The scientists Verhulst (1828) and Peal (1930) proposed the following growth model for a population living in an environment with limited resources (e.g. space, food, sunlight, etc):
= ,
where is the natural growth rate of the population when resources are limited and is a positive constant called carrying capacity of the environment.
(a) Using sign analysis, find all constant solution and discuss the general behavior of solution.
(b) Solve the equation when Compute the limit as of your solution.
(c) Solve the equation when Compute the limit as of your solution.
(d) Discuss why is called carrying capacity.
. Model : Logistic Model I
Differential Equation :
General Solution :
Graph :
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To answer these questions, we will first need to analyze and solve the differential equation given for the population growth model.
The differential equation is given by: dP/dt = r * P * (1 - P/K)
(a) Using sign analysis, we can determine the constant solutions and discuss the general behavior of the solution.
To perform sign analysis, we look at the signs of the terms involved in the equation to determine the behavior of the population growth.
- The term dP/dt represents the rate of change of the population over time. If it is positive, the population is increasing, and if it is negative, the population is decreasing.
- The term r represents the natural growth rate of the population when resources are limited. If r is positive, the population growth rate is positive, and if r is negative, the population growth rate is negative.
- The term P represents the current size of the population.
- The term K represents the carrying capacity of the environment, which is the maximum number of individuals that the environment can support.
Based on the signs of these terms, we can analyze the behavior of the solution:
- When P = 0, the population growth rate is positive (r > 0), resulting in an increasing population.
- When P = K, the population growth rate is negative (r < 0), resulting in a decreasing population.
- When P is between 0 and K, the population growth rate is positive (r > 0), but gradually decreases as the population approaches the carrying capacity.
- When P is greater than K, the population growth rate becomes negative (r < 0), leading to a decreasing population.
(b) To solve the equation, we can separate the variables and integrate:
dP/(P(1-P/K)) = r * dt
Integrating both sides:
∫ (K/(P(K-P))) * dP = ∫ r * dt
This integral can be solved using partial fraction decomposition to obtain a solution.
(c) Once we have the general solution obtained in part (b), we can substitute K = 0. Compute the limit as P approaches infinity of your solution.
By substituting K = 0 into the general solution, we can then find the limit as P approaches infinity. This limit will determine the maximum population size that the environment can support, which is known as the carrying capacity.
(d) The carrying capacity, represented by the constant K, is called so because it defines the maximum number of individuals that the environment can sustainably support. Beyond the carrying capacity, the limited resources in the environment cannot support further population growth, resulting in a decline in population size. Therefore, the carrying capacity determines the long-term capacity of the environment to support a stable population.