0.3 The Logistic Equation: Modeling Populations One model that incorporates limited growth is the logistic or Verhulst model. If we look at a population of fish, say cod, that live in the North Atlantic Ocean, without human inter- vention the population of cod can be modeled by the logistic equation. dP =rP dt P (1- // ) P K (1) where P(t) is in tons and t is in years, r and K are both positive constants. K is called the carrying capacity and represents the maximum population that the environment can support. 2. Logistic Constant-Harvest Model In the Constant-Harvest model fishermen are allowed to harvest a constant amount of fish during a fixed time period. The parameter h indicates the constant harvesting rate. dP dt =rP 1- (11) P - h h > 0. K 2 (2) (a) Our goal is to understand what happens to the equilibrium values as h increases. Use r = 1 and K = 1600 to find the equilibrium values, P+ and P_ of equation (2) for each of the values h = 90, 180, 270, and 360. You will need to solve a quadratic equation. (b) For each of the values of h above draw a phase diagram and classify the stability of the equilibrium values P+ and P. (c) What happens to P+ and P_ as h increases? (d) For which value of h is there only one critical value? Draw a phase line for this case. (e) What does the phase diagram for the case of one critical value mean in terms of the sustainability of the fishery? (f) What fraction of the carrying capacity is the h-value that yields a single critical value? (g) Use a slope field plotter to graph the slope field and solution curves for h = 90 and h=400. You will upload these graphs. (h) Explain what happens to fish population as h approaches h = 400. What danger does this represent to the fish population? Use this information to write a warning notice (one paragraph) to any Wildlife Management Agency that uses a constant harvest model.
0.3 The Logistic Equation: Modeling Populations One model that incorporates limited growth is the logistic or Verhulst model. If we look at a population of fish, say cod, that live in the North Atlantic Ocean, without human inter- vention the population of cod can be modeled by the logistic equation. dP =rP dt P (1- // ) P K (1) where P(t) is in tons and t is in years, r and K are both positive constants. K is called the carrying capacity and represents the maximum population that the environment can support. 2. Logistic Constant-Harvest Model In the Constant-Harvest model fishermen are allowed to harvest a constant amount of fish during a fixed time period. The parameter h indicates the constant harvesting rate. dP dt =rP 1- (11) P - h h > 0. K 2 (2) (a) Our goal is to understand what happens to the equilibrium values as h increases. Use r = 1 and K = 1600 to find the equilibrium values, P+ and P_ of equation (2) for each of the values h = 90, 180, 270, and 360. You will need to solve a quadratic equation. (b) For each of the values of h above draw a phase diagram and classify the stability of the equilibrium values P+ and P. (c) What happens to P+ and P_ as h increases? (d) For which value of h is there only one critical value? Draw a phase line for this case. (e) What does the phase diagram for the case of one critical value mean in terms of the sustainability of the fishery? (f) What fraction of the carrying capacity is the h-value that yields a single critical value? (g) Use a slope field plotter to graph the slope field and solution curves for h = 90 and h=400. You will upload these graphs. (h) Explain what happens to fish population as h approaches h = 400. What danger does this represent to the fish population? Use this information to write a warning notice (one paragraph) to any Wildlife Management Agency that uses a constant harvest model.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 15DE
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In the first picture I provided some background, please help with the problem in the second pict
Transcribed Image Text:0.3
The Logistic Equation: Modeling Populations
One model that incorporates limited growth is the logistic or Verhulst model. If we look at
a population of fish, say cod, that live in the North Atlantic Ocean, without human inter-
vention the population of cod can be modeled by the logistic equation.
dP
=rP
dt
P (1- // )
P
K
(1)
where P(t) is in tons and t is in years, r and K are both positive constants. K is called the
carrying capacity and represents the maximum population that the environment can support.
Transcribed Image Text:2. Logistic Constant-Harvest Model In the Constant-Harvest model fishermen are allowed
to harvest a constant amount of fish during a fixed time period. The parameter h indicates
the constant harvesting rate.
dP
dt
=rP 1-
(11)
P
- h
h > 0.
K
2
(2)
(a) Our goal is to understand what happens to the equilibrium values as h increases. Use
r = 1 and K = 1600 to find the equilibrium values, P+ and P_ of equation (2) for each
of the values h = 90, 180, 270, and 360. You will need to solve a quadratic equation.
(b) For each of the values of h above draw a phase diagram and classify the stability of the
equilibrium values P+ and P.
(c) What happens to P+ and P_ as h increases?
(d) For which value of h is there only one critical value? Draw a phase line for this case.
(e) What does the phase diagram for the case of one critical value mean in terms of the
sustainability of the fishery?
(f) What fraction of the carrying capacity is the h-value that yields a single critical value?
(g) Use a slope field plotter to graph the slope field and solution curves for h = 90 and
h=400. You will upload these graphs.
(h) Explain what happens to fish population as h approaches h = 400. What danger does
this represent to the fish population? Use this information to write a warning notice
(one paragraph) to any Wildlife Management Agency that uses a constant harvest
model.
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