285lect11 - 1 Lecture 11 1.1 Logistic equation and...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 11 1.1 Logistic equation and population models Let B ( t ) and D ( t ) denote the number of births and deaths that have oc- curred since t = 0 by the time t . The birth rate ( t ) = lim t ! 0 B ( t t ) B ( t ) P ( t t = 1 P ( t ) B 0 ( t ) The death rate ± ( t ) = lim t ! 0 D ( t t ) D ( t ) P ( t t = 1 P ( t ) D 0 ( t ) ; where P ( t ) is population at time t . So, ( t ) and ± ( t ) represent the number of births and death per unit of population and per unit of time. Population P [ ( t ) ± ± ( t tP ( t ) . So, P 0 = ( ± ± ) P: Notice that and ± may not be constants. Assumption (birth rate decreases as population increases): ( t ) = & 0 ± 1 P; where & 0 1 = const , & 0 > 0 1 > 0 : ± = ± 0 = const: Then P 0 = ( & 0 ± 1 P ± ± 0 ) P ) P 0 = kP ( M ± P ) ; where k = 1 ;M = & 0 ± ± 0 1 : The ODE P 0 = kP ( M ± P ) is called the logistic equation.
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

285lect11 - 1 Lecture 11 1.1 Logistic equation and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online