Math 121 - Calculus for Biology I Spring Semester, 2004 Logistic Growth - Worked Examples
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• Finding Equilibria
• Determining Stability

• Find all the equilibria for this model
• Determine the behavior of the solution near these equilibria
• Sketch a graph of the updating function and the identity map P_n+1 = P_n

• The equilibria satisfy

• To graph the updating function, find the P-intercepts and the vertex
  • The P-intercepts are 0 and 13,000
  • The vertex is at (6500, 4225)
• Below is the graph with the identity function and significant points shown on the graph

• Simulation of this model with an initial value of P₀ = 100 and 50 iterations

 

 

 

 

 

 

 

 

 

Example 2: Modify the example above and consider the discrete logistic growth model

P_n+1 = f₂(P_n) = 2.7P_n - 0.0001P_n².

 

• Find all the equilibria for this model
• Determine the behavior of the solution near these equilibria
• Sketch a graph of the updating function and the identity map P_n+1 = P_n

 

 

 

 

 

 

 

Solution: Substitute P_e for P_n and P_n+1 into the discrete logistic growth model

P_e = 2.7P_e - 0.0001P_e²

0 = 1.7P_e - 0.0001P_e² = P_e (1.7 - 0.0001P_e)

• The equilibria are

P_e = 0

P_e = 17000

 

 

 

• The derivative is

f₂'(P) = 2.7 - 0.0002P

• At P_e = 0, the derivative is

f₂'(0) = 2.7 > 1

• The solution monotonically grows away from this equilibrium

 

 

 

• At P_e = 17000, the derivative is

f₂'(17000) = 2.7 - 3.4 = -0.7

• Since -1 < f₂'(17000) < 0, the discrete logistic model oscillates and approaches this equilibrium
• This equilibrium is also stable

 

 

 

 

• To graph the updating function, find the P-intercepts and the vertex
  • The P-intercepts are 0 and 27,000
  • The vertex is at (13500, 18225)
• Below is the graph with the identity function and significant points shown on the graph

 

 

 

 

 

 

Simulation of this model with an initial value of P₀ = 100 and 20 iterations

 

 

 

 

 

 

 

Example 3: Another change in the discrete logistic growth model gives

P_n+1 = f₃(P_n) = 3.2P_n - 0.0001P_n²

 

• Find all the equilibria for this model
• Determine the behavior of the solution near these equilibria
• Sketch a graph of the updating function and the identity map P_n+1 = P_n

 

 

 

 

 

 

Solution: Substitute P_e for P_n and P_n+1 into the discrete logistic growth model

P_e = 3.2P_e - 0.0001P_e²

0 = 2.2P_e - 0.0001P_e² = P_e (2.2 - 0.0001P_e)

• The equilibria are

• The vertex occurs halfway between the p-intercepts, so

• The vertex is the maximum growth rate of the population, so the maximum this population can grow is 125 individuals/hr

• The graph of g(p) is shown below

 

 

c. From the graph above, it is clear that the growth rate is zero at 0 and 10,000, so the equilibria occur at

p_e = 0 and 10,000.

 

• The stability is still determined by differentiating the updating function, not the growth function,

• The derivative of the updating function is

• At P_e = 0, the derivative is f '(0) = 1.05 > 1. Thus, the solution is unstable and monotonically grows away from P_e = 0
• At P_e = 10000, the derivative is f '(10000) = 0.05. Since 0 < f '(10000) < 1, the discrete logistic model is stable and the solutions monotonically approach the equilibrium, P_e = 10000

• Population growth may be affected by immigration or emigration
• Consider a growth rate

• Each generation has a 71% growth rate
• 0.001p_n² are lost due to crowding
• 7 emigrate

p_n+1 = p_n + g(p_n) = p_n + 0.71 p_n - 0.001p_n² - 7,

where n is measured in generations.

a. Assume that p₀ = 100 and find the population for the next three generations, p₁, p₂, and p₃.

b. Find the p-intercepts and the vertex for g(p) and sketch a graph of g(p).

c. By finding when the growth rate is zero, determine all equilibria for this model.

