Getting Started With The Predator - Prey Model: Nullclines

Transcription

1 Getting Started With The Predator - Prey Model: Nullclines James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline The Predator - Prey Nullclines An Example The General Results

2 Abstract This lecture introduces the Predator - Prey Model and its nullclines. The Predator - Prey model is where x = a x b x y y = c y + d x y x(0) = x 0 y(0) = y 0 a > 0 is the birth rate of the food fish x(t). b > 0 is the death rate of the food fish due to being eaten by the predators. This death rate is modeled as a population interaction term so it has the form bxy. c > 0 is the rate at which the predators die. d > 0 is the birth rate of the predators. We assume predators need to eat to procreate and in order to eat, they must interact with the food fish. Hence, this birth rate is modeled using population interaction as dxy.

3 To figure out how the trajectories move for given initial conditions, we will use nullcline analysis once again. The x nullcline analysis begins with setting. This gives 0 = a x b x y = x (a by). Since this is a product, there are two possibilities: x = 0; the y axis and y = a b ; a horizontal line. The y nullcline analysis begins with setting. This gives 0 = c y + d x y = y ( c + dx). Since this is a product, there are two possibilities: y = 0; the and x = c d ; a vertical line. We need to determine the (+, +), (+, ), (, +) and (, ) x and y regions in order to see how the trajectories move. Here s an example: consider the predator - prey model x (t) = 2 x(t) 5 x(t) y(t) y (t) = 6 y(t) + 3 x(t) y(t) The x nullclines satisfy 2x 5xy = 0 or x(2 5y) = 0. Hence x = 0 or y = 2/5. Draw the lines x = 0, the y axis, and the horizontal line y = 2/5 in the xy plane. The factor 2 5y is positive when 2 5y > 0 or when y < 2/5. Hence, the factor is negative when y > 2/5. The factor x is positive when x > 0 and negative when x < 0. So the combination x(2 5y) has a sign that can be determined easily on the next graph.

4 y = 2 5 x(2 5y) x + x(2 5y) + x y axis x(2 5y) + x x(2 5y) + + x + The x nullclines. Each region has the product x(2 5y) in it. Underneath the factors are the algebraic signs for the factors. Their product determines the algebraic sign of x. Next, for our predator - prey model x (t) = 2 x(t) 5 x(t) y(t) y (t) = 6 y(t) + 3 x(t) y(t) the y nullclines satisfy 6y + 3xy = 0 or y( 6 + 3x) = 0. Hence y = 0 or x = 2. Draw the lines y = 0 (the and the vertical line x = 2 in the xy plane. The factor 6 + 3x is positive when 6 + 3x > 0 or when x > 2. Hence, the factor is negative when x < 2. The factor y is positive when y > 0 and negative when y < 0. So the combination y( 6 + 3x) has a sign that can be determined easily on the next graph.

5 y axis y( 6 + 3x) + y y( 6 + 3x) y + y( 6 + 3x) + + y + y( 6 + 3x) + y The y nullclines. Each region has the product y( 6 + 3x) in it. Underneath the factors are the algebraic signs for the factors. Their product determines the algebraic sign of y. x = 6/3 We combine the x and y nullcline information to create a map of how x and y change sign in the xy plane. Regions I, II, III and IV divide Quadrant I. We will show there are trajectories moving down the positive y axis and out along the positive. Thus, a trajectory that starts in Quadrant I with positive initial conditions can t cross the the trajectory on the positive or the trajectory on the positive y axis. Thus, the Predator - Prey trajectories that start in Quadrant I with positive initial conditions will stay in Quadrant I.

6 y = 2/5 V VI (, ) VII (,+) y axis II (, ) III VIII (+,+) I (,+) IV (+,+) IX x = 6/3 The and the equations determine regions in the xy plane. In each region, the algebraic sign of x and y are shown as an ordered pair. For example, in region I, x is negative and y is positive and so this is denoted by (, +). For the general predator - prey model x (t) = a x(t) b x(t) y(t) y (t) = c y(t) + d x(t) y(t) The x nullclines satisfy ax bxy = 0 or x(a by) = 0. Hence x = 0 or y = a/b. Draw the lines x = 0 (the y axis and the horizontal line y = a/b in the xy plane. The factor a by is positive when a by > 0 or when y < a/b. Hence, the factor is negative when y > a/b. The factor x is positive when x > 0 and negative when x < 0. So the combination x(a by) has a sign that can be determined easily on the next graph.

7 y = a b x(a by) x + x(a by) + x y axis x(a by) + x x(a by) + + x + The x nullclines. Each region has the product x(a by) in it. Underneath the factors are the algebraic signs for the factors. Their product determines the algebraic sign of x. Next, for our predator - prey model x (t) = a x(t) b x(t) y(t) y (t) = c y(t) + d x(t) y(t) the y nullclines satisfy cy + dxy = 0 or y( c + dx) = 0. Hence y = 0 or x = c/d. Draw the lines y = 0 (the and the vertical line x = c/d in the xy plane. The factor c + dx is positive when c + dx > 0 or when x > c/d. Hence, the factor is negative when x < c/d. The factor y is positive when y > 0 and negative when y < 0. So the combination y( c + dx) has a sign that can be determined easily on the next graph.

8 y axis y( c + dx) + y y( c + dx) y + y( c + dx) + + y + y( c + dx) + y The y nullclines. Each region has the product y( c + dx) in it. Underneath the factors are the algebraic signs for the factors. Their product determines the algebraic sign of y. x = c/d y = a/b V VI (, ) VII (,+) y axis II (, ) III VIII (+,+) I (,+) IV (+,+) IX x = c/d The and the equations determine regions in the xy plane. In each region, the algebraic sign of x and y are shown as an ordered pair. For example, in region I, x is negative and y is positive and so this is denoted by (, +).

9 Homework 66 For the following problems, do the x and y nullclines analysis in the whole plane separately and then assemble the combined picture for the whole plane. Use multiple colors as needed x (t) = 100 x(t) 25 x(t) y(t) y (t) = 200 y(t) + 40 x(t) y(t) x (t) = 1000 x(t) 250 x(t) y(t) y (t) = 2000 y(t) + 40 x(t) y(t) x (t) = 900 x(t) 45 x(t) y(t) y (t) = 100 y(t) + 50 x(t) y(t)