MAT 21B Lecture 18: Lecture 18

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22 Feb 2019

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MAT 21B Full Course Notes

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U sink doc dv for is integrals term in differentiable resulting expression . simpler product than of other two terms where integrable , How x to and sin choose ? we let. If u = du = sin cos x x a fxsinxdx. = ( ii. complicated . are integrable and differentiable . dv= v doc doc x. 2 sin x ) - jil . cos xdx. If we let u - x du - I doc doc ex du = sin x doc t. V= k x cos x fees t sin x da x t. Spca ) always integrate doc cos a choose a e. " doc p cx ) and du = Spca ) sin x doc is e a polynomial. " doc ( or , sin accord or in as cos a. Iz doc dv = doc v - se. Cx ) g how to f f- cx ) doc.

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A classical model in mathematical ecology is the LotkaâVolterra predator-prey model. Consider a simple ecosystem consisting of rabbits that have an infinite supply of food and foxes that prey on the rabbits for their food. This is modeled by a pair of nonlinear, first-order differential equations:

where t is time, r(t) is the number of rabbits, f(t) is the number of foxes, and Î± is a positive constant. If Î± = 0, the two populations do not interact, the rabbits do what rabbits do best, and the foxes die off from starvation. If Î± > 0, the foxes encounter the rabbits with a probability that is proportional to the product of their numbers. Such an encounter results in a decrease in the number of rabbits and (for less obvious reasons) an increase in the number of foxes. The solutions to this nonlinear system cannot be expressed in terms of other known functions; the equations must be solved numerically. It turns out that the solutions are always periodic, with a period that depends on the initial conditions. In other words, for any r(0) and f(0), there is a value t = t_p when both populations return to their original values. Consequently, for all t,

(a) Compute the solution with r₀ = 300, f₀ = 150, and Î± = 0.01. You should find that t_p is close to 5. Make two plots, one of r and f as functions of t and one a phase plane plot with r as one axis and f as the other.

(b) Compute and plot the solution with r₀ = 15, f₀ = 22, and Î± = 0.01. You should find that t_p is close to 6.62.

(c) Compute and plot the solution with r₀ = 102, f₀ = 198, and Î± = 0.01. Determine the period t_p either by trial and error or with an event handler.

(d) The point (r₀, f₀) = (1/Î±, 2/Î±) is a stable equilibrium point. If the populations have these initial values, they do not change. If the initial populations are close to these values, they do not change very much. Let u(t) = r(t)â1/Î± and v(t) = f(t) â 2/Î±. The functions u(t) and v(t) satisfy another nonlinear system of differential equations, but if the uv terms are ignored, the system becomes linear. What is this linear system? What is the period of its periodic solutions?

Hi, I need the full Matlab code and answers for all parts of the problem. Thanks!

General Math Terms A product B. digit C. numeral D. compass E. equation F. formula G. negative H. sum I. mean J. statistics K. positive L. percent M. straight-edge N. squareroot O. quotient P. prime numbers Q. difference R. addend S. minuend T. divisor Greater than zero: plus. A number being added to another number. A mathematical sequence whose verb is equal (=). The result of addition. The number doing the dividing in a division problem. An instrument used to construct straight lines. A mathematical expression of a natural law. The result of multiplication. The number from which another number is being subtracted. Any non-negative numeral less than ten. Less than zero: minus. An instrument used to construct circles and arcs of circles. A number that produces another number (it's square) when multiplied by itself. The result of subtraction. A symbol or name of a number. The average: the sum of the elements divided by the number of elements. The result of division. Parts per hundred: hundredths. Natural numbers greater than one that have no other factors except one and themselves. Set of numerical data. Geometry A. chord B. diameter C. radius D. perimeter E. cylinder F. surface area G. right angle H. square I. Hypotenuse J. obtuse angle K. vertex L point M. ray N. bisect O arc P. hexagon Q. pentagon R. polygon S. perpendicular T. rectangle U. rhombus V. triangle W. circumference X cube Y pyramid Z. sphere A six-sided polygon. The common end point of the sides of an angle. An angle measuring more than 90 degree. A long, round object with flat, circular ends. An angle of 90 degree. The side of a right triangle opposite the right angle. The total area of all the surfaces of an object. To multiply a number by itself. A location, the place where two lines cross or intersect. A closed geometric figure made up of three or more line segments. The perimeter of a circle: the distance around a circle. A shape having the same dimensions of length, height, and width. A half-line with a designated end point which extends in only one direction without stopping. A figure having a polygon base and triangular sides which meet in a point at the top. Divide an angle into two equal angles. A part of a circle

MAT 21B Lecture 18: Lecture 18

MAT 21B Full Course Notes

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