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

See similar textbooks

Similar questions

Consider the following real-world problem: Suppose that the population of Memphis is 22,400, and 60% of the population are susceptible to contract a serious illness from a very contagious bacteria that was introduced as a form of biological warfare. The remaining people have no reaction to the bacteria. Assume the susceptible people are equally mixed throughout the population. On January 1, 2022 (time t=0), the number N(t) of people who had contracted the disease was 180 and was increasing at the rate of 39 per day. Assume that N’(t) is proportional to the product of the numbers of those who have caught the disease and of those who have not. Write the differential equation that models the physical situation and solve for the exact answer.
From 2000 through 2009, the population of State A grew more slowly than that of State B. Models that represent the populations of the two states are given by P = 37.5t + 6075 State A P = 168.2t + 5138 State B where P is the population (in thousands) and t represents the year, with t = 0 corresponding to 2000. Use the models to estimate when the population of State B first exceeded the population of State A. (Round your answer to the nearest year.)
This exercise uses the population growth model. It is observed that a certain bacteria culture has a relative growth rate of 11% per hour, but in the presence of an antibiotic the relative growth rate is reduced to 5% per hour. The initial number of bacteria in the culture is 24. Find the projected population after 24 hours for the following conditions. (Round your answers to the nearest whole number.) (a) No antibiotic is present, so the relative growth rate is 11%.bacteria (b) An antibiotic is present in the culture, so the relative growth rate is reduced to 5%.bacteria
The same disease is spreading through two populations, say Pi and P2, with the same size. You may assume that the spread of the disease is well described by the SIR model. dSa - BaSaIa dt dla - dt dRa YaIa dt with Sa(0) + I.(0) + R.(0) = N where N denotes the fixed population size. The subscript a identifies the population P or P2. For example, if a = 1, the variables are related to P1. Assume that S1 (0) = S2 (0) and I1(0) = I½(0) and that no interventions such as quarantine or vaccination have been implemented. If the difference in the spread of the disease is due only to the poor over-all health of a population, which population has the best over-all health of the two populations? - Susceptible Population 1 - Susceptible Population 2 700 600 500 400 300 200 100 10 15 20 t P1. P2. Susceptible
1. A team of biologists stocked a lake with 100 fish of one species to conduct an experiment. They concluded that the rate of increase of the fish population is directly proportional to the population at that time, p(t) as well as to the difference between species' carrying capacity and the current population. The carrying capacity for this fish species in the lake was estimated to be 1000. It was observed that the number of the fish tripled in the first year. What would be the fish population in 5 years?