Get solution: Each of 1 through 6 can be interpreted as describing the interaction of

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t , and interpret the results in terms of the populations of the two species.dx/dt = x(1.5 x 0.5y) dy/dt = y(2 y 0.75x)

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t , and interpret the results in terms of the populations of the two species.dx/dt = x(1.5 x 0.5y) dy/dt = y(2 0.5y 1.5x)

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t , and interpret the results in terms of the populations of the two species.dx/dt = x(1.5 0.5x y) dy/dt = y(2 y 1.125x)

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t , and interpret the results in terms of the populations of the two species.dx/dt = x(1.5 0.5x y) dy/dt = y(0.75 y 0.125x)

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t , and interpret the results in terms of the populations of the two species.dx/dt = x(1 x y) dy/dt = y(1.5 y x)

Each of Problems 1 through 6 can be interpreted as describing the interaction of two species with populations x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to behave. (b) Find the critical points. (c) For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. (d) Sketch the trajectories in the neighborhood of each critical point. (e) Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. (f) Determine the limiting behavior of x and y as t , and interpret the results in terms of the populations of the two species.dx/dt = x(1 x + 0.5y) dy/dt = y(2.5 1.5y + 0.25x)

Consider the eigenvalues given by Eq. (39) in the text. Show that (1X + 2Y) 2 4(12 12)XY = (1X 2Y) 2 + 412XY. Hence conclude that the eigenvalues can never be complex.

Two species of fish that compete with each other for food, but do not prey on each other, are bluegill and redear. Suppose that a pond is stocked with bluegill and redear, and let x and y be the populations of bluegill and redear, respectively, at time t. Suppose further that the competition is modeled by the equations dx/dt = x(1 1x 1y), dy/dt = y(2 2y 2x). (a) If 2/2 > 1/1 and 2/2 > 1/1, show that the only equilibrium populations in the pond are no fish, no redear, or no bluegill. What will happen for large t? (b) If 1/1 > 2/2 and 1/1 > 2/2, show that the only equilibrium populations in the pond are no fish, no redear, or no bluegill. What will happen for large t?

Consider the competition between bluegill and redear mentioned in Problem 8. Suppose that 2/2 > 1/1 and 1/1 > 2/2, so, as shown in the text, there is a stable equilibrium point at which both species can coexist. It is convenient to rewrite the equations of Problem 8 in terms of the carrying capacities of the pond for bluegill (B = 1/1) in the absence of redear and for redear (R = 2/2) in the absence of bluegill. (a) Show that the equations of Problem 8 take the form dx dt = 1x  1 1 Bx 1 B y , dy dt = 2y  1 1 Ry 2 R x , where 1 = 1/1 and 2 = 2/2. Determine the coexistence equilibrium point (X, Y) in terms of B, R, 1, and 2. (b) Now suppose that a fisherman fishes only for bluegill with the effect that B is reduced. What effect does this have on the equilibrium populations? Is it possible, by fishing, to reduce the population of bluegill to such a level that they will die out?

Consider the system (2) in the text, and assume that 12 12 = 0. (a) Find all the critical points of the system. Observe that the result depends on whether 12 21 is zero. (b) If 12 21 > 0, classify each critical point and determine whether it is asymptotically stable, stable, or unstable. Note that Problem 5 is of this type. Then do the same if 12 21 < 0. (c) Analyze the nature of the trajectories when 12 21 = 0.

Consider the system (3) in Example 1 of the text. Recall that this system has an asymptotically stable critical point at (0.5, 0.5), corresponding to the stable coexistence of the two population species. Now suppose that immigration or emigration occurs at the constant rates of a and b for the species x and y, respectively. In this case Eqs. (3) are replaced by dx/dt = x(1 x y) + a, (i) dy/dt = y(0.75 y 0.5x) + b. The question is what effect this has on the location of the stable equilibrium point.

