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clase12handouts - Nonlinear models Logistic equation The...

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Nonlinear models: Logistic equation The exponential growth model is linear and predicts exponential growth of a population. This kind of growth may occur in the initial stages, but it cannot continue indefinitely. For long-range predictions, we need models that take into account the interaction of the population with its environment. Population growth levels off as a result of limited food supplies, increased diseases, crowding, and other factors. M.I. Bueno Differential Equations and Linear Algebra
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y 0 = ky y 0 = k ( y ) y . For most populations, the growth rate k ( y ) decreases as y increases. The simplest choice is k ( y ) = r - ay , a > 0, r > 0. Logistic equation : y 0 = ( r - ay ) y . or equivalently, y 0 = r ( 1 - a r y ) y = r ( 1 - y L ) y . The parameter r is called initial growth rate and L = r / a is called the carrying capacity . M.I. Bueno Differential Equations and Linear Algebra
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Observations The equation is autonomous. Therefore, all the solutions are horizontal translations of each others. Since y represents a population, y 0. The logistic equation has two equilibrium solutions: y = 0 and y = L . y = L is a stable equilibrium solution, y = 0 is unstable. All solutions approach y = L asymptotically. M.I. Bueno Differential Equations and Linear Algebra
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Systems of differential equations Population studies involving two or more interacting species lead to systems of two or more differential equations. Example: dx dt = x - 3 xy dy dt = 4 y + xy . . Similar situations arise in other areas. M.I. Bueno Differential Equations and Linear Algebra
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A solution of a system of two differential equations is a pair of functions x ( t ) and y ( t ) that simultaneously satisfies both equations. Consider the decoupled system dx dt = 2 x , dy dt = - 3 y . x ( t ) = c 1 e 2 t , y ( t ) = c 2 e - 3 t , is the general solution of the system. M.I. Bueno Differential Equations and Linear Algebra
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M.I. Bueno Differential Equations and Linear Algebra
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x 0 = 3 x - 2 y , y 0 = x , z 0 = - x + y + 3 z . x ( t ) = 2 c 2 e 2 t + c 3 e t y ( t ) = c 2 e 2 t + c 3 e t z ( t ) = c 1 e 3 t + c 2 e 2 t is the general solution. M.I. Bueno
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This note was uploaded on 03/14/2010 for the course MATH 3C taught by Professor Jacobs during the Fall '08 term at UCSB.

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clase12handouts - Nonlinear models Logistic equation The...

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