ch02_5 - Ch 2.5 Autonomous Equations and Population...

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Ch 2.5: Autonomous Equations and Population Dynamics In this section we examine equations of the form y' = f ( y ), called autonomous equations, where the independent variable t does not appear explicitly . The main purpose of this section is to learn how geometric methods can be used to obtain qualitative information directly from differential equation without solving it. Example (Exponential Growth): Solution: 0 , = r ry y rt e y y 0 =
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Logistic Growth An exponential model y' = ry , with solution y = e rt , predicts unlimited growth, with rate r > 0 independent of population. Assuming instead that growth rate depends on population size, replace r by a function h ( y ) to obtain y' = h ( y ) y . We want to choose growth rate h ( y ) so that h ( y ) 2245 r when y is small, h ( y ) decreases as y grows larger, and h ( y ) < 0 when y is sufficiently large. The simplest such function is h ( y ) = r ay , where a > 0. Our differential equation then becomes This equation is known as the Verhulst, or logistic , equation. ( 29 0 , , - = a r y ay r y
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Logistic Equation The logistic equation from the previous slide is This equation is often rewritten in the equivalent form where K = r / a . The constant r is called the intrinsic growth rate , and as we will see, K represents the carrying capacity of the population. A direction field for the logistic equation with r = 1 and K = 10 is given here. , 1 y K y r dt dy - = ( 29 0 , , - = a r y ay r y
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Logistic Equation: Equilibrium Solutions Our logistic equation is Two equilibrium solutions are clearly present: In direction field below, with r = 1, K = 10, note behavior of solutions near equilibrium solutions: y = 0 is unstable , y = 10 is asymptotically stable . 0 , , 1 - = K r y K y r dt dy K t y t y = = = = ) ( , 0 ) ( 2 1 φ
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Autonomous Equations: Equilibrium Solns Equilibrium solutions of a general first order autonomous equation y ' = f ( y ) can be found by locating roots of f ( y ) = 0. These roots of
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ch02_5 - Ch 2.5 Autonomous Equations and Population...

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