Question

You should be able to sketch a logistic growth curve for a population of snakes that colonized a Pacific island but then stabilized at about 1000 individuals. Describe how \(r\) changes along the length of the curve and suggest factors responsible for each change in \(r\).

Short answer

In the logistic growth model of a population of snakes colonizing an island with a carrying capacity of 1000, \(r\) starts high during the initial rapid growth phase due to abundant resources. As the population approaches the carrying capacity, \(r\) decreases due to increased competition and scarcity of resources. Eventually, \(r\) becomes zero when the population stabilizes at the carrying capacity. Factors affecting changes in \(r\) include abundance or scarcity of resources, competition, predation, and disease.

Step-by-step solution

Understand the Logistic Growth Model

The logistic growth model is a mathematical model of population growth where the growth rate decreases as the population nears its maximum sustainable capacity (\( K \)). The equation for logistic growth is: \[ \frac{dN}{dt} = rN\left (1 - \frac{N}{K} \right ) \] where: - \( N \) is the population size, - \( r \) is the intrinsic growth rate, - \( K \) is the carrying capacity.

Sketch the Logistic Growth Curve

The logistic growth curve begins with a rapid growth phase, followed by a slowing growth as the population approaches carrying capacity, and finally stabilizes at the carrying capacity. This results in an S-shaped curve. The value of \( r \) is highest during the initial rapid growth phase, when resources are abundant. Then, as resources become scarcer and competition increases, \( r \) decreases until the population stabilizes at the carrying capacity, at which point \( r \) becomes zero since the population is neither increasing nor decreasing.

Changes in \( r \) Along the Curve

In the initial phase when the population is low and growing rapidly, \( r \) is high due to abundant resources. As the curve flattens out and the growth slows, \( r \) decreases due to increased competition and scarcity of resources. It eventually becomes zero when the population stabilizes at \( K \).

Factors Affecting Changes in \( r \)

Several factors can contribute to the changes in \( r \) over time: - Abundance or scarcity of Resources: In the early stages, resources are plentiful, leading to a high \( r \). As the population nears the carrying capacity, resources become scarcer, leading to a reduced \( r \). - Competition: The more individuals, the more competition there is for resources, leading to a decrease in \( r \). - Predation: If predators are present, they could reduce the population size, impacting the value of \( r \). - Disease: The closer the population is to carrying capacity, the more likely disease may spread, decreasing \( r \).

Key concepts

Population Growth Model

The Population Growth Model is a mathematical representation of how a population changes over time. Understanding these models helps in predicting and managing wildlife populations and resources. One widely used model is the logistic growth model, which shows how populations expand in an environment with limited resources.

The model begins with an exponential growth phase, where the population size increases rapidly due to abundant resources and little competition. However, as the population grows, the rate of increase begins to slow due to a decrease in resources and increased competition. Eventually, the growth rate reaches zero, and the population size stabilizes at a so-called 'carrying capacity' of the environment. This behavior creates an S-shaped curve when plotted over time.

Carrying Capacity

Carrying capacity ( K ) refers to the maximum number of individuals in a population that an environment can sustain indefinitely without being degraded. It is a balance point where the number of organisms the environment can support equals the number present.

This concept is critical as it determines the maximum population size that an environment can hold without causing negative effects, such as resource depletion and habitat destruction. An environment's carrying capacity can change over time due to a variety of factors including climate changes, natural disasters, and human activities which can either improve or degrade the habitat quality.

Intrinsic Growth Rate

The intrinsic growth rate ( r ) represents the potential for growth of a population under ideal environmental conditions, with unlimited resources and without any growth restrictions imposed by predators, disease, or competition. It's a measure of how quickly the population can grow when no environmental resistance is present.

For instance, in the context of the logistic growth curve, the r is highest at the beginning when the population density is low, thus encountering minimal resistance to growth. As the population nears the carrying capacity, the intrinsic growth rate declines because the environmental resistance factors start to counterbalance the growth.

Environmental Resistance

Environmental resistance encompasses all the factors that limit the growth of a population and slow its approach to carrying capacity. These factors include lack of food, water or shelter, disease, predation, and competition within the population for resources.

As a population grows, it begins to encounter these limits, which is reflected in the logistic growth curve as the rate of population increase slows. This resistance can vary greatly from one environment to another and can also change within the same environment over time, impacting both the intrinsic growth rate and the carrying capacity.

Population Dynamics

Population dynamics involve the study of how and why populations change over time. It considers birth rates, death rates, immigration, and emigration to understand the fluctuations in population size and composition. Logistic growth is a fundamental concept within population dynamics because it captures the complex interplay between population growth and the environmental limitations.

The logistic growth curve specifically illustrates several stages of population dynamics, from the exponential growth phase to the point where growth is tempered by resource limits until the population stabilizes near the carrying capacity. These dynamics are crucial to wildlife management, conservation, and understanding the ecological balance of natural habitats.