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The exponential growth model simulates population growth where each individual produces new offspring at a constant reproductive rate, R, each generation. The population size increases dramatically over time following the equation NT+1 = NT + (1 + R) × NT. This model demonstrates unrestricted growth that cannot be sustained indefinitely in nature due to resource limitations.

The logistic growth model incorporates carrying capacity, K, which limits population growth as resources become scarce. Populations grow initially but slow as they approach carrying capacity, creating an S-shaped curve. Unlike exponential growth, logistic growth reflects realistic population dynamics where growth rate decreases as population size increases relative to available resources.

When a population exceeds carrying capacity, the population size decreases in the subsequent generation. The logistic equation accounts for this by reducing growth rate as population approaches or surpasses carrying capacity. This negative feedback mechanism prevents unlimited growth and stabilizes the population near the carrying capacity limit.

In the predator-prey model, prey populations increase when predators are scarce, but as predators consume more prey, predator populations grow. Increased predators then reduce prey populations, causing predators to subsequently decline from starvation. This creates cyclical population fluctuations where each species' growth depends on the other's population size, demonstrating interdependent population dynamics.

The growth rate R represents the proportion of new offspring produced by each individual per generation. Higher R values produce larger population sizes after a fixed number of generations. In exponential growth simulations, increasing R dramatically accelerates population expansion, demonstrating how reproductive rate directly determines population trajectory independent of carrying capacity constraints.

Initial population size influences the absolute population numbers achieved after a set number of generations in exponential growth models. Larger starting populations result in proportionally larger final populations when growth rate remains constant. However, the growth pattern shape remains similar regardless of initial size, as the exponential growth rate determines the trajectory.

The maximum growth rate, R max, determines how quickly populations approach carrying capacity in logistic growth models. Higher R max values cause populations to reach carrying capacity faster, while lower values result in slower approach. Carrying capacity itself sets the population limit, but R max controls the speed at which populations converge to that limit.