 

 

 

 

 

 

 

 

 

Solution:

a. The next three generations are

p₁ = p₀ + g(p₀) = 100 + 0.71(100) - 0.001(100)² - 7 = 154,

p₂ = 154 + 0.71(154) - 0.001(154)² - 7 = 233,

p₃ = 233 + 0.71(233) - 0.001(233)² - 7 = 337.

 

 

 

 

 

 

b. The growth function satisfies

g(p) = 0.71 p - 0.001p² - 7

g(p) = -0.001(p² - 710p + 7000)

g(p) = -0.001(p - 10)(p - 700).

 

 

The p-intercepts satisfy g(p) = 0, so

p = 10 or p = 700.

 

 

 

The vertex occurs halfway between the p-intercepts, so p = 355 and

g(355) = -0.001(345)(-345) = 119.

The maximum growth occurs when the population is 355 with a maximum growth of 119 individuals/generation.

 

 

 

 

 

 

The graph of g(p) is shown below showing the p-intercepts at p = 10 and 700 and the vertex.

c. From the graph above, it is clear that the growth rate is zero at p = 10 and 700, so the equilibria occur at p_e = 10 and 700.

 

 

 

 

 

 

 

 

 

 

 

Example 7 (U. S. Census with Logistic Growth Model)

• The Malthusian growth model did not work well for the U. S. census data over any extended period of time because the growth rate in general has been declining for most decades in U. S. history
• Can we apply the logistic growth model to the U. S. census data and get a better fit to the data and avoiding the problems of the time varying nonautonomous model developed at the end of the discrete Malthusian growth section?

• Consider the U. S. census data from 1790 to 2000
• How does one find the appropriate updating function for a logistic growth model?
• The best fit to a model uses the least squares best fit technique
• A least squares best fit to the U. S. census data was performed to find the best Malthusian growth model

• The best fit logistic growth model satisfies

• A nonautonomous model fitting the time varying growth satisfied

• The simulations of the Malthusian, logistic, and nonautonomous growth models are graphed below showing how well they compare to the actual U. S. census data
• It is readily apparent that the Malthusian growth model does not fit the data very well, growing too slow in the early history of the U. S. and growing too rapidly more recently
• The logistic growth model appears to fit the data fairly well
• The best fit is the nonautonomous model developed in the discrete Malthusian growth section.

• By plotting P_n+1 versus P_n, one can see how the data compares to the updating function for the logistic growth model
• The graph of this updating function with the data is seen below with the identity map P_n+1 = P_n:

 

 

 

 

• The logistic updating function very closely follows the census data except at a couple of points
• The equilibria occur at the intersection of the updating function and the identity map
• The slope of the updating function at a point of intersection determines the stability of that equilibrium

• The equilibria for the logistic growth model above are P_e = 0 or 313.8 million
• The derivative of the updating function is

• It follows that F '(0) = 1.3064, so P_e = 0 is unstable
• Similarly, F '(313.8) = 0.6936, so the equilbrium P_e = 313.8 million is stable
• The logistic growth model fit to the U. S. census data predicts that the population of the U. S. will growth monotonically, then level off at 313.8 million
• Note that this population is not much higher than the 2000 census value, while actual projections have the U. S. population rising to over 400 million by the middle of the next century

• The logistic growth model is easier to analyze than the nonautonomous model, but the nonautonomous model appears to fit the growth of the U. S. population better
• By extending the analysis of the nonautonomous growth model, we see that the growth continues until n = 25 (actually 24.7), then this model has the population beginning to decline
• Simulating this model for 25 decades (until 2040) finds that a maximum population of 318.8 million is reached
• Note that the nonautonomous model and the logistic growth model both seem to predict a similar maximum population for the U. S., and both are unrealistically low
• This shows that human populations have a more complicated dynamics than these models can predict with either time-varying or crowding factors
• Sociological and technical factors are especially difficult to incorporate into mathematical models
• Your ecology courses should help explain more details underlying the assumptions in these models, so explain a little better when the models are applicable and why they fail in other predictions
• It is unlikely that human population can continue its current course, but what will be the actual scenario?
• Mathematical modeling can provide reasonable estimates for short term growth and allows one to predict several different possibilities for longer term growth depending on the assumptions that are entered into the model