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The Verhulst model, also known as the logistic growth model, is commonly used to describe population growth in a confined environment. It considers the idea that populations grow rapidly when they are small but the growth rate decreases as the population size approaches the carrying capacity. This is formulated as:\[ P'(t) = r \times P(t) \times \left(1 - \frac{P(t)}{M}\right) \]where:

• \( P'(t) \) is the rate of change of the population.
• \( r \) is the intrinsic growth rate.
• \( M \) is the carrying capacity of the environment.
• \( P(t) \) is the population size at time \( t \).

When we set \( P'(t) = 0 \), the solutions we find are the equilibrium points. These occur at \( P(t) = 0 \) or \( P(t) = M \). This demonstrates two stable points: one at zero population, and another at population capacity. It's a simple yet powerful tool for understanding how populations stabilize in natural settings.

The Ricker model is another method of modeling population growth, primarily used in ecology for understanding stock-recruitment relationships. Unlike the Verhulst model, the Ricker model incorporates a diminishing return of population size due to overcrowding effects in the equation:\[ P'(t) = r \times P(t) \times \frac{Ae^{-P(t) / \beta} - 1}{A-1} \]Key variables include:

• \( A \), which modifies the effect of population size on growth.
• The exponential term \( e^{-P(t) / \beta} \), capturing how growth is hindered as the population grows.

Setting \( P'(t) = 0 \) results in equilibrium solutions at \( P(t) = 0 \) and \( P(t) = -\beta \ln(1/A) \). These indicate points where population ceases to speed up, either due to decline from scarcity or the growth limitation as it nears a certain size dictated by the interaction between population and resource availability.

The Beverton-Holt model is a popular choice for modeling species populations in ecology, focusing mainly on fish. It is represented by:\[ P'(t) = \frac{r \times P(t)}{1 + P(t) / \beta} \]Here:

• \( r \) is the growth rate constant.
• \( \beta \) affects the population scaling.

Solving \( P'(t) = 0 \) gives \( P(t) = 0 \) as the only solution, indicating the absence of alternative equilibrium points in the basic form. The Beverton-Holt model's simplicity makes it easy to understand the basic population dynamics, showing how populations balance growth against sustainable levels, particularly in fish stocks where recruitment could be represented by available habitat and breeding conditions.

The Gompertz model is frequently applied in the fields of population ecology and epidemiology. It describes diminishing growth rates as size increases, represented by:\[ P'(t) = -r \times P(t) \times \ln \left( \frac{P(t)}{\beta} \right) \]Parameters include:

• \( r \) dictating the growth rate.
• \( \beta \) is associated with the population size when growth starts to slow significantly.

Setting \( P'(t) = 0 \) gives two solutions: \( P(t) = 0 \) and \( P(t) = \beta \). This model highlights how growth decelerates exponentially as the population size nears the carrying capacity, being particularly useful for modeling growth dynamics where initial rapid expansion gives way to a plateau, like bacterial or tumor growth.