Solve each problem. Verhulst's Model for Bacteria Growth...

Solve each problem. Verhulst's Model for Bacteria Growth Refer to Exercise 97 . If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and growth will slow. According to Verhulst's model, the number of bacteria $N_{j}$ at time $40(j-1)$ in minutes can be determined by the sequence $$N_{j+1}=\left[\frac{2}{1+\frac{N}{K}}\right] N_{j}$$ where $K$ is a constant and $j \geq 1 .$ (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.) (a) If $N_{1}=230$ and $K=5000,$ make a table of $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Round values in the table to the nearest integer. (b) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Use the window $[0,20]$ by $[0,6000]$ (c) Describe the growth of these bacteria when there are limited nutrients. (d) Make a conjecture about why $K$ is called the saturation constant. Test the conjecture by changing the value of $K$ in the given formula.

Solve each problem.
Verhulst's Model for Bacteria Growth Refer to Exercise 97 . If the bacteria are not cultured in a medium with sufficient nutrients, competition will ensue and growth will slow. According to Verhulst's model, the number of bacteria
$N_{j}$ at time $40(j-1)$ in minutes can be determined by the sequence
$$N_{j+1}=\left[\frac{2}{1+\frac{N}{K}}\right] N_{j}$$
where $K$ is a constant and $j \geq 1 .$ (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.)
(a) If $N_{1}=230$ and $K=5000,$ make a table of $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Round values in the table to the nearest integer.
(b) Graph the sequence $N_{j}$ for $j=1,2,3, \ldots, 20 .$ Use the window $[0,20]$ by
$[0,6000]$
(c) Describe the growth of these bacteria when there are limited nutrients.
(d) Make a conjecture about why $K$ is called the saturation constant. Test the conjecture by changing the value of $K$ in the given formula.

Key Concepts

Discrete Dynamical Systems

Discrete dynamical systems study the evolution of variables that change in discrete steps over time. These systems are typically represented by recurrence relations, where each future state is defined as a function of the present state. This concept is critical in understanding how iterating a specific rule can model processes which evolve in discrete time intervals.

Recurrence Relations

Recurrence relations define sequences where each term is formulated as a function of preceding terms. They provide the mathematical framework for modeling iterative processes, making them essential in fields such as population dynamics, computer science, and economics. Solving a recurrence relation often involves identifying patterns, potential limits, or convergence behaviors in the sequence.

Logistic Growth Model

The logistic growth model is a type of recurrence relation that describes populations growing in an environment with limited resources. Unlike exponential growth, which assumes unlimited resources, the logistic model incorporates a feedback mechanism that slows growth as population size increases, thereby modeling realistic scenarios such as biological populations where resource competition occurs.

Carrying Capacity (Saturation Constant)

Carrying capacity, or saturation constant, represents the maximum population size that an environment can sustain due to limited resources. In logistic growth, as the population approaches the carrying capacity, the growth rate decreases, eventually stabilizing. This concept is fundamental in ecology and biology, providing insight into how external constraints shape the growth of populations.

Graphical Analysis of Sequences

Graphical analysis involves plotting the sequence to visually assess the behavior of the system over time, such as convergence, divergence, oscillation, or stability. This method helps in understanding the dynamics at a glance and in identifying trends, such as the transition from rapid growth to stabilization as a population approaches its carrying capacity under limited resource conditions.

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