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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 dt = rP (1-1/2) 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. 1. Logistic Model (a) Use the values r = 1 and K = 1600 find to the critical points of this autonomous DE. You will need to solve a quadratic equation. Label the larger root P+ and the smaller root P. This will prove helpful later on. (b) Use a phase line analysis (a one dimensional phase portrait) and find the stability of each critical point. (c) Use implicit differentiation to find an expression for the inflection point of P. What does the inflection point represent? (d) Use a slope field plotter such as the SLOPES APP by Tim Lucas (available or iOS and Android) to plot both the slope field of equation (1) and several solution curves. On your graph, include two solution curves for each of the 3 regions defined by the equilibrium values. Solutions in the P < 0 region do not make sense in the context of this problem but we will include the solutions for mathematical completeness. (e) Find the following limits: i. lim P(t) too ii. lim P(t) 0047 iii. lim P(t) t∞ when P(1600) = 400. = when P(0) = 1600. when P(5) = 1800.

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 dt = rP (1-1/2) 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. 1. Logistic Model (a) Use the values r = 1 and K = 1600 find to the critical points of this autonomous DE. You will need to solve a quadratic equation. Label the larger root P+ and the smaller root P. This will prove helpful later on. (b) Use a phase line analysis (a one dimensional phase portrait) and find the stability of each critical point. (c) Use implicit differentiation to find an expression for the inflection point of P. What does the inflection point represent? (d) Use a slope field plotter such as the SLOPES APP by Tim Lucas (available or iOS and Android) to plot both the slope field of equation (1) and several solution curves. On your graph, include two solution curves for each of the 3 regions defined by the equilibrium values. Solutions in the P < 0 region do not make sense in the context of this problem but we will include the solutions for mathematical completeness. (e) Find the following limits: i. lim P(t) too ii. lim P(t) 0047 iii. lim P(t) t∞ when P(1600) = 400. = when P(0) = 1600. when P(5) = 1800.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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 dt = rP (1-1/2) 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. 1. Logistic Model (a) Use the values r = 1 and K = 1600 find to the critical points of this autonomous DE. You will need to solve a quadratic equation. Label the larger root P+ and the smaller root P. This will prove helpful later on. (b) Use a phase line analysis (a one dimensional phase portrait) and find the stability of each critical point. (c) Use implicit differentiation to find an expression for the inflection point of P. What does the inflection point represent? (d) Use a slope field plotter such as the SLOPES APP by Tim Lucas (available or iOS and Android) to plot both the slope field of equation (1) and several solution curves. On your graph, include two solution curves for each of the 3 regions defined by the equilibrium values. Solutions in the P < 0 region do not make sense in the context of this problem but we will include the solutions for mathematical completeness. (e) Find the following limits: i. lim P(t) too ii. lim P(t) 0047 iii. lim P(t) t∞ when P(1600) = 400. = when P(0) = 1600. when P(5) = 1800.
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 dt = rP (1-1/2) 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. 1. Logistic Model (a) Use the values r = 1 and K = 1600 find to the critical points of this autonomous DE. You will need to solve a quadratic equation. Label the larger root P+ and the smaller root P. This will prove helpful later on. (b) Use a phase line analysis (a one dimensional phase portrait) and find the stability of each critical point. (c) Use implicit differentiation to find an expression for the inflection point of P. What does the inflection point represent? (d) Use a slope field plotter such as the SLOPES APP by Tim Lucas (available or iOS and Android) to plot both the slope field of equation (1) and several solution curves. On your graph, include two solution curves for each of the 3 regions defined by the equilibrium values. Solutions in the P < 0 region do not make sense in the context of this problem but we will include the solutions for mathematical completeness. (e) Find the following limits: i. lim P(t) too ii. lim P(t) 0047 iii. lim P(t) t∞ when P(1600) = 400. = when P(0) = 1600. when P(5) = 1800.
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