clase12handouts

# clase12handouts - Nonlinear models Logistic equation The...

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
M.I. Bueno Differential Equations and Linear Algebra
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 03/14/2010 for the course MATH 3C taught by Professor Jacobs during the Fall '08 term at UCSB.

### Page1 / 34

clase12handouts - Nonlinear models Logistic equation The...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online