Math 211 Math 211 Lecture #22 Systems of ODEs October 17, 2003 2 - PDF document

1 Math 211 Math 211 Lecture #22 Systems of ODEs October 17, 2003 2 Predator-Prey Populations Predator-Prey Populations • Consider a mixed population of predators (foxes) and prey (rabbits). • The prey, x ( t ) , flourish in the absence of the predators. • The predators, y ( t ) , depend on the prey as a food source, and would die out in the absence of the prey. • For predation to take place there must be an encounter between a predator and a prey. Return 3 Predator-Prey Model Predator-Prey Model The basic model is x ′ = r x · x and y ′ = r y · y, where r x and r y are the reproductive rates. • r x = a > 0 if y = 0 , and decreases as y increases. � r x = a − by. • r y = − c < 0 if x = 0 , and increases as x increases. � r y = − c + dx. • The system becomes: x ′ = ( a − by ) x y ′ = ( − c + dx ) y � This is called the Lotka Volterra model. • M ATLAB & pplane6 . Return Assumptions 1 John C. Polking

4 General System in 2D General System in 2D x ′ = f ( t, x, y ) y ′ = g ( t, x, y ) • Example 2: x ′ = y y ′ = − x • Solution 1: x 1 ( t ) = sin t and y 1 ( t ) = cos t � Verify by direct substitution. • Solution 2: x 2 ( t ) = cos t and y 2 ( t ) = − sin t � Verify by direct substitution. Return Predator Prey 5 General System in Higher D General System in Higher D x ′ 1 = f 1 ( t, x 1 , x 2 , . . . , x n ) x ′ 2 = f 2 ( t, x 1 , x 2 , . . . , x n ) . . . . . = . x ′ n = f n ( t, x 1 , x 2 , . . . , x n ) • The dimension of a system is the number of unknown functions = the number of equations. � The predator-prey model has dimension 2. • Example: A food chain. Return Planar system 6 Vector Notation — 2D Vector Notation — 2D • In 2D set u 1 ( t ) = x ( t ) & u 2 ( t ) = y ( t ) . Then � � � u ′ � u 1 ( t ) 1 ( t ) u ( t ) = and u ′ ( t ) = . u 2 ( t ) u ′ 2 ( t ) • For the right-hand side, set � � f ( t, u 1 , u 2 ) F ( t, u ) = . g ( t, u 1 , u 2 ) • Then x ′ = f ( t, x, y ) ⇔ u ′ = F ( t, u ) y ′ = g ( t, x, y ) Return 2 John C. Polking

7 2D Examples 2D Examples • The predator-prey model system can be written � u ′ � � � ( a − bu 2 ) u 1 u ′ = 1 = . u ′ ( − c + bu 1 ) u 2 2 • Example 2: x ′ = y � u ′ � � � u 2 u ′ = 1 = . ⇔ y ′ = − x u ′ − u 1 2 • These are autonomous systems. � The RHS has no explicit dependence on t . Return 8 Vector Notation — General Vector Notation — General • In higher dimensions, set ⎛ x 1 ( t ) ⎞ ⎛ f 1 ( t, x ) ⎞ x 2 ( t ) f 2 ( t, x ) ⎜ ⎟ ⎜ ⎟ x ( t ) = f ( t, x ) = ⎠ . ⎜ . ⎟ ⎜ . ⎟ . . ⎜ ⎟ ⎜ ⎟ . . ⎝ ⎠ ⎝ x n ( t ) f n ( t, x ) • The general system can be written x ′ = f ( t, x ) . • Example: A food chain. Return 9 Initial Value Problem Initial Value Problem x ′ = f ( t, x ) x ( t 0 ) = x 0 . • Each component of x ( t 0 ) must be specified. • Example 2: x ′ = y x (0) = 2 with y ′ = − x y (0) = 13 • PP model: Both the initial prey population and the initial predator population must be specified. Return 3 John C. Polking

10 Reduction of Higher Order Equation to a System Reduction of Higher Order Equation to a System For any higher order equation there is a first order system which is equivalent to it, in the sense that solutions of the system lead easily to solutions of the equation, and vice versa. • Reduces the study of higher order equations to the study of systems • Useful for the computation of solutions of higher order equations. Return 11 Example of Reduction Example of Reduction • Third-order equation: y ′′′ + 2 yy ′ = 3 cos t • Set x 1 = y , x 2 = y ′ , and x 3 = y ′′ . • Then x ′ 1 = x 2 x ′ 2 = x 3 x ′ 3 = 3 cos t − 2 x 1 x 2 • This system is not autonomous. Return 12 Geometric Interpretation of Solutions Geometric Interpretation of Solutions • Parametric plot � Tangent vectors • Vector fields • Phase plane • pplane6 for planar autonomous systems. 4 John C. Polking