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
Your question is still incomplete. You will have to type equations and whatever (a) is
(a) To find the constant solutions of the differential equation, we need to set the derivative equal to zero and solve for :
= 0
Assuming ≠ 0, we can divide both sides of the equation by to simplify:
= (1 - ) = 0
This equation has two possible solutions:
1 - = 0
This implies = 1, which corresponds to a constant solution.
- = 0
This implies = 0, which also corresponds to a constant solution.
Therefore, the constant solutions of the equation are = 1 and = 0.
To discuss the general behavior of the solution, we can analyze the sign of the derivative for different values of :
- If < 1, then - < 0, indicating that the population is decreasing. As approaches 0, the population approaches extinction.
- If = 1, then - = 0, indicating that the population remains constant.
- If > 1, then - > 0, indicating that the population is increasing. As approaches , the population approaches the carrying capacity .
(b) To solve the equation when = , we substitute = into the differential equation:
= -
Simplifying the equation, we get:
= -
This is a separable differential equation. Let's solve it by separating the variables:
dx = - dt
Integrating both sides:
ln(|x - |) = - t + C
Solving for :
|x - | = e^(-t+C) = Ce^(-t)
Taking the absolute value of both sides, we get:
x - = Ce^(-t)
Simplifying further:
x = Ce^(-t) +
Considering the initial condition x(0) = x₀, we can substitute these values into the equation to find C:
x₀ = Ce^(-0) +
Since e^0 = 1, we have:
x₀ = C +
Solving for C:
C = x₀ -
Substituting C back into the solution:
x = (x₀ - )e^(-t) +
(c) To solve the equation when = 0, we substitute = 0 into the differential equation:
= 0 -
Simplifying the equation, we get:
= 0
This is a separable differential equation. Let's solve it by separating the variables:
dx = 0 dt
Integrating both sides:
x = C
Therefore, the solution to the differential equation when = 0 is a constant value, given by x = C.
(d) The parameter is called the carrying capacity because it represents the maximum population size an environment can sustain in the long run, considering the available resources. As the population approaches the carrying capacity , the growth rate decreases, and the population stabilizes. The environment can only support a certain number of individuals, beyond which resources become limited and population growth slows down or stops altogether. This parameter is crucial in understanding the dynamics of population growth in relation to the available resources in the environment.