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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.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter5: A Survey Of Other Common Functions
Section5.1: Logistic Functions
Problem 17E: World Population The following table shows world population N, in billions, in the given year. Year...
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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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