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Logistic growth with a threshold: Most species have a survival threshold level, and populations of fewer individuals than the threshold cannot sustain themselves. If the carrying capacity is \(K\) and the threshold level is \(S\), then the logistic equation of change for the population \(N=N(t)\) is $$ \frac{d N}{d t}=-r N\left(1-\frac{N}{S}\right)\left(1-\frac{N}{K}\right) . $$ For Pacific sardines, we may use \(K=2.4\) million tons and \(r=0.338\) per year, as in Example 6.10. Suppose we also know that the survival threshold level for the sardines is \(S=0.8\) million tons. a. Write the equation of change for Pacific sardines under these conditions. b. Make a graph of \(\frac{d N}{d t}\) versus \(N\) and use it to find the equilibrium solutions. How do the equilibrium solutions correspond with \(S\) and \(K\) ? c. For what values of \(N\) is the graph of \(N\) versus \(t\) increasing, and for what values is it decreasing? d. Explain what can be expected to happen to a population of \(0.7\) million tons of sardines. e. At what population level will the population be growing at its fastest?

Step by step solution

01

Write the equation of change

Given the parameters for Pacific sardines, substitute into the logistic growth equation: \[-r N\left(1-\frac{N}{S}\right)\left(1-\frac{N}{K}\right)\] where \(r = 0.338\), \(K = 2.4\), and \(S = 0.8\). The equation becomes: \[\frac{d N}{d t} = -0.338 N\left(1-\frac{N}{0.8}\right)\left(1-\frac{N}{2.4}\right).\]

02

Analyze graph for equilibrium solutions

To find equilibrium solutions, set \(\frac{d N}{d t} = 0\): \[-0.338 N\left(1-\frac{N}{0.8}\right)\left(1-\frac{N}{2.4}\right) = 0.\] The solutions occur when: \(N = 0\), \(N = S = 0.8\), and \(N = K = 2.4\). These are the equilibrium solutions. By analyzing the graph of the derivative \(\frac{d N}{d t}\) vs \(N\), we see: - \(N = 0\) is unstable because any small population will grow or increase.- \(N = 0.8\) is stable as perturbations will revert back to equilibrium.- \(N = 2.4\) is stable, maximum carrying capacity.

03

Determine intervals of increase and decrease

The graph of \(N\) versus \(t\) is increasing where \(\frac{d N}{d t} > 0\), and decreasing where \(\frac{d N}{d t} < 0\). Analytically this happens:- Increase between \(0 < N < 0.8\) and \(0.8 < N < 2.4\).- Decrease when \(N > 2.4\).Thus, \(N\) increases towards \(0.8\) and \(2.4\), and decreases after \(2.4\).

04

Analyze population of 0.7 million tons

For \(N = 0.7\), since \(0 < 0.7 < 0.8\), \(\frac{d N}{d t} > 0\) which means the population is subjected to growth. As it is below the threshold \(S = 0.8\), it will tend to grow towards \(0.8\).

05

Find the fastest growth rate

The growth rate is maximized at the vertex of the parabola defined by the growth derivative. Taking the derivative of \(\frac{d N}{d t}\) with respect to \(N\) to locate this point analytically or graphically reveals the fastest growth rate occurs at the middle value between \(S\) and \(K\). Thus, the maximum growth occurs at \(N=\sqrt{S \cdot K} \approx \sqrt{0.8 \cdot 2.4} \approx 1.38\) million tons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carrying Capacity

In population dynamics, carrying capacity refers to the maximum population size of a species that a given environment can sustain indefinitely. It is denoted as \(K\). For instance, in our example of Pacific sardines, the carrying capacity is given as \(K = 2.4\) million tons. This means the oceanic environment can support a maximum of 2.4 million tons of sardines without any negative effects from overpopulation. Carrying capacity is influenced by several factors, including resource availability, environmental conditions, and space. When populations reach this size, resources become limited, and growth rates decrease, eventually stabilizing the population. In mathematical models like the logistic growth equation, the carrying capacity shapes the curve representing how the population grows over time.

Population Dynamics

Population dynamics studies how and why population sizes change over time and the effects these changes have on ecosystems. It takes into account factors such as birth rates, death rates, immigration, and emigration. For the Pacific sardines, important factors influencing dynamics include birth rates and natural resource limits.The logistic growth model we use with sardines is a classic example of population dynamics. It expresses how populations grow rapidly when resources are abundant (initial growth phase) and slow down as they near the carrying capacity. In the equation provided, the population \(N\) grows according to the rate \(r = 0.338\), but as \(N\) approaches either the threshold level \(S = 0.8\) or \(K = 2.4\), the growth rate adjusts to reflect available resources and population pressure.Understanding these dynamic processes allows ecologists to predict changes and manage conservation efforts effectively.

Equilibrium Solutions

Equilibrium solutions are points where population changes stabilize, meaning growth and decline pressures balance each other out. In the logistic equation, these points occur when \(\frac{dN}{dt} = 0\), indicating no change in population over time.For our Pacific sardines, the equilibrium points occur at \(N = 0\), \(N = S = 0.8\), and \(N = K = 2.4\). Each of these points represents a stable or unstable state:

• \(N = 0\): Unsustainable as any small increase leads to growth.
• \(N = 0.8\): Stable equilibrium as small deviations return to this state.
• \(N = 2.4\): Stable equilibrium representing the carrying capacity.

At these points of equilibrium, the population does not change unless influenced by significant external factors.

Threshold Level

The threshold level, \(S\), is a critical population size required for a population to sustain itself. For many species, this level represents the minimum population needed to avoid extinction due to factors like genetic diversity loss or societal disruptions.In the case of the sardines, the threshold level is \(S = 0.8\) million tons. Below this threshold, the population \(N\) finds it challenging to grow independently. The logistic growth model highlights how, below this threshold, \(\frac{dN}{dt} > 0\), indicating natural growth as the population tries to reach a stable state.This concept is essential in conservation biology, as it underscores the importance of maintaining populations above this critical level to ensure their long-term viability.