(a) To find the new critical point, we must solve the equations x(1 x y) + a = 0, (ii) y(0.75 y 0.5x) + b = 0. One way to proceed is to assume that x and y are given by power series in the parameter ; thus x = x0 + x1 + , y = y0 + y1 + . (iii) Substitute Eqs. (iii) into Eqs. (ii) and collect terms according to powers of . (b) From the constant terms (the terms not involving ), show that x0 = 0.5 and y0 = 0.5, thus confirming that, in the absence of immigration or emigration, the critical point is (0.5, 0.5). (c) From the terms that are linear in , show that x1 = 4a 4b, y1 = 2a + 4b. (iv) (d) Suppose that a > 0 and b > 0 so that immigration occurs for both species. Show that the resulting equilibrium solution may represent an increase in both populations, or an increase in one but a decrease in the other. Explain intuitively why this is a reasonable result

The system x = y, y = y x(x 0.15)(x 2) results from an approximation to the HodgkinHuxley5 equations, which model the transmission of neural impulses along an axon. (a) Find the critical points and classify them by investigating the approximate linear system near each one. (b) Draw phase portraits for = 0.8 and for = 1.5. (c) Consider the trajectory that leaves the critical point (2, 0). Find the value of for which this trajectory ultimately approaches the origin as t . Draw a phase portrait for this value of . Bifurcation Points. Consider the system x = F(x, y, ), y = G(x, y, ), (i) where is a parameter. The equations F(x, y, ) = 0, G(x, y, ) = 0 (ii) determine the x- and y-nullclines, respectively; any point where an x-nullcline and a y-nullcline intersect is a critical point. As varies and the configuration of the nullclines changes, it may well happen that, at a certain value of , two critical points coalesce into one. For further variation in , the critical point may once again separate into two critical points, or it may disappear altogether. Or the process may occur in reverse: For a certain value of , two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in , may split into two. A value of at which such phenomena occur is a bifurcation point. It is also common for a critical point to experience a change in its type and stability properties at a bifurcation point. Thus both the number and kind of critical points may change abruptly as passes through a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the systems solutions. In each of Problems 13 through 16: (a) Sketch the nullclines and describe how the critical points move as increases. (b) Find the critical points. (c) Let = 2. Classify each critical point by investigating the corresponding approximate linear system. Draw a phase portrait in a rectangle containing the critical points. (d) Find the bifurcation point 0 at which the critical points coincide. Locate this critical point and find the eigenvalues of the approximate linear system. Draw a phase portrait. (e) For >0 there are no critical points. Choose such a value of and draw a phase portrait.

The system x = y, y = y x(x 0.15)(x 2) results from an approximation to the HodgkinHuxley5 equations, which model the transmission of neural impulses along an axon. (a) Find the critical points and classify them by investigating the approximate linear system near each one. (b) Draw phase portraits for = 0.8 and for = 1.5. (c) Consider the trajectory that leaves the critical point (2, 0). Find the value of for which this trajectory ultimately approaches the origin as t . Draw a phase portrait for this value of . Bifurcation Points. Consider the system x = F(x, y, ), y = G(x, y, ), (i) where is a parameter. The equations F(x, y, ) = 0, G(x, y, ) = 0 (ii) determine the x- and y-nullclines, respectively; any point where an x-nullcline and a y-nullcline intersect is a critical point. As varies and the configuration of the nullclines changes, it may well happen that, at a certain value of , two critical points coalesce into one. For further variation in , the critical point may once again separate into two critical points, or it may disappear altogether. Or the process may occur in reverse: For a certain value of , two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in , may split into two. A value of at which such phenomena occur is a bifurcation point. It is also common for a critical point to experience a change in its type and stability properties at a bifurcation point. Thus both the number and kind of critical points may change abruptly as passes through a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the systems solutions. In each of Problems 13 through 16: (a) Sketch the nullclines and describe how the critical points move as increases. (b) Find the critical points. (c) Let = 2. Classify each critical point by investigating the corresponding approximate linear system. Draw a phase portrait in a rectangle containing the critical points. (d) Find the bifurcation point 0 at which the critical points coincide. Locate this critical point and find the eigenvalues of the approximate linear system. Draw a phase portrait. (e) For >0 there are no critical points. Choose such a value of and draw a phase portrait.

