 2. \(\frac{dV}{dt} = rV\). The solution for the prey population becomes \(V(t) = V(0)e^{rt}\), which means the prey population increases exponentially as time progresses. So, statement C is true. The explanation is that when the predator population is initially zero, the prey population will not have any predators to limit their growth, and their population will increase exponentially without bound.

Key concepts

Population Models

Population models are mathematical frameworks used to study how populations change over time. These models are essential in understanding the dynamics of populations, such as wildlife, humans, or even microorganisms. The basis of these models is to evaluate factors that influence population growth or decline.

Commonly, these factors include birth rates, death rates, immigration, and emigration. **Different types of population models:**

• Exponential Growth Model: Assumes unlimited population growth, with a constant rate of increase.
• Logistic Growth Model: Considers carrying capacity, the maximum population size that an environment can sustain.

These models help biologists and ecologists make predictions about population trends and understand the potential impact of changes in the environment or conservation efforts.

Growth Rate Function

The growth rate function is a critical component of population models. It defines the rate at which a population changes over a given period. It is essentially the derivative of the population size with respect to time, denoted as \( \frac{dP}{dt} \).

This function helps in understanding whether a population is increasing or decreasing. **Significance of Growth Rate Function:**

• If \(g(P) > 0\), the population is increasing because the rate of change is positive.
• If \(g(P) < 0\), the population is decreasing due to a negative rate of change.
• If \(g(P) = 0\), the population remains constant, neither increasing nor decreasing.

Interpreting the growth rate function allows scientists to predict the future size of a population and to understand the factors driving these trends.

Predator-Prey Models

Predator-prey models are specialized population models used to describe the interactions between two species: one as prey and the other as a predator. These models help to understand the cyclical relationship and dynamics between the two populations.

The Lotka-Volterra equations are the most classic example. They describe how the prey and predator populations influence each other's growth rates.
**Key equations include:**

• Prey growth: \( \frac{dV}{dt} = rV - cPV\), where \(V\) is prey, \(r\) is the growth rate, and \(c\) represents the interaction rate with predators.
• Predator growth: \( \frac{dP}{dt} = -aP + bPV\), where \(P\) is predators, \(-a\) is the death rate in absence of prey, and \(b\) is the growth rate due to consuming prey.

These models show that when predators are absent, prey populations grow exponentially, as seen in the issue where the initial predator population is zero.

Initial Value Problems

Initial value problems (IVPs) are mathematical problems used in differential equations where the solution is determined by an initial condition. For these problems, the primary goal is to find a function that satisfies the differential equation and the initial conditions provided.

IVPs are crucial in many real-world applications like engineering, physics, and even ecology. **Steps to solve IVPs include:**

• Specify the differential equation that represents the system's behavior.
• Define the initial condition or initial state of the system.
• Use mathematical techniques to solve the equation, incorporating the initial condition.

Once solved, an IVP provides a unique solution that defines the system's state over time based on its initial conditions, crucial for accurate predictions and analyses.