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There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the fac (1 - m/P). Thus the modified logistic model is given by the differential equation (a) Use the differential equation to show that any solution is increasing if m

There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the fac (1 - m/P). Thus the modified logistic model is given by the differential equation (a) Use the differential equation to show that any solution is increasing if m

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.EA: Extended Application Managing Renewable Resources
Problem 1EA: Suppose that a particular plot of land can sustain 500 deer and that the population of this...
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There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1 – m/P). Thus the modified logistic model is given by the differential equation -(1 -(1-) dP = kPI 1 dt P M (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m. (Ko(1 - 4)(1 - ). dP P m If --Select--- V then dP/dt = (+)(+)(+) = + P is ---Select--- ♥ . If |---Select--- ♥ then dP/dt = (+)(+)(-) = P is ---Select--- ♥ dt M (b) For the case where k = 0.08, M = 1000, and m = 150, draw a direction field and use it to sketch several solution curves. P P 3000) 1400 2500 1200 1000. 2000 800 1500- 600 1000 400 500 200 20 40 60 80 40 60 80 P 3000 t / 1400 2500 1200 2000. 1000, 800 1500 600, 1000 400, 500 200 20 40 60 80 20 40 60 80 20
Transcribed Image Text:There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1 – m/P). Thus the modified logistic model is given by the differential equation -(1 -(1-) dP = kPI 1 dt P M (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P < m. (Ko(1 - 4)(1 - ). dP P m If --Select--- V then dP/dt = (+)(+)(+) = + P is ---Select--- ♥ . If |---Select--- ♥ then dP/dt = (+)(+)(-) = P is ---Select--- ♥ dt M (b) For the case where k = 0.08, M = 1000, and m = 150, draw a direction field and use it to sketch several solution curves. P P 3000) 1400 2500 1200 1000. 2000 800 1500- 600 1000 400 500 200 20 40 60 80 40 60 80 P 3000 t / 1400 2500 1200 2000. 1000, 800 1500 600, 1000 400, 500 200 20 40 60 80 20 40 60 80 20
Describe what happens to the population for various initial populations. (Enter your answers using interval notation.) For Po E the population dies out. For Po E the population increases and approaches 1000. For Po E the population decreases and approaches 1000. What are the equilibrium solutions? (Enter your answers as a comma-separated list.) P = (c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population Po. P = (d) Use the solution in part (c) to show that if Po < m, then the species will become extinct. [Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.] If Po < m, then Po - m < 0. Let N(t) be the numerator of the expression for P(t) in part (c). Then N(0) = Po(M – m) ?v 0, and Po - m ? v 0 lim N(t) ?v 0. Since N is ---Select--- ?v 00 there is a number t lim M(Po - m)e(M – m)(k/M)t such that N(t) O and thus P(t) = 0. So the species will become extinct. %3D
Transcribed Image Text:Describe what happens to the population for various initial populations. (Enter your answers using interval notation.) For Po E the population dies out. For Po E the population increases and approaches 1000. For Po E the population decreases and approaches 1000. What are the equilibrium solutions? (Enter your answers as a comma-separated list.) P = (c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population Po. P = (d) Use the solution in part (c) to show that if Po < m, then the species will become extinct. [Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.] If Po < m, then Po - m < 0. Let N(t) be the numerator of the expression for P(t) in part (c). Then N(0) = Po(M – m) ?v 0, and Po - m ? v 0 lim N(t) ?v 0. Since N is ---Select--- ?v 00 there is a number t lim M(Po - m)e(M – m)(k/M)t such that N(t) O and thus P(t) = 0. So the species will become extinct. %3D
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