We studied the Lotka-Volterra model for predators and preys in class. There are systems of differential equations that model the growth and decline of populations of two competing species, they usually compete for their food supply. Let's see one! Let us work in a small region in of the Caribbean sea: Isla Paraiso. Let z be the population (in thousands) of the invasive lionfish that has become dominant in many places in the Caribbean, far from its natural Indo-Pacific waters. Let y be the population (in hundreds) of the local blacktip sharks. They both compete for the little fish around Isla Paraiso. The following system of differential equations models their populations: dr dt dy dt = x(2 - x - y) = y(3-2x - y) (a) What are the equilibrium solutions of the system? (b) What is the differential equation satisfied by y(r)? (c) Draw the phase portrait of this system.
We studied the Lotka-Volterra model for predators and preys in class. There are systems of differential equations that model the growth and decline of populations of two competing species, they usually compete for their food supply. Let's see one! Let us work in a small region in of the Caribbean sea: Isla Paraiso. Let z be the population (in thousands) of the invasive lionfish that has become dominant in many places in the Caribbean, far from its natural Indo-Pacific waters. Let y be the population (in hundreds) of the local blacktip sharks. They both compete for the little fish around Isla Paraiso. The following system of differential equations models their populations: dr dt dy dt = x(2 - x - y) = y(3-2x - y) (a) What are the equilibrium solutions of the system? (b) What is the differential equation satisfied by y(r)? (c) Draw the phase portrait of this system.
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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May I ask for the d part of this question? Please explain and answer it in clear writing. Thank you!
Transcribed Image Text:We studied the Lotka-Volterra model for predators and preys in class. There are systems of differential
equations that model the growth and decline of populations of two competing species, they usually
compete for their food supply. Let's see one!
Let us work in a small region in of the Caribbean sea: Isla Paraiso. Let z be the population (in
thousands) of the invasive lionfish that has become dominant in many places in the Caribbean, far
from its natural Indo-Pacific waters. Let y be the population (in hundreds) of the local blacktip sharks.
They both compete for the little fish around Isla Paraiso. The following system of differential equations
models their populations:
dr
dt
dy
dt
= x(2 - x - y)
= y(3-2x - y)
(a) What are the equilibrium solutions of the system?
(b) What is the differential equation satisfied by y(x)?
(c) Draw the phase portrait of this system.
(d) Explain why these equations make it mathematically possible, but extremely unlikely, for both
species to survive.
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