Consider the problem of predicting the population of two species, one of which is a predator, whose population at time t is x₂ (t), feeding on the other, which is the prey, whose population is x₁ (t). The two species competing for the same food supply. It is often assumed that, although the birthrate of each of species is simply proportional to the number of species alive at that time, the death rate of each species depends on the population of both species. We will assume that the population of a particular pair of species is described by the equations. dx₁ (t) dt dx₂ (t) dt = x₁(t) [4 - 0.0003x₁ (t) - 0.0004x₂(t)] = x₂(t) [2 -0.0002x₁ (t) - 0.0001x₂ (t)]. If it is known that the initial population of each species is 10,000. (a) Use Runge-Kutta Midpoint method (by hand) to approximate the solutions at t = 0.4 with h = 0.2. (b) Is there a stable solution to this population model? If so, for what values of x₁ and x₂ is the solution stable?

Consider the problem of predicting the population of two species, one of which is a predator, whose population at time t is x₂ (t), feeding on the other, which is the prey, whose population is x₁ (t). The two species competing for the same food supply. It is often assumed that, although the birthrate of each of species is simply proportional to the number of species alive at that time, the death rate of each species depends on the population of both species. We will assume that the population of a particular pair of species is described by the equations. dx₁ (t) dt dx₂ (t) dt = x₁(t) [4 - 0.0003x₁ (t) - 0.0004x₂(t)] = x₂(t) [2 -0.0002x₁ (t) - 0.0001x₂ (t)]. If it is known that the initial population of each species is 10,000. (a) Use Runge-Kutta Midpoint method (by hand) to approximate the solutions at t = 0.4 with h = 0.2. (b) Is there a stable solution to this population model? If so, for what values of x₁ and x₂ is the solution stable?

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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Question

Consider the problem of predicting the population of two species, one of which is a predator, whose
population at time t is x₂(t), feeding on the other, which is the prey, whose population is x₁ (t). The
two species competing for the same food supply. It is often assumed that, although the birthrate of each
of species is simply proportional to the number of species alive at that time, the death rate of each
species depends on the population of both species. We will assume that the population of a particular
pair of species is described by the equations.
dx, (t)
dt
dx₂ (t)
dt
=
= x₁(t)[4 — 0.0003x₁ (t) — 0.0004x₂(t)]
= x₂(t) |2 -0.0002x₁ (t) 0.0001x₂ (t)].
If it is known that the initial population of each species is 10,000.
(a) Use Runge-Kutta Midpoint method (by hand) to approximate the solutions at t 0.4 with h = 0.2.
(b) Is there a stable solution to this population model? If so, for what values of x₁ and x₂ is the solution
stable?

Expert Solution

Introduction

As per the question we are given a predictor-prey population model with initial conditions :

x₁(0) = x₂(0) = 10000

Now we have to find :

The population at time t = 0.4 using Runge-Kutta Midpoint Rule with step size h = 0.2
The stable solution to the dynamic system

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