Problem 1: Coupled Nonlinear Differential Equations in Predator-Prey ModelA predator-prey system is modeled by the following set of coupled nonlinear differential equations, where x(t) is the prey population and y(t)is the predator population at time t:\frac{dx}{dt} = x(1 - 0.1x) - 0.02xy\frac{dy}{dt} = -y(1 - 0.02x)The initial populations are given in the table below:Initial ConditionPrey Population x(0)x(0)x(0)Predator Population y(0)y(0)y(0)Condition 1409Condition 26012Condition 35010Solve the coupled nonlinear system for each set of initial conditions. Use numerical methods to compute the populations of prey and predators over time, say for t∈[0,50]t \in [0, 50]t∈[0,50].Plot the phase portrait of the system for the three different initial conditions.Interpret the behavior of the predator-prey dynamics, particularly focusing on the presence of limit cycles or extinction events.

We are solving a system of coupled nonlinear differential equations that model predator-prey…

Answered: Problem 1: Coupled Nonlinear…

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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Problem 1: Coupled Nonlinear Differential Equations in Predator-Prey Model

A predator-prey system is modeled by the following set of coupled nonlinear differential equations, where x(t) is the prey population and y(t)is the predator population at time t:

\frac{dx}{dt} = x(1 - 0.1x) - 0.02xy

\frac{dy}{dt} = -y(1 - 0.02x)

The initial populations are given in the table below:

Initial Condition	Prey Population x(0)x(0)x(0)	Predator Population y(0)y(0)y(0)
Condition 1	40	9
Condition 2	60	12
Condition 3	50	10

Solve the coupled nonlinear system for each set of initial conditions. Use numerical methods to compute the populations of prey and predators over time, say for t∈[0,50]t \in [0, 50]t∈[0,50].
Plot the phase portrait of the system for the three different initial conditions.
Interpret the behavior of the predator-prey dynamics, particularly focusing on the presence of limit cycles or extinction events.

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