5. In previous problems dealing with two species, one of the animals was the predator and the other was theprey. In this problem we recall the Bees & Flowers example and revisit systems of differential equationsdesigned to model two species that are either competitive (that is both species are harmed by interaction) orcooperative (that is both species benefit from interaction).(a) Which system of differential equations describes a situation where the two species compete and whichsystem describes pair of cooperative species? Explain your reasoning, making sure to note what the dif-ferent terms in each DE tell you about the population context.(A)(B)dx= -5x+2xydtdx= 3x|1dt1ху10-3:-4y+3xydtdydy= 2y(1yy[l- 10)-5*y1dt(b) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify yourequilibrium solutions algebraically.(c) Use your results from 5b to sketch in the long-term behavior of solutions for system (A) with initial con-ditions anywhere in the first quadrant of the phase plane. For example, describe the long-term behaviorof solutions if the initial condition is in such-and-such region of the first quadrant. Provide a sketch ofyour analysis in the xy-plane and justify any conjectures that you have about the long-term outcome forthe two populations based on the initial conditions.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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SOLVE STEP BY STEP BY HAND (LEGIBLE) Determine the STABILITY of the following systems of differential equations: = 4 -1 X. **()* e) X f) X 1 2 0 3 0x - (21) x 100 X.
Consider a system of differential equations describing the progress of a disease in a population, given by In our particular case, this is: a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y = x = 3-3xy - 1x y = 3xy - 2y where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. The y-nullclines are y = and x = b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1,9₁) and (x2, y2), where we order them so that x1 < x2. The equilibrium with the smaller x-coordinate is (x₁, y₁) : The equilibrium with the larger x-coordinate is (x2, y₂) = where A = 0 sin (a) c) The linearization of the system of differential equations at the equilibrium (ï1, Y₁) gives a system of the form (+) ¹ (~), f əx ∞ a Ω = A x' G E = F(x, y) for a vector-valued function…
In Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let’s modify those equations as follows: dR/dt = 0.08R (1 − 0.0002R) − 0.001RW dW/dt= −0.02W + 0.00002RW (a) According to these equations, what happens to the rabbit population in the absence of wolves?(b) Find all the equilibrium solutions and explain their significance.(c) The figure shows the phase trajectory that starts at the point (1000, 40). Describe what eventually happens to the rabbit and wolf populations.
Disease epidemics can be modelled by differential equations. Suppose we have two sub-populations of individuals in a city: individuals who are infected with some virus (I) and individuals who are susceptible to infection (S). We assume that if a susceptible person interacts with an infected person, there is a probability p that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. The differential equations that model these population sizes are TI - PSI, dS dt dI dt pSI - TI. (a) Assuming no one dies from the virus, the total population N is a constant number defined as N = S+I. Show in this case that you can reduce the above system to the single differential equation for I. (b) Find the general solution I(t) to the ODE found in part (a). (c) If I (0) = Io, give the equation for the unknown constant in terms of Io, N, p and r.
Solve the given system of differential equations by systematic elimination. (2D2 D-1)x - (2D + 1)y = 8 (D1)x+ Dy = -8 (x(t), y(t)) = = Need Help? Read It Watch It

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