The system x = y, y = y x(x 0.15)(x 2) results from an approximation to the HodgkinHuxley5 equations, which model the transmission of neural impulses along an axon. (a) Find the critical points and classify them by investigating the approximate linear system near each one. (b) Draw phase portraits for = 0.8 and for = 1.5. (c) Consider the trajectory that leaves the critical point (2, 0). Find the value of for which this trajectory ultimately approaches the origin as t . Draw a phase portrait for this value of . Bifurcation Points. Consider the system x = F(x, y, ), y = G(x, y, ), (i) where is a parameter. The equations F(x, y, ) = 0, G(x, y, ) = 0 (ii) determine the x- and y-nullclines, respectively; any point where an x-nullcline and a y-nullcline intersect is a critical point. As varies and the configuration of the nullclines changes, it may well happen that, at a certain value of , two critical points coalesce into one. For further variation in , the critical point may once again separate into two critical points, or it may disappear altogether. Or the process may occur in reverse: For a certain value of , two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in , may split into two. A value of at which such phenomena occur is a bifurcation point. It is also common for a critical point to experience a change in its type and stability properties at a bifurcation point. Thus both the number and kind of critical points may change abruptly as passes through a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the systems solutions. In each of Problems 13 through 16: (a) Sketch the nullclines and describe how the critical points move as increases. (b) Find the critical points. (c) Let = 2. Classify each critical point by investigating the corresponding approximate linear system. Draw a phase portrait in a rectangle containing the critical points. (d) Find the bifurcation point 0 at which the critical points coincide. Locate this critical point and find the eigenvalues of the approximate linear system. Draw a phase portrait. (e) For >0 there are no critical points. Choose such a value of and draw a phase portrait.

The system x = y, y = y x(x 0.15)(x 2) results from an approximation to the HodgkinHuxley5 equations, which model the transmission of neural impulses along an axon. (a) Find the critical points and classify them by investigating the approximate linear system near each one. (b) Draw phase portraits for = 0.8 and for = 1.5. (c) Consider the trajectory that leaves the critical point (2, 0). Find the value of for which this trajectory ultimately approaches the origin as t . Draw a phase portrait for this value of . Bifurcation Points. Consider the system x = F(x, y, ), y = G(x, y, ), (i) where is a parameter. The equations F(x, y, ) = 0, G(x, y, ) = 0 (ii) determine the x- and y-nullclines, respectively; any point where an x-nullcline and a y-nullcline intersect is a critical point. As varies and the configuration of the nullclines changes, it may well happen that, at a certain value of , two critical points coalesce into one. For further variation in , the critical point may once again separate into two critical points, or it may disappear altogether. Or the process may occur in reverse: For a certain value of , two formerly nonintersecting nullclines may come together, creating a critical point, which, for further changes in , may split into two. A value of at which such phenomena occur is a bifurcation point. It is also common for a critical point to experience a change in its type and stability properties at a bifurcation point. Thus both the number and kind of critical points may change abruptly as passes through a bifurcation point. Since a phase portrait of a system is very dependent on the location and nature of the critical points, an understanding of bifurcations is essential to an understanding of the global behavior of the systems solutions. In each of Problems 13 through 16: (a) Sketch the nullclines and describe how the critical points move as increases. (b) Find the critical points. (c) Let = 2. Classify each critical point by investigating the corresponding approximate linear system. Draw a phase portrait in a rectangle containing the critical points. (d) Find the bifurcation point 0 at which the critical points coincide. Locate this critical point and find the eigenvalues of the approximate linear system. Draw a phase portrait. (e) For >0 there are no critical points. Choose such a value of and draw a phase portrait.

Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2, except that some coefficients depend on a parameter . In each of these problems assume that x, y, and are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of your sketch may resemble different parts of Figure 9.4.5. (b) Find the critical points. (c) Determine the bifurcation points. (d) Find the Jacobian matrix J and evaluate it for each of the critical points. (e) Determine the type and stability property of each critical point. Pay particular attention to what happens as passes through a bifurcation point (f) Draw phase portraits for the system for selected values of to confirm your conclusions.

Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2, except that some coefficients depend on a parameter . In each of these problems assume that x, y, and are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of your sketch may resemble different parts of Figure 9.4.5. (b) Find the critical points. (c) Determine the bifurcation points. (d) Find the Jacobian matrix J and evaluate it for each of the critical points. (e) Determine the type and stability property of each critical point. Pay particular attention to what happens as passes through a bifurcation point (f) Draw phase portraits for the system for selected values of to confirm your conclusions.

Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2, except that some coefficients depend on a parameter . In each of these problems assume that x, y, and are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of your sketch may resemble different parts of Figure 9.4.5. (b) Find the critical points. (c) Determine the bifurcation points. (d) Find the Jacobian matrix J and evaluate it for each of the critical points. (e) Determine the type and stability property of each critical point. Pay particular attention to what happens as passes through a bifurcation point (f) Draw phase portraits for the system for selected values of to confirm your conclusions.