Problem 46 - Calculus for Biology and Medicine

Question

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0\)

Step-by-Step Solution

Answer

For \(x_0 = 0\), all \(x_t\) values remain zero. The graph is a flat line at zero.

Step 1: Understand the Logistic Equation

The discrete logistic equation is a mathematical model used to describe population dynamics. Given by \(x_{t+1} = R_{0} x_{t}(1-x_{t})\), it models the changing population \(x_{t}\) over time \(t\). The variable \(R_{0}\) is a growth rate parameter, and \(x_{0}\) is the initial population at \(t=0\). In this problem, \(R_{0} = 3.1\) and \(x_{0} = 0\).

Step 2: Calculate First Iteration

Calculate \(x_{1}\) using \(x_{t+1} = R_{0} x_{t}(1-x_{t})\). With \(x_{0} = 0\), we have:\[x_{1} = 3.1 \times 0 \times (1-0) = 0.\]The population remains zero at the next time step.

Step 3: Recognize the Pattern

Notice that when \(x_{t}=0\), it leads to \(x_{t+1} = 0\) because any number multiplied by zero results in zero. Therefore, \(x_{2}, x_{3}, \ldots, x_{20}\) will all remain zero.

Step 4: Graph the Solution

Since \(x_{t}\) evaluates to zero for all \(t\), plot \(x_{t}\) on the vertical axis and time \(t\) on the horizontal axis. The graph will be a horizontal line at zero, illustrating that the population does not change from its initial state.

Key Concepts

Population Dynamics

Population dynamics refer to the branch of life sciences that studies short and long term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes. In this context, the discrete logistic equation is fundamental as it models how populations grow and reach their carrying capacity, or the maximum population size that an environment can sustainably support. This particular equation, expressed as \( x_{t+1} = R_{0} x_{t}(1-x_{t}) \), assumes that the population changes at discrete time intervals and considers both the growth potential and the limitations imposed by environmental factors.

For instance, if the initial population \( x_0 \) starts at zero, and considering that any value multiplied by zero remains zero, each subsequent calculation of the population size over time will remain at zero. This can help you understand scenarios where the initial population is not sufficient enough to lead to growth, which might reflect real-world cases where resources or conditions are inadequate for sustaining life.

Growth Rate Parameter

The growth rate parameter, denoted as \( R_0 \) in the discrete logistic equation, plays a crucial role in determining how quickly a population can grow. This parameter represents the intrinsic growth rate of the population, indicative of how fast the population can increase without any limits initially imposed by environmental resistance. In the given problem, \( R_0 = 3.1 \), suggesting a high growth potential under optimal conditions.

However, this growth potential does not always equate to a rising population, as seen when \( x_0 = 0 \). Even with a theoretical model that allows rapid growth, actual growth is dependent on initial quantities and conditions, like availability of resources and space.

• A high \( R_0 \) can suggest faster growth in early stages before reaching saturation.
• An \( R_0 \) less than 1 can indicate a declining population.

Ultimately, \( R_0 \) helps predict possible outcomes of population changes under different settings.

Mathematical Model

A mathematical model is a systematic approach used in science to represent real-world systems using mathematical language and concepts. The discrete logistic equation is one such model that effectively represents the theory of population dynamics. These models allow scientists and researchers to predict future changes and trends within populations using mathematical expressions.

The equation \( x_{t+1} = R_{0} x_{t}(1-x_{t}) \) specifically models populations under constraints, taking into account growth that is initially exponential but ultimately limited by environmental capacities. This makes it useful not only in theoretical studies but also in practical applications like wildlife management, resource allocation, and ecological forecasting.

• Models help simplify complex biological systems.
• They provide insights into population stability and potential changes over time.

Though simplified, they offer valuable predictions that can be used for planning and decision-making in sustainability and conservation projects.