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7.4 Predator–Prey Equations We will denote by x and y the populations of the prey and predator, respectively, at time t. In constructing a model of the interaction of the two species, we make the following assumptions: 1. In the absence of the predator, the prey grows at a rate proportional to the current population; thus dx/dt = ax, a > 0, when y = 0. 2. In the absence of the prey, the predator dies out; thus dy/dt=−cy, c > 0, when x = 0. 3. The number of encounters between predator and prey is proportional to the product of their populations. Each such encounter tends to promote the growth of the predator and to inhibit the growth of the prey. Thus the growth rate of the predator includes a term of the form γ xy, whereas the growth rate of the prey includes a term of the form −αxy, where γ and α are positive constants.

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Predator–Prey Equations (Ctd.) We get the equations dx/dt = ax − αxy = x(a − αy), (1) dy/dt = −cy + γ xy = y(−c + γ x). The constants a, c, α, and γ are all positive; a and c are the growth rate of the prey and the death rate of the predator, respectively, and α and γ are measures of the effect of the interaction between the two species. In specific situations values for a, α, c, and γ must be determined by observation. Equations (1) are known as the Lotka– Volterra equations.

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Example Discuss the solutions of the system dx/dt = x(1 − 0.5y) = x − 0.5xy = F(x, y), dy/dt= y(−0.75 + 0.25x) = −0.75y + 0.25xy = G(x, y) for x and y positive. Answer The critical points of this system are the solutions of the algebraic equations x(1 − 0.5y) = 0, y(−0.75 + 0.25x) = 0.

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Example (ctd) Critical points and direction field for the predator–prey system.

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A phase portrait of the system.

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Variations of the prey and predator populations with time for the system in the example

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7.5 Periodic Solutions and Limit Cycles we discuss the possible existence of periodic solutions of two dimensional autonomous systems x' = f(x). (1) Such solutions satisfy the relation x(t + T ) = x(t) (2) for all t and for some nonnegative constant T called the period. The corresponding trajectories are closed curves in the phase plane.

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Periodic Solutions and Limit Cycles (Ctd.) Recall that the solutions of the linear autonomous system x' = Ax (3) are periodic if and only if the eigenvalues of A are pure imaginary. In this case the critical point at the origin is a center. We emphasize that if the eigenvalues of A are pure imaginary, then every solution of the linear system (3) is periodic, whereas if the eigenvalues are not pure imaginary, then there are no (nonconstant) periodic solutions.

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Example Discuss the solutions of the system (4)

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Terminology In this example, the circle r = 1 not only corresponds to periodic solutions of the system (4), but it also attracts other nonclosed trajectories that spiral toward it as t→∞. In general, a closed trajectory in the phase plane such that other nonclosed trajectories spiral toward it, either from the inside or the outside, as t→∞, is called a limit cycle. Thus the circle r = 1 is a limit cycle for the system (4). If all trajectories that start near a closed trajectory (both inside and outside) spiral toward the closed trajectory as t→∞, then the limit cycle is asymptotically stable. Since the limiting trajectory is itself a periodic orbit rather than an equilibrium point, this type of stability is often called orbital stability. If the trajectories on one side spiral toward the closed trajectory, while those on the other side spiral away as t→∞, then the limit cycle is said to be semistable. If the trajectories on both sides of the closed trajectory spiral away as t→∞, then the closed trajectory is unstable. It is also possible to have closed trajectories that other trajectories neither approach nor depart from—for example, the periodic solutions of the predator–prey equations in Section 7.4. In this case, the closed trajectory is stable.

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THEOREM 7.5.1 Let the functions F and G have continuous first partial derivatives in a domain D of the xy-plane. A closed trajectory of the system dx/dt = F(x, y) dy/dt = G(x, y) must necessarily enclose at least one critical (equilibrium) point. If it encloses only one critical point, the critical point cannot be a saddle point.

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THEOREM 7.5.2 Let the functions F and G have continuous first partial derivatives in a simply connected domain D of the xy- plane. If F x + G y has the same sign throughout D, then there is no closed trajectory of the system dx/dt = F(x, y) dy/dt = G(x, y) lying entirely in D.

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THEOREM 7.5.3 (Poincar´e– Bendixson Theorem) Let the functions F and G have continuous first partial derivatives in a domain D of the xy-plane. Let D 1 be a bounded subdomain in D, and let R be the region that consists of D 1 plus its boundary (all points of R are in D). Suppose that R contains no critical point of the system (15), dx/dt = F(x, y) dy/dt = G(x, y). If there exists a constant t0 such that x = φ(t), y = ψ(t) is a solution of the system (15) that exists and stays in R for all t ≥ t0, then either x = φ(t), y = ψ(t) is a periodic solution (closed trajectory), or x = φ(t), y = ψ(t) spirals toward a closed trajectory as t→∞. In either case, the system (15) has a periodic solution in R.

