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

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1 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 change is de°ned by ° 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. Let us °nd solution of the logistic equation. Z dP P ( M ± P ) = k Z dt: 1 P ( M ± P ) = 1 M ° 1 M ± P + 1 P ± : 1
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So, Z dP P ( M ± P ) = 1 M Z dP M ± P + 1 M Z dP P = 1 M [ln P ± ln j M ± P j ] = 1 M ln ² ² ² ² P M ± P ² ² ² ² :
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