Consider the equation $$ N_{++1}=N, \exp \left[r\left(1-N_{1} / K\right)\right] $$ This equation is sometimes called an analog of the logistic differential equation (May, 1975). The equation models a single-species population growing in an environment that has a carrying capacity \(K\). By this we mean that the environment can only sustain a maximal population level \(N=K\). The expression $$ \lambda=\exp \left[r\left(1-N_{1} / K\right)\right] $$ reflects a density dependence in the reproductive rate. To verify this observation, consider the following steps: (a) Sketch \(\lambda\) as a function of \(N\). Show that the population continues to grow and reproduce only if \(N

Let's understand the single-species population dynamics in the given equation. Here, we are looking at how the population size of a single species changes over time based on its growth rate and the environment's limits on that growth. The equation provided models this by incorporating factors that influence population changes such as the intrinsic growth rate and environmental resources. In mathematical terms, we are considering the equation:

\(N_{n+1} = N_n \exp\big[r\big(1 - \frac{N_n}{K}\big)\big]\).

This equation tells us how the population at the next time step \(N_{n+1}\) is determined by the current population size \(N_n\), an exponential growth factor, and the carrying capacity of the environment (K). By examining this equation, we can better understand how population sizes fluctuate under different conditions. This understanding is invaluable when managing wildlife reserves, agricultural planning, and similar areas that require knowledge of population trends.

Carrying capacity (\(K\)) is a critical concept in ecology. It represents the maximum population size that an environment can sustain indefinitely. This occurs due to limitations in resources such as food, habitat, and other essentials needed for survival. In our equation, the term \(K\) determines this limit.

When the population size \(N\) is below the carrying capacity, the population grows because the environment can support more individuals. Conversely, when \(N > K\), the environment cannot support the excess individuals, leading to a decline in the population. If \(N\) equals \(K\), the population size remains stable as the environment can exactly support that population size without inducing growth or decline.

This interplay is crucial for understanding the dynamics of populations in ecosystems and for making decisions related to conservation and resource management. A steady state is achieved at \(N = K\), where the birth rate equals the death rate, and the population stabilizes.

Density-dependent growth refers to the phenomenon where the growth rate of a population is related to its density. In the given equation, the growth factor is influenced by how close the population size is to the carrying capacity \(K\).

The term \( \lambda = \exp [r (1 - \frac{N}{K})] \) in the equation captures this density-dependent effect. When population \(N\) is much smaller than \(K\), \( \lambda \) is greater than 1, indicating that the population can grow rapidly. As \(N\) approaches \(K\), \( \lambda \) approaches 1, leading to a steady state with no net growth. When \(N\) exceeds \(K\), \( \lambda \) becomes less than 1, causing the population to decline.

This density-dependent control is a natural mechanism that ensures populations do not exceed the resources available in their environment. It acts as a feedback loop where high population density reduces the growth rate, stabilizing the population size over time. This concept helps ecologists and environmental planners design strategies for sustainable population management.

• Growth Rate (r): Determines how quickly a population can grow under ideal conditions.
• Population Size (N): Current number of individuals in the population.
• Carrying Capacity (K): Maximum population size that the environment can support.