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Example - van der Pol equation The van der Pol equation u'' − μ(1 − u 2 )u' + u = 0, (17) where μ is a nonnegative constant, describes the current u in a triode oscillator. Discuss the solutions of this equation.

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Trajectories of the van der Pol equation (17) for μ = 0.2.

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Plots of u versus t for trajectories of the van der Pol equation (17) for μ = 0.2.

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Trajectories of the van der Pol equation (17) for μ = 1

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Plots of u versus t for the trajectories for μ = 1

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Trajectories of the van der Pol equation (17) for μ = 5.

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Plot of u versus t for the outward spiraling trajectory for μ = 5.

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7.6 Chaos and Strange Attractors: The Lorenz Equations Lorenz equations. The nonlinear autonomous three-dimensional system dx/dt = σ(−x + y), dy/dt = rx − y − xz, dz/dt = −bz + xy. (1) The Lorenz equations also involve three parameters σ, r, and b, all of which are real and positive. The parameters σ and b depend on the material and geometrical properties of the fluid layer. For the earth’s atmosphere, reasonable values of these parameters are σ = 10 and b = 8/3 ; they will be assigned these values in much of what follows in this section.

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Figure 7.6.2. - A plot of x versus t for the Lorenz equations (1) with r = 28; the initial point is (5, 5, 5).

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FIGURE 7.6.3 Plots of x versus t for two initially nearby solutions of Lorenz equations with r = 28; the initial point is (5, 5, 5) for the dashed curve and is (5.01, 5, 5) for the solid curve.

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Strange Attractor & Chaos The attracting set in this case, although of zero volume, has a rather complicated structure and is called a strange attractor. The term chaotic has come into general use to describe solutions such as those shown in Figures 7.6.2 and 7.6.3.

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Plots of x versus t for three solutions of Lorenz equations with r = 21. (a) Initial point is (3, 8, 0). (b) Initial point is (5, 5, 5). (c) Initial point is (5, 5, 10).

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Projections of a trajectory of the Lorenz equations (with r = 28) in the xy-plane.

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Projections of a trajectory of the Lorenz equations (with r = 28) in the xz-plane.

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Chapter Summary Nonlinear two-dimensional autonomous systems have the form dx/dt= F(x, y), dy/dt= G(x, y), or, in vector notation, dx/dt= f(x). The first four sections in this chapter deal mainly with approximating a nonlinear system by a linear one. The last two sections introduce phenomena that occur only in nonlinear systems.

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Section 7.1 Critical points of the system x' = f(x) satisfy f(x) = 0. Formal definitions of stability, asymptotic stability, and instability of critical points are given. Stability and asymptotic stability are illustrated by an undamped and a damped simple pendulum, respectively, about its downward equilibrium position. Instability is illustrated by a pendulum, damped or undamped, about its upward equilibrium position. Examples illustrate basins of attraction and their boundaries, called separatrices.

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Section 7.2 If F and G are twice differentiable, then the nonlinear autonomous system x' = f(x) can be approximated near a critical point x0 by a linear system u ' = Au, where u = x − x0. The coefficient matrix A is the Jacobian matrix J evaluated at x0. Thus Theorem 7.2.2 states that the trajectories of the nonlinear system locally resemble those of the linear approximation, except possibly in the cases where the eigenvalues of the linear system are either pure imaginary or real and equal. Thus, in most cases, the linear system is a good local approximation to the nonlinear system.

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Section 7.3 Application: Competing Species The equations are often used as a model of competition, such as between two species in nature or perhaps between two businesses. Examples show that sometimes the two competitors can coexist in a stable manner, but sometimes one will overwhelm the other and drive it to extinction. The analysis in this section explains why this happens and enables you to predict which outcome will occur for a given system.

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Section 7.4 Application: Predator– Prey The predator–prey, or Lotka–Volterra, equations dx/dt = x (a − αy), dy/dt = y (−c + γ x) are a starting point for the study of the relation between a prey x and its predator y. The solutions of this system exhibit a cyclic variation about a critical point (a center) in the first quadrant. This type of behavior has sometimes been observed in nature.

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Section 7.5 Nonlinear systems, unlike linear systems, sometimes have periodic solutions, or limit cycles, that attract other nearby solutions. Several theorems specify conditions under which limit cycles do, or do not, exist. The van der Pol equation (written in system form) x' = y, y' = −x + μ(1 − x 2 ) y is an important equation that illustrates the occurrence of a limit cycle.

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Section 7.6 In three or more dimensions there is the possibility that solutions may be chaotic. In addition to critical points and limit cycles, solutions may converge to sets of points known as strange attractors. The Lorenz equations, arising in a study of the atmosphere, dx/dt= σ (−x + y), dy/dt= rx − y − xz, dz/dt= −bz + xy provide an example of the occurence of chaos in a relatively simple three-dimensional nonlinear system.

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