91Ó°ÊÓ

To find equilibrium points in the context of a discrete logistic model, you need to identify when the population size remains the same from one time point to the next. This means the population won't increase or decrease—it stays constant over time. Mathematically, this is expressed by setting \( N_{t+1} = N_t \). In our problem, this condition translates to the equation \( N = 2.5N - \frac{1}{100}N^2 \), which simplifies to a quadratic equation. Solving this equation, we find two equilibrium points: when \( N = 0 \) and \( N = 150 \).

• \( N = 0 \): This represents a situation where the population is extinct.
• \( N = 150 \): Here, the population stabilizes at this size.

Finding equilibria in logistic models involves solving polynomial equations, typically quadratic, that arise from setting the population change to zero.

Stability analysis involves determining whether small perturbations around an equilibrium point will grow or decay over time. In simpler terms, we check if the population will return to equilibrium or move away if slightly disturbed. To do this, we evaluate the derivative of the population function, \( f(N) = 2.5N - \frac{1}{100}N^2 \), which gives \( f'(N) = 2.5 - \frac{1}{50}N \).

• At \( N = 0 \), the derivative \( f'(0) = 2.5 \) is greater than 1, suggesting that even a small increase in population will grow, making this an unstable equilibrium.
• At \( N = 150 \), \( f'(150) = -0.5 \) has an absolute value less than 1, which indicates that any small disturbance will decay, confirming stability.

Stability in discrete models is essentially about checking whether the equilibrium can resist small changes.

Population dynamics deals with how populations change over time under various conditions. In the discrete logistic model, the change is controlled by growth rate and self-limitation due to competition or resource constraints. The formula \( N_{t+1} = 2.5N_t - \frac{1}{100}N_t^2 \) accounts for natural growth and limits it through a density-dependent term.
Understanding these dynamics helps in:

• Modeling realistic scenarios where population does not grow indefinitely but is regulated by carrying capacity.
• Predicting future population sizes and possible scenarios if environmental conditions change.

The logistic model is a cornerstone in understanding ecological systems and resource management.

In sequence calculation, we find specific values of the population at discrete time intervals given an initial condition. Starting with \( N_0 = 10 \), we use the recursive formula to find the next population sizes. The calculations proceed step by step as follows:

• \( N_1 = 24 \)
• \( N_2 = 54.24 \)
• \( N_3 = 106.18 \)
• And so on, until \( N_{10} = 150 \)

Each term is derived by applying the formula to the previous term, iteratively.
As observed, the sequence initially increases, reflecting rapid population growth, but stabilizes as it approaches the stable equilibrium point of \( N = 150 \). Calculating sequences allows us to forecast and study how close initial values will align with equilibrium in real-world conditions.