Suppose that the differential equation dy dt y(40-y)-300 models a logistic equation with harvesting, where y(t) represents a spotted grunter popu- lation (Figure 1) in the Knysna Lagoon at time t, measured in years. © SAIAB Figure 1: The spotted grunter. (Image courtesy of: https://wwfsassi.co.za/fish-detail/44/) Write down the (a) per capita growth rate of the spotted grunter population. (b) carrying capacity of the lagoon.
Suppose that the differential equation dy dt y(40-y)-300 models a logistic equation with harvesting, where y(t) represents a spotted grunter popu- lation (Figure 1) in the Knysna Lagoon at time t, measured in years. © SAIAB Figure 1: The spotted grunter. (Image courtesy of: https://wwfsassi.co.za/fish-detail/44/) Write down the (a) per capita growth rate of the spotted grunter population. (b) carrying capacity of the lagoon.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer Question 4b only
Transcribed Image Text:Question 4
Suppose that the differential equation
dy
dt
models a logistic equation with harvesting, where y(t) represents a spotted grunter popu-
lation (Figure 1) in the Knysna Lagoon at time t, measured in years.
= y(40-y)- 300
© SALAB
Write down the
Figure 1: The spotted grunter. (Image courtesy of: https://wwfsassi.co.za/fish-detail/44/)
(a) per capita growth rate of the spotted grunter population.
(b) carrying capacity of the lagoon.
(c) rate at which the spotted grunter are harvested from the lagoon.
(d) Determine the
(i) critical points of Eq. (3).
(ii) limiting and threshold solutions from Eq. (3).
(iii) stability of the critical points by drawing a phase diagram for Eq. (3).
(e) (i) Taking y(0) = 11 and a step size of h = 1/12, apply four iterations of the Forward
Euler Formula to approximate the solution of Eq. (3). Apply two decimal place rounding
where applicable.
(ii) How do your results from (e) (i) compare with what the phase diagram for Eq. (3)
reveals?
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